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-z gj<OU1 60447 >m OSMANIA UNIVERSITY LIBRARY Call No. * & t * 5* ^> Action No 3, ~ Author Tin.- , This bof>k should be returned on or before the date last marked below. THE FACTORIAL ANALYSIS OF HUMAN ABILITY THE FACTORIAL ANALYSIS OF HUMAN ABILITY By SIR GODFREY THOMSON D.C.L., D.Sc., Ph.D. Bell Professor of Education in the University of Edinburgh UNIVERSITY OF LONDON PRESS LTD. WARWICK SQUARE, LONDON, E.C.4 KIRST PRINTED ..... 1939 SECOND EDITION ..... 1946 THIRD EDITION ..... 1948 FOURTH EDITION . ... 1950 AGENTS OVERSEAS AUSTRALIA, NEW ZEALAND AND SOUTH SEA ISLANDS W. S. SMART P.O. Box 120, SYDNEY, N.S.W. Showroom 558, George Street. CANADA CLARKE, IRWIN & Co., LTD., 103, St. Clair Avenue West, TORONTO, 5. EGYPT AND SUDAN DINO JUDAH NAHUM P.O. Box 940, CAIRO. Showioom 44, Shana Shenf Pasha. FAR EAST (including China and Japan) DONALD MOORE 22, Orchard Road, SINGAPOIU INDIA ORIENT LONGMANS LTD., BOMBAY: Nicol Road, Ballard Estate. CAICUTTA- 17, Chittaranjan Ave. MADRAS: 36A, Mount Road. SOUTH AFRICA I f B. TIMMINS P O. Box 94, CAPE TOWN Show'oom 58-60, Long Street. Printed & Bound in England for the UNiVERbiiT OF LOXDO.N PKE>N LTD., by HAZELL, WATSON & VINEY, LTD., Aylesbury and London CONTENTS PAQK Preface to the First Editior , xiii Preface to the Second Edition ..... xiv Preface to the Third Edition xv Preface to the Fourth Edition . . . . . xv PART I. THE ANALYSIS OF TESTS CHAPTER I. THE THEORY OF Two FACTORS 1. Factor tests ....... 3 2. Fictitious factors ...... 4 3. Hierarchical order ...... 5 4. G saturations ....... 8 5. A weighted battery . . . . . .10 6. Oval diagrams . . . . . . .11 7. Tetrad-differences 12 8. Group factors . . . . . . .14 9. The verbal factor 15 10. Group-factor saturations . . . . .17 11. The bifactor method 19 12. Vocational guidance . . . . . .19 CHAPTER II. MULTIPLE-FACTOR ANALYSIS 1. Need of group factors ...... 20 2. Rank of a matrix and nurnber of factors . . 20 3. Thurstone's method used on a hierarchy. . . 23 4. Second stage of the " centroid " method . . .25 5. A three -factor example . . \ . .26 6. Comparison of analysis with diagram ... 32 7. Analysis into two common factors .... 33 8. Alexander's rotation ...... 30 9. Unique communalities ...... 38 vi CONTENTS PAGE CHAPTER III. THE SAMPLING THEORY 1. Two views. A hierarchical example as explained by one general factor ...... 42 2. The alternative explanation. The sampling theory 45 3. Specific factors maximized ..... 48 4. Sub-pools of the mind ...... 50 5. The inequality of men ...... 53 CHAPTER IV. THE GEOMETRICAL PICTURE 1. The fundamental idea . . . . . .55 2. Sectors of the crowd . . . . . .56 3. A third test added. The tripod .... 57 4. A fourth test added ...... 59 5. Two principal components ..... 60 6. Spearman axes for two tests ..... 61 7. Spearman axes for four tests . . . . .62 8. A common-factor space of two dimensions . . 63 9. The common-factor space in general ... 64 10. Rotations 64 CHAPTER V. HOTELLING'S " PRINCIPAL COMPONENTS " 1. Another geometrical picture ..... 66 2. The principal axes ...... 68 3. Advantages and disadvantages .... 69 4. A calculation ....... 70 5. Acceleration by powering the matrix . . .74 6. Properties of the loadings ..... 76 7. Calculation of a man's principal components. Esti- mation unnecessary ...... 78 PART II. THE ESTIMATION OF FACTORS CHAPTER VI. ESTIMATION AND THE POOLING SQUARE 1. Correlation coefficient as estimation coefficient . 83 2. Three tests 84 3. 'The straight sum and the pooling square. . . 85 CONTENTS vii PAGE 4. The pooling square with weights .... 86 5. Regression coefficients and multiple correlation . 87 6. Aitken's method of pivotal condensation . . 89 7. A larger example ...... 91 8. The geometrical picture of estimation ... 95 9. The " centroid " method and the pooling square . 98 10. Conflict between battery reliability and prediction . 100 CHAPTER VII. THE ESTIMATION OF FACTORS BY REGRESSION 1. Estimating a man's " g " ..... 102 2. Estimating two common factors simultaneously . 107 3. An arithmetical short cut . . . . .110 4. Reproducing the original scores . . . .112 5. Vocational advice with and without factors . .114 6. Why, then, use factors at all ? . . . .116 7. The geometrical picture of correlated estimates . 120 8. Calculation of correlation between estimates . .124 CHAPTER VIII. MAXIMIZING AND MINIMIZING THE SPECIFICS 1. A hierarchical battery . . . . . .130 2. Batteries of higher rank . . . . .181 3. Error specifics . . . . . . .132 4. Shorthand descriptions . . . . .133 5. Bartlett's estimates of common factors . . .134 6. Their geometrical interpretation . . . .136 7. A numerical example . . . . . .138 PART III. THE INFLUENCE OF SAMPLING AND SELECTION OF THE PERSONS CHAPTER IX. SAMPLING ERROR AND THE THEORY OF Two FACTORS 1. Sampling the population of persons . . .143 2. The normal curve . . . . . .145 3. Error of a single tetrad-difference . . . .150 viii CONTENTS PAGE 4. Distribution of a group of tetrad-differences . . 151 5. Spearman's saturation formula . . . .153 6. Residues . . . . . . . .155 7. Reference values for detecting specific correlation . 156 CHAPTER X. MULTIPLE-FACTOR ANALYSIS WITH FALLIBLE DATA 1. Method of approximating to the communalities . 161 2. Illustrated on the Chapter II example . . .163 3. Error specifics . . . . . . .167 4. Number of factors . . . . . .168 CHAPTER XI. THE INFLUENCE OF UNIVARIATE SELECTION ON FACTORIAL ANALYSIS 1. Univariate selection . . . . . .171 2. Selection and partial correlation . . .173 3. Effect on communalities . . . . .175 4. Hierarchical numerical example . . . .176 5. A sample all alike in Test 1 180 6. An example of rank 2 . . . . . .182 7. A simple geometrical picture of selection . .184 8. Random selection . . . . . .185 9. Oblique factors . . . . . . .186 CHAPTER XII. THE INFLUENCE OF MULTIVARIATE SELECTION 1 . Altering two variances and the co variance . .187 2. Aitken's multivariate selection formula . . .188 3. The calculation of a reciprocal matrix . . .190 4. Features of the sample co variances . . .192 5. Appearance of a new factor . . . . .193 PART IV. CORRELATIONS BETWEEN PERSONS CHAPTER XIII. REVERSING THE ROLES 1. Exchanging the roles of persons and tests . .199 2. Ranking pictures, essays, or moods . . . 201 3. The two sets of equations ..... 202 CONTENTS ix PAGE 4. Weighting examiners like a Spearman battery . . 203 5. Example from The Marks of Examiners . . . 206 6. Preferences for school subjects .... 208 7. A parallel with a previous experiment . . . 210 8. Negative loadings . . . . . .211 9. An analysis of moods . . . . . .211 CHAPTER XIV. THE RELATION BETWEEN TEST FACTORS AND PERSON FACTORS 1. Rurt's example, centred both by rows and by columns 218 2. Analysis of the co variances . . . . .214 3. Factors possessed by each person and by each test . 216 4. Reciprocity of loadings and factors . . .217 5. Special features of a doubly centred matrix . .218 6. An actual experiment . . . . . .221 PART V. THE INTERPRETATION OF FACTORS CHAPTER XV. THE DEFINITION OF g 1. Any three tests define a "g" . . . .225 2. The extended or purified hierarchical battery . . 226 3. Different hierarchies with two tests in common . 228 4. A test measuring " pure g " . . . . . 230 5. The Hey wood case 231 6. Hierarchical order when tests equal persons in number ........ 234 7. Singly conforming tests ..... 236 8. The danger of " reifying " factors . . . .240 CHAPTER XVI. " ORTHOGONAL SIMPLE STRUCTURE " 1. Simultaneous definition of common factors . . 242 2. Need for rotating the axes . . . . .243 3. Agreement of mathematics and psychology . . 244 4. An example of six tests of rank 3 . . . . 245 5. Two-by-two rotation ...... 247 6. New rotational method ..... 249 7. Finding the new axes . . . . . .251 8. Landahl preliminary rotations . . - .255 x CONTENTS PAGE 9. A four-dimensional example . . . . .258 10. Ledermann's method of reaching simple structure . 260 11. Leading to oblique factors ..... 261 CHAPTER XVII. LIMITS TO THE EXTENT OF FACTORS 1. Boundary conditions in general .... 262 2. The average correlation rule . . . . .263 3. The latent-root rule 267 4. Application to the common-factor space. . . 268 5. A more stringent test . . . . . .270 CHAPTER XVIII. OBLIQUE FACTORS, AND CRITICISMS 1. Pattern and structure ...... 272 2. Three oblique factors 273 3. Primary factors and reference vectors . . . 277 4. Behind the scenes ...... 280 5. Box dimensions as factors ..... 282 6. Criticisms of simple structure .... 284 7. Reyburn and Taylor's alternative method . . 287 8. Special orthogonal matrices . . . . .291 9. Identity of oblique factors after univariate selection 292 10. Multivariate selection and simple structure . . 294 11. Parallel proportional profiles .... 295 12. Estimation of oblique factors .... 296 CHAPTER XIX. SECOND-ORDER FACTORS 1. A second-order general factor . . ... 297 2. Its correlations with the tests .... 298 3. A g plus an orthogonal simple structure . . .301 CHAPTER XX. THE SAMPLING or BONDS 1. Brief statement of views ..... 303 2. Negative and positive correlations .... 304 3. Low reduced rank ...... 306 4. A mind with only six bonds ..... 307 5. A mind with twelve bonds . . . . .311 CONTENTS xi PAGE 6. Professor Spearman's objections to the sampling- theory ........ 313 7. Contrast with physical measurements . . .315 S. Interpretation of g and specifics on the sampling theory . . . 317 9. Absolute variance of different tests . . .318 10. A distinction between g and other common factors . 319 CHAPTER XXI. THE MAXIMUM LIKELIHOOD METHOD OF ESTIMATING FACTOR LOADINGS 1. Basis of statistical estimation .... 321 2. A numerical example . . . . . .321 3. Testing significance . . . . . .324 4. The standard errors of individual residuals . . 326 5. The standard errors of factor loadings . . . 327 6. Advantages and disadvantages . . . .321) CHAPTER XXII. SOME FUNDAMENTAL QUESTIONS 1. Metric 330 2. Rotation 331 3. Specifics 336 4. Oblique factors 339 Addenda 341 Bifactor analysis ....... 343 Variances and co variances of regression coefficients . 350 The geometrical picture ..... 353 Methods of estimating communalities . . . 354 Mathematical Appendix ...... 357 References 383 Index ......... 391 LIST OF DIAGRAMS PAGE 1. Two ovals, illustrating correlation by overlap . 11 2. The same, with different variances . . .11 3. Four ovals, overlap of four tests . . .11 4. Histogram of tetrad-differences . . . .14 5. A pair of four-oval diagrams giving the same 15 6. correlations . . . . . . .15 7. Three four-oval diagrams all giving the same 8. correlations, one with three common factors, 9. the others with two . . . .26, 34, 38 10. 11. Four-oval diagrams comparing the two -factor 12. theory and the sampling theory 13. 44 14. Correlation as cosine of angle . . . .55 15. Two principal components . . . .60 16. Spearman axes for four tests .... 62 17. Common-factor space of two dimensions . . 63 18. Elliptical density-contour ..... 07 19. Estimation by regression ..... 96 20. Correlation of estimated factors . . . .121 21. Illustrating Bartlett's estimates . . . .136 22 . A normal curve . . . . . 1 46 jo/ Two diagrams illustrating selection . . 1 84 r 25 1 2fi Two-by-two rotation .... 247, 248 27. Extended vectors 250 28. The ceiling triangle . . . . . .251 29. More common factors than three . . .260 30. Oblique factors 275 31. Ceiling triangle after univariate selection . . 293 PREFACE TO THE FIRST EDITION THE theory of factorial an aly sis is mathematical in nature, but this book has been written so that it can, it is hoped, be read by those who have no mathematics beyond the usual secondary school knowledge. Readers are, however, urged to repeat some at least of the arithmetical calculations for themselves. It is probable that the subject-matter of this book may seem to teachers and administrators to be far removed from contact with the actual work of schools. I would like therefore to explain that the incentive to the study of factorial analysis comes in my case very largely from the practical desire to improve the selection of children for higher education. When I was thirteen years of age and finishing an elementary school education, I won a " scholar- ship " to a secondary school in the neighbouring town, one of the early precursors of the present-day " free places " in England. I have ever since then been greatly impressed by the influence that event has had on my life, and have spent a great deal of time in endeavouring to improve the methods of selecting pupils at that stage and in lessening the part played by chance. It was inevitable that I should be led to inquire into the use of intelligence tests for this purpose, and inevitable in due course that the possibilities of factorial analysis should also come under consideration. It seemed to me that before any practical use could be made of factorial analysis a very thoroughgoing examina- tion of its mathematical foundations was necessary. The present book is my attempt at this. ... It may seem remote from school problems. But much mathematical study and many calculations have to precede every improvement in engineering, and it will not be otherwise in the future with the social as well as with the physical sciences. GODFREY H. THOMSON MORAY HOUSE, UNIVERSITY OF EDINBURGH, November 1938 xiv PREFACE PREFACE TO THE SECOND EDITION THE former chapter on Simple Structure has been entirely rewritten and expanded into three chapters on Orthogonal Simple Structure, Oblique Factors, and Second-order Factors, with a corresponding expansion of section 19 of the mathematical appendix. A chapter on estimating factor loadings by the method of maximum likelihood and a corresponding section of the mathematical appendix have been supplied by Dr. D. N. Lawley. Many smaller changes have been made in the other chapters, which have in some cases been supplemented by Addenda at the end of the book. I owe Dr. Lawley thanks for much other assistance as well as for the above chapter, and I am again indebted to Dr. Walter Ledermann for stimulating dis- cussions of several points and especially for suggesting, by a remark of his, the geometrical interpretation of " structure " and " pattern." He and Mr. Emmett have again read the proofs, and the latter has made the now index. GODFREY H. THOMSON MORAY HOUSE, UNIVERSITY OF EDINBURGH, July 1945 PREFACE TO THE THIRD EDITION CONDITIONS in the printing trade have again made it desirable to break as few pages as possible in making changes, and the page-numbering is unaltered, in the main, up to page 292, where three sections have had to be in- serted on the identity of oblique factors after univariate selection, on multivariate selection and simple structure, and on parallel proportional profiles. A small addition has been made in the mathematical appendix at the end of section 19, and a new section 9a added on relations between two sets of variates, corresponding to a new section in the text, on pages 100 and 101, on the conflict between battery reliability and prediction. Deletions have been made near page 150 to make room for a fuller explanation of PREFACE xv " degrees of freedom " and for the transfer to the text of the Addendum in the second edition about Fisher's z. Some changes and additions have been made in pages 165- 170 concerning the number of significant factors in a cor- relation matrix, and a number of smaller changes here and there throughout. GODFREY H. THOMSON MORAY HOUSE, UNIVERSITY OF EDINBURGH, February 1948 PREFACE TO THE FOURTH EDITION Two sections are added to Dr. Lawley's chapter (XXI), giving formulae for the standard errors of individual residuals, and of factor loadings, when maximum likeli- hood methods have been used. A section, 10a, on Leder- mann's shortened calculation of regression coefficients for estimating a man's factors now appears in the mathe- matical appendix (where it had inadvertently been omitted previously). I have added a section on estimating oblique factors to Chapter XVIII, and in section 19 of the mathematical appendix, I give the modifications necessary in Ledermarm's shortened calculation when oblique factors are in question. Other changes here and there arc only slight. GODFREY H. THOMSON MORAY HOUSE, UNIVERSITY OF EDINBURGH, January 1949 All science starts with hypotheses in other words, with assumptions that are unproved, while they may be, and often are, erroneous ; but which are better than nothing to the searcher after order in the maze of pheno- mena. T. H. HUXLEY PART I THE ANALYSIS OF TESTS To simplify and clarify the exposition, errors due to sampling the population of persons are in Parts I and II assumed to be non-existent. CHAPTER I THE THEORY OF TWO FACTORS 1. Factor tests. The object of this book is to give some account of the " factorial analysis *' of ability, as it is called. In actual practice at the present day this science is endeavouring (with what hope of success is a matter of keen controversy) to arrive at an analysis of mind based on the mathematical treatment of experimental data obtained from tests of intelligence and of other qualities, and to improve vocational and scholastic advice and prediction by making use of this analysis' in individual cases. r It is a development of the " testing " movement the movement in which experimenters endeavour to devise tests of intelligence and other qualities in the hope of sorting mankind, and especially children, into different categories for various practical purposes ; educational (as in directing children into the school courses for which they are best suited) ; administrative (as in deciding that some persons are so weak-minded as to need lifelong institutional care) ; or vocational, etc. There are many psychologists who would deny that from the scores in such tests, or indeed from any analysis, we can (ever) return to a full picture of the individual ; and without entering into any discussion of the fundamental controversy which this denial reveals, everyone who has had anything to do with tests will readily agree that this is certainly so at present in practice. But the tester may be allowed to try to make his modest diagram of the individual better, more useful, and if possible simpler. Now, the broadest fact about the results of " tests " of air sorts, when a large number of them is given to a large number of people, is that every individual and every test is different from every other, and yet that there are certain rather vague similarities which run through groups of people or groups of tests, not very well marked off from 3 4 THE FACTORIAL ANALYSIS OF HUMAN ABILITY one another but merging imperceptibly into neighbouring groups at their margins. To describe an individual ac- curately and completely one would have to administer to him all the thousand and one tests which have been or may be devised, and record his score in each, an impossible plan to carry out, and an unwieldy record to use even if obtained. Both practical necessity and the desire for theoretical simplification lead one to seek for a few tests which will describe the individual with sufficient accuracy, and possibly with complete accuracy if the right tests can be found. If, as has been said, there is some tendency for the tests to fall into groups, perhaps one test from each group may suffice. Such a set of tests might then be said to measure the " factors " of the mind. 2. Fictitious factors. Actually the progress of the " factorial " movement has been rather different, and the factors are not real but as it were fictitious tests which represent certain aspects of the whole mind. But con- ceivably it might have taken the more concrete form. In that case the " factor tests " finally decided upon (by whom, the reader will ask, and when " finally " ?) would be a set of standards which, like any other standards, would have to be kept inviolate, and unchanged except at rare intervals and for good reasons. Some tendency towards this there has been. The Binet scale of tests is almost an international standard, and there is a general agreement that it must not be changed except by certain people upon whose shoulders Binet's mantle has fallen, and only seldom and as little as possible even by them. But the Binet scale is a very complex entity, and rather represents many groups of tests than any one test. By " factor tests " one would more naturally mean tests of a " pure " nature, differing widely from one another so as to cover the whole personality adequately. And since actual tests always are more or less mixed, it is understandable why " factors " have come to be fictitious, not real, tests, to be each approximated to by various combinations of real tests so weighted that their unwanted aspects tend to cancel out, and their desired aspects to reinforce one another, the team approximating to a measure of the pure " factor." THE THEORY OF TWO FACTORS 5 But how, the reader will ask, do we know a " pure " factor, how are we to tell when the actual tests approximate to it ? To give a preliminary answer to that question we must go back to the pioneer work of Professor Charles Spearman in the early years of this century (Spearman, 1904). The main idea which still, rightly or wrongly, dominates factorial analysis was enunciated then by him, and practically all that has been done since has been either inspired or provoked by his writings. His discovery was that the " coefficients of correlation " between tests tend to fall into " hierarchical order," and he saw that this could be explained by his famous " Theory of Two Factors." These technical terms we must now explain. 3. Hierarchical order. A coefficient of correlation is a number which indicates the degree of resemblance between two sets of marks or scores. If a schoolmaster, for example, gives two examination papers to his class, say (1) in arith- metic and (2) in grammar, he will have two marks for every boy in the class. If the two sets of marks are identical the correlation is perfect, and the correlation coefficient, denoted by the symbol r 12 , is said to be + 1. If by some curious chance the one list of marks is exactly like the other one upside down (the best boy at arithmetic being worst at grammar, and so on), the correlation is still perfect, but negative, and r lz = 1. If there is absolutely no resemblance between the two lists, r 12 = 0. If there is a strong resemblance, but falling short of identity, r 12 may equal -9 ; and so on. There is a method (the Bravais- Pearson) of calculating such coefficients, given the list of marks.* " Tests " can obviously be correlated just like * The " product-moment formula " is sum (a?!^ 2 ) 12 ~~ <v/{ sum ( x i 2 ) X sum ( x z z ) where x { and a? 2 are the scores in the two tests, measured from the average (so that approximately half the scores are negative), and the sums are over the persons to whom the scores apply. The quantity a 2 = sum (a;, 2 ) 1 number of persons is called the variance of Test 1, and a its standard deviation. If the scores in each test are not only measured from their average, but 6 THE FACTORIAL ANALYSIS OF HUMAN ABILITY examinations, and a convenient form in which to write down the intercorrelations of a number of tests is in a square chequer board with the names of the tests (say a, b, c . .) written along the two margins, thus : a b c d e / a 48 24 54 42 30 b 48 . 32 72 56 40 c 24 32 . 36 28 20 d 54 72 36 63 45 e 42 5(5 28 63 35 f 30 40 20 45 35 Totals 1-98 2-48 1-40 2-70 2-24 1-70 It was early found that such correlations tend to be positive, and it is of some interest to see which of a number of tests correlates most with the others. This can be found by adding up the columns of the chequer board, when we see in the above example that the column referring to Test d has the highest total (2-70). The tests can then be rearranged and numbered in the order of these totals, thus : 1 2 3 4 5 6 d ' b e a / c 1 d 72 63 54 45 36 2 b 72 56 48 40 32 3 e 63 56 . 42 35 28 4 a 54 48 42 30 24 5 / 45 40 35 30 20 6 c 36 32 28 24 20 After the tests have been thus arranged, the tendency which Professor Spearman was the first to notice, and which are then divided through by their standard deviation, they are said to be standardized, and we represent them by ^ and 2 2 . About two-thirds of them, then, lie between plus and minus one. With such scores Pearson's formula becomes __ sum of the products z^z 2 1 number of persons p In theoretical work, an even larger unit is used, namely o\/p. With these units, the sum of the squares is unity, and the sum of the products is the correlation coefficient. The scores are then said to be normalized, but note that this does not mean distributed in a " normal " or Gaussian manner. THE THEORY OF TWO FACTORS 7 he called " hierarchical order," is more easily seen. It is the tendency for the coefficients in any two columns to have a constant ratio throughout the column. Thus in our example, if we fix our attention on Columns a and /, say, they run (omitting the coefficients which have no partners) thus : 54 -45 48 -40 42 -35 24 -20 and every number on the right is five-sixths of its partner on the left. Our example is a fictitious one, and the tendency to hierarchical order in it has been made perfect in order to emphasize the point. It must not be supposed that the tendency is as clear in actual experimental data. Indeed, at the time there were some who denied altogether the existence of any such tendency in actual data. Those who did so were, however, mistaken, although the tendency is not as strong as Professor Spearman would seem originally to have thought (Spearman and Hart, 1912). The follow- ing is a small portion of an actual table of correlation coeffi- cients* from those days (Brown, 1910, 309). (Complete tables must, of course, include many more tests ; in recent work as many as 57 in one table.) 1 2 3 4 5 6 1 t 78 45 27 59 30 2 78 . 48 28 51 24 3 45 48 , 52 40 38 4 27 28 52 . 41 38 5 59 51 40 41 . 13 6 30 24 38 38 13 * In this, as in other instances where data for small examples are taken from experimental papers, neither criticism nor comment is in any way intended. Illustrations are restricted to few tests for economy of space and clearness of exposition, but in the experiments from which the data are taken many more tests are employed, and the purpose may be quite different from that of this book. 8 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 4. G saturations. This tendency to " hierarchical order " was explained by Professor Spearman by the hypothesis that all the correlations were due to one " factor " only, present in every test, but present in largest amount in the test at the head of the hierarchy. This factor is his famous " ," to which he gave only this algebraic name to avoid making any suggestions as to its nature, although in some papers and in The Abilities of Man he has permitted himself to surmise what that nature might be. Each test had also a second factor present in it (but not to be found elsewhere, except indeed in very similar varieties of the same test), whence the name, " Theory of Two Factors " really one general factor, and innumerable second or specific factors. It will be proved in the Mathematical Appendix* that this arrangement would actually give rise to " hierarchical order." Meanwhile this can at least be made plausible. For if Test d has that column of correlations (the first in our table) with the other tests solely because it is saturated with so-and-so much g ; and if Test b has less g in it than d has, it seems likely enough that fe's column of correlations will all be smaller in that same proportion. We can, moreover, find what these " saturations ' with g are. For on the theory, each of our six tests contains the factor g, and another part which has nothing to do with causing correlation. Moreover, the higher the test is in the hierarchical ranking, the more it is " saturated " with. Imagine now a fictitious test which had no specific, a test for g and for nothing else, whose saturation with g is 100 per cent., or 1-0. This fictitious test would, of course, stand at the head of the hierarchy, above our six real tests, and its row of correlations with each of those tests (their 46 saturations ") would each be larger than any other in the same column. What values would these saturations take ? Before we answer this, let us direct our attention to the diagonal cells of the " matrix " of correlations (as it is called a matrix is just a square or oblong set of numbers), cells which we have up to the present left blank. Since each number in our matrix represents the correlation of the two tests in whose column and row it stands, there should * Para. 3 ; and see also Chapter XI, end of Section 2, page 175. THE THEORY OF TWO FACTORS g 1 2 3 4 5 6 g 1 r i 9 r ** r 3. r 4. *5f r e. 1 r lff . 72 63 54 45 36 2 T *9 -72 . 56 48 40 32 3 r s ff -63 56 ". 42 35 28 4 T 4ff 54 48 42 . 30 24 5 T 5g 45 40 35 30 . 20 6 r e 36 32 28 24 20 . be inserted in each diagonal cell the number unity 9 repre- senting the correlation of a test with its own identical self. In these ^//correlations, however, the specific factor of each test, of course, plays its part. These self-correlations of unity are the only correlations in the whole table in which specifics do play any part. These " unities," there- fore, do not conform to the hierarchical rule of propor- tionality between the columns. But the case is different with the fictitious test of pure g. It has no specific, and its self-correlation of unity should conform to the hierarchy. If, therefore, we call the " saturations " of the other tests r lg9 r 2g9 r 3g9 r^ g9 r 5g9 and r 6g9 we see that we must have, as we come down the first two columns within the matrix r lg _ -72 __ -63 -54 __ -45 -36 and similar equations for each other column with the g column, which together indicate that the six " saturations " are ~~ -9 -8 -7 -6 -5 -4 Furthermore, each correlation in the table is the product of two of these saturations. Thus 72 = -9 X -8 42 = -7 X -6 **34 = T 3 ff X r lg The six tests can now be expressed in the form of equations: % = . 9 g + -436 5l 800*4 866s 5 917s, F.A. 1 10 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Herein, each z represents the score of some person in the test indicated by the subscript, a score made up of that person's g and specific in the proportions indicated by the coefficients. The scores are supposed measured from the average of all persons, being reckoned plus if above the average and minus if below ; and so too are the factors g and the specifics. And each of them, tests and factors, is " standardized," i.e. measured in such units that the sum of the squares of all the scores equals the number of persons. This is achieved by dividing the raw scores by the " standard deviation." The saturations of the specifics are such that the sum of the squares of both saturations comes in each test to unity, the whole variance of that test. Thus 436 5. A weighted battery. This brief outline of the Theory of Two Factors must for the moment suffice. It is enough to enable the question to be answered which at the end of our Section 2 led to the digression. " How," the reader asked, " do we know a pure factor, how are we to tell when the actual tests approximate to it ? " In the Two-factor Theory the important pure factor was g itself, and a test approximated to it the more, the higher it stood in the hierarchy. Its accuracy of measurement of g was indicated by its " saturation." And a battery of hier- archical tests could be weighted so as to have a combined saturation higher than that of any one member, each test for this purpose being weighted (as will be shown in Chapter ' VII) by a number proportional to - ^ , where r^ is the I Tig g saturation of Test i (Abilities, p. xix). Although g remained a fiction, yet a complex test, made up of a weighted battery of tests which were hierarchical, could approach nearer and nearer to measuring it exactly, as more tests were added to the hierarchy. Each test added would have to conform to the rule of proportionality in its correlations with the pre-existing battery. If it did not do so it would have to be rejected. The battery at any stage would form a kind of definition of g, which it ap- THE THEORY OF TWO FACTORS 11 preached although never reached. And a man's weighted score in such a battery would be an estimate of his amount of g, his general intelligence. The factorial description of a man was at this period confined to one factor, since the specific factors were useless as description of any man. For one thing, they were innumerable ; and for another, being specific, they were only able to indicate how the man would perform in the very tests in which, as a matter of fact, we knew exactly how he had performed. 6. Oval diagrams. It is convenient at this point to introduce a diagrammatic illustration which will be useful in the less technical part of this book, although like all illustrations it must be taken only as such, and the analogy must not be pushed too far. If we represent the two abilities, which are measured by tests, by two over- lapping ovals as in Figure 1, then the amount of the overlap can be made to represent the degree to which these tests are corre- lated. If we call the whole area Figure 1. of each oval the " variance " of Figure 2. that ability, we shall be intro- ducing the reader to another technical term (of which a de- finition was given in the footnote to page 5). Here it need mean nothing more than the whole 66 amount " of the ability. The overlap we shall call the " co- variance." If the two variances Figure 3 are each equal to unity, then the co variance is the correlation coefficient. To make the diagram quantitative, we can indicate in figures the contents of each part of the variance, as in the instance shown, which gives a correlation of ^, or -6. If the separate parts of each variance (i.e. of each oval) do not add up to the same quantity, but to v^ and v Z9 say, then the co variance (the amount in the overlap) must be 12 THE FACTORIAL ANALYSIS OF HUMAN ABILITY divided by V^i^ in order to give the correlation. Thus, Figure 2 represents a correlation of 3 -f- \/(4 X 9) = *5. No attempt is made in the diagrams to make the actual areas proportional to the parts of the variance, it is the numbers written in each cell which matter. The four abilities represented by four tests can clearly overlap in a complicated way, as in Figure 3, which shows one part of the variance (marked g) common to all four of the tests ; four parts (left unshaded) each common to three tests ; six parts (shaded) each common to two tests ; and four outer parts (marked s) each specific to one test only. The early Theory of Two Factors adopted the hypothesis that, except for very similar varieties of the one test, none of the cells of such a diagram had any contents save those marked g and s, the general and the specific factors. The " variance " of each ability was in that theory completely accounted for by the variance due to g, and the variance due to s. 7. Tetrad-differences. In Section 3 it was explained that the discovery made by Professor Spearman was that the correlation coefficients in two columns tend to be in the same ratio as we go up and down the pair of columns. That is to say, if we take the columns belonging to Tests b and /, and fix our attention on the correlations which b and / make with d and e, we have : b f 72 45 56 35 d e where -72 = --56 45 ~~ ^35 This may be written 72 X -35 -45 X -56 = and in this form is called a " tetrad-difference." In symbols this one is V e/ - rjr* = Spearman's discovery may therefore be put thus : " The tetrad-differences are, or tend to be, zero." It is clear that THE THEORY OF TWO FACTORS 18 this will be so if, as we said was the case in the Theory of Two Factors, each correlation is the product of two cor- relations with g. For then the above tetrad-difference becomes which is identically zero. The present-day test for hier- archical order in a correlation matrix is to calculate all the tetrad -differences (always avoiding the main diagonal) and see if they are sufficiently small. If they are, then the correlations can be explained by a diagram of the same nature as Figure 3, by one general factor and specifics. It is, of course, not to be expected in actual experimenting that the tetrad-differences will be exactly zero ; no experi- ment on human material can be as accurate as that. What is required is that they shall be clustered round zero in a narrow curve, falling off steadily in frequency as zero is departed from. The number of tetrad-differences increases very rapidly as the number of tests grows, and in an actual experimental battery the tetrads arc very numerous indeed. In the small portion of a real correlation table given above (page 7), with six tests, there are 45 tetrad-differences,* and in this instance they are distributed as follows (taking absolute values only and disregarding signs, which can be changed by altering the order of the tests) : From -0000 to -0999, 28 tetrad-differences. From -1000 to -1999, 13 tetrad-differences. From -2000 to -2796, 4 tetrad-differences. This distribution of tetrads can be represented by a " histogram " like that shown in Figure 4, which explains itself. It is clear that some criterion is required by which we can know whether the distribution of tetrad-differences, after they have been calculated, is narrow enough to justify us in assuming the Theory of Two Factors. This criterion is explained in Part III of this book. One form of it consists in drawing a distribution curve to which, on grounds of sampling, the tetrad-differences may be expected to con- form. Any tetrad -differences which seem to be too large * Not all independent. 13 28 28 13 3 -2 -I O -I Figure 4. -2 -3 14 THE FACTORIAL ANALYSIS OF HUMAN ABILITY to be accounted for by the Theory of Two Factors are then examined, to see whether the tests giving them have any special points of resemblance, in content, method, or other- wise, which may explain why they disturb the hierarchy. 8. Group factors. As time went on it became clear that the tendency to zero tetrad- differences, though strong, was not universal enough to permit an explanation of all correla- tions between tests in terms of g and specifics, with a few slight " disturbers " in the form of slightly overlapping specifics. It became necessary to call in group factors, which run through many though not through all tests, to explain the deviations from strict hierarchical order. The Spearman school of experimenters, however, tend always to explain as much as possible by one centra] factor, and to use group factors only when necessitated. They take the point of view that a group factor must as it were establish its right to existence, that the onus of proof is on him who asserts a group factor. As a tiny artificial illustration, a matrix of correlation coefficients : 1 2 3 4 would be examined, and its three tetrad-differences found to be : zero -15 15 Inspection shows that the correlation r 23 is the cause of the discrepancies from zero, and the experimenter trained in 1 2 3 4 ^ 5 5 5 5 . > -8 -5 5 8 , 5 5 -5 5 % THE THEORY OF TWO FACTORS 15 the Two-factor school would therefore explain these correlations by a central factor running through them all, plus a special link joining Tests 2 and 3, as in Figure 5. There are innumerable other possible ways of explaining these same correlations. For example, the linkages be- tween the tests might be as in Figure 6, which gives exactly the same correlations. This lack of uniqueness is something which must always be borne in mind in studying factorial analysis. There are always, as here, in- numerable possible analyses, and the final decision between them has to be made on some other grounds. The decision may be psychological, as when for ex- ample in the above case an experimenter chooses one of the possible diagrams because it best agrees with his psychological ideas about the tests. Or the decision may be made on the ground that we should be par- simonious in our invention of 44 factors," and that where one general and one group factor will serve we should not invent five group factors as required by Figure 6. Both diagrams, however, fit the correlational facts exactly, and so also would hundreds of other diagrams which might be made. As has been said, the two- factor tendency is to take the diagram with the largest general factor (and the largest specifics also) and with as few group factors as possible. 9. The verbal factor. In this way the Theory of Two Factors has gradually extended the " two " to include, in addition to g and specifics, a number of other group factors, still, however, comparatively few. These group factors bear such names as the verbal factor v, a mechanical factor m, an arithmetic factor, perseveration> etc. The charafc- Figure 6. 16 THE FACTORIAL ANALYSIS OF HUMAN ABILITY teristic method of the Two-factor school can be well seen, without any technical difficulties unduly obscuring the situation, in the search for a verbal factor. The idea that, in addition to a man's g (which is generally thought of as something innate) there may be an acquired factor of verbal facility which enables him to do well in certain tests, is a not unnatural one. A battery of tests can be assembled, of which half do, and half do not, employ words in their construction or solution. The correlation matrix will then have four quadrants, the quadrant V containing the correlations of the verbal tests am6ng themselves, the quadrant P the correlations of the non-verbal or, say, pictorial tests, and the quadrants C containing the cross- correlations of the one kind of test with the other. If the whole table is sufficiently " hierarchical," there is no evidence for a group factor v or a group factor p. If either of these factors exists, there will be differences to be noticed between the six kinds of tetrad which can be chosen, namely : P P (1) v p x . (4) x V V p p p p v p X .(5) p x x (3) p x x v p p X (6) X A tetrad like 1, with two verbal tests along one margin and two pictorial tests along the other, will be found in THE THEORY OF TWO FACTORS 17 quadrant C. Neither a factor common to the verbal tests only, nor one common to the pictorial tests only, will add anything to any of the four correlations in such a tetrad- difference, which may be expected, therefore, to tend to be zero. If the tetrads in C seem to do so, the other tetrads can be examined. Tetrad 2 is taken wholly from the V quadrant. In it the verbal factor, if any is present, will reinforce all the four correlations, and should not therefore disturb very much the tendency to a zero tetrad -difference. (Reinforced correlations are marked by x in the diagrams.) The same is true of Tetrad 3 taken wholly from the P quadrant. Tetrads 4 and 5 have each two of their cor- relations reinforced, by the v factor in 4 and by the p factor in 5, but in each case in such a way as not to change very much the tetrad -difference. It is when we come to tetrads like 6, which have one correlation in each of the four quadrants, that the presence of either or both factors should show itself strongly : for the two reinforced correla- tions here occur on a diagonal, and inflate only the one member of the tetrad-difference T T T T 1 vv' pp ' vp' pv If, then, a verbal factor, and also a pictorial factor, are present, the tendency for the tetrad-differences to vanish should become less and less strong as we consider tetrads of the kinds 1, 2 and 3, 4 and 5, and especially 6, where the tetrad-differences should leap up. If only the verbal factor is present, tetrad-differences of the kind 3 should vanish rather more than those of the kind 2. But it will not be easy to distinguish between either suspected factor, and both. Tetrads like 6, however, should give conclusive evidence of the presence of one or the other, if not both. Methods like this were employed by Miss Davey (Davey, 1926), who found a group factor, but not one running through all the verbal tests, and by Dr. Stephenson (Stephenson, 1931), whose results indicated the presence of a verbal factor.* 10. Group-factor saturations. Just as the g saturations * T. L. Kelley had already found by other methods strong evidence of a verbal factor (Kelley, 1928, 104, 121 et passim). 18 THE FACTORIAL ANALYSIS OF HUMAN ABILITY of tests can be calculated, so also can the saturation of a test with any group factor it may contain. The general method of the Two-factor school is first to work with batteries of tests which give no unduly large tetrad- differences, and which also appear to satisfy one's general impression that they test intelligence. From such a battery, of which the best example is that of Brown and Stephenson (B. and S., 1933), the g saturations can be calculated.* Each test has, however, also its specific, which, so long as it is in the hierarchical battery, is unique to it and shared with no other member of the battery. A test may now be associated with some other battery of different tests, and with some of these it may share a part of its former specific, as a group factor which will increase its correlation beyond that caused by g. The excess correla- tion enables the saturation of the test with this group factor to be found the details are too technical for this chapter and the specific saturation correspondingly reduced. Finally, the tester may be able to give the composition of a test as, let us say (to invent an example) Tig + -40t; + -34n + -47s where g is Spearman's g, v is Stephenson's verbal factor, n is a number factor, and s is the remaining specific of the test. The coefficients are the " saturations " of the test with each of these ; that is, the correlations believed to exist between the test and these fictitious tests called factors. The squares of these saturations represent the fractions of the test-variance contributed by each factor, and these squares sum to unity, thus : Saturation Squared g -5041 v -1600 n -1156 s . . . -2209 1-0006 * For the sake of clarity the text here rather oversimplifies the situation. The battery of Brown and Stephenson contains In fact a rather large group factor as Well as g and specifics. THE THEORY OF TWO FACTORS 19 11. The bif actor method. Holzinger's Bif actor Method (Holzinger, 1935, 1937a) may be looked upon as another natural extension of the simple Two-factor plan of analysis. It endeavours to analyse a battery of tests into one general factor and a number of mutually exclusive group factors. A diagram of such an analysis looks like a " hollow stair- case," thus : I Test g h k 1 X X 2 X X 3 X X 4 X X 5 X X 6 X X 7 X 8 X 9 X Here factor g runs through all, as is indicated by the column of crosses. Factors h, k, and I run through mutu- ally exclusive groups of tests each. The saturations with g can be calculated from sub-batteries of tests which form perfect hierarchies, by selecting only one test from each group (in every possible way). After these are known, the correlation due to g can be removed, and then the saturations due to each group factor found.* 12. Vocational guidance. It will clearly be an aim of the experimenter along all these lines to obtain if possible single real tests, or failing that weighted batteries of tests, which approximate as closely as possible to the factors he has found, or postulated ; and with these to estimate the amount of each factor possessed by any man, and also (by giving such tests to tried workmen or school pupils) to estimate the amount of each factor required by different " occupations " (including higher education) with a view to vocational and educational selection and guidance. * See also the Addendum, page 348* CHAPTER II MULTIPLE-FACTOR ANALYSIS 1. Need of group factors. The two-factor method of analysis, described in the last chapter, began with the idea that a matrix of correlations would ordinarily show perfect hierarchical order if care was taken to avoid tests which were " unduly similar," i.e. very similar indeed to one another. If such were found coexisting in the team of tests, the team had to be " purified " by the rejection of one or other of the two. Later it became clear that this process involves the experimenter in great difficulty, for it subjects him to the temptation to discover " undue simi- larity " between tests after he has found that their correla- tion breaks the hierarchy. Moreover, whole groups of tests were found to fail to conform ; and so group factors were admitted, though always, by the experimenter trained in that school, with reluctance and in as small a number as possible. It had, however, become quite clear that the Theory of Two Factors in its original form had been super- seded by a theory of many factors, although the method of two factors remained as an analytical device for indicating their presence and for isolating them in com- parative purity. Under these circumstances it is not surprising that some workers turned their attention to the possibility of a method of multiple-factor analysis, by which any matrix of test correlations could be analysed direct into its factors (Garnett, 1919a and b). It was Professor Thurstone of Chicago who saw that one solution to this problem could be reached by a generalization of Spearman's idea of zero tetrad -differences. 2. Rank of a matrix and number of factors. We saw that when all the tetrad-differences are zero, the correlations can all be explained by one general factor, a tetrad being 20 MULTIPLE-FACTOR ANALYSIS 21 formed of the intereorrelations of two tests with two other tests, thus : 3 4 1 2 and the tetrad-difference being Thurstone's idea, though rather differently expressed by him ( Vectors, Chapter II), can be based on a second, third, fourth . . . calculation of certain tetrad-differences of tetrad-differences . To explain this, let us consider the correlation co- efficients which three tests make with three others : 4 5 6 1 ^14 7*15 7*16 2 7*24 7*25 ^26 3 7*34 7-35 7*36 This arrangement of nine correlation coefficients might have been called a " nonad," by analogy with the tetrad, Actually, by mathematicians, it is called a " minor deter- minant of order three " or more briefly a three-rowed minor ; a tetrad is in this nomenclature a " minor of order two." We can now, on the above three-rowed determinant, perform the following calculation. Choose the top left coefficient as " pivot," and calculate the four tetrad- differences of which it forms part, namely : These four tetrad-differences now themselves form a tetrad which can be evaluated. If it is zero, we say that the three-rowed determinant with which we started " vanishes." Exactly the same repeated process can be carried on with larger minor determinants. For example, the minor of order four here shown vanishes : 22 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 34 72 46 60 (26) 32 38 42 36 62 44 62 66 45 58 63 ( -0408) 0016 pivotal 0204 0044 0444 -0300 t.d.'s are ( -00021216) -00031824 and then -00028288 -00042432 and finally zero This process of continually calculating tetrads is called " pivotal condensation." The reader should be given a word of warning here, that the end result of this form of calculation, if not zero, has to be divided by the product of certain powers of the pivots, to give the value of the deter- minant we began with. A routine method (Aitken, 1937a) of carrying out pivotal condensation, including division by the pivot at each step, is described in Chapter VI, pages 89 ff.* We can in this way examine the minors of orders two, three, four (and so on) of a correlation matrix, always avoiding those diagonal cells which correspond to the correlation of a test with itself. We may come to a point at which all the minors of that order vanish. Suppose these minors which all vanish are the minors of order five. We then say that the " rank " of the correlation matrix is four (with the exception of the diagonal cells). There then exists the possibility that the " rank " of the whole corre- lation matrix can be reduced to four by inserting suitable quantities in the diagonal cells (see next section). The " rank " of a matrix is the order of its largest f non-vanish- * If the process gives, at an earlier stage than the end, a matrix entirely composed of zeros, the rank of the original determinant is correspondingly less, being equal to the number of condensations needed to give zeros. t " Largest " refers to the number of rows, not to the numerical value. MULTIPLE-FACTOR ANALYSIS 28 ing minor. Thurstone's discovery was that the tests could be analysed into as many common factors as the above reduced rank of their correlation matrix the rank, that is to say, apart from the diagonal cells plus a specific in each test. He also invented a method of performing the analysis. 3. Thurstone's method used on a hierarchy. Thurstone's rule 'about the rank includes Spearman's hierarchy as a special case, for in a hierarchy the tetrads that is, the minors of order two vanish. The rank is therefore one, and a hierarchical set of tests can be analysed into one common factor plus a specific in each. A simple way of introducing the reader to Thurstone's hypothesis and also to his " centroid " method * of finding a set of factor satura- tions will be to use it first of all on the perfect Spearman hierarchy which we cited as an artificial example in our first chapter. Tests I 2 3 4 5 6 72 72 63 -56 54 -48 45 -40 36 32 63 54 45 36 56 48 40 32 . 42 35 28 42 . 30 24 35 30 , 20 28 24 20 , The first step in Thurstone's method, after the rank has been found, is to place in the blank diagonal cells numbers which will cause these cells also to partake of the same rank as the rest of the matrix, numbers which, for a reason which will become clear later, are called " communalities." In our present Spearman example that rank is one 9 i.e. the tetrads vanish. The communalities, therefore, must be such numbers as will make also those tetrads vanish which include a diagonal cell : this enables them to be calculated. Let us, for example, fix our attention on the communality of the first test, which we will designate h^ (the reason for the " square " will become apparent later). Then the tetrad formed by Tests 1 and 2 with Tests 1 and 3 is : * We shall see why it is called the " centroid " method in Section 9 of Chapter VI, after we have learned to use a " pooling square." 24 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 1 3 1 2 V -63 72 -56 and the tetrad-difference has to vanish. Therefore 56V - -72 X -63 = .-.hS = -81 Similarly all the communalities can be calculated, and found to be 81 -64 -49 -36 -25 -16 (The observant reader will notice that they are the squares of the " saturations " of our first chapter ; but let us con- tinue with Thurstone's method as though we had not noticed this.) Thurstone's method of finding the saturations of each test with the first common factor is then to insert the com- munalities in the diagonal cells and add up the columns * of the matrix, thus : Original Correlation Matrix (.81) 72 63 54 45 36 72 (64) 56 48 40 32 63 56 (-49) 42 35 28 54 48 42 (36) 30 24 45 40 35 30 (25) 20 36 32 28 24 20 (16) 3-51 3-12 2-73 2-34 1-95 1-56 15-21 The column totals are then themselves added together (15-21) and the square root taken (3-90). The " satura- * This, the " centroid " method of finding a set of loadings, is not in any way bound up with Thurstone's theorem about the rank and the number of common factors. It can be used, for example, with unity in each diagonal cell, in which case it will give as many factors as there are tests and saturations somewhat resembling those given by Hotelling's process described in Chapter V : and vice versa Hotel- ling's process could be used on the matrix with communalities inserted. MULTIPLE-FACTOR ANALYSIS 25 tions " of the first (and here the only) common factor are then the columnar totals divided by this square root, namely 3-51 3-12 2-73 2-34 1-95 1-56 3-90 3-90 3-90 3-90 3-90 3-90 or -9 '8 -7 -6 -5 -4 as in the present instance we already know them to be. (Very often in multiple-factor analysis the " saturation " of a test with a factor is called the " loading," and this is a convenient place to introduce the new term.) As applied to the hierarchical case, this method of finding the saturations or loadings had been devised and employed many years previously by Cyril Burt, though it is not quite clear how he would have filled in the blank diagonal cells (Burt, 1917, 53, footnote, and 1940, 448, 462). It should be explained that in actual practice Thurstone and his followers do not calculate the minor determinants to find the rank and the communality, for that would be too laborious. Instead they adopt the approximation of inserting in each diagonal cell the largest correlation coefficient of the column (see Chapter X). 4. The second stage of the " centroid " method. If there is more than one common factor, the process goes on to another stage. Even with our example we can show the beginning of this second stage, which consists in forming that matrix of correlations which the first factor alone would produce. This is done by writing the loadings along the two sides of a chequer board and filling every cell of the chequer board with the product of the loading of that row with the loading of that column, thus : First-factor Matrix 9 8 7 6 5 4 9 81 72 63 54 45 36 8 72 64 56 48 40 32 -7 63 56 49 42 35 28 6 54 48 42 36 30 24 5 45 40 35 30 25 20 4 / 36 32 28 24 20 16 26 THE FACTORIAL ANALYSIS OF HUMAN ABILITY This is the " first-factor matrix," which gives the parts of the correlations due to the first factor. This matrix has now to be subtracted from the original matrix to find the resi- dues which must be explained by further common factors. In our present example the first-factor matrix is identical with the original matrix and the residues are all zero. Only the one common factor is therefore required. (Of course, the reader will understand that in a real experimental matrix the residues can never be expected to be exactly zero : one is content when they are near enough to zero to be due to chance experimental error.) Had the rank of our original matrix of correlations been, however, higher than one, there would have been a matrix of residues. Let us now make an artificial example with a larger number of common factors, say three, which we can after- wards use to illustrate the further stages of Thurstone's method. We can do this in an illuminating manner by the aid of the oval diagrams described in Chapter I. 5. A three-factor example. In Figure 7, a diagram of the overlapping variances of four tests, let us insert three common factors and specifics to complete the variance of each test to 10 (to make our arithmeti- cal work easy). No factor here is common to all the four tests. The factor with a variance of 4 runs through Tests 1, 2, and 3. That with a variance 3 runs through Tests 2, 3, and 4. That with a variance 2 runs through Tests 1 and 4. The other factors are specifics. The four test variances being each 10, the correlation coefficients are written down from the overlaps by inspection as : Figure 7. 1 2 3 4 1 (-6) 4 4 2 2 4 (-7) 7 3 3 4 7 (7) 3 4 -2 3 3 (-5) MULTIPLE-FACTOR ANALYSIS 27 Moreover, we can put into our matrix the communalities corresponding to our diagram. Each communality is, in fact, that fraction of the variance of a test which is not specific. Thus '6 of the variance of Test 1 is " communal," 4 being specific or "selfish." In this way we have the matrix above, with communalities inserted. We can now pretend that it is an experimental matrix, ready for the application of Thurstone's method, as follows : (-6) 4 4 2 4 (?) 7 3 Original 4 7 (-7) 3 experimental 2 3 3 (5) matrix. 1-6 2-1 2-1 1-3 = 7-1 = 2-6646 2 6005 7881 7881 4879 = 2-6646* (3606) 4733 4733 2930 4733 (6211) 6211 3845 First-factor 4733 6211 (6211) 3845 matrix. 2930 3845 3845 (2380) 1st Loadings 6005 7881 7881 4879 Here it is seen that the loadings of the first factor, when cross-multiplied in a chequer board, give a firsts-factor matrix which is not identical with the original experimental matrix, unlike the case of the former, hierarchical, matrix. Here (as we who made the matrix know) one factor will not suffice. We subtract the first-factor matrix from the original experimental matrix to see how much of the correlations still has to be explained, and how much of the " communalities " or communal variances. The latter were- 6 and of these amounts the first factor has explained 3606 -6211 -6211 -2880 If we subtract the first-factor matrix, element by element, from the original experimental matrix, we get the residual matrix : * This check should always be applied. To avoid complication it is not printed in the later tables. It applies to the loadings with their temporary signs (see below). 28 THE FACTORIAL ANALYSIS OF HUMAN ABILITY (-2394) -0733 - -0733 -0930 - -0733 (-0789) -0789 -0845 First residual - -0733 -0789 (-0789) - -0845 matrix. - -0930 - -0845 - -0845 (-2620) To this matrix we are now going to apply exactly the same procedure as we applied to the original experimental matrix, in order to find the loadings of the second factor. But we meet at once with a difficulty. The columns of the residual matrix add up exactly * to zero ! This always happens, and is indeed a useful check on our arithmetical work up to this point, but it seems to stop our further progress. To get over this difficulty we change temporarily the signs of some of the tests in order to make a majority of the cells of each column of the matrix positive. The practice adopted by Thurstone in The Vectors of Mind is to change the sign of the test with most minuses in its column and row, and so on until there is a large majority of plus signs. This is the best plan. Copy the signs on a separate paper, omitting the diagonal signs, which never change. Since some signs will change twice or thrice, use the convention that a plus surrounded by a ring means minus, and if then covered by an X means plus again. Near the end, watch the actual numbers, for the minus signs in a column may be very small. The object is to make the grand total a maximum, and thus take out maximum variance with each factor. We shall here, however, for simplicity adopt his easier rule given in A Simplified Factor Method, i.e. to seek out the column whose total regardless of signs is the largest, and then temporarily change the signs of variables so as to make all the signs in that column positive. The sums of the above columns, regardless of sign, are 4790 -3156 -3156 -5240 and therefore we must change the signs of tests so as to make all the signs in Column 4 positive ; that is, we must change the signs of the first three tests .f Since we change * When enough decimals have been retained. In practice there may be a discrepancy in the last decimal place. f Changing the sign of Test 4 would here have the same result, but for uniformity of routine we stick to the letter of the rule. MULTIPLE-FACTOR ANALYSIS 29 the three row signs, as well as the three column signs, this will leave a block of signs unchanged, but will make the last column and the last row all positive. We now have : 2394 - -0733 - -0733 (-)-0930 - -0733 0789 0789 (-)-0845 First residual -0733 0789 0789 (-)-0845 matrix with (-)-0930 (-)-0845 (-)-0845 2620 changed signs. 1858 1690 1690 5240 = 1-0478 = 1-0236 2 2nd 1815 1651 1651 5119 With temporary Loadings signs. 1815 0329 0300 0300 0929 1651 0300 0273 0273 0845 Second-factor 1651 0300 0273 0273 0845 matrix. 5119 0929 0845 0845 2620 2065 -1033 -1033 0001 - -1033 0516 0516 Second residual - -1033 0516 0516 , matrix. 0001 . . On the matrix with these temporarily changed signs we then operate exactly as we did on the original experimental matrix, and obtain second-factor loadings which (with temporary signs) are 1815 1651 1651 5119 The second-factor matrix, that is, the matrix showing how much correlation is due to the second factor, is then made on a chequer board still using the temporary signs, and subtracted from the previous matrix of residues (with its temporary signs, not with its first signs) to find the residues still remaining, to be explained by further factors. In the present instance we see that the whole variance of the fourth test entirely disappears, and also all the correla- tions in which that test is concerned.* This test, therefore, is fully explained by the two factors already extracted . Only the first three test variances remain unexhausted, and their correlations. Again the columns of the residual * When enough decimals are retained. We shall treat the 0001 as zero. 30 THE FACTORIAL ANALYSIS OF HUMAN ABILITY matrix sum exactly to zero. Following our rule, the signs of Tests 2 and 3 have to be temporarily changed before the process can continue. After these changes of sign the second residual matrix is as follows, and the same operation as before is again performed on it : 2065 (-)-1033 (-)-1033 . Second residual ( )-1033 -0516 -0516 . matrix with signs (-)-1033 -0516 -0516 . temporarily . changed. 4131 -2065 -2065 . = -8261 = -9089 2 &rd Loadings -4545 -2272 -2272 . with temporary signs. With these third-factor loadings we can now calculate the variances and correlations due to the third factor : and we find these are exactly equal to the second residual matrix. On subtracting, the third residual matrix we obtain is entirely composed of zeros. (In a practical example we should be content if it was sufficiently small.) We thus find (as our construction of the artificial tests entitled us to expect) that the matrix of correlations can be completely explained by three common factors. After the analysis has been completed, some care is needed in returning from the temporary signs of the load- ings to the correct signs. The only safe plan is to write down first of all the loadings with their temporary signs as they came out in the analysis. In our present example these happen to be all positive, though that will not always occur. Loadings with Temporary Signs Test 1 2 3 4 / // III 6005 -1815 -4545 7881 -1651 -2272 7881 -1651 -2272 4879 -5119 Now, in obtaining Loadings II the signs of Tests 1, 2, and 3 were changed. We must, therefore, in the above table reverse the signs of the loadings of these three tests in MULTIPLE-FACTOR ANALYSIS 81 Column II and each later column. Then in obtaining Loadings III the signs of Test 2 and 3 were changed ; that is, in our case changed back to positive. The loadings with their proper signs are therefore as shown in the first three columns of this table : Loadings of the Factors (Signs Replaced) Test / II III Specific 1 6005 -1815 -4545 6324 . 2 7881 -1651 + -2272 5477 . 3 7881 -1651 + -2272 -5477 , 4 4879 5119 . 7071 In this table each column of loadings, for the common factors after the first, adds up to zero. The loading of the specific is found from the fact that in each row the sum of the squares must be unity, being the whole variance of the test. The inner product * of each pair of rows gives the correlation between those two tests (Garnett, 1919a). Thus r 12 = -6005 X -7881 + -1815 X -1651 - -4545 X -2272 = -4000 in agreement with the entry in the original correlation matrix. With artificial data like the present, the analysis results in loadings which give the correlations back exactly. It will be seen that all the signs in any column of the table of loadings can be reversed without making any change in the inner products of the rows ; that is, without altering the correlations. We would usually prefer, there- fore, to reverse the signs of a column like our Column III, so as to make its largest member positive. The amount which each factor contributes to the variance of the test is indicated by the square of its loading in that * By the " inner product " of two series of numbers is meant the sum of their products in pairs. Thus the inner product of the two sets: abed and A B C D is aA -f bB + cC + dD 82 THE FACTORIAL ANALYSIS OF HUMAN ABILITY test. The sum of the squares of the three common-factor loadings gives the " communality " which we originally deduced from Figure 7 and inserted in the diagonal cells of our original correlation matrix. These facts can be better seen if we make a table of the squares of the above loadings : Variance contributed by Each Factor Test I II III Communality Specific Variance Total 1 3606 0329 2065 6000 4000 1 2 6211 0273 0516 7000 3000 1 3 6211 0273 0516 7000 3000 1 4 2380 2620 5000 5000 1 Total 1-8408 3495 3097 2-5000 1 -5000 4 6. Comparison of the analysis with the diagram. The reader has probably been turning from this calculation of the factor loadings back to the four-oval diagram with which we started (page 26), to detect any connection ; and has been disappointed to find none. The fact is that the analysis to which the Thurstone method has led us is, except that it too has three common factors, a different analysis from that which the original diagram naturally invites. That diagram gave for the variance due to each factor the following : Variance contributed by Each Factor Test / II III Communality Specific Variance Total 1 4 2 6 4 1 2 4 3 . 7 3 1 3 4 3 . 7 3 1 4 3 2 5 5 1 Totals 1-2 9 4 2-5 1-5 4 MULTIPLE-FACTOR ANALYSIS 33 and the factor loadings are the positive square roots of these. Loadings of the Factors Test I 2 3 / II III 6325 4472 Specifics 6324 6325 -5477 . . -5477 '6325 -5477 . . . -5477 4 i . -5477 -4472 . . . -7071 The only points in common between the two analyses are that they both have the same communalities (and therefore the same specific variances) and the same number of com- mon factors. The Thurstone analysis has two general factors (running through all four tests), while the diagram had none : and the Thurstone analysis has several negative loadings, while the diagram had none. We shall see later that Thurstone, after arriving at this first analysis, en- deavours to convert it into an analysis more like that of our diagram, with no negative loadings and no completely general factors. This is one of the most difficult yet essential parts of his method. 7. Analysis into two common factors. When we began our analysis of the matrix of correlations corresponding to Figure 7, we simply put the communalities suggested by that figure into the blank diagonal cells. That served to illustrate the fact that the Thurstone method of calculation will bring out as many factors as correspond to the com- munalities used, here three factors. But it disregarded (intentionally for the purpose of the above illustration) a cardinal point of Thurstone 's theory that we must seek for the communalities which make the rank of the matrix a minimum, and therefore the number of common factors a minimum. We simply accepted the communalities sug- gested by the diagram. Let us now repair our omission and see if there is not a possible analysis of these tests into fewer than three common factors. There is no hope "of reducing the rank to one, for the original correlations give 34 THE FACTORIAL ANALYSIS OF HUMAN ABILITY two of the three tetrads different from zero, and we may (in an artificial example) assume that there are no experi- mental or other errors. But there is nothing in the experi- mental correlations to make it certain that rank 2 cannot be attained. With only four tests (far too few, be it remembered, for an actual experiment) there is no minor of order three entirely composed of experimentally obtained correlations. It may then be the case that communalities can be found which reduce the rank to 2. Indeed, as we shall see presently, many sets of communalities will do so, of which one is shown here : (26) -4 -4 -2 4 (-7) -7 -3 4 -7 (-7) -3 2 -3 -3 (-15) These communalities '26, *7, -7, and -15 make every three-rowed minor exactly zero. For example, the minor (26) -4 -2 4 . (-7) -3 -2 -3 (-15) becomes by " pivotal condensation " : 026 and finally It must, therefore, be possible to make a four-oval diagram, showing only two common factors, and indeed 3. 4. Figure 8. more than one such diagram can be found. One is shown in Figure 8. MULTIPLE-FACTOR ANALYSIS 35 This gives exactly the correct correlations. For ex- ample r 23 = 12+2 12 = ^ = 7 20 * o 80) 40 It also gives the communalities *26, -7, -7, 15. For example, in Test 1, variance to the amount of 12 out of 45 is communal, and 12/45 = -26. The insertion of these communalities, therefore, in the matrix of correlations ought to give a matrix which only two applications of Thurstone's calculation should com- pletely exhaust. The reader is advised to carry out the calculation as an exercise. He will find for the first-factor loadings 5000 -8290 -8290 -3750 and if in the first residual matrix, following our rule, he changes temporarily the signs of Tests 2 and 3, the second- factor loadings will be 1291 -1128 -1128 -0968 The second residual matrix will be found to be exactly zero in each of its sixteen cells. The variance (square of the loading) contributed by each factor to each test is then in this analysis : Test 1 Variance contributed by Each Factor I II Communality Specific Variance Total 2500 0167 2667, 7333 1 2 6873 0127 7000 3000 1 3 6873 0127 7000 3000 1 4 1406 0094 1500 8500 1 Totals 1 -7652 0515 1-8167 2-1833 4 If we now compare these analyses, we see that the three common factors of the previous analysis " took out," as 36 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the factorial worker says, a variance of 2*5 of the total 4, leaving 1*5 for the specifics. The present analysis leaves 2-1833 for the specifics, which here form a larger part of the four tests. 8. Alexander's rotation. We saw in Section 6 that the Thurstone method there led to an analysis which was different from the analysis corresponding to the diagram with which we began. That is also the case with the present analysis into two common factors the very fact that it gives the second factor two negative loadings shows this, for the diagram (Figure 8) corresponds to positive loadings only. We said, too, in Section 6 that a difficult part of Thurstone's method was the conversion of the loadings into new and equivalent loadings which are all positive. This will form the subject of a later and more technical chapter ; but a simple illustration of one method of conversion (or " rotation " as it is called, for a reason which will become clear later) can be given from our present example. It is a method which can be used only if we have reason to think that one of our tests contains only one common factor (Alexander, 1935, 144). Let us suppose in our present case that from other sources we know this fact about Test 1. The centroid analysis has given us the loadings shown in the first two columns of this table : Test Unrelated Loadings Communality Rotated Loadings Rotated Loadings I II I* II* /** //** 1 2 3 4 5000 -1291 8290 -1128 8290 -1128 3750 -0968 2667 7000 7000 1500 5164 7746 -3162 7746 -3162 3873 4781 -1952 8367 8367 3586 -1464 The communalities are also shown ; they are the sums of the squares of the loadings. If now we know or decide to assume that Test 1 has really only one common factor, and if we want to preserve the communalities shown, then the MULTIPLE-FACTOR ANALYSIS 37 loading of factor I* in Test 1 must be the square root of 2667, namely -5164. The loadings of factor I* in the other three tests can now be found from the fact that they must give the corre- lations of those tests with Test 1, since Test 1 has no second factor to contribute. The loadings shown in column I* are found in this way : for example, -7746 is the quotient of -5164 divided into r ia (-4), and -3873 is similarly r u (-2) divided by *5164. The contributions of factor I* to the communalities are obtained by squaring these loadings. In Test 1, we already know that factor I* exhausts the communality, for that is how we found its loading. We discover that in Test 4, factor I* likewise exhausts the communality, for the square of 3873 is -1500. The other two tests, however, have each an amount of communality remaining equal to 1000 (i.e. -7000 -7746 a ). The square root of -1000, therefore (*3162), must be the loading of factor II* in Tests 2 and 3. The double column of loadings ought now to give all the correlations of the original correlation matrix, and we find that it does so. Thus, e.g. r 23 = -7746 X -7746 + -3162 X -3162 = -7000 and r 24 = -7746 X -3873 == -3000 Moreover, the analysis into factors I* and II* corre- sponds exactly to Figure 8. For example, the loading of factor II* in Test 2 in that diagram is the square root of 2/20 (-3162) ; and the loading of factor I* in Test 4 is the square root of 12/80 (-3873). If, however, the experimenter had reasons for thinking that Test 2 (not Test 1) was free from the second common factor, his " rotation " of the loadings would have given a different result, shown in the table opposite in columns I** and II**. This set of loadings also gives the correct communalities and the experimental correlations, but does not correspond to Figure 8. A diagram can, however, be constructed to agree with it (Figure 9), and the reader is advised to check the agreement by calculating from the diagram the loadings of each factor, the communalities of each test, and the correlations. 88 THE FACTORIAL ANALYSIS OF HUMAN ABILITY We have had, in Figures 7, 8, and 9, three different analyses of the same matrix of correlations. If with Thurstone we decide that analyses must always use the minimal number of common factors, we will reject Figure 7. Between Figures 8 and 9, however, this principle makes no choice. Much of the later and more technical part of Thurstone 's method is taken up with his endeavours to lay down conditions which will make the analysis unique. 9. Unique communalities. The first requirement for a unique analysis is that the set of communalities which gives the lowest rank should be unique, and this is not the case with a battery of only four tests and minimal rank 2, like our example. There are many different sets of com- munalities, all of which reduce the matrix of correlations of our four tests to rank 2. If, for example, we fix the first communality arbitrarily, say at -5, we can condense the determinant to one of order 3 by using -5 as a pivot (as on page 22) except that the diagonal of the smaller matrix will be blank : Figure 9. (-5) -4 4 2 4 7 3 19 07 4 7 19 07 2 3 3 07 07 We can then fill the diagonal of the smaller matrix with numbers which will make each of its tetrads zero, namely 19 -19 -0258 and then, working back to the original matrix, find the communalities 5 -7 -7 -1316 MULTIPLE-FACTOR ANALYSIS 89 which make its rank exactly 2. We can .similarly insert different numbers for the first communality and calculate different sets of communalities, any one set of which will reduce the rank to 2. In this way we can go from 1*0 down to 0*22951 for the first communality without obtain- ing inadmissible magnitudes for the others. Some sets are given in the following table * : 1 2 3 4 Sum 1-0 7 7 12963 2-52963 7 7 7 13030 2-23030 5 7 7 13158 2-03158 3 7 7 14 1-84 26 7 7 15 1-816 256 7 7 1583 1-8143 25 7 7 16 1-816 24 7 7 20 1-84 23 7 7 7 2-33 22951 7 7 1-0 2-62951* If, however, we search for and find a fifth test to add to the four, which will still permit the rank to be reduced to 2, this fifth test will fix the communalities at some point or other within the above range. Suppose that this test gave the correlations shown in the last row and column : 1 . 4 4 2 4 . 7 3 4 7 . 4 2 3 3 5 5883 2852 2852 2 3 3 1480 5883 2852 2852 1480 If we now try to find communalities to reduce this matrix to rank 2 (as can be done), we find only the one set 7 -7 -7 -13030 -5 The reader can try this by assigning an arbitrary value for * The circumstance that the communalities of Tests 2 and 3 remain fixed and alike is due to these tests being identical except for their specific. This lightens the arithmetic, but would not occur in practice. 40 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the first one,* and then condensing the matrix on the lines employed above, when he will always find some obstacle in the way unless he chooses -7. Try, for example, -5 for the first communality : (5) : 4 4 2 5883 4 . 7 3 2852 4 7 3 2852 2 3 3 . 1480 5883 2852 2852 1480 - (X) 19 07 -09272 19 07 -09272 07 07 . - -04366 -09272 -09272 -04366 e Now, if the upper matrix is to be of rank 2, the second condensation must give only zeros (see footnote, page 22). But if we fix our attention on different tetrads in the lower matrix which contain the pivot x 9 we see that they give, if they have to be zero, incompatible values for x. Thus from one tetrad we get x = -19, from another x = -14866. With 5 as first communality, rank 2 cannot be attained. With five tests (or more), if rank 2 can be attained at all, it can only be by one unique set of communalities. Just as it took three tests to enable the saturations with Spearman's g to be calculated, so it takes five tests to enable communalities due to two common factors to be calculated. For larger numbers of common factors, the number of tests required to make the set of communalities unique is shown in the following table (Vectors, 77). The lower numbers are given by the formula (2r + lH ^ r Factors 1 2 3 4 5 6 7 8 9 10 11 12 n Tests 3 5 6 8 9 10 12 13 14 15 17 18 * Alternatively, the communalities (which are now unique) can be found by equating to zero those three-rowed minors which have only one element in common with the diagonal (Vectors, 86). In this connection see Ledermann, 1937. MULTIPLE-FACTOR ANALYSIS 41 If we were actually confronted with the matrix of correla- tions shown on page 39, and asked what the communalities were which reduced it to the lowest possible rank, we would find it very unsatisfactory to have to guess at random and try each set ; and our embarrassment would be still greater if there were more tests in the battery, as would actually be the case in practice. There would also be sampling error (which in this our preliminary description of Thurstone's method we are assuming to be non-existent). Under these circumstances, devices for arriving rapidly at approximate values of the communalities are very desirable. The plan adopted by Thurstone will be described in Chapter X, to which a reader who wants rapid instruction in his methods of calculation should next turn. NOTE, 1945. With six tests the communalities which reduce to rank 3 are not necessarily unique, for there are, or there may be, two sets of them. See Wilson and Worcester, 1939. I think the ambiguity, which is not practically important, only occurs when n is exactly equal to the quantity at the foot of the opposite page, e.g. when r = 3, 6, 10, etc. CHAPTER III THE SAMPLING THEORY 1. Two views. A hierarchical example as explained by one general factor. The advance of the science of factorial analysis of the mind to its present position has not taken place without opposition, and it is the purpose of the pre- sent chapter to give a preliminary description of some objections which have been frequently raised by the present writer (Thomson, 1916, 1919a, 19356, etc.) and which indeed he still holds to, although there has been of late years a considerable change of emphasis in the inter- pretations placed upon factors by the factorists themselves, which have tended to remove his objections. Briefly, the opposition between the two points of view would dis- appear if factors were admitted to be only statistical coefficients, possibly without any more " reality " than an average, or an index of the cost of living, or a standard deviation, or a correlation coefficient though, on the other hand, it may be admitted that some of them, Spearman's g for example, may come to have a very real existence in the sense of being both useful and influential in the lives of men. There seems to be room for some form of integration of a number of apparently antithetical ideas regarding the way in which the mind functions, and the sampling theory which the writer has put forward * seems in particular to show that what have been called " monarchic," " oli- garchic," and u anarchic " doctrines of the mind (Abilities, Chapters II-V) are very probably only different ways of describing the same phenomena. The contrast perhaps one should say the apparent * For a general statement see Brown and Thomson, 1921, Chapter X, and Thomson, 19356, and references there given. A somewhat similar point of view has in more recent years been taken in America by R. C. Tryon, 1932a and 6, and 1935. 42 THE SAMPLING THEORY 43 contrast between the factorial and the sampling points of view * can be best seen by considering the explanation of the same set of correlation coefficients by both views. As we have consistently done, so far, in this part of our book, we shall again suppose that there are no experi- mental or sampling errors we shall consider them abundantly in due course and to simplify the argument we shall take in the first place a set of correlation coefficients whose tetrads are exactly zero, which can therefore be completely " explained " by a general factor g and specifics, as in this table : 1 2 3 4 1 746 646 527 2 740 . 577 471 3 646 577 . 408 4 527 471 408 . We can more exactly follow the argument if we employ the vulgar fractions of which these are the decimal equivalents, namely the following, each divided by 6 : I 1 2 3 4 v/20 V 15 V 10 V 12 V 3 V 15 V^ 2 \/6 4 V 10 A/8 V& In this form the tetrad-differences are all obviously zero by inspection. These correlations can therefore be ex- plained by one general factor, as in Figure 10, which gives them exactly. We have here a general factor of variance 30 which is the sole cause of the correlations, and specific factors of variances 6, 15, 30, and 60. The variances of the four * Two papers by S. C. Dodd (1928 and 1929) gave a very full and competent comparison of the two theories up to that date. The present writer agrees with a great deal, though not with all, of what Dodd says ; but see the later paper (Thomson, 19356) and also Chapter XX of this book. 44 THE FACTORIAL ANALYSIS OF HUMAN ABILITY (45) Figure 10. Figure 12. (72) Figure 13. " tests " are 36, 45, 60, and 90. The " communalities " and " specificities " are : Test 1 2 3 4 Totals Communality . 30 36 30 45 30 60 30 90 -.,, Specificity 6 36 15 45 30 60 60 90 S ' Totals 1 1 1 1 4 These communalities can be calculated from the corre- lation coefficients, for it will be remembered (Chapter I, Section 4) that when tetrad-differences are exactly zero, each correlation coefficient can be expressed as the THE SAMPLING THEORY 45 product of two correlation coefficients with g (two " saturations "). Thus 7*23 == ^2g^3g Therefore !jL 2 -!2L = V ig^fy) v Vfe) _ r 2 TK (VV) lg the square of the saturation of Test 1 with g. And when there is only one common factor, the square of its satura tion is the communality. The quantity r 12 r 13 /r23, therefore, means, on this theory of one common factor, the communality, or square of the saturation with g, of the first test. Its value in our example is 30/36, or five-sixths. 2. The alternative explanation. The sampling theory. The alternative theory to explain the zero tetrad- differences is that each test calls upon a sample of the bonds which the mind can form, and that some of these bonds are common to two tests and cause their correlation. In the present instance we have arranged this artificial example so that the tests can be looked upon as samples of a very simple mind, which can form in all 108 bonds (or some multiple of 108).* The first test uses five-sixths of these (or 90), the second test four-sixths (or 72), the third three- sixths (54), and the fourth two-sixths (or 36). These fractions are the same in value as the communalities of the former theory. Each of them may be called the " richness " of the test. Thus Test 1 is most rich, and draws upon five-sixths of the whole mind. The fractions r ij r ikl r jk> which in the former theory were " communali- ties," are in the sampling theory " coefficients of rich- ness." They formerly indicated the fraction of each test's variance supplied by g; they indicate here the fraction which each test forms of the whole " mind " (but see later, concerning " sub-pools "). * There is nothing mysterious about the number 108. It is chosen merely because it leads to no fractions in the diagram. Any latge ntiraber would do. .. ...... ..... 46 'THE FACTORIAL ANALYSIS OF HUMAN ABILITY Now, if our four tests use respectively 90, 72, 54, and 36 of the available bonds of the mind, as indicated in Figure 11, then there may be almost any kind of overlap between two of the tests. Any of the cells of the diagram may have contents, instead of all being empty except for g and the specifics. If we know nothing more about the tests except the fractions we have called their "richnesses," we cannot tell with certainty what the contents of each cell will be ; but we can calculate what the most probable contents will be. If the first test uses five-sixths and the second test four-sixths of the mind's bonds, it is most probable that there will be a number of bonds common to both tests 5 4 equal to - X -, or 20/36ths of the total number. That is, 6 6 the four cells marked a, b, c, d in the diagram, the cells common to Tests 1 and 2, will most likely contain 20 X 108 = 60 bonds 36 between them. By an extension of the same principle we can find the most probable number in each cell. Thus c, the number of bonds used in all four of the tests, is most probably 5 X 4 X 8 X 2 X 108 = 10 bonds. 6666 In this way we reach the most probable pattern of overlap of the four tests shown in Figure 12. And this diagram gives exactly the same correlations as did Figure 10. Let us try, for example, the value of r 2S in each diagram. In Figure 10 we had r = - 30 - - V 12 - 577 23 X 60) In Figure 12 the same correlation is _ 20 + 10 + 4_+2 _ V12 _ 23 ~^ " ~~ This form of overlap, therefore, will give zero tetrad - differences, just as the theory of one general factor did. More exactly, this sampling theory gives zero tetrad- THE SAMPLING THEORY 47 differences as the most probable (though not the certain) connexion to be found between correlation coefficients (Thomson, 1919a). If we let p l9 p 2 , p 3 , and p^ represent fractions which the four tests form of the whole pool of N bonds of the mind, then the number common to the first two tests will most probably be pip 2 N, and the correlation between the tests We therefore have, in any tetrad, quantities like the following : 3 4 1 2 and the tetrad-difference is, most probably (Thomson, 1927a, 253) This may be expressed by saying that the laws of proba- bility alone will cause a tendency to zero tetrad-differences among correlation coefficients. In another form, which will be useful later, this statement can be worded thus : The laws of probability or chance cause any matrix of correlation coefficients to tend to have rank 1, or at least to tend to have a low rank (where by rank we mean the maximum order among those non- vanishing minors which avoid the principal diagonal elements). It is, in the opinion of the present writer, this fact a result of the laws of chance and not of any psychological laws which has made conceivable the analysis of mental abilities into a few common factors (if not into one only, as Spearman hoped) and specifics. Because of the laws of chance the mind works as if it were composed of these hypothetical factors g, v 9 n, etc., and a number of specific factors. The causes may be " anarchic," meaning that they are numerous and un connected , yet the result is " monarchic," or at least " oligarchic," in the sense that it may be so described provided always that large specific factors are allowed. 48 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Of course, if the tetrad-differences actually found among correlation coefficients of mental tests were really exactly zero, or so near to zero that the discrepancies could be looked upon as " errors " due to our having tested a particular set of persons who did not accurately represent the whole population, then the theory of only one general factor would have to be accepted. For it gives exactly zero tetrad-differences, whereas the sampling theory only gives a tendency in that direction. But in actual fact it is only a tendency which is found, and matrices of correla- tion coefficients do not give zero tetrad-differences until they have been carefully purified by the removal of tests which " break the hierarchy." It has not proved very difficult to arrive at such purified teams of hierarchical tests. That is to be expected on the Sampling Theory, according to which hierarchical order is the most probable order. In the same way one would not have to go on throwing ten pennies for long before arriving at a set which gave five heads and five tails, for that is the most probable (yet not the certain) result. 3. Specific factors maximized. The specific factors play, in the Spearman and Thurstone methods of factorization, an important r61e, and our present example can be used to illustrate the fact, which is not usually realized, that both these methods maximize the specifics (Thomson, 1938c) by their insistence on minimizing the number of general factors. In Figure 10, of the whole variance of 4, the specific factors contribute 1*667, or 41*7 per cent. In Figure 12, they contribute only 10 + 4 + 2 + 1 = .2315, or 5-8 per cent. 90 72 54 36 1,080 * Apart from certain trivial exceptions which do not occur in practice, it is generally true that minimizing the number of common factors maximizes the variance of the specifics. Numerous other analyses of the above correlations can be made (Thomson, 19350), but they all give a variance to the specifics which is less than 1 -667. Here, for example, in Figure 13 (pagte 44), is an analysis whic'h has no > THE SAMPLING THEORY 49 factor but six other common factors, and which gives a total specific variance of I 5 + .1 + A + = ~ 3 ~- = -3056, or 7-6 per cent. 90 72 54 1,080 r The same principle, that reducing the number of common factors tends to increase the variance of the specifics, can be seen illustrated in Figures 5 and 6 (Chap- ter I, page 15). Figure 6 has five common factors, and the proportion which the specific variance bears to the whole four tests is + + + = 0-4, or 10 per cent. 10 10 10 10 In Figure 5 there are only two common factors, and the specific variance has risen to _ -j_ - + + = 1-4, or 35 per cent. 10 10 10 10 Again, in Figures 7, 8, and 9 (Chapter II, pages 26, 34, and 38) the same phenomenon can be observed. In Figure 7, with three common factors, the specific variances form 37 '5 per cent, of the four tests ; in Figures 8 and 9, with only two common factors, the specific variances form 54 '6 per cent. Now, specific factors are undoubtedly a difficulty in any analysis, and to have the specific factors made as large and important as possible is a heavy price to pay for having as few common factors as possible. Spearman, it is true, in his earlier writings, and in Chapter IX of The Abilities of Man, boldly accepts the idea of specific factors ; that is, factors which play no part except in one activity only, or in very closely allied acti- vities. His analogy of " mental energy " (g) and " neural machines " (the specifics) always makes a considerable appeal to an audience. On that analogy the energy of the mind is applicable in any of our activities, as the electric energy which comes into a house is applicable in several different ways : in a lighting-bulb, a radio set, a cooking- stove, a heater, possibly an electric razor, etc. Some of the spe'cifie machines which us'e th efe'etfje et^e^gy ne'ed 50 THE FACTORIAL ANALYSIS OF HUMAN ABILITY more of it than do others, just as some mental activities are more highly saturated with g. If it fails, they all cease to work ; if it weakens, they all work badly. Yet when it is strong, they do not all work equally well : the electric carpet-sweeper may function badly while the electric heater functions well, because of a faulty connec- tion in the (specific) carpet-sweeping machine ; while Jones next door (enjoying the same general electric supply) possesses no electric carpet-sweeper. So two men may have the same g, but only one of them possess the specific neural machine which will enable him to perform a certain mental task. The analogy is attractive, and, it must be agreed, educationally and socially useful. There is no objection to accepting it so far. But with the complication of group factors it begins to break down. Most activities are found to require the simultaneous use of several " machines." There does not seem so sharp a distinction between the machines and the general energy. Moreover, the general energy, if there be such a thing, of our person- alities is commonly held to be of instinctive and emotional nature rather than intellective, while g, whatever else it is, is commonly thought of as closely connected with intelligence. That specific factors are a difficulty seems to be recog- nized by Thurstone. u The specific variance of a test/' he writes (Vectors, 63), " should be regarded as a challenge," and he looks forward to splitting a specific factor up into group factors by brigading the test in question with new companion tests in a new battery. It seems clear that the dissolution of specifics into common factors is unlikely to happen if each analysis is conducted on the principle of making the specific variances as large as possible. We must, however, leave this point here, to return to it in a later chapter of this book. 4. Sub-pools of the mind. A difficulty which will occur to the reader in connexion with the sampling theory is that, when the correlation between two tests is large, it seems to imply that each needs nearly the whole mind to perform it (Spearman, 1928, 257). In our example the correlation bpjbween Te.sts 1 and 2 was -746, a correlation npj: infre- THE SAMPLING THEORY 51 quently reached between actual tests. It is, for instance, almost exactly the correlation reported by Alexander between the Stanford-Binet test and the Otis Self- administering test (Alexander, 1935, Table XVI). Does this, then, mean that each of these tests requires the activity of about four-sixths or five-sixths of all the " bonds " of the brain ? Not necessarily, even on the sampling theory. These two tests are not so very unlike one another, and may fairly be described as sampling the same region of the mind rather than the whole mind, so that they may well include a rather large proportion of the bonds found in that region. They may be drawn, that is, from a sub-pool of the mind's bonds rather than from the whole pool (Thomson, 19356, 91 ; Bartlett, 1937a, 102). Nor need the phrase " region of the mind " necessarily mean a topographical region, a part of the mind in the same sense as Yorkshire is part of England. It may mean something, by analogy, more like the lowlands of England, all the land easily accessible to everybody, lying below, say, the 300-foot contour line. What the " bonds " of the mind are, we do not know. But they are fairly certainly associated with the neurones or nerve cells of our brains, of which there are probably round about ten thousand million in each normal brain. Thinking is accompanied by the excitation of these neurones in patterns. The simplest patterns are instinctive, more complex ones acquired. Intelligence is possibly associated with the number and complexity of the patterns which the brain can (or could) make. A " region of the mind " in the above paragraph may be the domain of patterns below a certain complexity, as the lowlands of England are below a certain contour line. Intelligence tests do not call upon brain patterns of a high degree of complexity, for these are always associated with acquired material and with the educational environment, and intelligence tests wish to avoid testing acquirement. It is not difficult to imagine that the items of the Stanford-Binet test call into some sort of activity nearly all the neurones of the brain, though they need not thereby be calling upon all the patterns which those neurones can form. When a teacher is 52 THE FACTORIAL ANALYSIS OF HUMAN ABILITY demonstrating to an advanced class that " a quadratic form of rank 2 is identically equal to the product of two linear forms," he is using patterns of a complexity far greater than any used in answering the Binet-Simon items. But the neurones which form these patterns may not be more numerous. Those complicated patterns, however, are forbidden to the intelligence tester, for a very intelligent man may not have the ghost of an idea what a " quadratic form "is. Within the limits of the comparatively simple patterns of the brain which they evoke, it seems very possible that the two tests in question call upon a large proportion of these, and have a large number in common. The hope of the intelligence tester is that two brains which differ in their ability to form readily and clearly the comparatively simple patterns required by his test will differ in much the same way if, given the same educational and vocational environment, they are later called upon to form the much more complex patterns there found. As has been indicated, the author is of opinion that the way in which they magnify specific factors is the weak side of the theories of a single general factor or of a few common factors. That does not mean, however, that a description of a matrix of correlations in terms of these theories is inexact. Men undoubtedly do perform mental tasks as if they were doing so by means of a comparatively small number of group factors of wide extent, and an enormous number of specific factors of very narrow range but of great importance each within its range. Whether a description of their powers in terms of the few common factors only is a good description depends in large measure on what purpose we want the description to subserve. The practical purpose is usually to give vocational or educational advice to the man or to his employers or teachers, and a discussion of the relative virtues of different theories in this respect must wait until we have considered the somewhat technical matter of " estimation " in later chapters. We shall there see that factors, though they cannot improve and indeed may blur the accuracy of vocational estimates, may, however, faciKt&te the.m .whetfe dtherwi^e they would haVe bfesn THE SAMPLING THEORY 53 impossible, as money facilitates trade where barter is impossible. As a theoretical account of each man's mind, however, the theories which use the smallest number of common factors seem to have drawbacks. They can give an exact reproduction of the correlation coefficients. But, because of their large specific factors, they do not enable us to give an exact reproduction of each man's scores in the original tests, so that much information is being lost by their use. Reproduction of the original scores with complete exacti- tude can only be achieved by using as many factors as there are tests. But it can be done with considerable accuracy by a few of Hotelling's factors (called " principal components "), which will be described later. It will be seen from considerations such as these that alternative analyses of a matrix of correlations, even although they may each reproduce the correlation coeffi- cients exactly, may not be equally acceptable on other grounds. The sampling theory, and the single general factor theory, can both describe exactly a hierarchical set of correlation coefficients, and they both give an explana- tion of why approximately hierarchical sets are found in practice. In a mathematical sense, they are alternatives. But as Mackie has shown (Mackie, 19286), a psychologist who believes that the " bonds " of the sampling theory have any real existence, in the sense, say, of being represented in the physical world by chains and patterns of neurones, cannot without absurdity believe in the similarly real existence of specific factors. The analogue to Spearman's g, on the sampling theory, is simply the whole mind. " How, then," (as Mackie asks) " can we have other factors independent of such a factor as this ? " Only by the formal device of letting the specific factor include the annulling of the work done by the other part of the mind, a legitimate mathematical procedure but not one compatible with actual realities. Either, then, we must give up the factors of the two -factor theory, or the bonds of the sampling theory, as realities. We cannot keep both as realities, though we may employ either mathematically. 5. The inequality of men. Professor Spearman has 54 THE FACTORIAL ANALYSIS OF HUMAN ABILITY opposed the sampling theory chiefly on the ground that it would make all correlations equal (and zero), and involve the further consequence that all men are equal in their average attainments (Abilities, 96), if the number of elementary bonds is large, as the sampling theory requires. Both these objections, however, arise from a misunder- standing of the sampling theory, in which a sample means " some but not all " of the elementary bonds (Thomson, 1935&, 72, 76). As has been explained, tests can differ, on this theory, in their richness or complexity, and less rich tests will tend to have low, more complex tests will tend to have high correlations, at any rate if the " bonds " tend to be all-or-none in their nature, as the action of neurones is known to be. Neurones, like cartridges, either fire or they don't. And as for the assertion that the theory makes all men equal, there is no basis whatever for the suggestion that it assumes every man to have an equal chance of possessing every element or bond. On the con- trary, the sampling theory would consider men also to be samples, each man possessing some, but not all, both of the inherited and the acquired neural bonds which are the physical side of thought. Like the tests, some men are rich, others poor, in these bonds. Some are richly endowed by heredity, some by opportunity and education ; seme by both, some by neither. The idea that men are samples of all that might be, and that any task samples the powers which an individual man possesses, does not for a moment carry with it the consequences asserted of equal correlations and a humdrum mediocrity among human kind. CHAPTER IV THE GEOMETRICAL PICTURE* 1. The fundamental idea. The student reading articles on factorial analysis is continually coming across geometrical and spatial expressions. For example, in Section 8 of our Chapter II we spoke of " rotating " the loadings of Thurstone's " centroid " method until they fulfil certain conditions. These geometrical expressions arise from the fact that the mathematics of mental testing is the same in its formal aspect as the mathematics of multi-dimensional space, and it is the object of the present chapter to explain this in elementary terms. Some degree of understanding of this is essential for the worker with tests, and it is not difficult when divested as far as possible of the algebraic symbols in which it is usually clothed. The fundamental idea is that the correlation between two tests can be pictorially represented by the angle between two lines which stand for the two tests, and which pass through a point, thus forming an X with its legs stretching ever so far in both ways. The point where the lines cross represents a man who has the average score on both tests. Other points on the lines represent standardized scores in the tests which are more or less removed from the average an arrowhead can be placed on each line to represent the positive direction, as in Figure 14. If the lines taken in the direction Figure 14. of these arrowheads make only a small angle with one another, they represent tests which are highly correlated. As the correlation decreases, this angle increases. When the correlation is zero, the angle * See Addendum, page 353. 55 56 THE FACTORIAL ANALYSIS OF HUMAN ABILITY is a right angle. If the angle becomes obtuse, the corre- lation is negative. Any point on the paper then represents a person by his two standardized scores in these two tests, obtained by dropping perpendiculars on to the two lines representing the tests. If we were to measure a large number of persons by each of these two tests say, ten thousand persons and place a dot on the paper for each person as represented by his two scores, we would naturally find that these dots would be crowded most closely together round the point where the test lines (or test vectors, as they are technically called) cross, where the average man is situated. The ten thousand dots would look, in fact, like shot marks on a target of which the bull's-eye was the average man at the cross-roads of the test vectors. The density of the dots would fall off equally to the north, south, east, and west of this point. Their " contours of density," as we say, would be circles. Circles, because any line through the imaginary man-who-is-average-in-everything repre- sents a conceivable test, and the standard deviation is everywhere represented by the same unit of length. The dots would look exactly like a crowd which, equally in all directions, was surrounding a focus of attraction at the crossing-point of the tests. 2. Sectors of the crowd. On the diagram are shown also two dotted lines, perpendicular respectively to the two test vectors. Persons who are standing on one of these dotted lines have exactly the average score in the test to which it is perpendicular. Two of the sectors of the crowd are distinguished by shading in the diagram. Let us fix our attention on the northern shaded sector, which includes the two positive directions of the test vectors, marked by the arrowheads. Everybody in this sector of the crowd has a score above the average in both tests. Similarly, in the other shaded sector of the crowd, everybody has a score below the average in both tests. Both these sectors of the crowd contribute to the correlation between the tests, since everybody in these sectors does well in both, or badly in both. The people in the white sectors of the crowd, however, THE GEOMETRICAL PICTURE 57 have scores above the average in one test and below the average in the other. They diminish the correlation be- tween the tests. Those in the western white sector have scores above the average in Test X , but below the average in Test Y ; and vice versa for those in the eastern white sector. If the arrowheads X and Y are brought nearer together (while the people in the circular crowd remain standing still), so that the angle between the test vectors is diminished, the dotted lines will move so as to diminish the white sectors which lie between them, and the correlation will increase. When the test vectors are close together, one coinciding with the other, the white sectors will have dis- appeared and the correlation will be perfect. When the test vectors are at right angles, the white sectors will be quadrants, the crowd will be half " black " and half " white," and the correlation zero. Beyond the right-angle position, there will be more white than black, and a negative correlation . It is clear, then, that the angle between the test vectors inversely represents the correlation between the tests. It can be shown (but we shall take it on trust) that the cosine of the angle is equal to the correlation (Garnett, 1919a ; Wilson, 1928a). If we wish, therefore, to draw two vectors for two tests whose correlation we know, we consult a table of trigonometrical ratios, to find the angle whose cosine is equal to the correlation coefficient, and draw the lines accordingly. 3. A third test added. The tripod. If we now wish to draw the vector of a third test, we must similarly consult the trigonometrical table to find from its correlation coefficients the angles it makes with the two former tests. We shall then usually discover that we cannot draw it on our paper, but that it has to stick out into a third dimension. It will only lie in the same plane as the other two if either the sum or the difference of its angles with them equals the angle between the first two tests. Usually this is not the case, and the vectors of these three tests will require three-dimensional space. They will look like a tripod extended upwards as well as downwards. If the correla- 58 THE FACTORIAL ANALYSIS OF HUMAN ABILITY tions are high, the tripod's legs will be close together ; if low, they will be far apart. This tripod analogy will make plausible to the reader the assertion that some sets of correlation coefficients cannot logically coexist. For the legs of a tripod cannot take up positions at any angles. If two of the angles are very small, the third one cannot be very large. The sum of any two of the angles must at least equal the third angle. And so on. For example, the following matrix of correlations is an impossibility : 123 1 2 3 1-00 -34 -77 34 1-00 -94 77 -94 1-00 Here Tests 1 and 2 are highly correlated with Test 3, so highly that they cannot possibly have only a correlation coefficient of *34 with each other. The angles corre- sponding to the above coefficients (taken as cosines) are : 1 2 3 1 70 40 2 70 20 3 | 40 20 and the fact that 40 + 20 is less than 70 shows that the matrix is impossible. When the symmetrical matrix of correlations is an impossible one, which could not really occur, it will be found that either the determinant itself, or one of the pivots in the calculation explained in Chapter II, Section 2, is negative. Let us carry out the calculation for the above matrix : (1-00) -34 -77 34 1-00 94 77 -94 1-00 (8844) 6782 6782 4071 Determinant = -0999 This test serves also for larger matrices. THE GEOMETRICAL PICTURE 59 Let us, however, return to our tripod of three vectors which by their angles with one another represent the corre- lations of three tests the legs of the tripod being the negative directions of the tests, let us assume, and their continuation upward past their common crossing-point the positive directions, though this is not essential. The point where the three vectors cross represents the average man, who obtains the average score (which we will agree to call zero) on each of the three tests. Any other point in space represents a man whose scores in the three tests are given by the feet of perpendiculars from this point on to the three test vectors. If, again, we sup- pose that ten thousand persons have undergone these three tests, the space round the test vectors will be filled with ten thousand points, which will be most closely crowded together near the average man at the crossing- point (or " origin ") of the vectors, and will form a spherical swarm falling off in density equally in all directions from that point. 4. A fourth test added. One test was represented by a line. Two tests by two lines in a plane. Three tests by three lines in ordinary space. Suppose now we have a fourth test, look up its angles with the pre-existing three tests, and try to draw its line or vector, adding a fourth leg to the tripod. Just as the third test would not usually lie in the plane of the first two, but required a third dimen- sion to project out into, so the fourth test will not usually be capable of being represented in the three-space of the three tests. Its angles with them will not fit unless we add a fourth dimension. Here, of course, the geometrical picture, strictly speaking, breaks down. But it is usual and mathematically helpful to continue to speak as though spaces of higher dimensions really existed. In a " space " of four dimensions we can imagine four test vectors crossing at a point, their angles with one another depending upon the correlations. We can imagine a " spherical " swarm of dots representing persons. And when we add more tests, we can similarly imagine spaces of 5, 6 ... n dimensions to accommodate their test vectors. The reader should not allow the im- 60 THE FACTORIAL ANALYSIS OF HUMAN ABILITY possibility of visualizing these spaces of higher dimensions to trouble him overmuch. They are only useful forms of speech, useful because they enable us to refer concisely to operations in several variables which are exactly analogous to familiar operations in the real space in which we live such as " rotating " a line or a set of lines round a pivot. 5. Two principal components. Let us now express the ideas we have used in the preceding three chapters in terms of this geometrical picture. Independent factors will be represented by vectors at right angles to one another (we shall for the most part be concerned only with independ- ent, i.e. uncorrelated factors, though at a later stage we shall have something to say about correlated or " oblique " factors). Analysing a set of tests into independent factors means, in terms of our geometrical picture, referring their test vectors to a set of rectangular vectors as axes of co-ordinates the Greek equivalent " orthogonal " is gen- erally used in this connexion instead of " rectangular.' 5 Let us explain this first of all in the simplest case, that of two tests, represented by their vectors in a plane, at the angle corresponding to their correlation. In this case, the most natural way of drawing orthogonal co-ordinates on the paper is to place one of them (see Figure 15) half-way between the test vectors, and the other, of course, at right angles to the first. These factor vectors correspond, in fact, to Hotelling's " principal " components," to which we shall return later. Of these two factors (or components) OA is as near as it can be to both test vectors . ' - it is the "first principal corn- Figure 15. r r ponent. We pictured, before, a swarm of ten thousand dots on the paper, each representing a person by his scores in the two tests, found by dropping perpendiculars from his dot to the two vectors. Instead of describing each point (each person, that is) by the two test scores, it is clear that we THE GEOMETRICAL PICTURE 61 could describe it by the two factor scores the feet of perpendiculars on to the factor vectors. It is also clear that, as far as this purpose goes, we might have taken our factor vectors or factor axes anywhere, and not necessarily in the positions OA and OB, provided they went through the point and were at right angles. In other words, we can " rotate " OA and OB round the point 0, and any position is equally good for describing the crowd of persons. Either of the tests, indeed, might be made one of the factors. The positions shown in Figure 15 are advantageous only if we want to use only one of our factors and discard the other, in which case obviously OA is the one to keep, as it lies as near as possible to both test vectors.* The scores along OA are the best possible single description of the two test results. That is the distinguishing virtue of Hotelling's " first principal component." 6. Spearman axes for two tests. The orthogonal axes chosen by Spearman for his factors are, however, none of the positions to which OA and OB can be rotated in the plane of the paper. Besides, Spearman has three factors, and therefore three axes, for two tests, namely the general factor and the two specific factors, and we cannot have three orthogonal axes or factor vectors on a sheet of paper. The Spearman factors must, for two tests, lie in three- dimensional space, like the three lines which meet in the corner of a room. If we rotate the OA and OB of Figure 15 out of the plane of the paper (say, pushing A below the surface of the paper, and, say, raising B above it), we shall clearly have to add a third axis, at right angles to OA and OB, to enable us to describe the tests and the persons who remain on the paper. There are now three axes to rotate ; and they must rotate rigidly, remaining at right angles to one another. The point at which Spearman stops the rotation, and decides that the lines then represent the " best " factors, is a position in which one of the axes is * Persons will, in fact, be placed in the same order of merit by their factors A as they are placed in by their average scores on the two tests, but this is not the case with the Hotelling first component of larger numbers of tests. Figure 16. 62 THE FACTORIAL ANALYSIS OF HUMAN ABILITY at right angles to Test X, and another is at right angles to Test Y. The third axis then represents g. 7. Spearman axes for four tests. We are accustomed to depicting three dimensions on a flat sheet of paper, and so we can, in Figure 16, represent the Spearman axes g, s l9 and s z for two tests. And since we have begun to depict other dimensions, by means of per- spective, on a flat sheet, let us continue the process and by a kind of super-perspective imagine that the lines s 3 , s l9 and any others we may care to add, re- present axes sticking out into a fourth, a fifth, and higher dimensions. Figure 16 thus re- presents the five Spearman axes for four tests, of which only the vector of the first test is shown (in its positive half only). All the five lines g, s i9 $ 2 , 3, and s 4 must be imagined as being each at right angles to all the others in five-dimen- sional space. The vector of Test 1, shown in the diagram, lies in the plane or wall edged by g and Si. It forms acute angles with g and with s l9 the cosines of which angles are its saturations with g and s t respectively. If it had been highly saturated with g 9 it would have leaned nearer to g and farther away from s^ The other three axes, s 29 $a> and $ 4 , are all at right angles to the wall or plane in which Test 1 lies. They have, therefore, no correlation with Test 1, no share in its composition. Test vector 2 similarly lies in the wall edged by g and s 29 test vector 3 in that edged by g and s 3 . The axis g forms a common edge to all these planes. If the battery of tests is hierarchical that is, if the tetrad- differences are all zero then all the tests of the battery can be depicted in this way, each in its own plane at right angles to all the other planes, no test vector being in the spaces between the " walls." The four test vectors themselves, of course, are only in a four-dimensional space (a 4-space we shall say, for THE GEOMETRICAL PICTURE 63 brevity). Just as, when we were discussing Figure 15, we said that Spearman used three axes which were all out of the plane of the paper, so here in Figure 16, with four test vectors (only one shown) in a 4-space, Spearman uses five axes in a space of one dimension higher than the number of tests. For n hierarchical tests, Spearman's factors are in an (n + l)-space. If along each test vector we measure the same distance as a unit, then perpendiculars from these points on to the g axis will give the saturations of the tests with g as fractions of this unit distance. The four dots on the g axis in Figure 16 may thus be taken as representing the test vectors projected on to the " common-factor space," which is here a line, a space of one dimension only. Thurstone's system is like Spearman's except that the common-factor space is of more dimensions, as many as there are common factors. Figure 17 shows the Thurstone axes for four tests whose matrix of correlation coefficients can be reduced to rank 2. 8. A common-factor space of two dimensions. Here there are two common factors, a and b, and four specifics, Si, s 2 , s 3 , and s 4 . All the six axes representing these factors in the figure are to be imagined as existing in a 6-space, each at right angles to all the others. The common -factor space is here two-dimensional, the plane or wall edged by a and b to make it stand out in the figure, a door and a window have been sketched upon it. In Spearman's Figure 16, each test vector lay in a plane defined by g and one of the specific axes. Here in Figure 17, each test vector lies in a different 3-space. These v . ~ , . Figure 17. different 3-spaces have nothing in common with one another except the plane ab, the wall with the door and window in the diagram. In Figure 16 the projections of the test vectors on to the common-factor space were lines which all coincided in direction (though they were of different lengths), for 64 THE FACTORIAL ANALYSIS OF HUMAN ABILITY there the common-factor space was a line. Here the common-factor space is a plane, and the projections of the four test vectors on to that plane are shown in the figure by the lines on the " wall." These lines, if they are all pro- jections of vectors of unit length, wilj by their lengths on the wall represent the square roots of the communalities. 9. The common-factor space in general. When there are r common factors, the common -factor space is of r dimen- sions, and the whole factor space (including the specifics) is of (n + r) dimensions. The test vectors themselves are in an n-space ; their projections on to the common-factor space are crowded into an r-space, and are naturally at smaller angles with one another than the actual test vectors are. These angles between the projected test vectors do not, therefore, represent by their cosines the correlations be- tween the tests. The angles are too small for that, and the cosines, therefore, too large. But if we multiply the cosine of such an angle by the lengths of the two projections which it lies between, we again arrive at the correlation. Thus in Figure 17, the angle between the lines 1 and 3 on the wall is less than the angle between the actual test vectors 1 and 3 out in the 6-space, of which the lines on the wall are the projections. But the lengths of the lines 1 and 3 on the wall are less than the unit length we marked off on the actual vectors, being in fact the roots of the com- munalities. If we call these lengths on the wall h^ and & 3 , then the product hji^ times the cosine of the projected angle again gives the correlation coefficient. 10. Rotations. It will be remembered that Thurstonc, after obtaining a set of loadings for the common factors by his method of analysis of the matrix of correlations, " rotates " the axes until the loadings are all positive and he also likes to make as many of them as possible zero. It is instructive to look at this procedure in the light of our geometrical picture from which the phrase " rotating the factors " is taken. It should be emphasized first of all that such rotation of the common-factor axes in Thur- stone's system must take place entirely within the com- mon-factor space, and the common-factor axes must not leave that space and encroach upon the specifics. In THE GEOMETRICAL PICTURE 65 Figure 16, therefore, no rotation, in Thurstone's sense, of the g axis can be made (since the common -factor space is a line), except indeed reversing its direction and measuring stupidity instead of intelligence. In Figure 17 the common -factor space is a plane, and the axes a and b can be rotated in this plane, like the hands of a clock fixed permanently at right angles to one another. When the positive directions of a and b enclose all the vector projections, as they do in our figure, then all the loadings are positive. The position shown would, there- fore, fulfil this desire of Thurstone 's. Moreover, one of the loadings could be made zero, by rotating a and b until a coincides with line 1 (when b will have no loading in Test 1), or until b coincides with line 4 (when a will have no loading in Test 4). When there are three common factors, the common- factor space is an ordinary 3-space. The three common- factor axes divide this space into eight octants. Rotating them until all the loadings are positive means until all the projections of the test vectors are within the positive octant. This will always be nearly possible if the corre- lations are all positive. Moreover, it is clear that we can always make at any rate some loadings zero. In the common-factor 3-space we can move one of the axes until it is at right angles to two of the test projections, in which tests that factor will then have no loading. Keeping that axis fixed, we can then rotate the other two axes round it, seeking for a position where one of them is at right angles to some test. The number of zero loadings obtainable will clearly be limited unless the configuration of the test vectors happens to lend itself to many zeros. We shall see later that Thurstone seeks for teams of tests which do this. Although Thurstone makes his rotations exclusively within the common-factor space, keeping the specifics sacrosanct at their maximum variance, there is, of course, nothing to prevent anyone who does not hold his views from rotating the common-factor axes into a wider space, and increasing the number of common-factor axes at the expense of the specific variance, until ultimately we reach as many common factors as we have tests, and no specifics. F.A. 3 CHAPTER V HOTELLING'S "PRINCIPAL COMPONENTS" 1. Another geometrical picture. The geometrical picture of the last chapter, however, is not the only form of spatial analogy which can be used for representing the results of mental tests, nor indeed was it the first in the field, though it is the most powerful. The earlier, and perhaps more natural, plan of representing two tests was by two lines at right angles, instead of at an angle depending on their correlation as in Chapter IV. Using the two lines at right angles, and the two test scores as co-ordinates, each person could, in this form of diagram also, be represented by a point on the paper, and his two scores by the feet of perpendiculars from that point on to the test axes. But if, on such a diagram, we mark the points of ten thousand persons, these will, of course, not be distributed in the same circular symmetrical fashion as in Figure 14 (page 55). If we look at Figure 14, we can see what would happen to the crowd of persons if we were to pull the test vectors farther and farther apart * until finally they were at right angles. The shaded northern sector of the crowd is com- posed of persons whose scores are above average in both tests, and this sector is bounded by the two dotted lines which are at right angles to the test vectors. As the angle between the test vectors grows larger, the two dotted lines in question close towards one another, and this shaded section of the crowd is driven northward. Simultaneously the other shaded section is driven southward. When the test vectors reach a position at right angles to one another, the dotted line at right angles to X falls along F, and the other along X 9 and we have Figure 18. The crowd is no longer distributed in a circular fashion round the origin. * It is understood that they continue to stand for the same tests, with the same correlation, though the latter is no longer represented by the cosine of the angle between the vectors. 66 HOTELLING'S " PRINCIPAL COMPONENTS" 67 It now bulges out to the north and south, in the quadrants where the two test scores are either both positive or both negative, and its lines of equal density, formerly circles, have become ellipses. In this form of diagram, it is this ellipticity of the crowd which shows the presence of correla- tion between the tests. If the tests are highly correlated, the ellipses will be long and narrow ; if they are less correlated, they will be plumper; if there is no correlation, they will be circles ; if there is negative correlation, they will be longer the other way, i.e. from east to west in our diagram. In our former figures in Chapter Figure 18. IV, the space of the diagram, whether plane, solid, or multi-dimensional, was peopled by a " spherical " crowd whose density fell away equally in all directions from the origin, while correlation between tests was indicated by the angles between their test vectors. In the present chapter, all the test vectors are at right angles, and the space is peopled by a crowd whose density falls off differently in different directions unless there is no correlation present. If we add a third test to the two in Figure 18, its axis, in the present system, has to be at right angles to the first two. The former spherical swarm of persons (of Chapter IV) has become now an ellipsoidal swarm, like a Zeppelin, with proportions determined by the correlations. If these are positive, its greatest length will be in the direction of the positive octant of space (that octant in which all scores are above average, i.e. positive), and the opposite negative octant. Its waist-line will not, as a rule, be circular, but elliptical. The ellipse of Figure 18 has two principal axes, a major axis from north to south, and a minor axis at right angles to it from east to west. The ellipsoid of three tests has three principal axes ; the " ellipsoid " (for we continue to use the term) for n tests will be in n-dimensional space and will have n principal axes. It is these principal axes of 68 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the ellipsoids of equal density which are the " principal components " of Hotelling's method (Hotelling, 1933). They are exactly equal in number to the tests, but usually the smaller ones are so small as to be negligible, within the limits of exactitude reached by psychological experiment. 2. The principal axes. Finding Hotelling's principal components, therefore, consists in finding those axes, all at right angles to one another, which lie one along each principal axis of the ellipsoids of equal density of the population of persons tested. In Figure 18, for example, one of them lies north and south, the other east and west. The crowd of persons can then be described in terms of these new axes, in terms of factors, that is, instead of in terms of the original tests. These factors are uncorrelated, for the crowd is symmetrically distributed with regard to them, though not in a circular manner. This brings us to one more thing that has to be done to these factors before they become Hotelling's principal components : they have to be measured in new units. The original test scores were, we have tacitly assumed in making our diagrams, measured in comparable units, namely each in units of its own standard deviation. But the factors arrived at by a mere rotation to the principal axes, in an elliptically distributed crowd, are no longer such that the standard deviation of each is represented by the same distance in the diagram. If in Figure 18 all the points representing people are pro- jected on to a horizontal east-and-west factor (Factor II), the feet of these perpendiculars are obviously more crowded together than the corresponding points would be on a north-and-south factor (Factor I). On this diagram, therefore, the standard deviation of Factor II is represented by a shorter distance than is the standard deviation of Factor I. To make these equal, we would have to stretch our paper from east to west, or compress it from north to south, until the crowd was again circular, during which procedure the test vectors would have to move back to the position of Figure 14 to keep the crowd's test scores equal to their projections, and we are then back at the space of Chapter IV. The " ellipsoidal " space of this present chapter, in fact, is used only until the principal HOTELLING'S "PRINCIPAL COMPONENTS" 69 axes of the ellipsoid are discovered, after which, by a change of units along each principal axis, it is made into a " spherical " space again. In the preceding paragraph, the reader may feel a difficulty which has been known to trouble students in class. If, he may say, we stretch Figure 18 from east to west till the ellipse is a circle, that ought to separate the arrows of the test vectors still farther. Yet you say they will return to the positions shown in Figure 14 ! The mistake lies in thinking that stretching the space the plane of the paper in Figure 18 till the ellipsoid is spherical will move the test vectors with the space. The points representing persons move with the space ; indeed, they are the space. But the test vectors are not rigidly at- tached to the space. Each test vector must be such that every person's point, projected on to it, gives his score. If the points move about, as they do when we stretch the paper, the test vector must move so that this remains true, and in our case that means moving nearer together as the crowd becomes more circular. It is just the reverse of the process by which we obtained Figure 18 from Figure 14. 3. Advantages and disadvantages. The advantage of Hotelling's factors can be best appreciated while the crowd of persons is in the ellipsoidal condition. Hotelling's first factor (or " component," as he calls it) runs along the greatest length of the crowd, and gives the best single description of a person's position. If we know all his factor scores, we know exactly where he is in the crowd. If we have to search for him, we would rather be told his position on the long axis, and search along the short ones, than be told his position on any other axis instead. If there are, say, twenty tests, there will be twenty principal axes ranging from longest to shortest, and twenty Hotelling components.* But the first four or five of these will go a long way towards defining a man's position in the tests, * All that is here said about principal components refers to the case, which is that considered by Hotelling, in which the method of calculation about to be described is applied to the matrix of correlations with unities (or possibly with reliabilities) hi the diagonal cells. The method, as a means of calculation, however, could be 70 THE FACTORIAL ANALYSIS OF HUMAN ABILITY and will do so better than any other equally numerous set of factors, whether of Hotelling's or of any other system. In this respect Hotelling's factors undoubtedly stand foremost. They will not, however, reproduce the correla- tions exactly unless they are all used, whereas in Thurstone's system a few common factors can, theoretically, do this, though in actual practice the difference of the two systems in this respect is not great. The chief disadvantage of Hotelling's components is that they change when a new test is added to the battery. When a new test is added to a Spearman battery, provided that it conforms to the hierarchy, g does not change in nature, though its exactness of measurement is changed. Whether Thurstone's com- mon factors will remain invariant in augmented batteries, and whether they will also do so when differently selected samples of people are tested, are questions we shall con- sider at a later stage in this book. 4. A calculation. The actual calculation of the loadings of Hotelling's components requires, for its complete under- standing, a grasp of the method of finding algebraically the principal axes of an ellipsoid, a problem which will be found dealt with in three dimensions in any textbook on solid geometry. We give an account of this, for n dimen- sions, in the Appendix. Here we shall only explain Hotelling's ingenious iterative method of doing this arithmetically, by means of an example, for which we shall use the matrix of correlations already employed in Chapter II to illustrate Thurstone's method (see opposite page). We have inserted unities in the diagonal cells, for Hotelling's procedure does not contemplate the assumption of specific factors (much less maximized specifics) except possibly that part of a specific which is due to error, in which case what are called " reliabilities " (actual correla- tions of two administrations of the test) would be used in the diagonal. Hotelling's arithmetical process then begins with a guess used to obtain loadings for the common factors after Thurstone's communalities have been inserted, instead of the "centroid" method. The advantage over the centroid method would be that absolutely the maximum variance would be " taken out " by successive factors. HOTELLING'S " PRINCIPAL COMPONENTS " 71 1-0 4 4 2 8 -78 -775 4 1-0 7 3 1-0 1-00 1-000 4 7 1-0 3 1-0 1-00 1-000 2 3 3 1-0 7 -65 -637 80 32 32 16 40 1-00 70 30 40 70 1-00 30 14 21 21 70 1-74 2-23 2-23 1-46 780 312 312 156 400 1-000 700 300 400 700 1-000 300 130 195 195 650 1-710 2-207 2-207 1-406 at the proportionate loadings of the first principal com- ponent. Practically any guess will do a bad guess will only make the arithmetic longer. We have guessed *8, 1, 1, ?, the numbers to be seen on the right of the matrix, because these numbers are roughly proportional to the sums of the four columns, and such numbers usually give a good first guess. Each row of the matrix is then multiplied by the guessed number on its right, giving the matrix below the first one, beginning with -80. We then take, as our second guess, numbers proportional to the sums of the columns of this matrix, namely 1-74 2-23 2-23 1-46 giving -78 1 1 -65 That is, we divide the sums of the columns by their largest member, and use the results as new multipliers. They are seen placed farther on the right of the original matrix. It is unusual for two of them to be of the same size that is a peculiarity of our example. It is always the original matrix whose rows are multiplied by each improved set of multipliers. The above set gives the next matrix shown, that beginning with '780, and the sums of its columns 72 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 1-710 2-207 2-207 1-406 give a third guess at the multipliers, namely 775 1 1 -637 And so the reiteration goes on, and the reader, who is advised to carry it a stage farther at least, would find if he persevered that the multipliers would change less and less. If he went on long enough, he would reach this point (usually, however, far fewer decimals are sufficient) : 1-0 4 4 2 4 1-0 7 3 4 7 1-0 3 2 3 3 1-0 -772865 -309146 -309146 -154573 400000 1-000000 -700000 -300000 400000 -700000 1-000000 -300000 125962 -188943 -188943 -629811 772865 1 -000000 1-000000 629811 1-698827 2-198089 2-198089 1-384384 giving -772865 1 1 -629813 that is, totals in exactly the same proportion as the multi- pliers. These final multipliers (or earlier ones if the experi- menter is content with less exact values) are then propor- tionate to the loadings of the first Hotelling component in the four tests. They have, however, to be reduced until the sum of their squares equals the largest total, 2-198089, which is called the first " latent root " of the original matrix. This is done by dividing them by the square root of the sum of their squares mid multiplying them by the square root of the latent root. They then become 662 857 857 540. The next step in Hotelling's process is similar to one with which we have already become familiar in Thur- stone's method. The parts of the variances and correla- tions due to this first component are calculated and sub- tracted from the original experimental matrix. These variances and correlations due to the first component are : HOTELLING'S "PRINCIPAL COMPONENTS 1 73 662 857 857 540 662 439 567 567 357 857 567 734 734 462 Matrix due to 857 567 734 734 462 first principal 540 357 462 462 291 component. 561 -167 -167 - -157 3 -18 Residual - -167 266 - -034 - -162 -4 -38 matrix -167 - -034 266 - -162 -4 -38 -157 - -162 -162 709 1-0 1-00 168 - -050 -050 -047 067 -106 013 065 067 013 -106 065 -157 -162 -162 709 145 -305 -305 792 The residual matrix is then treated in exactly the same way as the original matrix, the beginnings of the process being shown above. The guessed multipliers, proportional to the sums of the columns, are not so near the truth this time, for the first one, which we have guessed at #, and which reduces after one operation to 18, goes on reducing until it becomes negative, the final values of these second loadings being as shown in the appropriate column of the following table, which also gives the loadings of the third and fourth factors, obtained in the same way. The vari- ances and correlations due to each factor in turn are subtracted from the preceding residual matrix and the new residual matrix analysed for the next factor : Factor I 11 111 IV Sum of Squares Test 1 662218 -323324 675967 , 1 ., 2 856836 - -135197 - -312332 - -387298 1 3 856836 - -135197 - -312332 387298 1 * 539645 826092 162323 1 Sum of squares * 2-198090 823526 678383 300000 4 Percentages 55-0 20-6 16-9 7-5 100 * These four quantities are, in the Hotelling process, what are called the '* latent roots " of the matrix. 74 THE FACTORIAL ANALYSIS OF HUMAN ABILITY An alternative method of finding principal components, due to Kelley, is to deal with the variables two at a time. The pair first chosen are rotated in their plane until they are uncorrelated. Then the same is done to another pair, and so on, the new uncorrelated variables being in turn paired with others, until finally all correlations are zero. (Kelley, 1935, Chapters I and VI.) A chief advantage is that the components are obtained pari passu 9 and not successively ; also, in certain circumstances where Hotel- ling's process converges very slowly, Kelley's is quicker. The end results are the same. 5. Acceleration by powering the matrix. In a later paper Hotelling pointed out that his process of finding the load- ings of the principal components can be much expedited by analysing, not the matrix of correlations itself, but its square, or fourth, eighth, or sixteenth power, got by repeated squaring (Hotelling, 19356). Squaring a sym- metrical matrix is a special case of matrix multiplication (see Chapter VII, Section 8) : it is done by finding the " inner products " (see footnote, page 31) of each pair of rows, including each row with itself, and setting the results down in order. Applying this to the correlation matrix : 1-0 4 4 2 4 1-0 7 3 4 7 1-0 3 2 -3 3 1-0 we see that the inner product of the first row with itself is 1*36 ; of the first row with the second, 1-14 ; and so on. Setting these down in order, we get for the matrix squared : 1-36 1-14 1-14 64 1-14 1-74 1-65 89 1-14 1-65 1-74 89 64 89 89 1-22 HOTELLING'S "PRINCIPAL COMPONENTS 1 75 Exactly the same process is applied to this, beginning with guessed multipliers, as we applied to the original matrix. The multipliers, however, settle down twice as rapidly towards their final values, which are the same here as there. We have finally : 1-36 1-14 1-14 64 772865 1-14 1-74 1-65 89 1 -000000 1-14 1-65 1-74 89 1-000000 64 89 -89 1-22 629811 1-051096 881066 881066 494634 1-140000 1-740000 1 '650000 890000 1-140000 1-650000 1-740000 890000 -403079 560532 560532 768369 3-734175 4-831598 4-831598 3-043003 Ratio -772865 1 -000000 1-000000 629812 The " latent root," however, or largest total, 4-831598, is the square of the former latent root, 2-198090, so that its square root must be taken before we complete finding the loadings. In exactly the same way the squared matrix may be again squared, and again and again, before we analyse it. The more we square it, the quicker the Hotelling iteration process works. The end multipliers are always the same, but the " root " is the same power of the root we need as is the matrix of the original matrix. A still further acceleration of the process is due to Cyril Burt, who observed that as the matrix is repeatedly squared it becomes more and more nearly hierarchical, including the diagonal cells (Burt, 1937a). This is due to the largest factor increasingly predominating as it is " powered," especially if the largest latent root is widely separated from the others. In consequence, the square roots of the diagonal cells become more and more nearly in the ratio of the Hotelling multipliers, and form an excellent first guess for the latter. When our matrix is squared twice again, giving the eighth power, it becomes : 76 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 108-78 140-67 140-67 88-54 140-67 182-03 182-03 114-61 140-67 182-03 182-03 114-61 88-54 114-61 114-61 72-38 and the square roots of its diagonal members are 10-429 13-492 13-492 8-508 which are in the ratio 7730 1 1 -6306 very near indeed to the Hotelling final multipliers 772865 1 1 -629811 Hotelling gives a method of finding the residues, for the purpose of calculating the next factor loadings, from the " powered " matrix. But it may be so nearly perfectly hierarchical that this fails unless an enormous number of decimals have been retained, and it is in practice best to go back to the original matrix and obtain the residues from it. Their matrix can in turn be squared, and so on. Other and very powerful methods of acceleration will be found described in Aitken, 1937ft. 6. Properties of the loadings. If all the Hotelling com- ponents are calculated accurately, their loadings ought completely to exhaust the variance of each test ; that is, the sum of the squares of the loadings in each row should be unity. The sum of the squares of the loadings in each column equals the " latent root " corresponding to that column, and the sum of the four latent roots is exactly equal to the number of tests. Each latent root represents the part of the whole variance of all the tests which has been " taken out " by that factor. Thus the first factor " takes out " 55 per cent., the first two factors together 75-6 per cent., of the variance of the original scores. The four factors account for all the variance. If we turn back to Chapter II, where we made a Thurstone analysis of this same battery of four tests into two common factors and four specifics (six factors in all), we see, in the table on page 35, that the two first Thurstone factors HOTELLING'S " PRINCIPAL COMPONENTS" 77 "take out " 1-7652 and -0515 respectively that is, 44-1 per cent, and 1-3 per cent, of the four tests much less than the two first Hotelling factors account for. Because of this, the two first Hotelling factors will reproduce the original scores much better than the two Thurstone factors will. On the other hand, the two Thurstone factors reproduce the correlations exactly, while it takes all four Hotelling factors to do this. The correlations which correspond to the loadings given in the table on page 73 are obtained by finding the " inner product " of each pair of rows. Applying this to the table we find the correlation r 24 , say, to be 856836 X -539645 -135197 X -826092 -312332 X -162323 -387298 X zero = -300000 In this way, as we said above, the loadings of the four Hotelling factors will exactly reproduce the correlations we began with. If, however, we have stopped the analysis after we have found only two principal components (or factors), these two would have reproduced the correlations only approximately. For example, for r 24 we should only have -856836 X -539645 -135197 X -826092 = -350702 instead of -300000 Before we leave the table of Hotelling loadings, we may note that the signs of any column of the loadings can be reversed without changing either the variances or the correlations. Reversing the signs in a column merely means that we measure that factor from the opposite end, as we might rank people either for intelligence or stupidity and get the same order, but reversed. We will usually desire to call that direction of a factor positive which most conforms with the positive direction of the tests them- selves, and therefore we will usually make the largest loading in each column positive. All the loadings of Hotelling's first factor are, in an ordinary set of tests, positive. Of the other loadings, about half are negative. Thurstone's first analysis, it will ite renitembeited, also -gave a number t>f negative loadings 78 THE FACTORIAL ANALYSIS OF HUMAN ABILITY to the factors after the first, but he rotated his factors until these disappeared. 7. Calculation of a man's principal components. Esti- mation unnecessary. The Hotelling components have one other advantage over other kinds of factors that we did not mention in Section 3. They can be calculated exactly from a man's scores (provided that unities and not communalities are used in the diagonal, as was the case in Hotelling' s exposition), whereas Spearman or Thurstone factors can only be estimated. This is because the Hotel- ling components are never more numerous than the tests, whereas the Thurstone or Spearman factors, including the specifics, are always more numerous than the tests. For the Hotelling components, therefore, we always have just the same number of equations as unknowns, whereas we have more unknowns than equations in the Spearman- Thurstone system. We have hitherto given the analysis of tests into factors in the form of tables of loadings, or matrices of loadings, as we may call them, adopting the mathematical term. But we can alternatively write them out as " specification equations," as we shall call them. Thus the table on page 73 would be written % = -6622187! ~ '323324y 2 + -675967y 3 ZB = -8568367! - -135197y 2 - -312332y 3 - -387298y 4 2 3 = -8568367! ~ '135197y 2 - -312332y 3 -f -387298y 4 92y2 -f '162323y 3 Here z ly z Z9 s 8 , and * 4 stand for the scores in the four tests, measured in standard units ; that is, measured from the mean in units of standard deviation. The factors Yi Y2> YSJ an d y 4 are also supposed to be measured in such units. These specification equations enable us to calculate any man's standard score in each test if we know his factors, and since there are just as many equations as factors, they can be solved for the y's and enable us to calculate, conversely, any man's factors if we know his scores in the tests. The solution to these Hotelling equa- tions for the y's happens to be peculiarly simple, as we shall pfoVe in the Appendix* Section 7. It is as follows* HOTELLING'S "PRINCIPAL COMPONENTS" 79 yi = ( '662218% + -856836*2 + -8568362 3 + -539645z 4 ) - 2-198090 n = (_ .323324% - -1351972; 2 - -13519723 + -826092* 4 ) - -823526 y 3 = ( .675967*! - -312332z 2 - -3123322 3 + -162323z 4 ) - -678383 y 4 = ( - -387298*2 + -38729823 ) - -300000 The table on page 73, therefore, serves a double purpose. Read horizontally it gives the composition of each test in terms of factors. Read vertically it gives the composition of each factor in terms of tests, if we divide the result by the root at the foot of the column.* Suppose, for example, that a man or child has the fol- lowing scores in the four tests 1-29 -36 -72 1-03. This is evidently a person above the average in each test, since the scores are all positive. His factors will be obtained by substituting these scores for the 2 5 s in the above equations, with the result Yi = 1-062504 y 2 = -349441 y 3 = 1 -034624 Y* = -464757 (Of course, in practical work six decimal places would be absurd. They are given here because we are using this artificial example to illustrate theoretical points, in place of doing algebraic transformations, and they need, there- fore, to be exact.) If these values for the factors are now inserted in the specification equations opposite, the scores z in the test will be reproduced exactly (1-29, -36, -72, and 1-03). Notice, too, that if we have stopped our analysis at less than the full number of Hotelling factors, we can never- theless calculate these factors for any person exactly. As soon as we have the first column of the table on page 73, we can calculate YI for anyone whose scores z we know. Had we done this with the person whose scores are given * If the analysis has been performed with " reliabilities " in the diagonal cells instead of units, the statement in the text still holds (Hotelling, 1933, 498). If on correlations corrected for " attenua- tion," the matter is m'ore complicated (ibid. 4&'9-02). 80 THE FACTORIAL ANALYSIS OF HUMAN ABILITY above, we should have summarized his ability in these four tests by the one statement Yi = 1-062504 This would have been an incomplete statement, but it is the best single statement that can be arrived at. If we attempt to reproduce the scores z from this one factor alone, we can use only the first term in each of the specifica- tion equations on page 78. These give for the scores 704 -910 -910 -573 instead of 1-29 -36 -72 1-03, the true values, a pretty bad shot, as the reader will agree. But bad as it is, it is better than any other one factor will provide, as we shall show later after we have considered how to estimate Spearman and Thurstone factors. It will be seen from these first chapters that the different systems of factors proposed by different schools of " fac- torists " have each their own advantages and disadvan- tages, and it is really impossible to decide between them without first deciding why we want to make factorial analyses at all. This fundamental question we will devote some pages to in later chapters. But there are still several things we must do in preparation, and we turn next to a matter which has wider applications than in factorial analysis, namely the method of estimating one quality from measurements of other qualities with which it is correlated. This, for example, is the problem before those who give vocational advice to a man after putting him through various t6sts, or who give educational advice (or more peremptory instructions) to English children of eleven years of age after examining them in English, arithmetic, and perhaps with an " intelligence test," sorting them into those who may attend a secondary school, those who go to a central school, and those who remain in an elementary school. PART II THE ESTIMATION OF FACTORS To simplify and clarify the exposition, errors due to sampling the population of persons are in Parts I and II assumed to be non-existent. CHAPTER VI ESTIMATION AND THE POOLING SQUARE 1. Correlation coefficient as estimation coefficient. A corre- lation coefficient indicates the degree of resemblance between two lists of marks : and therefore it also indicates the confidence with which we can estimate a man's position in one such list x if we know his position in the other y. If the correlation between two lists is perfect (r = 1), we know that his standardized score * in the one list is exactly the same as in the other (x = y). If the correlation between the two lists is zero (r = 0), then the knowledge of a man's position in the one list tells us nothing whatever about his position in the other list. If we are compelled to make an estimate of that, we can only fall back on our knowledge that most men are near the average and few men are very good or very bad in any quality. We have, therefore, most chance of being correct if we guess that this man is average in the unknown test. (x = 0. The average mark we have agreed to call zero ; marks above average, positive ; marks below average, negative.) In the first case, when r = 1, we are justified in equating his unknown score x to his known score y x = y In the second case, when r = 0, we are compelled by our ignorance to take refuge in x = or average. Both these statements can be summed up in the one statement x = ry where the circumflex mark over the x is meant to indicate * A test score always means a standardized score unless the contrary is stated. But estimates are not in standard measure in general. 83 84 THE FACTORIAL ANALYSIS OF HUMAN ABILITY that this is an estimated, not a measured, value. If, now, we consider a case between these, where the correlation is neither perfect nor zero, it can be shown that this equation still holds, provided each score is measured in standard deviation units. Since r is always a fraction, this means that we always estimate his unknown x score as being nearer the average than his known y score. That is because we know that men tend to be average men. If this man's y score is high, say = 2 (two standard deviations above the average), and if the correlation between the qualities x and y is known to be r = -5, we guess his position in the x test as being fc^ry = >5x2 = I i.e. only one standard deviation above the average. This is a guess influenced by our two pieces of knowledge, (1) that he did very well in Test j/, which is correlated with Test a?, and (2) that most men get round about an average score (zero). It is a compromise, an estimate. It will often be wrong ; indeed, very seldom will it be exactly right. But it will be right on the average, it will as often be an underestimate as an overestimate, in each array of men who are alike in y. The correlation coefficient, then, is an estimation coefficient for tests measured in standard deviation units. 2. Three tests. Suppose now that we have three tests whose intercorrelations are known, and that a man's scores on two of them, y and z, are known. We wish to estimate what his score will most probably be in the other test, x. x need not be a test in the ordinary sense of the word, but may be an occupation for which the man is a candidate or entrant. According as we use his known y or his known z score, we shall have two estimates for his x score. To fix our ideas, let us take definite values for the correla- tions, say : x y z X 1-0 -7 -5 V '7 1-0 3 z -5 3 1-0 ESTIMATION AND THE POOLING SQUARE 85 The two estimates for his x are then < = -7y x = -5z and of these we shall have rather more confidence in the estimate associated with the higher correlation. But we ought to have still more confidence in an estimate derived from both y and z. Such an estimate could use not only the knowledge that y and z are correlated with x, but also the knowledge that they are correlated to an extent of r = -3 with each other. Just to take the average of the above two separate estimates will not utilize this knowledge, nor will it utilize the fact that the estimate from y (r = *7) is more worthy of confidence than the estimate from z (r = -5). What we want is to know how to combine the two scores y and z into a weighted total (by + cz) which will have the highest possible correlation with x. Such a correlation of a best-weighted total with another test is called a multiple correlation. From such a weighted total of his two known scores we could then estimate the man's x score more accurately than from either the y or the z score alone. It must use all the information we have, including our information that y and z correlate to an amount r = '3. 3. The straight sum and the pooling square. In order to answer this question, we shall first consider the problem of finding the correlation of the straight unweighted sum of the scores y + z with x. This is the simplest form of a problem to which a general answer was given by Professor Spearman (Spearman, 1913). We shall put his formula into a very simple form, which we may call a pooling square. In our present instance we want to find the correlation of y + z with x (all of these being, we are assuming, measured in standard deviation units). We divide the matrix of correlations by lines separating the " criterion " x from the " battery " y + z thus : 1-0 7 5 7 1-0 3 5 3 1-0 86 THE FACTORIAL ANALYSIS OF HUMAN ABILI1Y x y z x y z In each of the quadrants of this pooling square (with unities in the diagonal, be it noted) we are going to form the sum of all the numbers, and we shall indicate these sums by the letters : A C C B (where C is the sum of the Cross-correlations between the battery y + z and the criterion x, which can be regarded as a second battery of one test only). Then the correlation of x with y + z is equal to C = -744 which in our present example is = - ~= V(l) X (1 + -3 + -3 + 1) V 2 ' 6 so that the battery (y + z) has a rather better correlation (744) with x than has either of its members (-7 and -5). From the straight sum of the man's scores in the two tests y and z we can therefore in this case get a better estimate of his score in x than we could get from either alone. 4. The pooling square with weights. We want, however, to know whether a weighted sum of y and z will give a still higher combined correlation with x. With sufficient patience, we could answer this by trial and error, for the pooling square enables us to find almost as easily the correlation of a weighted battery with the criterion.* Let us, for example, try the battery 3y + * For this purpose * The pooling square can also be used to find the correlations or covariances of weighted batteries with one another. Elegant developments are Hotelling's ideas of the most predictable criterion (1085a) and of vector correlation (1936). ESTIMATION AND THE POOLING SQUARE 87 we write the weights along both margins of the pooling square : 1-0 7 5 3 7 1-0 .3 1 5 3 1-0 and multiply both the rows and the columns by these weights before forming the sums A, B, and C. The result of the multiplications in our case is : 1-0 2-1 5 2-1 9-0 9 5 9 1-0 1-0 2-6 and we therefore have correlation = 2-6 2-6 = -757 11-8 VII -8 a higher value than *744 given by the simple sum. So we have improved our estimation of the man's x score, and estimates made by taking By + would correlate *757 with the measured values of x. 5. Regression coefficients and multiple correlation. Similarly we could try other weights for y and z and search by trial and error for the best. There is, however, a general answer to this question, namely that the best weights for y and z are proportional to certain minor determinants of the correlation matrix. The weight for y is proportional to the minor left when we cross out the criterion column and the y row, the weight for z is proportional to minus the minor left when we similarly cross out the criterion column and the z row. The matrix of correlations with the criterion column deleted being: 7 1-0 3 5 3 1-0 88 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the weight for y is therefore proportional to : = -55 = -29 and that for z is proportional to : 7 -5 1-0 -3 that is, they are as -55 : -29. To make these weights not merely proportional but absolute values we must divide each of them by the minor left when the row and column concerned with the " criterion " x are deleted, namely : 1-0 -3 3 1-0 = -91 so that these absolute best weights, for which the technical name s or regression coefficients," are -55 -29 - V -\ -- Z 9I y ^ -91 -6044z/ + -3187* We are inviting the reader to take this method of calculat- ing the regression coefficients on trust ; but he can at least satisfy himself that when applied to the pooling square they give a higher correlation of battery with criterion than any other weights do. The result of multiplying the y column and row by -6044, and the z column and row by -3187, is the following : 6044 -3187 6044 3187 1-0 7 5 7 1-0 3 5 3 1-0 1-0000 4231 1593 4231 1593 3653 0578 -0578 1015 1-0000 5824 5824 5824 Multiple correlation 5824 = -763 = r m , say, which V'5824 is higher than any other weighting will produce, if the reader cares to try others. Notice the peculiarity of the pooling ESTIMATION AND THE POOLING SQUARE 89 square with regression coefficients as weights, that C = B (5824 = -5824). We can deduce that the inner product of the regression coefficients with the correlation coefficients gives the square of the multiple correlation 604 X -7 + -319 X -5 = -583 = r ro a Indeed, we can take this as forming one reason for using 604 and *319, and not any other numbers proportional to them, although the latter would give the same order of merit. We want our estimates of x not merely to be as highly correlated with the true values of x as is possible, but also to be equal to them on the average in the long run, in the sense that our overestimations will, in each array of men who have the same y and z 9 be as numerous as our underestimations, and this is achieved by using not merely -55 and 29 as weights, but -55 -f- -91, and *29 ~- -91. 6. Aitken's method of pivotal condensation. When there are more than two tests y and z in the battery, the applica- tion of the above rules becomes increasingly laborious. It is desirable, therefore, to have a routine method of calcu- lating regression coefficients which will give the result as easily as possible even in the case of a team of many tests. The method we shall adopt (Aitken, 1937a) is based upon the calculation of tetrads, as already used in our Chapter II. We shall first calculate the above regression coefficients again by this method. Delete the criterion column in the matrix of correlations, transfer the criterion row to the bottom, and write the resulting oblong matrix in the top left-hand corner of the sheet of calculations, preferably on paper ruled in squares : Check Column A (1-0) -3 3 1-0 7 -5 ~] _j 3 3 1-2 B (91) 3 -1 21 1-00 29 3297 1-0989 7 2308 99 C 604 -319 923 90 THE FACTORIAL ANALYSIS OF HUMAN ABILITY On the right of the oblong matrix of correlation coeffi- cients we rule a middle block of columns of the same number, here two, and on the right of all a check column. The columns of the middle block we fill with a pattern of minus ones diagonally as shown, leaving the other cells empty,* including the bottom row. In the check column we write the sum of each row. The top left-hand number of all we mark as the " pivot." Slab B of the calculation is then formed from slab A by writing down, in order as they come, all the tetrad-differences of which the pivot in A is one corner. Thus the first row of slab B is calculated thus IX 1 -3 X -3 = -91 IX -3 X ( 1) = -3 1 X ( 1) -3 X = 1 1 X -3 -3 X -3 = -21 and the row is checked by noting that -21 is the sum of the others. Immediately below this first row a second version of it is written, with every member divided by the first (91). This is to facilitate the calculation of slab C by having unity again as a pivot. The second row of slab B is then formed, beginning with 1 X -5 -T X -3 = -29 Throughout the whole calculation, except for the division of the first row, only one operation needs to be performed, namely the computing of tetrad-differences, beginning with the pivot. The same operation is then repeated to give slab C, using the modified first row of B 9 with pivot unity. This procedure goes on, slab after slab, until no numbers remain in the left-hand block. There being only three tests in all in our example, this happens at slab C. The middle block then gives the regression coefficients -604 and 319, with their proper signs, all ready for use. Throughout the calculation the check column detects any blunder in each row. When the number of tests in the battery is large, the * The dots represent zeros. ESTIMATION A&D THE POOLING SQUARE 91 calculation of the regression coefficients is a laborious business, but probably less so by this method than by any other. It will be clear to the reader that so long a calculation is not worth performing unless the accuracy of the original correlation coefficients is high. Only very accurate values can stand such repeated multiplication, etc., without giving untrustworthy results (Etherington, 1932). In other words, regression coefficients have a rather high standard error. 7. A larger example. Next we give in full the calculation of the regression coefficients in a slightly larger example, though one still much smaller than a practical scheme of vocational advice would involve. Here 2 is the "occu- pation," and z l9 z 29 3, and # 4 are tests. To give the example an air of reality, these and their intercorrelations are taken from Dr. W. P. Alexander's experimental study, Intelligence, Concrete and Abstract (Alexander, 1985). They were * : z t Stanford-Binet test ; z 2 Thorndike reading test ; z 3 Spearman's analogies test in geometrical figures ; 2 4 A picture-completion test. But the occupation is a pure invention, for purposes of this illustration only. The correlation matrix is : ZQ Zi Z Z #3 #4 2o 1-00 72 58 41 63 Z l 72 1-00 -69 49 39 Z 2 58 69 1-00 38 19 *3 41 49 38 1-00 27 2 4 63 39 19 27 1-00 The fact that we possess these correlations means that we have given these tests to a sufficiently large number of * In this, as in other instances where data for small examples are taken from experimental papers, neither criticism nor comment is in any way intended. Illustrations are restricted to few tests for economy of space and clearness of exposition, but in the experiments from which the data are taken many more tests are employed, and the purpose may be quite different from that of this book. 92 THE FACTORIAL ANALYSIS OF HUMAN ABILITY persons whose ability in the occupation is also known. The occupation can be looked upon as another test, in which marks can be scored. In an actual experiment, obtaining marks for these persons' abilities in the occupa- tion is in fact one of the most difficult parts of the work. We can now find by Aitken's method the best weights for Tests z 1 to 4 to make their weighted sum correlate as highly as possible with Z Q . To make the arithmetic as easy as possible to follow in an illustration, the original correla- tion coefficients are given to two places of decimals only, and only three places of decimals are kept at each stage of the calculation. The previous explanation ought to enable the reader to follow. As an additional help, take the explanation of the value -153 in the middle of slab Z>. It is obtained thus from slab C 1 X -158 -050 X -106 = -153 and is typical of all the others. Except for the division of each first row, only one kind of operation is required through the whole calculation, which becomes quite mechanical. The numbers shown on the left in brackets are the reciprocals of '524, -757, -826, used as multipliers instead of dividing by the latter numbers, in obtaining the modified first rows. The process continues until the left- hand block is empty, when the regression coefficients appear in the middle block (see opposite page).* The result is that we find that the best prediction of a man's probable success in this occupation is given by the regression equation Z = -390*! + 2222 2 + -018*3 + -43l2 4 We give a candidate the four tests, reduce his scores * The product of all the unconverted pivots, 1 x -524 x -757 x 826, is the value -328 of the determinant : 1-00 -69 -49 -39 69 1-00 -38 -19 49 -38 1-00 -27 39 -19 -27 1-00 For a different method of finding the regression coefficients, with certain advantages, see Addendum, page 350. ESTIMATION AND THE POOLING SQUARE COMPUTATION OF REGRESSION COEFFICIENTS Aitken's Modified Method with Each Pivot converted to Unity Check (1) 69 49 39 -1 1-57 69 1 38 19 . -1 . ; 1-26 A 49 38 1 27 1 . 1-14 39 19 27 1 1 85 72 58 41 63 . 2-34 (1 -908) ( 524) 042 - 079 690 -1 177 1 000 080 - 151 1-317 -1-908 338 i B 042 760 079 490 . 1 371 079 079 848 390 . . -1 238 083 057 349 720 1-210 (1-321) (757) 085 435 -080 -1 357 1-000 112 575 -106 -1-321 472 C 085 836 494 - -151 . -1 265 050 362 -611 -158 1-182 (1 D\ E -211) 826) 1-000 356 445 -160 112 -1 225 539 582 -194 153 136 066 -1-211 272 1-158 390 222 018 431 1-061 Regression Coefficients to standard measure by dividing by the known standard deviation of each test, insert these standard scores into this equation, and obtain an estimated score for him in the occupation. Thus the following three young men could be placed in their probable order of efficiency in this occupation from their test scores : Standard Scores in Tom % *, % * *, 7 -2 - -5 31 Dick -4 -1 3 - -8 - -47 Harry 2 -8 6 1-3 83 94 THE FACTORIAL ANALYSIS OF HUMAN ABILITY The multiple correlation of such estimates with the true values would be obtained by inserting the four correlation coefficients 72 -58 -41 -63 instead of the s's in the regression equation, and taking the square root, thus 390 X -72 + -222 X -58 + -018 X -41 + -431 X -63 = -68847 = r* Finally, we can, as we did in the former example, use the regression weights on a pooling square and see if we obtain this same multiple correlation of r m = -83 : 390 -222 -018 -431 1-00 72 58 -390 72 1-00 69 222 58 69 1-00 018 41 49 38 -431 63 39 19 41 63 69 49 39 1-00 88 19 38 1-00 27 19 27 1-00 It will be remembered that we have to multiply each row and column by its appropriate weight, and then sum all the numbers in each quadrant. The easiest way of doing this in large pooling squares is to multiply the rows first, then add the columns and multiply the totals by the column weights, finally adding these products, thus : Multiply the rows : Sums 390 222 018 431 1-0000 72 58 41 63 2808 1288 0074 2715 3900 1532 0088 1681 2691 2220 0068 0819 1911 0844 0180 1164 1521 0422 0049 4310 6885 7201 5798 4093 6302 ESTIMATION AND THE POOLING SQUARE 95 If we had kept all decimals these columnar sums would, since we are using regression coefficients as weights, have been exactly equal to the top row. With the actual figures shown, on multiplying the column totals and adding them, we find that the pooling square condenses to : 1-0000 6885 6885 6885 6885 QQ , P r = - = -83 as before. V-6885 8. The geometrical picture of estimation. Before we close this chapter it will be illuminating to consider what esti- mation of occupational ability means in terms of the geometrical picture of Chapter IV. Consider the illustra- tion used in the earlier pages of the present chapter, with the matrix : X y z X 1-0 7 -5 y 7 1-0 3 z 5 3 1-0 Here x is the criterion, y and z are the tests. Each of them can be represented by a line vector, as explained in Chapter IV, with angles between these vectors such that their cosines are the above correlations. The three vectors will then be in an ordinary space of three dimensions. The two tests y and z themselves have, of course, vectors which lie in a plane : any two lines springing from the same point as origin lie in a plane. These are the two tests to which we subject the candidate, whose probable score in x we are then going to estimate. His two scores OY and OZ in y and z enable us to assign to this man a point P on the yz plane, a point so chosen that its projec- tions on to the y and z vectors give the scores made by him in those tests (see Figure 19). But we cannot say that this is his point in the three-dimensional space ofx> y, and z. His point in that space may be anywhere on a line P'PP* 90 THE FACTORIAL ANALYSIS OF HUMAN ABILITY at right angles to the plane yz. For from anywhere on that line, projections on to y and z fall on the points Y and Z. Yet the projection on to the vector x, which gives his score in the criterion test #, depends very much on the position of his point on the line P'PP". All the people represented by points on that line have the same scores in y and z but different scores in x, and our man may be any one of them. Before deciding what to do in these circumstances, let us consider this set of people P'PP'' in more detail. It will be remembered that the whole population of persons is represented by a spherical swarm of points, crowded together most closely round about the origin O, and falling off in density equally in all directions from that point. Every test vector is a diameter of this sphere, and the plane containing any two test vectors divides the spherical swarm into equal hemispheres. It follows that a line like P'PP" is a chord of the sphere at right angles to a diameter (the line OP), and consequently that it is peopled symmetrically on both sides of P, both upwards along PP' in our figure, and downwards along PP*, the ESTIMATION AND THE POOLING SQUARE 97 men on the line being most crowded near the point P itself. The average man of the array of men P'PP" (who are all alike in their scores in the two tests y and z) is therefore the man at P, and since we do not know exactly where our candidate's point is along P'PP", we take refuge in guessing that he is the average man of his group- and is at the point P itself. From P, therefore, we drop a perpen- dicular on to the vector x 9 and take the distance OX as representing his estimated score in that test. This geo- metrical procedure corresponds exactly to the calculation we made, as a little solid trigonometry will show the mathematical reader. The non -mathematical reader must take it on trust, but the model may illuminate the calcula- tion. In our numerical example, taking the angles whose cosines are the correlations, the angle between y and z is about 72 1, that between x and z is 60, and that between x and y about 46. It is worth the reader's while to draw y and z on a sheet of paper on the table, and to represent x by a knitting-needle rising at an angle above the table, making roughly angles of 46 with y and 60 with z. Any point P on the paper represents a person's scores in y and z 9 scores shared by all persons vertically above and below P. The projection of P on to the knitting-needle is rf, the estimate. It is the average of all the different scores x that a person with scores OY and OZ can have. The estimate will only be certain if the knitting-needle itself is on the table ; it will be less and less certain, the more the knitting-needle is inclined to the table. In Section 3 of Chapter IV we noted that the angles which three test vectors make with each other are impossible angles, if the determinant of the matrix of correlations becomes negative. Ordinarily, that determinant is posi- tive. In our present example we have, for example : 1-0 -7 -5 7 1-0 -3 5 -3 1-0 = -38 Such a determinant, however, though it cannot be negative, can be zero, namely in the cases where the two smaller angles exactly equal the largest. In that case the 98 THE FACTORIAL ANALYSIS OF HUMAN ABILITY three vectors lie in one plane the knitting-needle has sunk until it too lies on the table. In that case alone, when the determinant is zero, the " estimation " is certain, and all the people in the line P'PP" have not only the same scores in y and z 9 but also the same scores in x. The vanishing of the above determinant therefore shows that this is so. And in more than three dimensions, although we can no longer make a model, the vanishing of the determinant : A ^*01 ^*02 ^*03 ^*0/i ^*01 * ^*12 ^*13 ^"ln /*/*"!/ /* '02 '12 '23 2*i i ^. A sa y> ^03 ^13 ^23 I ^3 shows that the criterion Z Q can be exactly estimated from the team z l9 z z . . . z n . In fact, the multiple correlation r m , which we have already learned to calculate in another way, can also be calculated as AGO where A is the whole determinant, and A o is the minor left after deleting the criterion row and column. This expression clearly becomes equal to unity when A = 0. In our small example x, y, z, we have = V 1 - 2i = A/1 = V'5824 = . = -38 AOO = ' 91 .763 as we already know it to be from page 88. 9. The " centroid " method and the pooling square. The pooling square, which we have learned to use in this chapter, enables us to see more clearly the nature of the factors first arrived at by Thurstone's " centroid " method. It will be remembered that in Chapter II, page 23, in a footnote we promised an explanation of this name " cen- ESTIMATION AND THE POOLING SQUARE 99 troid " (or centre of gravity) method as applied to the calculations of factor loadings. Let us suppose that the tests Zi, 2 2 z 39 and # 4 have the correlations shown, and let us by the aid of a pooling square find the correlation of each of them with the average of all. This means giving each test an equal weight in pooling it. Equal z 2 Weights z 3 Equal Weights Z 1 Z 2 Z 3 *4 r. A The correlation of z l with the average of all is then obtained from the above pooling square, which condenses to: r ' r 19 + r 14 Sum of all the cells of the table of corre- lations. and the correlation coefficient is _ Vabove sum This, however, is exactly Thurstone's process applied to a table with full communalities of unity. The first Thurstone factor obtained from such a table is simply for each individual the average of his four test scores, and the method is called the " centroid " method, because " cen- troid " is the multi-dimensional name for an average (Vectors, Chapter III; and see Kelley, 1935, 59). The vector, in our geometrical picture, which represents the first Thurstone factor, is in the midst of the radiating 100 THE FACTORIAL ANALYSIS OF HUMAN ABILITY vectors which represent the tests, like the stick of a half- opened umbrella among the ribs. It does not, however, make equal angles with the test vectors unless these all make equal angles with each other. If several of them are clustered together, and the others spread more widely, the factor will lean nearer to the cluster. In the foregoing explanation the communalities have been taken as unity, and the factor axis was pictured in the midst of the test vectors. If smaller communalities are used, the only difference is that a specific component of each test is discarded, and the first-factor axis must be pictured as in the midst of the vectors representing the other components of the tests. It can be shown that when communalities less than unity are used, if we bear in mind that the communal components of the tests are not then standardized, the pooling square gives the correlations exactly as -before, if we use communalities instead of units in the diagonal. The first centroid factor is the average of the communal parts of the tests. The later factors in their turn are, in a sense, averages of the residues. There are, however, some complications, the first being that the average of the residues just as they stand is zero. The manner in which Thurstone circum- vents this has already been described in Chapter II. 10. Conflict between battery reliability and prediction. Weighting a battery of tests, to give maximum correlation with a criterion (for example, to give the best prediction of future vocational or educational success), will alter the reliability of the battery, that is the correlation of the weighted battery with a similarly weighted duplicate bat- tery. The best weights for prediction will usually differ from the weights which give maximum battery reliability, so that there is a conflict between two " bests." Thomson (19406) has described how to find the best weights for battery reliability, as a special case of Hotel- ling's " most predictable criterion " (Hotelling, 1935a, and see Thomson, 1947, and M. S. Bartlett, 1948), and Peel (1947) has given a simpler formula than Thomson's (see page 367 in the Mathematical Appendix, section 9a). If there are ESTIMATION AND THE POOLING SQUARE 101 only two tests in the battery, with reliabilities r u , r 22 and correlating with another r 12 , then Peel's formula gives as the maximum attainable reliability the largest root [i of the equation ~ P >22 - = that is (r>(l - r l2 2 ) - [z(r n + r 22 - 2r 12 2 ) + (r u r 22 - r 12 2 ) = 0. If, for example, r 12 = *5, r u = -7 and r 22 = *8, the quadratic has roots '843 and '490, and a battery reliability of '843 is attainable by using weights proportional to either row of the above determinant with (x = -843, taken reversed and with alternate signs, that is '0785 and *1431 or -0431 and -0785 or 1 and 1-8 approximately. If as a check we set out a pooling square for the two bat- teries it will be 1 1-8 1 1-8 1 1-8 1 1-8 1-0 5 1-0 5 8 7 5 1-0 5 5 8 5 1-0 and if we multiply the rows and columns by the weights shown, and add together the quadrants, this reduces to 6-04 5-092 5-092 6-04 giving a battery self-correlation or reliability of- 5-092 6-04 843 as expected. Since there is the conflict mentioned above between such weights which increase battery reliability, and weights to make the battery agree as well as possible with a criterion, it would be an advance to possess some reasonably simple form of calculation to find weights to make the best com- promise (see Thomson, 19406, pages 864-5). CHAPTER VII THE ESTIMATION OF FACTORS BY REGRESSION 1. Estimating a man's "g." So far, our discussion of estimation in Chapter VI has had nothing immediate to do with factorial analysis. We are next, however, going to apply these principles of estimation to the problem of estimating a man's Spearman or Thurstone factors, given his test scores. As we have already explained in Chapter V, there is no need to " estimate " Hotelling's factors ; they can be calculated without any loss of exactness because they are equal in number to the tests : and even if we analyse out only a few of them, they can be exactly calculated for a man from his test scores. When we say exactly here, we mean that the factors are known with the same exactness as the test scores which are our data. Spearman or Thurstone factors, however, are more numerous than the tests, and can therefore only be " estimated." Two men with the same set of test scores may have different Thurstone factors. All we can do is to estimate them, and since the test scores of the two men are the same, our estimates of their most probable factors will be the same. The problem does not differ essentially from the estimation of occupational success or of ability in any " criterion " test. The loadings of a factor in each test give the z row and column of the correlation matrix. Let us first consider the case of a hierarchical battery of tests, and the estimation of g, taking for our example the first four tests of the Spearman battery used as illustra- tion in Chapter I, with these correlations : Z l Z 2 Z, Z* *1 1-00 72 63 54 %2 72 1-00 56 48 Z 3 63 56 1-00 42 z* 54 48 42 1-00 102 * THE ESTIMATION OF FACTORS BY REGRESSION 303 These correspond, in the analogy with the ordinary cases of estimation of the first chapter of this part, to the tests given to a candidate. In those cases, however, there was a real criterion whose correlations with the team of tests were known, and formed the z row and column of the matrix. Here the "criterion" isg, and it cannot be measured directly ; it can only be estimated in the manner we are now about to describe. We have here, therefore, no row and column of experimentally measured correlations for the criterion z or g in the present case (Thomson, 1934&, 94). From the hierarchical matrix of inter- correlations of the tests, however, we can calculate the " saturation " or " loading " of each test with the hypo- thetical g, and use these for our criterion column and row of correlations. We thus arrive at the matrix : Zo * t 2, Z 3 Z, 1-00 90 80 70 60 90 1-00 72 63 54 80 72 1-00 56 48 70 63 56 1-00 42 60 54 48 42 1-00 and we want to know the best-weighted combination of the test scores %i to 2 4 in order to correlate most highly with z = g. The problem is now the same as one of ordinary estimation of ability in an occupation, and the mathematical answer is the same. We can, for example, use Aitken's method of finding the regression coefficients, although in this case, because of the hierarchical qualities of the matrix, there is, as we shall shortly see, an easier method. It is, however, illuminating for the student actually to work out the regression coefficients as in an ordinary case of estimation, as shown on the next page. If, therefore, we know the scores z l9 z 29 2 3 , and # 4 which a man has made in these four tests, we can estimate his g by the equation (see overleaf) g = -55312, + -2595*2 + *16023 3 + -1095* 4 104 THE FACTORIAL ANALYSIS OF HUMAN ABILITY (1-00) -72 -63 -54 72 1-00 -56 48 63 -56 1-00 -42 54 -48 42 1-00 90 -80 -70 -60 -1-00 -1-00 -1-00 -1-00 1-89 1-76 1-61 1-44 3-00 (2-0764)(-4816) -1064 -0912 72 -1-00 3992 1-0000 -2209 -1894 1064 -6031 -0798 0912 -0798 -7084 1520 -1330 -1140 1-495 -2-0764 63 . -1-00 54 . . -1-00 90 ... 8289 4193 4190 1-2994 (1*7258) (-5796) -0596 4709 -2209 -1-00 3311 1-0000 -1028 0597 -6911 0994 -0852 8124 -3811 -1-7253 4037 -1894 . -1-00 6728 -3156 5712 3438 1-1730 (1-4599) (-6850) 3552 -1666 -1030 -1-00 3097 1-0000 0750 5186 -2432 -1504 -1-4599 5920 -2777 -1715 4521 1-1162 5531 -2595 -1602 -1095 Regression Coefficients 1 -0823 The multiple correlation of such estimates in a large number of cases with the true values of g will be by analogy with our former case given by r m a= -5531 X -90 + -2595 X -80 + -1602 X -70 + -1095 X -60 = -883 r m = -940 We must remember, however, that such a correlation here is rather a fiction. We had in the former case the possi- bility of comparing our estimates with the candidate's eventual performance in the occupation or criterion 2 . Here we have no way of knowing g ; we only have the estimates. As before, we can check the whole calculation by a pooling square, thus : THE ESTIMATION OF FACTORS BY REGRESSION 105 5531 -2595 -1602 -1095 1-00 90 80 70 60 5531 90 1-00 72 63 54 2595 80 72 1-00 56 48 1602 70 63 56 1-00 42 1095 60 54 48 42 1-00 Multiplying by the row weights and summing the columns condenses this to : 5531 -2595 -1602 -1095 1-000 883 90 80 70 60 900 -800 -700 -600 and multiplying by the column weights gives : 1-000 883 883 883 showing that our calculation was exact to three places. Estimating g from a hierarchical battery is therefore, mathematically, exactly the same problem as estimating any criterion, and can be done arithmetically in the same way. Because of the special nature of the hierarchical matrix of correlations, however, with its zero tetrad- differences, there is an easier way of calculating the estimate of g, due to Professor Spearman himself (Abilities, xviii). For its equivalence mathematically to the above see Thomson (19346, 94-5) and Appendix, paragraph 10. Meanwhile we shall illustrate it by an example which will at least show that it is equivalent in this instance. The calculation is best carried out in tabular form, and is based entirely on the saturations or loadings of the tests with g, which are also their correlations with g. 106 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Regression Test r * V j r 2 V r ig Coefficients 10 v r t<r \,r iff 1 + S " 1 - V 1 9 81 19 4-2632 4-7368 5533 2 8 64 36 1-7778 2-2222 2596 3 7 49 51 9608 1-3725 1603 4 6 36 64 5625 9375 1095 S = 7-5643 1 + S = 8-5643 = -1168 The result, with much less calculation, is the same. The quantity S is of some importance in this formula. It is formed in the fourth column of the table, from which it will be seen that S = V 4- 1 r 2 2 1 r\ 2 ' 'n 1 ~ - r ? 1 - r ig It is clear that S will become larger and larger as the number of tests is increased. Now, we saw that the square of the multiple correlation r m is obtained when we multiply each of the weights by r ig and sum the products. That is to say r m 2 = Z (weight X saturation) 1 +S S 1 This fraction will be the nearer to unity, the larger S is ; and we can make S larger and larger by adding more and more (hierarchical) tests to the team. Thus in theory we can make a team to give as high a multiple correlation with g as we desire. It will also be noticed, however, from our table that the tests with high g saturation make THE ESTIMATION OF FACTORS BY REGRESSION 107 much the largest contribution to S, and therefore to the multiple correlation (see Piaggio, 1933, 89). 2. Estimating two common factors simultaneously. We have seen in the preceding section how to estimate a man's g from his scores in a hierarchical team of tests, and in this we shall consider the broader question of estimating factors in general. Thus in Chapter II the four tests with correlations : 1 . -4 4 2 2 4 , 7 3 3 4 7 . 3 4 ! 2 3 -3 t were analysed into two common factors and four specifics with the loadings (see Chapter II, page 36). Common Factors ' / // Specific Factors 5164 . -8563 2 -7746 -3162 . -5477 3 -7746 -3162 . . -5477 4 -3873 . ... -9220 Any one column of these loadings can be used as the criterion row in the calculation by Aitken's method, and the regression coefficients calculated with which to weight a man's test scores in order to estimate that factor for him. If, as is probable, we want to estimate both common factors, we can do the two calculations together, as shown at top of next page. Both rows of loadings are written below the matrix of intercorrelations, and then pivotal condensation automatically gives both sets of regression coefficients, with only one extra row in each slab of the calculation, as on the next page. If, therefore, we have a man's scores (in standard measure) in these four tests, our estimate of his Factor I will be (see overleaf) 1787*! + -3932* 2 + -3932z 3 + -1156* 4 108 THE FACTORIAL ANALYSIS OF HUMAN ABILITY L-0) 4 4 -2 -1-0 1-0 4 1-0 -7 -3 -1-0 14 4 -7 1-0 -3 -1-0 1-4 2 -3 -3 1-0 -1-0 8 5164 -7746 -7746 -3873 . 2-5429 3162 -3162 . . 6324 (84) -54 -22 40 -1-0 1-0 1-00 -6429 -2619 4762 -1-1905 1-1905 54 -84 -22 40 -1-0 1-0 22 -22 -96 20 . -1-0 6 5680 -5680 -2840 5164 1-9365 3162 -3162 . . 6324 (4928) -0786 1429 -6429 -1-0 3571 1-0000 -1595 2900 1-3046 -2-0292 7246 0786 -9024 0952 -2619 . -1-0000 3381 2028 -1352 2459 -6762 1-2603 1129-0828 -1506 -3764 2560 (-8899 0724 -1594 -1594 -1-0000 2811 1-0000 0814 -1791 -1791 -1-1237 3159 1029 1871 -4116 -4116 1-1134 - -1008 -1833 -2291 -2291 1742 1787 -3932 -3932 -1156 1-0809 -1751 -2472 -2472 --1133 2060 Regression Coefficients 1 and estimates made in this way will have a multiple correlation r m with the " true " values of the factor, in a number of different candidates, given by r m * = -1787 X -5164 + -3932 X -7746 + -3932 X -7746 + -1156 X -3873 = -7462 r m = -864 Similarly, the multiple correlation of the estimate of the second factor with the " true " values can be found to be r m = -395 The two factors are not, therefore, estimated with equal accuracy by the team. As with ordinary estimation, the whole calculation can be checked by a pooling square. This check for the second factor is as follows : THE ESTIMATION OF FACTORS BY REGRESSION 109 1751 -2472 -2472 -1133 1-0000 3162 3162 1751 1-0 -4 4 2 2472 3162 4 1-0 7 3 2472 3162 4 7 1-0 3 1133 9 2 3 3 1-0 Multiplying the rows gives : - -1751 -2472 -2472 - -1133 1-00000 3162 3162 Sums of columns 07816 07816 - -17510 - 09888 09888 - -02266 - 07002 24720 17304 03399 - -07004 - 17304 24720 - -03399 - 03502 07416 07416 11330 15633 31621 31621 Multiplying then by the column multipliers, and adding, we get : 1-00000 15633 15633 15633 where the equality of the three quadrants shows that our regression weights were correct : and the multiple corre- lation is V 15633 = * 395 - We have now found the regression equations for esti- mating the two common factors by treating each in turn as a " criterion." It is also possible to estimate a man's specific factors in the same way. Indeed, we might have written the loadings of the specific factors as four more rows below the common-factor loadings in the first slab and calculated thfcir regression coefficients all in thfe one 110 THE FACTORIAL ANALYSIS OF HUMAN ABILITY calculation. But it is easier to obtain the estimate of a man's specific by subtraction (compare Abilities, 1932 edition, page xviii, line 10). For example, we know that the second test score is made up as follows Z 2 = -7746/i + -3162/ 2 + -5477s 2 where /i and / 2 are the man's common factors and s 2 his specific. We have estimated his /i and / 2 , and we know his 2 2 ; so we can estimate his $ 2 from this equation. The estimates of all a man's factors, to be consistent with the experimental data, must satisfy this equation and similar equations for the other tests. If the estimate of the specific is actually made by a regression equation, just like the other factors, it will be found to satisfy this require- ment.* From the estimates of all a man's factors, there- fore, including any specifics, we can reconstruct his scores in the tests exactly. From only a few factors, however, even from all the common factors, we cannot reproduce the scores exactly, but only approximately. 3. An arithmetical short cut (Ledermann,1938a, 1939). If the number of tests is appreciably greater than the number of common factors, the following scheme for computing the regression coefficients will involve less arithmetical labour than the general formulae expounded in Chapter VI and applied to the factor problem in this chapter.f For illustration, we shall use the data of the preceding section (page 108), although in that example the number of tests (four) exceeds the number of common factors (two) only by two, which is too small an amount to demonstrate * It is interesting to note that we know the best relative loadings of the tests to estimate a specific by regression without needing to know how many common factors there are, or whether indeed any specific exists or not. (Wilson, 1934. For the same fact in more familiar notation, see Thomson, 19860, 43.) f This short cut, in the form here given, is only applicable to orthogonal factors. For oblique factors, which are described below in Chapter XVIII, modifications are necessary in Ledermann's formulse, for which see Thomson (1949) and the later part of section 19/of the Mathematical Appendix^ page 378* THE ESTIMATION OF FACTORS BY REGRESSION 111 fully the advantages of the present method. The common- factor loadings and the specifics of the four tests form a 4x2 matrix and a 4 X 4 matrix respectively, thus : 5164 7746 -3162 7746 -3162 3873 8563 5477 5477 9220 the matrix M Q being identical with the first two columns, and the matrix M l with the last four columns of the table on page 107. Before the data are subjected to the com- putational routine process, which will again consist in the pivotal condensation of a certain array of numbers, some preliminary steps have to be taken : (i) the loadings of each test are divided by the square of its specific, and the modified values are then listed in a new 4x2 matrix : 7042 2-5820 2-5820 4556 1-0540 1-0540 J = e.g. 2-5820 = (-7746) -f- (-5477) 2 1-0540 = (-3162) -7- (-5477) 2 (ii) Next, the inner products (see footnote on page 81) of every column of M in turn with every column of M i are calculated and arranged in a 2 X 2 matrix : 5401 1-63291 6329 -6665 J i.e. the first row of this matrix contains the inner products of the first column of M with all the columns of M i, similarly the second row of J contains all those inner products which involve the second column of M 09 e.g. 4-5401 = -5164 X -7042 + -7746 X 2-5820 + -7746 X 2-5820 + -3873 X -4556 6665 = -3162 X 1-0540 + -3162 X 1-0540 If there had been r common factors the matrix J would have been an r X r matrix. The arithmetic is simplified 112 THE FACTORIAL ANALYSIS OF HUMAN ABILITY by the fact that J is always symmetrical about its diagonal, so that only the entries on and above (below) the diagonal need be calculated, (iii) Finally, each element on the diagonal of J is augmented by unity, giving, in the notation of matrix calculus, the matrix : 5-5401 1 -6329 1-6329 1 -6665 This matrix is now " bordered " below by the matrix MOI, and on the right-hand side by a block of minus ones and zeros in the usual way. The process of pivotal condensation then yields the same regression coefficients as were obtained on page 108. 6-1730 5-5401 1-6329 1-0000 1-6329 7042 2-5820 2-5820 4556 2947 1-6665 1-0540 1-0540 1-1853 -1-0000 - -1805 -1-0000 2947 - 1 -0000 1-1142 2-2994 7042 3-6360 3-6360 4556 4800 1-0000 2486 -8437 i 4050 - -2075 1271 - -0804 2931 4661 7591 2931 4661 ! 7591 -1343 0822 - -0520 1787 - -1751 0036 efficients 3932 2473 6404 3932 2473 6404 1156 - -1133 0023 4. Reproducing the original scores. Let us imagine a man who in each of the four tests in our example obtains a score of + 1 ; that is, one standard deviation above the average. We choose this set of scores merely to make the arithmetic of the example easy. The regression estimates of his two common factors are /! = -17872! + -3932*2 + -3932* 3 + -1156*4 THE ESTIMATION OF FACTORS BY REGRESSION 113 Inserting his scores z t = z z = % 3 = 4 = 1 into these equations we get for the regression estimates of his factors /! = 1-0807 / 2 = -2060 that is, we estimate his first factor to be rather more than one standard deviation, his second factor to be about one-fifth of a standard deviation, above the average. Now, the specification equations which give the composi- tion of the four tests in terms of the factors are z v = -5164/i . + -8563$! Z 2 = -7746/! + -3162/2 + -5477$ 2 * 3 = -7746/i + -3162/2 + -5477s 3 s 4 = -3873/j . + -9220s 4 If we insert the above estimates/! and/ 2 in lieu of /j and / 2 , we get for this man's scores *! == -5581 + -8563^ z 2 = -9022 + -5477^2 z a = -9022 + -5477^ ^ 4 = -4186 + -9220^ We know his four scores each to have been + 1, and if we had also worked out the estimates of his specifics by the regression method we should have found that they added just enough to the above equations to make each indeed come to + 1. We can, therefore, find his estimated specifics more easily from the above equations, as in this case 1 -5581 -8563 -9022 ""5477 = -5161 = -1786 and so for s 3 and $ 4 , subtracting the contribution of the common factors from the known score (here + 1 in each case) and dividing by the specific loading. The regression estimates of the factors, made by the system we have so far been considering, are as a matter of fact not the only estimates which have been proposed. The alternative system has certain advantages, to be explained later. The regression estimates are thfc best in 114 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the sense, as we said when deducing them, that they give the highest correlation, taken over a large number of men, between the estimates and the true values of a criterion when the latter can be separately ascertained. Just what this correlation means, however, when there is no possibility of ascertaining the " true " values (for factors, when they outnumber the tests, only can be estimated) it is not so easy to say. The regression estimates of the factors, as calculated in the present chapter, have one other great advantage, that they are consistent with the ordinary estimation of voca- tional ability made without using factors at all, as can best be shown by means of the example of Section 7 of Chapter VI. 5. Vocational advice with and without factors. In that example we had an " occupation " z , and four tests %i9 #2> 3? and 2 4 ; and in Chapter VI, without using factors at all, we arrived at the following estimation of a man's success or " score " in the occupation (which is, after all, only a test like the others, though a long-drawn-out one) Z = -3902! + -222s 2 + -OlSSg + -43l2 4 Now let us suppose that the matrix of correlations of these five tests (including the occupation as a test) had been analysed, by Thurstone's method or any other, into common factors and specifics the matrix is given in Chapter VI, page 91. Indeed, the four tests proper were so analysed by Dr. Alexander in the monograph from which we took their correlations, and the analysis below is based on his. The " occupation " s is a pure fiction made for the purpose of this illustration, but we can easily imagine it also being analysed in exactly the same way as a test. The table of loadings of the factors, to which we may as well give Dr. Alexander's names of g (Spearman's g), v (a verbal factor), and jP (a practical factor), is as follows : g v F Specific Occupation Z -55 -45 -60 -37 Stanford-Binet z^ -66 -52 -21 -50 Reading test z 2 -52 -66 . -54 Geometrical analogies z 3 -74 . . -67 Picture completion z 4 - -37 . '71 -60 THE ESTIMATION OF FACTORS BY REGRESSION 115 With this table of loadings "in our possession we might have given vocational advice to a man in a roundabout way. Instead of inserting his scores in z l9 * 2 , * 3 , and * 4 in the equation (see page 93). * = -390*! + -222*2 + -018*3 + -431* 4 we might have estimated his factors g 9 v 9 and F from his scores in the four tests, and then inserted these estimated factors in the specification equation of the occupation 2 = *55g + *45z; + *6QF ~\- -37s (ignoring the specific s Q9 which cannot be estimated from 2i 5 z 29 * 3 , and * 4 ). Had we done so 9 we should have arrived at exactly the same numerical estimate of his * as by the direct method (Thomson, 1936a, 49 and 50). The actual estimation of the factors g 9 v 9 and F from the four tests will form a good arithmetical exercise for the student. The beginning and end of the calculation of the regression coefficients is shown here, following exactly the lines of the smaller example on page 108 of this chapter : | Check -1 . . . 1-57 -1 . . 1-26 -1 . 1-14 1 -85 ! 2-29 1-18 21 . . -71 . . . . -92 This reduces by pivotal condensation step by step to the three sets of regression coefficients : forg -300 -095 -532 -095 for v -353 -581 -352 -153 for P ] -121 - -148 206 -747 The result is to give us three equations for estimating g 9 v 9 and F from a man's scores in the four tests, viz. g = -300*! + -095*2 + -532*3 + -095* 4 v = -353*! + -581*3 -352*3 -153* 4 F = -121*! -148*2 -206*3 + -T47* 4 Now let us assume a set of scores z l9 * 2 , * 8 , * 4 for a man, and see what the estimate of his occupational ability is by 1-00 69 -49 39 69 1-00 38 19 49 38 1-00 27 -39 19 27 1-00 66 52 74 37 -52 -66 . . 116 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the two methods, the one direct without using factors, the other by way of factors. Suppose his four scores are *1 ^2 2s *4 2 -4 -7 -6 The estimates of his factors g, v, and F will therefore be | = -300 X -2 + -095 x (- -4) + -532 X -7 + -095 x -6 = -451 V = -353 X -2 + -581 X ( -4) -352 X -7 -153 X -6 = -500 F = -121 X -2 - -148 X (- -4) - -206 X -7 + -747 x -6 = -387 If now we insert these estimates of his factors into the specification equation of the occupation, ignoring its specific, we get for our estimate of his occupational success : z<> = -55 X -451 + -45 X ( -500) + -60 X -387 = -255 that is, we estimate that he will be about a quarter of a standard deviation better than the average workman. This by the indirect method using factors. By the direct method, without using factors at all, we simply insert his test scores into the equation = .390*! + -222*0 + -018*3 + -431*4 and obtain g = -390 X -2 + -222 X ( -4) + -018 X -7 + -431 X -6 = -260 exactly the same estimate as before for the difference in the third decimal place is entirely due to "rounding off" during the calculation. The third decimal place of the direct calculation is more likely to be correct, since it is so much shorter. 6. Why, then, use factors at all ? The reader may now ask, " What, then, is the use of estimating a man's factors at all ? " Well, in a case analogous to that of the present example, it is quite unnecessary to use factors at all, and there is no doubt that a great many experimenters have rushed to factorial analysis with quite unjustifiable hopes of somehow getting more out of it than ordinary methods of vocational and educational advice can give without mentioning factors. But we must not go to the other extreme and " throw out the baby with the bath- water." There may be other reasons for using factors, aj>art from THE ESTIMATION OF FACTORS BY REGRESSION 117 vocational advice. And even in giving such advice, which really means describing men and occupations in similar terms, so that we can see if they fit one another or not, it may be that factors have some advantages not disclosed by the above calculation. This man whom we have used above, for example, may be described either in terms of his scores in four fairly well-known tests, or in terms of the factors g, v 9 and F. By the former method his description is : Stanford-Binet test -2, slightly above average Thorndike reading test -4, distinctly below average Spearman's geometrical analogies . . . -7, good Picture-completion test 6, good This description already suggests to us that he is a man of average intelligence or rather better, of not much schooling, and with a bit of a gift for seeing shapes, and similarities in them. From the correlations of the occupation with these four tests we know that it most resembles the first and last tests and least resembles the third. We can probably draw the conclusion that this man will be above average in it ; and we can draw this conclusion accurately if we calculate the regression equation z<> = -390*! + -222*2 + -01823 + -43l2 4 As a description of the man, however, the above table suffers from the fact that the four tests are correlated with one another. We feel a certain clarity in the description in terms of factors, because these are independent of one another and uncorrelated. This man whom we are at present considering is alternatively described, in terms of factors, as : Factor Estimated Amount g -451 v -500 F -387 that is, a quite intelligent (g) and practical (F) man with, however, not much ability in using and understanding words (v). There is a certain air of greater generality about the factors than there is about the particular tests 118 THE FACTORIAL ANALYSIS OF HUMAN ABILITY from which they have been deduced, and v they give definition and point to mental descriptions, or at least they seem to do so. Yet some of these " advantages " of using factors begin to look less bright when looked into more carefully. We said that one advantage is that factors are independent and un correlated. So they are, if their true values are known. But we only know their estimates, and these are correlated, as we shall illustrate shortly. If we use factors it is clear that we must, if we value the advantage of independence, seek to obtain estimates which are as little correlated with one another as possible. There have been proposals to use factors which are really correlated ; not merely correlated when their estimates are taken, but correlated in their true measures. What advantage can these have over the actual correlated tests ? The funda- mental advantage hoped for by the factorist seems to be that the factors (correlated or un correlated) may turn out to be comparatively few in number, and may thus replace a multitude of tests and innumerable occupations by a description in these few factors. The student whose knowledge of the subject is being obtained from this book is not yet equipped to discuss adequately the very funda- mental questions raised in this section, to which we shall return several times in later chapters. One last point in favour of factors may, however, be expanded somewhat here. We said a couple of sentences back that factorists hope to give adequate descriptions of men and of occupa- tions in terms of a comparatively small number of factors. This, if achieved, would react on social problems somewhat in the same way as the introduction of a coinage influences trade previously carried on by barter. A man can ex- change directly five cows for so many sheep, so much cloth, and a new ploughshare ; but the transaction is facilitated if each of these articles is priced in pounds, shillings, and pence, or in dollars and cents, even though the end result is the same. And so perhaps with the " pricing " of each man and each occupation in terms of a few factors. But the prices must be accurate ; and the analyses of THE ESTIMATION OF FACTORS BY REGRESSION 119 tests and occupations into factors, still more the calculation of quantitative estimates of these factors, are as yet very inaccurate, and perhaps are inherently subject to uncer- tainty. A fluctuating and doubtful coinage can be a positive hindrance to trade, and barter may be preferable in such circumstances. We showed in Section 5 above that a direct regression estimate of a man's ability in an occupation gives identically the same result as an estimate via the roundabout path of factors, so that at least when the direct regression estimate is possible there can be no quantitative advantage in using factors. When, however, is the direct regression estimate possible, and when is it impossible ? To make the direct regression estimate we require the complete table of correlations of the tests with one another and with the occupation, and we have to know the candidate's scores in the tests. This implies that these same tests have been given to a number of workers whose proficiency in the occupation is known, for otherwise we would not know the correlations of the tests with the occupation. Under these ideal circumstances any talk of factors is certainly unneces- sary so far as obtaining a quantitative estimate is concerned. But suppose these ideal conditions do not hold ! These tests which we have given to the candidate have never been giveij, at any rate as a battery, to workers in the occupation, and their correlations with the occupation are unknown ! This situation is particularly likely to arise in vocational advice or guidance as distinguished from vocational selection. In the latter we are, usually on behalf of the employer, selecting men for a particular job, and we are practically certain to have tried our tests on people already in the job, and to be in a position to make a direct estimation without factors. But in vocational guidance we wish to gauge the young person's ability in very many occupations, and it is unlikely that just this battery of tests that we are using has been given to workers in all these different jobs. In that case we cannot make a direct regression estimate of our candidate's probable proficiency in every occupation. Can we, then, obtain an estimate in any other way ? 120 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Other ways are conceivable, but it must at the outset be emphasized that they are bound to be less accurate than the direct estimate without factors. Although this battery of tests has not been given to workers in the occupation, perhaps other tests have, and by the aid of that other battery a factor analysis of the occupation has perhaps been made. If our tests enable the same factors to be estimated, we can gauge the man's factors and thence indirectly his occupational proficiency. Unfortunately, the " if " is a rather big one. Are factors obtained by the analysis of different batteries of tests the same factors ; may they not be different even though given the same name ? We shall discuss this very important point later, but meanwhile let us suppose that we have reasonable confidence in the identity of factors called by the same name by different workers with different batteries. Then the probable course of events would be something like this. An experimenter, using whatever tests he thinks practicable and suitable, analyses an occupation into factors. Another experimenter, at a different time and place, is asked to give advice to a candidate for that occupation. Using whatever tests he in his turn has available, he assesses in this candidate the factors which the previous experimenter's work leads him to think are necessary in the occupation, and gives his advice accordingly. The factors have played their part as a go-between, like a coinage. All depends on the confidence we have in the identity of the factors. We shall see later that there is only too much reason to think that the possibility of this confidence being misplaced has hardly been sufficiently realized by many over-enthusiastic factorists. And even if the common factors are identical, there remains the danger that the " specific " of the occu- pation may be correlated with some of the " specifics " of the tests, a fact which cannot be known unless the same tests have been given to workers in the occupation. 7. The geometrical picture of correlated estimates. Of the swarm of difficulties and doubts raised by these remarks we shall choose one to deal with first. We said that even although we make our analysis of the tests we use into uncorrelated factors, the estimates of these factors will be THE ESTIMATION OF FACTORS BY REGRESSION 121 correlated. This can best be appreciated if we consider what the estimation of factors means in terms of the geometrical picture of Chapter IV, which we also used in Chapter VI, Figure 19 (page 96). In this latter figure we were illustrating the straightforward process of esti- mating a " criterion " x, given a man's scores in two tests y and z. We saw that these two scores did not tell us exactly the man's position in the three-dimensional space of x, y, and z, but only told us that he stood somewhere along a line P'PP" at right angles to the plane of yz. In default of his exact point, we took the point P, which is where the average man of the array P'PP" stands, and by projection from it on to the vector x found an estimate OX of his x score. Exactly the same picture will serve for the estimation of a factor, if we suppose the vector x to be now the vector of a factor (say a) whose angles with y and z are known for the loadings of y and z with a are their cosines. Now, suppose that we are referring these two tests y and z to three uncorrelated factors. It is immaterial whether any of these factors are specifics, for a specific is estimated exactly like any other factor. We shall call them simply a, b, and c. Since the three factors are uncorrelated, they are represented in the geometrical picture by orthogonal (i.e. rectangular) axes, as shown in Figure 20. The vectors a and b are at right angles to each other in the plane of the paper, while the vector c is at right angles to both of them, standing out from the paper. These axes are to be imagined as continued backwards in their negative directions also, but only their positive portions are shown, to Figure 20. avoid confusing the diagram. The vectors y and z, also shown only in their positive directions, represent the two tests, and the angle between them represents by its cosine their correlation with one another. These two vectors y and z are not in 122 THE FACTORIAL ANALYSIS OF HUMAN ABILITY any of the planes ab, be, or ca, but project into the space between them. The three orthogonal planes ab, be, and ca divide the whole of three-dimensional space into eight octants, and if as is usual the final positions chosen for a, b, and c are such that all loadings are positive, the positive, directions of y and z will project into the positive octant as shown in the figure, in which the vector z is coming out of the paper more steeply than y is. The two vectors Oy and Oz define a plane, on which a circle has been drawn, which in the figure appears as an ellipse, since the plane yz is not in the paper but inclined to it. In the three-dimensional space defined by abc, the population of all persons is represented by a spherical swarm of points dense at O, more sparse as the distance from O increases in any direction. From any point in this space, perpendiculars can be dropped to a, b, c, y, and z, and the distances from O to the feet of these perpendiculars represent the amount of the factors a, b, and c possessed by the person whom that point represents, and his scores in the two tests. Conversely, a knowledge of his three factors would enable us to identify his point by erecting three perpendiculars and finding their meeting-point. But a knowledge of his scores in y and z does not enable us to identify his point, but only to identify a line P'PP", anywhere on which his point may lie. In the figure, let OY and OZ represent a person's scores in y and z. Then on the plane yz we may draw perpendiculars meeting at P. But the point representing the person whose scores are OY and OZ need not be at P; it can be anywhere in P'PP" at right angles to the plane yz, for wherever it is on this line, perpendiculars from it on to y and z will fall on the points Y and Z. In estimating factors from tests we have to choose one point on P'PP" from which to drop perpendiculars on to a, b, and c, and we choose P because the man at P is the average man of the array of men P'PP". Thus when we are estimating factors, all our population is represented by points on the plane yz (the plane on which in the figure the circle is drawn which THE ESTIMATION OF FACTORS BY REGRESSION 123 looks like an ellipse), although really they should be represented by a spherical swarm of dots. When the population is truly represented by its spherical swarm of dots, the axes a, b, and c represent uncorrelated factors. But when the spherical swarm of dots has been collapsed or projected on to the diametrical plane yz this is no longer the case. By taking only points in the plane yz from which to estimate factors in a three-dimensional space we have passed as it were from the geometrical picture of Chapter IV to the geometrical picture used in the first portion of Chapter V, where correlation between rectangular axes was indicated by an ellipsoidal distribu- tion of the population points. We have introduced correlation between the estimates of a, b, and c, because we have distorted the distribution of the population from a three-dimensional sphere to a flat circle on the plane yz, that is to an ellipsoid, for in a space of three dimensions the circle is an ellipsoid with two axes equal and the third one zero. Consider, for example, the particular point P shown in the figure. From it, projections on to a, b, and c are all positive, the man with scores OY and OZ in y and z is estimated to have all three factors above the average, which adds to their positive correlation. But in actual fact, since P may really lie anywhere along P'PP", a line which does not remain for its whole length in the positive octant abc 9 the man may really have some of his factors positive and some negative. If, together with the population, the rectangular axes a, b, and c are also projected on to the plane yz, these projections will not all be at right angles obviously they cannot, for three lines in a plane cannot all be at right angles to one another. The angles between these projections of the factor axes on to the test plane represent the correla- tions between the estimated factors. Our illustration has been only in two and three dimen- sions, for clearness and to permit of figures being drawn. Similar statements, however, are true of more tests and more factors, where the spaces involved are of dimensions higher than three. If there are n tests, the n test vectors define an n-space, analogous to the yz plane of Figure 20, 124 THE FACTORIAL ANALYSIS OF HUMAN ABILITY If these n tests have been analysed into r common factors and n specifics, n + r factors in all, the factor axes will define an (n + f) space analogous to the three-dimensional abc space of Figure 20. A man's n scores in the tests define his position P in the n-space of the tests, but he may be anywhere in a space P'PP", of r dimensions, at right angles to the test space, analogous to the line P'PP" in Figure 20. We take the point P to represent him faute de mieux, and project the distance OP on to the factor axes to get his estimated factors. These estimated factors are correlated with one another, and if we project the n + r factor axes from the (n + r)-space on to the n-space of the tests, the angles between these shadow vectors represent the correlations between the estimates. 8. Calculation of correlation between estimates. Arith- metically, these correlations are easily calculated from the inner products of (b) 9 the loadings of the estimated factors with the tests (page 115), with (a), the loadings of the tests with the factors (page 114). Moreover, this gives us the opportunity to explain in passing what is meant by " matrix multiplication." The matrix of loadings of the four tests with the three common factors is (page 114) : M = 66 52 74 37 52 66 21 71 and the matrix of the loadings of the three estimated factors with the four tests is (page 115) : 300 -095 -532 -095 353 -581 -352 -153 121 -148 -206 -747 i Then the matrix of variances and covariances of the estimated factors is jr in which formula we must explain how we form the product of two matrices. By the product of two matrices THE ESTIMATION OF FACTORS BY REGRESSION 125 we mean the new matrix formed of the inner products of the rows of the left-hand matrix with the columns of the right-hand matrix, set down in the order as formed. Thus, in forming the product : 300 -095 -532 -095 353 -581 -352 -153 121 -148 --206 -747 676 -219 -130 218 -567 -034 127 -035 -556 66 52 74 37 52 66 21 71 K the first element *676 of K is the inner product of the first row of N with the first column of M 300 X -66 + -095 X -52 + -532 X -74 + -095 X -37 == -676 In the same way, every element in K is formed. The element -034, in the second row and third column of K, is the inner product of the second row of N with the third column of M 353 X -21 + -581 X zero -352 X zero -153 X -71 = -034 If our arithmetic throughout the whole calculation of these loadings had been perfectly accurate, the matrix K would have been perfectly symmetrical about its diagonal. The actual discrepancies (as -127 and -130) are a measure of the degree of arithmetical accuracy attained.* The matrix K thus arrived at gives by its diagonal elements '676, -567, and *556, the variances of the three estimated factors (that is, the squares of their standard deviations), and by its other elements their covariances in pairs (that is, their overlap with one another). The correlation of any two estimated factors is equal to (see Chapter I, Figure 2) * A trial will show the reader that the product NM is quite different from the product MN. This is the only fundamental difference between matrix algebra and ordinary algebra. 126 THE FACTORIAL ANALYSIS OF HUMAN ABILITY co variance (ij) 13 V variance (i) x variance (j) From K we can therefore form the matrix of correlations of the estimated factors. It is : 1-000 -353 -212 353 1-000 -061 212 -061 1-000 wherein -353, for example, is -219 ~ -y/O 676 X * 567 )- Although, therefore, the " true " factors g and v are un- correlated, their estimates ^ and v are correlated to an amount -353. The "true" factors g, v, and F are in standard measure, but their estimates g, v, and F have variances of only -676, -567, and -556 instead of unity. These variances, be it noted in passing, are equal also to the squares of the correlations between g and , v and v, F and P. Not only are the estimates of the common factors correlated among themselves ; they are correlated with the specifics, so that the estimates of the specifics are not strictly specific. As a numerical illustration we may take the hierarchical matrix used in Section 1, pages 102 ff., four tests of the larger hierarchical matrix used in Chapters I (page 6) and II (page 23). 1-00 72 63 54 72 1-00 56 48 63 56 1-00 42 54 48 -42 1-00 The regression estimate of g from this battery is, as we found on page 104) = -553% + -259*2 + -160*3 + '109* 4 The regression estimates for the four specifics can also be found, either by a full calculation like that of page 108, or by the simpler method of subtraction of page 110. Thus, to estimate s x in our present example we know that THE ESTIMATION OF FACTORS BY REGRESSION 127 = -9 + -486* Also we know that the estimates g and same equation will satisfy the that is s = 1 -436 On inserting the expression for g into this we get $! = 1-1522! -535*2- -33323- -225*4 and similarly s 2 = -737*! + 1'3133 2 -215*3 '145* 4 3 = -542* x -253* 2 + 1 -242*3 '106* 4 S 4 = -415*! -194*2 -121*3 -|- 1-169* 4 We have now both N, the matrix of loadings of the estimated factors , s i9 s 29 s S9 ^ 4 with the four tests, and M, which we already know, the matrix of loadings of the four tests with the five factors g, s l9 s 29 s s , and $ 4 , namely : M = 9 8 7 6 436 600 714 -800 From their product NM we obtain the matrix K of variances and covariances of the estimated factors, namely : 553 -259 -161 -109 1-152 -535 -333 -225 - -737 1-313 - -215 - -145 - -542 - -253 1-242 - -106 -415 - -194 - -121 1-169 9 8 7 6 -436 -600 714 800 880 -241 -155 -115 -087 241 -502 - -321 - -238 - -180 150 - -321 -788 - -154 - -116 116 -236 -152 -887 -085 088 -181 -116 -086 -935 = K 128 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Again, we have a check on the accuracy of our arith- metic, for K will, if we have been accurate, be exactly symmetrical about its principal diagonal, i.e. its diagonal running from north-west to south-east. The largest dis- crepancy in our case is between -150 and -155. Moreover, since in this case K includes all the factors, we have another check which was not available when we calculated a K for common factors only : the sum of the elements in the principal diagonal (called the " trace," or in German the " Spur ") here must come out equal to the number of tests. In our case we have 880 + -502 + -788 + -887 + -935 = 3-992 and there are four tests. These elements which form the trace of K are, it will be remembered, the variances of the estimates ^, s i9 s 2 , s s , and $ 4 . So that we see that the total variances of the five factors is no greater than the total variance (viz. 4) of the four tests in standard measure. This is only another instance of the general law that we cannot get more out of anything than we put into it (at any rate, not in the long run). From K we can at once calculate the correlation of the estimated factors. Adjusting the slight arithmetical de- partures from symmetry, we get : 1-000 -362 -184 -131 -096 362 1-000 -510 -354 -263 184 -510 1-000 -183 -135 131 -354 -183 1-000 -094 096 -263 -135 -094 1-000 from which we see that is correlated with each of the estimated specifics positively, while the latter are correlated negatively among themselves, in this (a hierarchical) example. We have then this result, that although we set out to analyse our battery of tests into independent uncorrelated factors, the estimates which we make of these factors are correlated with one another, and instead of being in THE ESTIMATION OF FACTORS BY REGRESSION 129 standard measure have variances, and therefore standard deviations, less than unity. We could, of course, make them unity by dividing all our estimates by their calculated standard deviation. But that would make no change in their correlations. The cause of all this is the excess of factors over tests, and consequently this drawback the correlation of the estimates depends upon the ratio of the number of factors to the number of tests. The extra factors are the common factors, for there is a specific to each test, and therefore with the same number of common factors the correlation between the estimates will decrease as the number of tests in the battery increases. Just as in the hierarchical case one of the tasks of the experimenter is to find tests to add to the number in his battery without destroying its hier- archical nature, so in the case of a Thurstone battery which can be reduced to rank 2, 3, 4 ... or r, a task will be to add tests to the battery which with suitable communalities will leave the rank unchanged and the pre- existing communalities unaltered, in order that the common factors may be the more accurately estimated, and the estimates be more nearly uncorrelated. With Thurstone batteries of tests, therefore, we arrive at the same necessity to " purify " any extended battery as we spoke of in Chapter II, Section 1, in the hierarchical case. Indeed, the need will be greater, for larger batteries will be required to reach the same accuracy of estimation with more extra factors. F.A. 6 CHAPTER VIII MAXIMIZING AND MINIMIZING THE SPECIFICS 1. A hierarchical battery. In Section 3 of Chapter III a brief reference was made to the fact that the Spearman Two-factor Method, and Thurstone's Minimal Rank Method, of factorizing batteries of tests maximize the variance of the specific factors, by reason of minimizing the number of common factors. In the present chapter we shall inquire further into this aspect, and describe a method of estimating factors (Bartlett, 1935, 1937), which in its turn endeavours to minimize the specifics again. First take the case of the analysis of a hierarchical battery. As was illustrated in Chapter III, the analysis of such a battery into one general factor only, and specifics, gives the maximum variance possible to the specifics. The combined communalities of the tests are less in the two- factor analysis than in any other analysis. In the matrix of correlations after it has been reduced to the lowest possible rank, the communalities occupy the principal diagonal : 7*24 7*34 7*24 7*34 The mathematical expression of the above fact is that the trace of the reduced correlation matrix, i.e. the sum of the cells of the principal diagonal, is a minimum. It ,is true that certain exceptions to this statement are mathematically possible, but their occurrence in actual psychological work is a practical impossibility. They have been investigated by Ledermann (Ledermann, 1940), who finds, in the case of the hierarchical matrix, that an excep- 130 MAXIMIZING AND MINIMIZING THE SPECIFICS 131 tion is only possible when one of the g saturations is greater than the sum of all the others. When the battery is of any size, this is most unlikely to occur : and almost always, when it did occur, the large saturation of one test would turn out to be greater than unity, which is not permissible (the Hey wood case).* 2. Batteries of higher rank. The same general statement as the above, that the specifics are maximized, is also true of Thurstone's system, of which its predecessor (Spearman's two-factor system) is a special case. The communalities which give the matrix its lowest rank are in sum less than any other diagonal elements permissible. If numbers smaller than the Thurstone communalities are placed in the diagonal cells, the analysis fails unless factors with a loading of V~~ * are employed (Vectors, page 103), and such factors are, of course, inadmissible. Here again there are possibly cases where the lowest rank is not accompanied by the lowest trace (i.e. the lowest sum of the communalities). But here again it seems cer- tain that if such cases do exist, they are mathematical curiosities which would never occur in practice. As an illustration the reader may use the example of Chapter II, Section 9 : 5 5883 2852 2852 1480 As we there saw, this matrix can be reduced to rank 2 by the unique set of communalities- 1 7 -7 -7 -13030 -5 and we found there that, if we wanted to attain rank 2, we could not, for example, reduce the first communality to -5. We can, however, reduce the first communality to *5 if * See Chapter XV, Section 5, page 281. 1 2 3 4 1 , 4 4 2 2 4 , 7 3 3 4 7 . 3 4 2 3 3 t 5 5883 2852 2852 1' 132 THE FACTORIAL ANALYSIS OF HUMAN ABILITY we are willing to accept a higher rank than 2, that is, if we are willing to accept more common factors than two. But we find in that case that the remaining communalities necessarily rise so as to annul, and more than annul, the saving in communality achieved on the first test. We find ourselves bound to take the second communality more than the former ?, or inadmissible consequences ensue. We have a certain latitude in its choice, but there is a lower limit somewhere between -7 and '8 below which it makes the matrix inadmissible. Let us take -8 as the second communality (having thus still made a gross saving on the former communalities of -7 and -7) and calculate the remaining communalities, now fixed, which give rank 3. We can do this by the same process of pivotal condensation used in Chapter II, Section 9, making this time the matrix consist of nothing but zeros after three condensations (for rank 3) and then working back to the communalities. We find for the five communalities 5 -8 -65474 -14592 -80786 with a sum of 2-90852 for the total communality (or trace) compared with the total of 2-73030 with rank 2. Our attempt to save communality by reducing that of the first test from -7 to -5 and letting the rank rise has been foiled. The minimum rank carries with it, in all practically possible cases, the minimum communality and the maxi- mum specific variance. Minimizing the number of common factors maximizes the specific variance. 3. Error specifics. That some of the variance of a test will probably be unique to that particular test given on that particular occasion is clear ; there will be an error specific. But not all errors in testing will produce unique or specific factors. The errors will include sheer blunders, such as mistakes in recording results ; sampling errors due to the particular set of persons tested ; and variable chance errors in the performances of the individuals. The first can with care be reduced to infinitesimal proportions. Sampling errors will be discussed in Chapter X, and we will only say here that they will in many or most cases produce not specific but common factors. The variable chance MAXIMIZING AND MINIMIZING THE SPECIFICS 133 errors in the performances of the individual may be unique to each test, but often they too will run through several tests, as when a candidate has a slight toothache, or is elated by good news, or disturbed by a street organ all of which things may affect several tests if they are adminis- tered on the same day. The " unreliability " of a test, due to variable chance errors, is caused by factors which are unique not to the test but to the occasion. Tests a and b performed to-day, and repeated as Tests a' and V to- morrow, may have reliabilities less than unity, yet the chance errors of to-day may link a and 6, and the chance errors of to-morrow may link a' and b'. Nevertheless some of the error variance will doubtless be unique, but surely nothing like the amount of specific variance due to the Thurstone principle of minimizing the number of com- mon factors can be due to this. There remains the true specific of each test. It does not seem unreasonable to suppose that such exist, though it is not easy to imagine them existing before the test is given. The ordinary idea of specific factors would be tricks learned by doing that particular test, as a motor-car or a rifle may have and usually does have idiosyncrasies unknown to the stranger. But it seems questionable whether a method of analysis is justifiable which makes specific factors play so large a part. 4. Shorthand descriptions. It is to be observed that an analysis using the minimal number of common factors, and with maximized specific variance, is capable of reproducing the correlation coefficients exactly by means of these few common factors, and in the case of an artificial example will actually do so ; while in the case of an experimental example including errors, it will do so at least as well as any other method. If this is our sole purpose, therefore, the Thurstone type of analysis is best, since it uses fewest factors. But the few common factors of a Thurstone analysis do not enable us to reproduce the original test scores from which we began, they do not enable us to describe all the powers of our population of persons very well. With the same number of Hotelling's " principal components " as 134 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Thurstone has of common factors we could arrive at a better description of the scores, though a worse one of the correlations. The reader may reply that he does not want factors for the purpose of reproducing either the original scores or the original correlations, for he possesses these already ! But what we really mean, and what it is very convenient to have, is a concise shorthand description, and the system we prefer will depend largely on our motives, whether we have a practical end in view or are urged by theoretical curiosity. The chief practical incentive is the hope that factors will somehow enable better vocational and educational predictions to be made. Mathematically, however, as we have seen, this is impossible. If the use of factors turns out to improve vocational advice it will be for some other reason than a mathematical one. For vocational or educational prediction means, mathemati- cally, projecting a point given by n oblique co-ordinate axes called tests on to a vector representing the occupation, whose angles with the tests are known, but which is not in the w-space of the tests. The use of factors merely means referring the point in question to a new set of co- ordinate axes called factors, a procedure which cannot define the point any better and, unless care is taken, may define it worse, nor does the change of axes in any way facilitate the projection on to the occupation vector. Moreover, the task of carrying out prediction with the aid of factors is rendered more difficult by the circumstance that the popular systems use more factors than there are tests, so that the factors themselves have to be estimated. In addition, it is usual to estimate only the common factors, throwing away the maximum amount of variance unique to each test, maximized by insisting on as few com- mon factors as possible. If there is any guarantee that these abandoned portions of the test variance are un- correlated with the occupation to be predicted, no harm is done. But the circumstances under which this guarantee can be given are precisely those circumstances under which a direct prediction without the intervention of factors can easily be made. 5. Bartletfs estimates of common factors. Since, then, MAXIMIZING AND MINIMIZING THE SPECIFICS 135 the Thurstone system suffers, from a practical point of view, from this handicap of throwing away all information which can possibly be ascribed, rightly or wrongly, to specific factors, there is a peculiar interest in the proposal (M. S. Bartlett, 1935, 193Ta, 1938) to estimate the common factors, not by the regression method of the previous chapter, but by a method * which minimizes the sum of the squares of a man's specific factors (already maximized by the principle of using few common factors). The way in which Bartlett's estimates differ from regression estimates of factors can be very clearly seen by thinking in terms of the geometrical picture already used in earlier chapters (see Figures 14 to 20). When the factors outnumber the tests, the vectors representing the former are in a space of higher dimensions than the test space. The individual person is represented in the test space by a point, namely that point P whose projections on to the test vectors give his test scores. We do not know a representative point for this individual in the complete factor space, however. His representative point Q may be, for all we know, anywhere in the subspace which is perpendicular to the test space and intersects with it at P. In these circumstances the regression method takes refuge in the assumption that this individual is average in all qualities of which we know nothing ; that is, in all qualities orthogonal to our test space. It therefore assumes P to be his point also in the factor space, and projects P on to the factor axes to get the factor estimates for him. Bartlett's method is, in the present writer's opinion, equivalent to a different assumption about the position of the point Q. Within the complete factor space there is a subspace which contains the common factors. Of all the positions open to the point Q, Bartlett's method chooses that one which is nearest to the common-factor space, and from thence projects on to the common-factor vectors. This is equivalent to making the assumption that this man is not average in the qualities about which we know nothing, * See Appendix, paragraph 13. 136 THE FACTORIAL ANALYSIS OF HUMAN ABILITY but instead possesses in those unknown qualities just those degrees of excellence which bring his representative point to the chosen point Q. Both the regression method and Bartlett's method make assumptions about qualities which are quite unknown to us, and are quite uncorrelated with the tests we know. The regression assumption is that the man is average in these, Bartlett's assumption is that he is not average ; and because men are most frequently near the average, the regression assumption seems more likely to be correct. The other assumption can be justified only by its utility in attaining special ends ; it cannot be the most generally useful assumption. 6. Their geometrical interpretation. All this can be most clearly seen (because a perspective diagram can be made) in the case of estimating one general factor g only, the hierarchical case. A figure like Figure 19 will illustrate this case, if we take y and z there to be two tests and x to be the g vector (see Figure 21). The man's representative point in the yz plane is P. But we do not know his re- presentative point Q in solid three-dimensional space, only that it is somewhere on the line P'PP". The regression method assumes that it is actually at P, the average, and projects P itself on to the g line to get the estimate OX of g. Bartlett's method, on the other hand, assumes that Q is at that point on P'PP" where it most nearly approaches the g line, that is, somewhere near the position P' in our diagram. Bartlett's estimate of g is then represented by OX'. Now, any point on the line p'pp" ? when projected on to the test vectors y and z, gives the same two test scores Y and Z. There is, in general, no point on the line g which does this exactly. But clearly X' 9 of all the points on g, Figure 21. MAXIMIZING AND MINIMIZING THE SPECIFICS 137 will be the point whose projections most nearly fall on Y and Z, for X 1 is as near as possible to the line P'PP". That is, the projection of X' on to the plane of the tests falls as near to the point P as is possible. In other words, if we ignore the specifics entirely and use only the estimated g in the specification of y and z, Bartlett's estimate comes as near as is possible to giving us back the full scores OY and OZ. If the regression estimate OX is projected on to the lines y and z, it will obviously give a worse approxima- tion much worse in our figure to OY and OZ. The regression method, in order to recover as much as possible of the original scores, would have to make a second estimate of them. For the estimates of g repre- sented by quantities like OX are not in standard measure. Before projecting the point X on to the lines y and z, therefore, to recover the original scores as far as possible, the regression method would alter the scale of its space along the g vector until the quantities like OX were in standard measure. This would not only change the posi- tion of X on the line, it would change the angles which the lines in the figure make with one another ; and would change them exactly in such a manner that, in the new space, the projection of OX on to y and z would fall exactly where the Bartlett projections from X' fall in the present space (Thomson, 1938a). There is, therefore, no final difference in excellence between the two methods in the matter of restoring the original scores as fully as possible, but the regression method takes two bites at the cherry. On the other hand, the regression estimates can be put straight into the speci- fication equation of an occupation which is known to require just these common factors, whereas here it is the Bartlett method which has to have a second shot. Both methods have to change their estimate of g when a new test is added to the battery. For the man is not very likely to have, in the specific of this new test, either the average value previously assumed by the regression method, or the special value assumed by the Bartlett method. But he is more likely to have the former than the latter, so the Bartlett estimates will change more F.A.S* 138 THE FACTORIAL ANALYSIS OF HUMAN ABILITY than do the regression estimates as the battery grows. Ultimately, when the number of tests becomes infinite, the two forms of estimate will agree. 7. A numerical example. In the case of estimates of one general factor g from a hierarchical battery, the Bartlett estimates differ from the regression estimates only in scale. They put the candidates in the same order of merit for g as do the regression estimates, but give them a greater scatter, making the high g's higher and the low g's lower. The formula is 1 r JV 2 ^ s i - v instead of Spearman's i-qhs r r^7^ (see page 106) ' With more than one common factor, the connexion between the two kinds of estimate is not so simple (Appen- dix, Section 13). The mathematical reader will be able to calculate the Bartlett factor estimates from the matrix formulae given in the Appendix. We shall here calculate them, for the example of Chapter VII, Section 5, from the regression estimates there given, and their matrix of variances and co variances given in Section 8 of that chapter. For if the matrix of regression loadings be represented by N, and the matrix of variances and co variances of the regression factors by K 9 then the matrix of Bartlett load- ings can be shown (Bartlett, 1938) to be This matrix multiplication can be carried out by Aitken's pivotal condensation also. For it has been shown (Aitken, 1937a) that the pivotal condensation of a pattern of three matrices arranged thus : Y - Z X gives, when by repeated condensations all numbers have been removed from the left-hand block, the triple product MAXIMIZING AND MINIMIZING THE SPECIFICS 139 X Y" 1 Z. We shall therefore obtain the Bartlett loadings for estimating the factors from the tests if we condense K - N where / is the unit matrix which has unity in each cell of the principal diagonal and zeros elsewhere. The matrices K * and N are taken from pages 125 and 124, and the whole calculation is as follows (to three places of decimals only, to facilitate the arithmetic for readers who wish to check it) : Check Column 674 -218 127 -300 095 532 095 003 1-000 -323 188 445 141 -789 -141 -004 (5) 218 -567 -035 -353 581 352 153 321 127 --035 556 -121 148 206 -747 134 1-000 , . . . . 1-000 . 1-000 . . . , . 1-000 1-000 1-000 497 -076 256 550 524 184 322 (3) 1-000 -153 -515 1-107 1-054 370 650 (49) -076 532 064 166 306 -729 135 -323 -188 445 141 789 141 1-005 1-000 . . . . . 1-000 1-000 - - 1-000 520 -103 082 386 -701 184 1-000 -198 158 742 -1-348 354 237 279 217 1-129 261 1-215 153 515 1-107 V054 -370 351 1-000 1-000 232 -180 1-305 -058 1-299 545 1-083 1-168 -164 297 (6) 198 158 742 1-348 646 The Bartlett estimates of the factors, therefore, which * Slightly corrected to make it symmetrical. 140 THE FACTORIAL ANALYSIS OF HUMAN ABILITY we shall distinguish from the regression estimates by turning the circumflex accent upside down, are g = -232*! -180*2 + 1 -30523 -058* 4 v = -545*! + 1 -083*2 1-168*3 '164* 4 F = -198% -158*2 -742*3 + l-348* 4 In Chapter VII, Section 5, we imagined a man whose scores in the four tests, in standard deviations, were *! * 2 2 -4 * 4 6 and calculated the regression estimates of his three factors, g, v 9 and F. By inserting his test scores in the above equations we can find, for comparison, the Bartlett esti- mates of his factors, shown in the following table : Factors Regression estimates Bartlett estimates This illustrates the tendency of the Bartlett estimates to be farther from the average than the regression estimates are. g V F tes 451 997 -500 1-240 387 392 PART III THE INFLUENCE OF SAMPLING AND SELECTION OF THE PERSONS CHAPTER IX SAMPLING ERROR AND THE THEORY OF TWO FACTORS 1. Sampling the population of persons. In the previous pages we have seldom mentioned sampling errors. There is an implicit reference to them in Chapter I, where a portion of an actual experimental matrix of correlations is shown as a contrast to the artificial ones used in the text ; and later in that chapter there is a closer approach to the difficulties caused by sampling errors. But apart from this, and perhaps one or two other references, the exposition in Parts I and II is entirely free from any consideration of them. The examples are made and worked as if on every occasion the whole population of people concerned had been accurately tested. The advantage of this is that it makes the theoretical principles stand out clearly, unobscured by the sampling difficulty. As a result, to mention one important point, it is thus made clear that the difficulties of estimating factors, described in Chapters VII and VIII, have nothing directly to do with sampling the population, but are due to having more factors than tests. It is true that an abso- lutely clean cut between an exposition which considers sampling errors, and one which disregards them, cannot be made. For sampling errors introduce error factors, and thereby swell the total number of factors. But even were the whole population of persons tested, factors which out- number the tests would remain " indeterminate," as it is sometimes expressed, meaning that they can only be estimated, not measured exactly. Another kind of sampling, however, does exist in Parts I and II, a sampling of the tests. We have there assumed that the whole population of persons is tested, but we have not supposed that they were plied with the whole population ' of tests. It is( difficult perhaps to say* what 143 144 THE FACTORIAL ANALYSIS OF HUMAN ABILITY " the whole population of tests " means, but at any rate it is clear that in Parts I and II we were using only a few, not all possible tests. There is thus in our subject a double sampling problem, and this makes it very difficult. In the present section of this book (Part III) we shall consider the effects of sampling the population of persons. The general idea underlying the notion of a sampling error is not a difficult one. Take, for example, the average height of all living Englishmen who are of full age. This could, if need be, be ascertained by the process of measuring every living Englishman of full age. Actually this has never been done, and when anyone makes a statement such as " The average height of Englishmen is 67^ inches," he is basing it upon a sample only. This sample may not be an unbiased one. Indeed, samples of Englishmen whose height has been officially recorded are heavily loaded with certain classes of Englishmen for example, prisoners in gaol, and unemployed young men joining the army of preconscription days. The average height of such men may well differ from that of all Englishmen. But when we speak of sampling error, we do not mean error due to the sample being known to be a biased one. Even if the sample of Englishmen used to find the average height of their race were, as far as could be seen, a perfectly fair sample, containing the proper proportion of all classes of the community and of all adult ages, etc., it yet would not necessarily yield an average exactly equal to that of all Englishmen. Several apparent replicas of the sample would yield different averages. It is these differences, between statistics gathered from different but equally good samples, that we mean by sampling errors. It is worth while calling attention at this point to a general fact which will be found of importance at a later stage of this book. The true average height of Englishmen is only so by definition, and does not in principle differ from the average of a sample. We had to define the popu- lation we had in mind as " all living Englishmen of full age." This is a perfectly well-marked body of men. But it is itself in its turn only a sample : a sample of all living Europeans, or all living men. It is, indeed, altering daily SAMPLING ERROR AND TWO-FACTOR THEORY 145 and hourly as men die or reach the age of 21, and each generation is a sample of those that have been and may be. Those who reach the age of 21 are only some, and therefore only a sample, of those born. And even those born are only a sample of those who might have been born had times been better or had there been no war, or a tax on bachelors. So the idea of sampling is a relative one, and the " complete population " from which we take samples is a matter of definition only. The mathematical problem in connexion with sampling which it is desirable to solve if possible for each statistic is to find the complete law of its distribution when it is derived from each of a large number of samples of a given size. Mathematically this is often very difficult, and frequently we have to be content with a formula which gives its approximate variance if certain assumptions are allowed and certain small quantities are neglected. Sampling problems are of two kinds, direct and inverse. The easier kind of problem is to say what the distribution of a statistic will be in samples of a given size when we know all about the true values in the whole population : the more difficult kind is to estimate what the true value of a statistic is in a complete population when we know its observed value in certain samples. They differ as do problems of interpolation and extrapolation. As an example of the direct kind of problem let us suppose that we actually knew the height of every adult Englishman of full age. We could then, on being told a certain sample of p Englishmen averaged such and such a height, calculate the probability that this sample was a random sample, a probability that would obviously grow less as the average of the sample departed from the average of the whole population. It would also depend on the size of the sample, for if a very large sample deviates far from the true average, it is less likely to be random, more likely to have some reason for the difference, than a small sample with the same average would have. 2. The normal curve. By the distribution of a certain variable in the population we mean the curve (usually expressed as an equation) showing its frequency of dcctir- 146 THE FACTORIAL ANALYSIS OF HUMAN ABILITY rence for each possible value. Thus the curve in Figure 22 might show the distribution of height in living adult Englishmen, by its height above the base line at each point. More men (represented by the line MN) have the average height, 67J inches, than have the height 73 inches, the frequency of the latter being shown by the line PQ. The shaded area represents all men whose height is 73 inches or more, and its ratio to the area under the whole curve is the probability that an Englishman taken absolutely at random will have a height of 73 inches or more. Very often distributions are, at any rate approximately, of a certain shape called the " normal curve." The normal curve has a known equation, it is symmetrical about its mid point, and with the aid of published tables can be drawn accurately (or reproduced arithmeti- cally) if we know the mid point M (which is the average of the measurements) and a certain distance ST or S'T (which is equal to the standard deviation of the measurements). S and S' are the points where the curve changes from being convex to being concave. If the distribution of a variable, say the heights of adult Englishmen, is " normal," then the distribution of the means of samples of p Englishmen's heights will also be normal, but will be more closely concentrated about the point M than are the measurements of individuals : in point of fact, its variance will be p times smaller, its standard deviation thus ^/p times smaller. That is to say, if we take sample after sample of 25 Englishmen each time, and for each sample record the average height, the means thus accumulated will be distributed in a curve of the same shape as that of Figure 22, but narrower from side to side, so that SS' would be one-fifth (V 25 ) of what it is in Figure 22, which is the distribution of single measurements; I | III! i I 61 62 63 64 65 66 67 68 6970 71 72 73 74 7576 Figure 22. SAMPLING ERROR AND TWO-FACTOR THEORY 147 If a sample were made with some special end in view, such as ascertaining whether red-headed men tend to be tall, we would decide whether we had detected such a tendency by calculating the probability that a mean such as our red-headed sample showed, or a mean still further away from M 9 would occur at random. For this purpose we would compare the deviation of our sample from M with the standard deviation of the distribution of such samples, obtained by dividing the standard deviation of individuals by the square root of p, the number in the sample. The ratio of the deviation found, to the standard deviation, is the criterion, and the larger it is the more likely is it that red -headed men really do tend to be tall. For most practical purposes we take a deviation of over three times the standard deviation as " significant." Sometimes the reader will find significance questions discussed in terms of the " probable error " instead of the standard deviation. The probable error is best considered as a conventional reduction of the standard deviation (or standard error, as it is sometimes called) to two-thirds of its value (more exactly, to '67449 of its value). Not only would the average height, or the average weight, of the sample of red -headed men differ from sample to sample. Statistics calculated in more complex ways from the measurements will also vary from sample to sample, as, for example, the variance of height, or the variance of weight, or the correlation of height and weight. Let us consider first the variance of the heights. In the whole population this is calculated by finding the mean, expres- sing every height as a plus or minus deviation from the mean, squaring all these deviations, and dividing the sum by the number in the population. This is also how we would find the variance of the sample if we really want the variance of the sample. But if we want an estimate of the variance in the whole population, and the sample is small, it is better to divide by one less than the number in the sample. A glimpse of the reason for this can be got by considering the case of the smallest possible sample, namely, one man. Here the mean of the sample is the one height that we have mfe'asur&d, and the 148 THE FACTORIAL ANALYSIS OF HUMAN ABILITY deviation of that measurement from the mean of the sample is zero. The formula if we divide by the number in the sample (one) will give zero for the variance and that is correct for the sample. But it would be too bold to estimate the variance of the whole population from one measurement : if we divide by one less than the sample we get variance = 0/0, that is, we don't know, which is a wiser statement.* More generally we can begin to understand the reason for dividing by (p 1) instead of by p by the following considerations. The quantity we want to estimate is the mean square deviation of the measurements of the whole population, the deviations being taken from the mean of that whole population. We do not, however, know that true mean, and therefore in a sample we are reduced to using the mean of the sample, which except by a miracle will not exactly coincide with the true or population mean. The conse- quence is that the sum of the squares we obtain is smaller than it would have been had we known and used the true mean. For it is a property of a mean that the sum of the squares of deviations from it is smaller than of deviations from any other point. Consider for example the numbers 2, 3 and 7. Their mean is 4, and the sum of the squares about 4 is ( 2) 2 + ( I) 2 + 3 2 = 14 * It is important to remember that sampling the population is not the only source of error in the measurement of statistics, e.g. the correlation coe fficient . All sorts of influences may disturb it . These will usually " attenuate " the correlation coefficient, i.e. tend to bring it nearer to zero, as can be seen when we consider that a perfect correlation only can be reduced by error. But they will not always do so, and if the errors in the two trait measurements are themselves correlated, they may even increase the true correlations in a majority of cases. An estimate of the amount of variable error present can be made from the correlation of two measurements of the same trait on the same group, a correlation called the " reliability," which should be perfect if no variable errors are present. Spearman's cor- rection for attenuation (see Brown and Thomson, 1925, 156) is based upon this. Like all estimates, the correction for attenuation is correct, even if the errors are uncorrelated, only on the average and not in each instance, and it should never be used unless it is small. If it is large, the experiments' are u unreliable " and shbulft be improved. SAMPLING ERROR AND TWO-FACTOR THEORY 149 About any other point this sum will be greater than 14. About 5, for example, the sum is ( 3) 2 + ( 2) 2 + 2 2 = 17 About 2 the sum is O 2 + I 2 + 5 2 - 26 It follows that the sum of the squares we obtained by using the sample mean was as small as possible, and in the immense majority of cases smaller than the sum about the true mean. It is to compensate for this that we divide by (p 1) instead of by p. These elementary considerations do not of course indi- cate just why this procedure should, in the long run, ex- actly compensate for using the sample mean. Why not (p 2), one might say, or (p 3) ? It is not possible, in an elementary account like the present, to answer this. Geometrical considerations, however, throw some further light on the problem. The p measurements of the sample may be thought of as existing in a certain space of (p 1) dimensions. For example, two points define a line (of one dimension), three points define a plane (of two dimensions) and so on. The true mean of the whole population is not likely to be within that space, whereas the mean of the sample is. The deviations we have actually squared and summed are therefore in a space of one dimension less than the space containing the true mean. One " degree of free- dom " has been lost by the fact that we have forced the lines we are squaring to exist in a space of (p 1) di- mensions instead of permitting them to project into a p - space. Hence the division by (p 1) instead of p. This principle goes further. For each statistic which we calculate from the sample itself and use in our subsequent calculations, we lose a " degree of freedom." The standard error of a variance u, if the parent popula- tion from which the samples are drawn is normally distri- buted, is estimated as where p is the number of persons in the sample. The 150 THE FACTORIAL ANALYSIS OF HUMAN ABILITX standard error of a correlation coefficient r is, with the same condition, estimated as 1 r 2 V(p - 1) The use of this standard error, however, should be dis- continued (unless the sample is large and r small). Fisher (1938, page 202) has pointed out that the use of the formula for the standard error of a correlation coefficient is valid only when the number in the sample is large and when the true value of the correlation does not approach 1. For in small samples the distribution of r is not normal, and even in large samples it is far from normal for high correlations. The distribution of r for samples from a population where the correlation is zero differs markedly from that where the correlation is, say, 0-8. This means that the use of a standard error for testing the significance of correlation coefficients should, except under the above conditions, be discouraged. To get over the difficulty Fisher transforms r into a new variable z given by It is not, however, necessary to use this formula, as com- plete tables have been published for converting values of r into the corresponding values of z. As r goes from 1 to + 1, z goes from oo to -f- o> and r = Q corresponds to z = 0. The great advantage of using z as a variable instead of r is that the form of the distribution of z depends very little upon the value of the correlation in the population from which samples are drawn. Though not strictly normal, it tends to normality rapidly as the size of the sample is increased, and even for small samples the assumption of normality is adequate for all practical purposes. The standard deviation of z may in all cases be taken to be l/Vp 3, where p is the number of persons in the sample. 3. Error of a single tetrad-difference. For our discussion of the influence of sampling on the factorial analysis of SAMPLING ERROR AND TWO-FACTOR THEORY 151 tests one of the most important quantities to know is the standard error of the tetrad-difference. There has been much debate concerning the proper formula for this. (See Spearman and Holzinger, 1924, 1925, 1929 ; Pearson and Moul, 1927 ; Wishart, 1928 ; Pearson, Jeffery, and Elder- ton, 1929 ; Spearman, 1931.) That generally employed is formula (16) in the Appendix to Spearman's The Abilities of Man : Standard error of r 13 r 24 ^ 2 3 r i4 2 ni [Spearman and r 2 (l - r 12 - r 34 + r 1 ) + (1 - 2r 2 )* 2 Holzinger's L J formula (16).] where N is the number of persons in the sample,* r is the mean of the four correlation coefficients, and s 2 is their mean squared deviation (variance) from r. The probable error is -6745 times the above. A worked example will be found on page xii of Spearman's Appendix, using (which is all one can do) the observed values of the r's. It will be remembered that in Section 7 of Chapter I we stated Spearman's discovery in the form " tetrad- differences tend to be zero." If tetrad -differences in the whole population, however, were all actually zero, they would not remain exactly zero in samples, and it is only samples that are available to us. We are faced, therefore, with a twofold problem, (a) We have to decide, from the size of the tetrad-differences actually found in our sample, whether the sample is compatible with the theory that the tetrad-differences are zero in the whole population. But (b) we should also go on to consider whether the sample is equally compatible with the opposed hypothesis that the tetrad-differences are not zero in the whole population, leaving a verdict of "not proven." (See Emmett, 1936.) 4. Distribution of a group of tetrad-differences. The actual calculation, for every separate tetrad-difference, of its standard error by Spearman and Holzinger's formula (16) is, however, an almost impossibly laborious task. In * We use p to mean the number of persons in this book, but are retaining N here and in " formula 16A " below to preserve the usual appearance of these well-known and much-used expressions. 152 THE FACTORIAL ANALYSIS OF HUMAN ABILITY a table of correlations formed from n tests there are n(n l)/2 correlation coefficients, and n(n l)(n 2) (n - 3)/8 different (though not independent) tetrad- differences. Any one particular correlation-coefficient is concerned in (n 2)(n 3) different tetrad-differences, and any one test in (n I)(n 2)(n 3)/2 different tetrad-differences. Thus with ten tests there are 630 tetrad-differences, and with twenty tests 14,535 tetrad- differences. In the latter case, any one test is concerned in 2,907. Under these circumstances, it is natural to look for a more wholesale method than that of calculating the standard error of each tetrad-difference. The method adopted by Spearman is to form a table of the distribution of the tetrad-differences, and compare this distribution with that of a normal curve centred at zero and with standard deviation given by 2 - * - - [Spearman and Hol- - rt* - - _ r) Z i ng er's formula (16 A ).] where N number of persons in the sample, r = the mean of all the r's in the whole table, s 2 = their mean squared deviation from r. n n 4 n 6 R = 3r . -- 2r 2 . - and n 2 n 2' n = number of tests. Numerous examples of the comparison of " histograms " of tetrad-differences with normal curves whose standard deviation is found by (16A) are given in Spearman's The Abilities of Man. This method of establishing the hypo- thesis, that the tetrad-differences are derived by sampling from a population in which they are really zero, is open to the same doubt as was explained in the simpler case of one tetrad-difference. The comparison can prove that the tetrad-differences observed are compatible with that hypothesis. It does not in itself prove that they are compatible with that hypothesis only; and, as Emmett has shown in the article already mentioned, the odds are commonly rather against this. SAMPLING ERROR AND TWO-FACTOR THEORY 158 The usual practice, moreover, is to " purify " the battery of tests until the actual distribution of tetrad-differences agrees with (16A), so that in effect all that is then proved is that a team can be arrived at which can be described in terms of two factors. This, although a more modest claim than has often been made, and certainly less than is implicitly understood by the average reader, is never- theless a matter of some importance. Not all teams of tests can be explained by one common factor ; but it is not very difficult to find teams which can. There is little doubt in the minds of most workers that a tendency towards hierarchical order actually exists among mental tests. 5. Spearman's saturation formula. It will be remem- bered from Section 4 of Chapter I that the calculation of the g saturation of each test forms an important part of the Spearman process. We saw there that in a hierarchical matrix each correlation is the product of the two g satura- tions of the tests, for example Since this is so, each g saturation can be calculated from the correlations of a test with two others, and their inter-correlation. Thus to find r lg we can take Tests 2 and 3 as reference tests, when we have 7*12^3 _ r ig T 2 ff ^Jlg^Sff __ 2 / lg ^*23 ^2g Tty When the matrix is really hierarchical, and there are no sampling errors present, it is immaterial which two tests we associate with Test 1 in order to find its g saturation. We have, in fact, in that case 7*12 ^13 __ ?*14 ^15 ___ ^*12^Jjl5 ___. . ?*23 ftt* ?*25 But even if the correlations, measured in the whole population, were really exactly hierarchical, sampling errors would make these fractions differ somewhat . from one another, and we are faced with the problem of deciding which value to accept for the g saturation. The average of all possible fractions like the above would be one very 154 THE FACTORIAL ANALYSIS OF HUMAN ABILITY plausible quantity to take but is laborious to compute. Spearman therefore adopts a fraction _+ r u ^ HF" r etc. + etc. whose numerator is the sum of the numerators, and whose denominator is the sum of the denominators, of the single fractions. This combined fraction he computes in a tabular manner which we will next describe, by the algebraically equivalent formula ~T 2A l [Spearman's formula (21), Appendix, Abilities of Man.] The quantities A i9 A 29 etc., are the sums of the rows (or columns) of the matrix of correlations without any entries in the diagonal cells. (The arithmetical example is con- fined to five tests to economize space) : 1 2 3 4 5 A A* 1 m 50 34 33 24 1-41 1-988 2 50 . 56 32 15 1-53 2-341 3 34 56 . 13 35 1-38 1 -904 4 33 32 13 . 29 1-07 1-145 5 24 15 35 29 1-03 1-061 T = 6-42 T is the sum of all the A's, and therefore of all the correlations in the table (where each occurs twice). A new table is now written out, with each coefficient squared, and its rows summed to obtain the quantities A' : 1 2 3 4 5 A' 1 % 250 116 109 058 533 2 250 . 314 102 023 689 3 116 314 . 017 123 570 4 109 102 017 . 084 312 5 058 023 123 084 288 The calculation of all the saturations is then best per- formed in a tabular manner, thus : SAMPLING ERROR AND TWO-FACTOR THEORY 155 A*-A' g A* A' A 2 A' 2A T-2A Satu- ration 1 1-988 533 1-455 2-82 3-60 4042 66 2 2-341 689 1-652 3-06 3-36 4917 70 3 1-904 570 1-334 2-76 3-66 3645 60 4 1-145 312 833 2-14 4-28 1946 44 5 1-061 288 773 2-06 4-36 1773 42 where the last column is the square root of the preceding. The reader should calculate the six different values of r lg from the original table by the formula (r tj . r ik /r jk )* 9 for comparison with the value 66 obtained above. He will find 55 -72 -93 89 48 52 with an average of -68. 6. Residues. If the correlations which would arise from these saturations or loadings are calculated, and subtracted from the observed correlations, we obtain the residues which have then to be examined to see if they are small enough to be attributable to sampling error. In the following double table of correlations are set out the ob- served correlations uppermost, and those calculated from the g saturations below. The difference is the residue, which may be plus or minus : g Loadings -66 -70 60 44 42 66 70 60 44 42 t 50 34 33 24 46 ,40 29 28 50 9 -56 32 15 46 42 31 29 34 56 . 13 35 40 -42 26 25 33 32 13 . 29 29 31 26 18 24 15 35 29 B 28 29 25 18 156 THE FACTORIAL ANALYSIS OF HUMAN ABILITY The lower numbers are the products of the two saturations. In this case the residues range from -14 to + *14 and at first sight appear in many cases to be too large to be neglected in comparison with the original correlations. To check this impression, consider the correlation '56 and the value '42 from which it is supposed to depart only by sampling error, a deviation of -14. Fisher's z corres- ponding to r *42 is -45, and that corresponding to r = 56 is z = -63, so that the z deviation is -18. The standard deviation of z for 50 cases is 1 -f- V^7 = -15. The devia- tion is little larger than one standard deviation and cannot therefore be called significant. But as the reader will ob- serve, this conclusion is due more to the large size of the standard error than to the small size of the residue. The residue is here attributable to sampling error, because the latter is so large. But because the latter is large it does not follow that the large residue is certainly due to it. A test of the second kind is needed here (but is hardly ever ap- plied) to determine the odds for or against the alternative hypothesis, that the residue is not due to sampling error. The lack of tests of this second kind, as has already been emphasized in discussing tetrad-differences, is one of the most serious blemishes in the treatment of data during factorial analysis. If we are willing to allow 10 per cent, of the correlation coefficient as being a negligible quantity (a very generous concession), then the chance of our experi- mental value '56 having come by sampling from outside the area -42 i '042 is (with 50 cases in the sample) still quite considerable, about 5 to 1 for. These odds do not justify us in feeling confident that -56 does come from outside 42 *042. But much less do they justify us in feeling that it comes from inside that region. 7. Reference values for detecting specific correlation. If, after a calculation like that described, one of the residues is found to be too large to be explicable by sampling error, the excess of correlation over that due to g is attributed to " specific correlation," meaning correlation due to a part of their specific factors being not really unique but shared by these two tests. In the case of our numerical example, SAMPLING ERROR AND TWO-FACTOR THEORY 157 if the number of subjects tested had been larger, the standard errors of the coefficients would have been smaller, and some of the discrepancies between the experimental values and those calculated from the g saturations would have been too large to be overlooked, but would have had to be attributed to specific correlation. In such a case, the g loadings would, of course, be wrong and would have to be recalculated from the battery after one of the tests con- cerned in the specific correlation was removed from it. Later, the other test could be replaced in the battery instead of the first, and thus its g saturation found. The difference between the experimental correlation of the two, and the product of their g saturations, with a standard error dependent on the size of the sample, would be then attributed to their specific linkage. If two tests, v and w, are thus suspected of having a specific link as well as that due to g, it is clear that the smallest battery of tests which could be used in the above manner to detect that link would be one of two other tests, x and y, say, to make up a tetrad : v x V ' r vy and these two " reference " tests would have to be known to have no specific links with each other or with the two suspected tests. The example which gave rise to Figure 5 (see Chapter I, page 15) illustrates this. Tests 2 and 8 there are, let us suppose, those with a suspected specific link. The tetrad-difference to be examined by means of Spearman's formula (16) is that which has r 23 as one corner. In such a case, where the two reference tests 1 and 4 are known to have no link except g with one another, or with the other two tests, two of the possible tetrad-differences ought to be larger than three times the standard error given by formula (16), and equal to one another, while the third tetrad-difference should be zero (or sufficiently near to zero, in practice) (Kelley, 1928, 67). The g saturation of each of the tests under examination 158 THE FACTORIAL ANALYSIS OF HUMAN ABILITY for specific correlation can be found by grouping it with the two reference tests. Thus in the case of our Figure 5, we have 2 _ r !2 ?*24 __ '5 X '5 _ ' 20 -- _____ -- - - - j r 14 -5 2 - ^3 ^4 _ '5 X -5 'So - ______ - - ^ - Therefore the correlation between 2 and 3 which is due to g is *V ^ = V' 5 x V' 5 = * 5 and the difference between this and -8, the actual value, is the part to be explained by the specific factor shared by these two tests. The difference of *3 is not what is called the specific correlation itself, it should be remarked, but only its numerator. By specific correlation is meant the correlation between the two " specific " parts of the linked tests, due to these not being entirely unique, but having a part in common. How to calculate this we shall see after considering the effect of selection on correlation, in Chapter XI, end of Section 2 (page 175). When there are several reference tests available, all believed to have no link except g with one another or with the two tests suspected of specific overlap, there will be a number of ways of picking two of them to obtain the tetrad required to decide the matter, and the results will, because of sampling and other errors, be discrepant. Under these circumstances Spearman has devised an interesting procedure for amalgamating the results into one, which we can describe with the aid of the Pooling Square. Instead of using two single tests, let us in the first place imagine that the n tests available as reference tests are divided into (Mf\ - ) , and that the correlations of these pools with one another, and with the suspected tests, are used to form the tetrad. Following Spearman's notation in paragraph 9 of the Appendix to The Abilities of Man, we shall call the suspected tests v and w, and the SAMPLING ERROR AND TWO-FACTOR THEORY 15& two pools the x pool and the y pool. We then want the tetrad of correlation coefficients : V x poo 1 w y pool *. T vy TXU r *y of which r vw is known experimentally. The others we can find by using pooling squares. Take first r^. We have (writing three tests in each reference pool instead of - ) : 2 x l x z x s 2/4 2/5 2/6 *1 1 T T 14 *16 *1 iT 2 r !2 * r 23 % r 25 r 26 x* r, 3 r 23 1 T U ^35 ^36 2/4 r !4 ^24 ^34 1 ^45 r 46 2/5 r 16 r 20 r 35 ^45 1 ^56 2/6 ^16 ^28 ^36 ^46 ^56 1 and the correlation r xy of the two pools with one another is (Chapter VI, Section 2) Here the quantities r a , r 6 , and r c are the mean values of the correlation coefficients (excluding the units) to be found in the quadrants of the pooling square, thus : Now, there is clearly an arbitrary factor left in this procedure, inasmuch as the division of the n available tests into an x pool and a y pool can be made in many different ways, in each of which the mean values r a , r b9 and f c will be slightly different. To obviate this, Spearman takes the mean value f of all the n reference tests with one another 160 THE FACTORIAL ANALYSIS OF HUMAN ABILITY instead of each of these three means, upon which the formula for r xy simplifies to n . r 2 1 + G-o and it is this value which he uses in the tetrad. Similarly, the correlation of the test w with the x pool can be found by a pooling square : W w X l X* X s 1 r- wl r w , r a3 J- ^*12 ^*13 /*12 1 ^23 r r- 23 1 Its value is n . n Here, for the same reason as before, not only do we use the average inter-correlation of all the reference tests for r, but for f w we use the average of the correlations of the test w with all the reference tests and not merely with the x pool, for the x pool could be any half of them. Similarly the correlation r^ is found. Thus to form the tetrad all that we need do is to find : r, the average correlation of all the reference tests with one another ; r w , the average correlation of all the reference tests with w ; r v9 the average correlation of all the reference tests with v ; and substitute in the formulae. A numerical example is given by Spearman on page xxii of his Appendix. CHAPTER X MULTIPLE-FACTOR ANALYSIS WITH FALLIBLE DATA 1. Method of approximating to the communalities. The influence of sampling errors on multiple-factor analysis is in general similar to that on the tetrad method. Sampling errors blur the picture. They make it both difficult for us to see the true outlines and easy to entertain hypotheses which cannot be disproved, though often they cannot be proved either, by the data. With artificial data like the examples used in Chapter II it may be laborious, but is not impossible, to find the actual rank of the matrix with various communalities, and thus to arrive by trial at the minimum rank. But when sampling errors are present, or any kind of errors, the question becomes at once immensely more difficult. We have seen in the previous chapter something of the diffi- culty of deciding from the size of the tetrad-differences when the rank of a matrix may justifiably be regarded as one. Such methods have not been used for higher ranks. The labour of calculating all three-rowed, four-rowed, or larger minors, setting out their distribution and comparing it with that to be anticipated from true zero values plus sampling error, is too great, and the mathematical difficulty not slight. What has been done is to judge of the rank by the inspection of the residues left after the removal of so-and-so many common factors, e.g. at the end of so-and- so many cycles of Thurstone's process, just as in Section 6 of the preceding chapter we examined the residues left after one common factor was removed. But we must first show how Thurstone meets the difficulty of the unknown communalities. Thurstone has described many ways of estimating the communalities, and articles still issue from his laboratory on this subject (see in the Addendum on page 854 a brief F.A.-6 161 162 THE FACTORIAL ANALYSIS OF HUMAN ABILITY account of a paper by Medland). He points out, however, that if the number of tests is fairly large, an exact estimate is not very important, and can in any case be improved by iteration, using the sums of squares of the loadings for a new estimate. His practice is to use as an approximate communality the largest correlation coefficient in the column (Vectors, 89). That this is plausible can be seen from a con- sideration of the case where there is only one factor, when the communality of Test 1 would be r 12 r 13 /r 23 , which is likely to be roughly equal to either r 12 or r 13 if these correlate highly with Test 1 and probably therefore with each other. We shall illustrate this approximate method of Thur- stone's on the same example as we used near the end of Chapter II, for the sake of comparison and for ease in arithmetical computation, even although that example is really an exact and artificial one unclouded by sampling error. Inserting then the highest coefficients in each column we get : (5883) -4 -4 -2 -5883 4 (-7) -7 -3 -2852 4 -7 (-7) -3 -2852 -2 -3 -3 (-3) -1480 5883 -2852 -2852 -1480 (-5883) 2-1766 2-3852 2-3852 1-2480 1-8950 = 10-0900 3-1765 2 First Loadings ^-6852 -7509 -7509 -3929 -5966 The communalities which really give the minimum rank are, as we saw in Section 9 of Chapter II 7 -7 -7 -1303 -5 and the correct first-factor loadings obtained by their use 7257 -7564 -7564 -3420 -5729 With a large battery the difference between the loadings obtained by the approximation and by the correct com- MULTIFACTOR ANALYSIS WITH FALLIBLE DATA 163 munalities would be much less . For the * ' centroid ' ' method depends on the relative totals of the columns of the correla- tion matrix ; and when there are twenty or more tests, these relative totals will not be seriously changed by the exact value given to the communality in the column. When the number of tests is large, the influence of the one communality in each column is swamped by the influence of the numerous correlations. The process now goes on as in Chapter II, and the resid- uals left after subtraction of the first-factor matrix check by summing in each column to zero, as there. Before, however, proceeding any farther, in this approxi- mate method we delete the quantities in the diagonal (the residues of the guessed communalities) and replace them by the largest coefficient in the column regardless of its sign, which we change to plus in the diagonal cell if it is negative in its own cell. The reason for this is apparent, especially when, as may and does happen, the existing diagonal residues are negative, which is theoretically impossible. For although the guessing of the first communalities does not in a large battery make much difference to the first- factor loadings, it may make a big difference to the diagonal residues. If the battery is very large indeed, our first- factor loadings would come out much the same, even if we entered zero for every communality, but the diagonal residues would then all be negative. In short, the diagonal residues are much the least trustworthy part of the calcu- lation when approximate communalities are used, and it is better to delete them at each stage and make a new approximation. 2. Illustrated on the Chapter II example. To make this clearer, the whole approximate process is here set out for our small example as far as the second residual matrix. The explanations printed alongside the calculation will make each stage clear. It is important to form the residual matrices exactly as instructed, as otherwise the check of the columns summing to zero will not work. In practice, certainly if a calculating machine were being used, several of the matrices here printed for clearness would be omitted ; for example, with a machine one would go straight from 164 THE FACTORIAL ANALYSIS OF HUMAN ABILITY A to C, while D and E would be made by actually altering C itself: (5883) 4 4 2 5883 4 (7) 7 3 2852 Largest r of 4 7 (7) 3 2852 column inserted 2 3 3 (3) 1480 in diagonal cell. 5883 2852 2852 1480 (5883) Loadings 1 2-1766 2-3852 2-3852 1-2480 1-8950 = 10-0900 = 3-1765 2 6852 -7509 -7509 -3929 -5966 = 3-1765 6852 (4695) 5145 5145 2692 4088 B 7509 7509 5145 5145 (5639) 5639 5639 (-5639) 2950 2950 4480 4480 First-factor TY-ir-f'||V i 3929 2692 2950 2950 (-1544) 2344 5966 4088 4480 4480 2344 (-3559) (1188) -1145 -1145 - -0692 1795 -1145 (1361) 1361 0050 - -1628 First residual C -1145 1361 (1361) 0050 - -1628 matrix. -0692 0050 0050 (1456) - -0864 A - B 1795 -1628 - -1628 - -0864 (-2324) 0001 - -0001 - -0001 0000 - -0001 Columns check to zero. (1795) -1145 -1145 - -0692 1795 Largest r of each -1145 (-1628) -1361 0050 - -1628 column (regard- D -1145 1361 (-1628) 0050 - -1628 less of sign) in- - -0692 0050 0050 (-0864) -0864 serted in each 1795 - -1628 -1628 -0864 (1795) diagonal cell. 6572 5812 5812 2520 7710 Sum disregard- ing signs. (1795) 1145 1145 0692 1795 Signs of Tests 2, I 1145 (1628) 1361 0050 1628 3, and 4 changed E 1145 1361 (-1628) 0050 1628 to make largest 0692 0050 0050 (0864) 0864 column (-7710) 1795 1628 1628 0864 (1795) all positive. Algebraic Sum 6572 5812 -5812 2520 7710 = 2-8426 ! =1-6860 2 Loadings Il\ -3898 -3447 -3447 -1495 -4573 (With temporary | signs.) MULTIFACTOR ANALYSIS WITH FALLIBLE DATA 165 3898 (1519) 1344 1344 0583 1783 3447 1344 (1188) 1188 0515 1576 Second-factor F 3447 1344 1188 (1188) 0515 1576 matrix, using 1495 0583 0515 0515 (0124) 0683 temporary signs 4573 1783 1576 1576 0683 ( 2091) (-0276) - -0199 - -0199 0109 0012 -0199 (0440) 0173 - -0465 0052 Second residual G -0199 0173 (0440) -0465 0052 matrix. 0109 - -0465 - -0465 (0640) 0180 E - F 0012 0052 0052 0180 (- 0296) - -0001 -0001 0001 -0001 0000 Columns check to zero. Notes. It is fortuitous that all the entries in E are positive. Usually some will be negative. In the check for the residual matrices, a discrepancy from zero in the last figure is often to be expected, even of three or four units in a large matrix. Note the negative value occurring in a diagonal cell in G. Further stages would be carried on in the same way. But at each stage the residues will be examined to see if further analysis is worth while, by methods indicated in Section 4 below. Meanwhile let us assume in the present example that no more factors need be extracted. The matrix of loadings of common factors thus arrived at is, after we have replaced the proper signs in Loadings II: Test 6852 7509 7509 3929 5966 Approximate Method True Values 11 Communality Communality 3898 - -3447 - -3447 - -1495 4573 6214 6827 6827 1767 5651 7000 7000 7000 1303 5000 2-7286 2-7303 The communalities *6214, etc., are the sums of the squares of the two loadings. For comparison with the 166 THE FACTORIAL ANALYSIS OF HUMAN ABILITY approximate communalities thus obtained there are shown the true values, which in this artificial case are known to us (see Chapter II, Section 9). This is for instructional purposes only the comparison is not intended as any criticism of Thurstone's method of approximation. As has been explained, this method is used only on large batteries, and it is a very severe test indeed to employ it on a battery of only five tests. We might now go back and begin our whole calculation again, using the communalities '6214, etc., arrived at by the first approximation. This does not seem often to be done in practice, most workers being content with the approximation first arrived at. If we repeat the calcula- tion again and again with our present example, on each occasion using as communalities the sum of the squares of the loadings given by the preceding calculation, we get the following sets of closer and closer approximation to the true communalities : * V kr? V V V mu- 5883 7000 7000 3000 5883 ion 6214 6827 6827 1767 5651 ion 6381 6970 6970 1477 5392 ,ion 6535 7043 7043 1397 5253 7000 7000 7000 1303 5000 First trial commu- nalities Next approximation Next approximation Next approximation True values The example has served to show how to work Thurstone's method of approximating to the communalities. It should be emphasized again that, being composed of only five tests, it is not a suitable example to employ in criticism of that method, and it is not so used here, but only as an illustra- tion. Being an artificial example, and not really overlaid with sampling error, it has had the advantage of allowing us to compare the approximations with the true values. But it must be remembered that a real experimental * It should be pointed out that iteration of each factor extraction separately will not give the same result as iteration of all. Iteration of the first factor will give the best approximation to rank one in the correlation matrix ; iteration of factor II the best approximation to rank one in the residues ; and so on. But this is not the same thing as approximating to the lowest rank of the whole matrix. MULTIFACTOR ANALYSIS WITH FALLIBLE DATA 167 matrix is not likely to have an exact low rank to which approximation can converge as here. In that case the approximations will presumably give an indication of the low rank which the matrix nearly has, which it might be made to have by adjustments in its elements within the limits of their sampling errors. We might, indeed, have dealt with this method in Chapter II, quite unconnected with sampling errors, regarding it as a method of finding the communalities by successive approximations It has, however, been left to the present chapter because in actual practice it is asso- ciated with the difficulty of finding communalities because of sampling error, and also is not generally used as a repetitive process. The labour of repeating the whole calculation with new approximations to the communalities has been a deterrent, and the further fact that with large batteries the improvement produced is very small. Usually, therefore, the experimenter is content with the factor loadings first obtained. It is a great drawback of the method, especially in this form, that any mathematical expression of the standard errors of the resulting loadings is almost impossible, by reason of the chance nature of the approximations made at each stage (see McNemar, 1941). On the other hand, the method does give load- ings which will imitate the experimental correlations to any desired degree of exactness, and does so with not very laborious arithmetic. 3. Error specifics. We shall consider next the influence of sampling errors upon the specific factors of tests. We have hitherto used the term " specific " to mean all that part of a test ability which is unique to that test. There is a tendency, however, to confine the term specific factor to that non-communal part of the test ability which is not due to any kind of error, and to use " uniqueness " for both the true specific and the error specifics. Now it is not at all obvious that sampling errors in the correlation coefficients will produce only unique factors. Rather the contrary. In general, they will produce new common factors, for the sampling errors of correlation coefficients are themselves correlated. Pearson and Filon 168 THE FACTORIAL ANALYSIS OF HUMAN ABILITY gave the formulae for such correlation in 1898. The corre- lation coefficient of the sampling errors of r 12 and r 13 (where one of the tests occurs in each correlation) is roughly some- what less than r 23 , for positive correlations. The correla- tion coefficient of the sampling errors of r 12 and r 34 , on the other hand, is a much smaller quantity of the second order only. The result of this is (Thomson, 1919a, 406) that sampling errors tend to produce, not irregular ups and downs of the correlations, but a ridged effect, with a general upward, or a general downward, tendency. In other words, the error factors are, or include, common factors. Some of the unique variance of the tests may be due to sampling errors : but so will some of the communality of the tests. 4. The number of common factors. As was indicated in Section 2 above, it is necessary to examine the residues left after each factor is removed, to see if it is worth while continuing. When the residues are so small that they may merely be due to random error (sampling or other error) it would seem to be futile to continue. There are certain snags here, however, connected with the skew sampling distribution of a correlation coefficient unless the true value is quite small, and with the fact mentioned in the preceding paragraph that sampling errors in correlation coefficients are themselves correlated with one another. The earliest method was to compare each residue with the standard error of the original correlation coefficient and cease factorizing when the residues all sank below twice these standard errors. But, as we have said on page 150, the use of the formula for the standard error of r is now frowned upon because of the skewness of the distribution. Moreover, sampling errors in the correlation coefficients, being themselves correlated, produce further factors ; and the above-mentioned test tended to stop the analysis too soon (Wilson and Worcester, 1939). These further factors must be taken out in order to give elbow room for rotation of the axes to some psychologically significant position. For the error factors are not concentrated in the last cen- troid or other factors taken out, but have been entangled with all the eentroids. Usually more factors have to be MULTIFACTOR ANALYSIS WITH FALLIBLE DATA 169 taken out than can be expected, on rotation, to yield meaningful psychological factors, but all the dimensions are required nevertheless for the rotations. In geometrical terms, some of the dimensions of the common factor space will be due to sampling error, but not the particular di- mensions indicated by the directions of the last factors to be extracted. In terms of Hotelling's plan, the whole ellipsoid is distorted ; its small major axes are not neces- sarily due entirely to sampling, nor its large ones free from it. A %* method is described by Wilson and Worcester (1939, 139) which is, however, laborious when the number of tests is large. See also Burt (1940, 338-40). Lawley (1940, 76 et seq.) repeated Wilson and Worcester's criticism and developed an accurate criterion described in Chapter XXI. It is, however, only legitimate when the factor loadings have been found by Lawley's application of the method of maximum likelihood. There have been various suggestions, mainly empirical, for an easily applied criterion to decide when to stop factorizing. Thurstone (1938, 65 et seq.) discusses some of the earlier ones. Ledyard Tucker's criterion is that the ratio of the sums of the absolute values of the residuals, including the diagonal used, just after and just before the extraction of a factor must be less than (n l)/(n -f 1) where n is the number of tests. Coombs' criterion depends upon the number of negative signs left among the residuals after everything has been done to reduce them by sign-changing, in the centroid process. If they are few, another factor may be extracted. More exactly, the permissible number is given in this table : Number of tests . 10 15 20 25 30 Negative signs . . 31 79 149 242 358 Standard error . 5 7 10 12 15 A fuller table is given in Coombs' article (1941). An example of the use of these two will be found in Blakey (1940, 126). Quinn McNemar (1942), who considers both of the above inadequate, gives a formula which includes 2V the 170 THE FACTORIAL ANALYSIS OF HUMAN ABILITY size of the sample. He takes out factors until cjj reaches or falls below l/\/2V, where <ii = a, -r (1 M h2 ), a, = st. dev. of the residuals after s factors, M h2 = mean communality for s factors. Others go on until the distribution of the residuals ceases to be significantly skew (Swineford, 1941, 378). Reyburn and Taylor (1939) divide the residuals by the probable errors of the original coefficients, and plot a distribution of the results disregarding sign. If it is significantly different from a normal curve of the same area and with standard deviation 1-4825, they take out more factors. Swineford (1941, 377) finds the correlation between the original correlations and the corresponding residuals and takes out factors till it is not significant. Hotelling's principal components lend themselves to more exact treatment. Hotelling himself (1933, 437-41) discusses the matter of the number which are significant. Davis (1945) shows how to find the reliability of each prin- cipal component from the reliabilities of the tests, and finds that it may happen that a later component is more reliable than an earlier one. The effect of sampling errors on factors and factorial analyses is indeed a very complex business, and it is advis- able to discuss how deliberate sampling of the population (whether human selection or natural selection) modifies analyses. CHAPTER XI THE INFLUENCE OF UNIVARIATE SELECTION ON FACTORIAL ANALYSIS* 1. Univariate selection. All workers with intelligence tests know, or ought to know, that the correlations found between tests, or between tests and outside criteria, depend to a very great extent indeed upon the homogeneity or heterogeneity of the sample in which the correlations were measured. If, to take the usual illustration, we measure the correlation between height and weight in a sample of the population which includes babies, children, and grown- ups, we shall obviously get a very high result. If we confine our measurement to young people in their 'teens, we shall usually get a smaller value for the coefficient of correlation. If we make the group more homogeneous still, taking, say, only boys, and all of the same race and exactly the same age, the correlation of height and weight will be still less.*]* Through all these changes towards greater homogeneity in age, the standard deviation (or its square, the variance) of height has also been sinking, and the standard deviation of weight also. The formulae which describe these changes (in samples normally distributed, at any rate) were given in 1902 by Professor Karl Pearson, and when the selection of the persons forming the sample is made on the basis of one quality only, these formulae can be put into the following very simple form. Let the standard deviations of (say) four qualities be in the complete population we must, of course, in each case define what we mean by the complete population, as for example all living adults who were born in Scotland given by S 1? S 2 , S 3 , and S 4 , and their correlations by * Thomson, 1937 and 1938&. | Greater homogeneity need not necessarily, in the mathematical sense, decrease correlation, and occasionally it does not do so in actual psychological experiments. But it almost always does so. 171 172 THE FACTORIAL ANALYSIS OF HUMAN ABILITY RU, RU> etc. Now let a selection of persons be made who are more homogeneous in the first quality say, in an intelligence test which has been given to them all so that its standard deviation in the sample is only c^, and write The smaller p is, the more homogeneous the group is in intelligence-test score. If we write q l will be larger, the greater the shrinkage in intelligence score-scatter from S x to a l9 We shall call q the " shrink- age " of the quality No. 1 in the sample. The other qualities 2, 3, and 4, being correlated with the first, will tend to shrink with it, and their expected shrink- ages q Z9 #3, and q^ can be calculated from the formula q i = qjt u For the sort of reason indicated earlier in this paragraph, the correlations of the four qualities which we are for simplicity in exposition assuming to be positively correlated in the whole population will also alter, according to the formula PiPj A numerical example will illuminate these formulae. Let us define our " whole population " as all the eleven -year- old children in Massachusetts, and let us suppose (the numbers are entirely fictitious) that the standard devia- tions of all their scores in four tests are : 1. Stanford-Binet test 16-5 = 2 l5 2. The X reading test 24-9 = S 2 , 3. The Y arithmetic test 27-3 = S,, 4. The Z drawing scale 14-2 = 2 4 , while the correlations between these four, in a State-wide survey, are (these are the R correlations) : THE INFLUENCE OF UNIVARIATE SELECTION 178 1 2 3 4 1 . 69 75 32 2 69 54 18 3 75 54 , 06 4 32 18 06 4 Now let a sample of Massachusetts eleven -year-olds be taken who are less widely scattered in intelligence, with a standard deviation in their Stanford-Binet scores of only 10-2. How will all the other quantities listed above tend to alter in this sample ? We have, using the formulae quoted, the following Pl= =[^ = .618 ?1 = ^/(l -618 2 ) = -786 and from q i = qiRu we have the other shrinkages q, and thence the coefficients p and the new standard deviations 9 P 786 -542 618 -840 10-2 20-9 590 -252 808 -968 22-1 13-7 The formula for r^ then enables us at once to calculate the correlations to be expected in the sample, namely : i 1 2 34 1 . 509 574 204 2 509 . 325 054 3 574 325 . 113 4 204 054 113 The greater homogeneity in the sample has made all the correlation coefficients smaller, and has indeed made r 84 become negative. 2. Selection and partial correlation. If a sample is made completely homogeneous in the Stanford-Binet test, clearly pi = and qi = 1. The s&me formulae then give us : 174 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 1234 P a I 69 -75 -32 524 -438 -904 13-0 11-9 12-8 and the resulting correlation coefficients, which in this case are called " coefficients of partial correlation for constant Stanford-Binet score," are, by the same formula : 1234 1 2 3 4 098 -086 098 . -455 -086 -455 The correlations of the Stanford-Binet test with the others are given by the formula as 0/0, that is, indeter- minate. That they are really zero is seen from the fact that when p l is taken as not quite zero, but very small, these correlations come out by the formula as very small. They vanish with p t . In this special case of " partial correlation," where the directly selected test is so stringently selected that everyone in the sample has exactly the same score in it, our formula has a more familiar form. For since ft = 9iRu and q l = 1 in this case of complete shrinkage we have ft = RU and Pi = V(l - #i* 2 ) so that our formula becomes the usual form of a partial correlation coefficient. Its more conventional notation is, calling the test which is made constant tefct k instead of Test 1 THE INFLUENCE OF UNIVARIATE SELECTION 175 If the " test " which is held constant is the factor g, this becomes - V) \/(i - V) which is called the " specific correlation " between i andj. As we said in Section 7 of Chapter IX, its numerator is the " residue " left after removing the correlation due to g. If g is the sole cause of correlation, holding g constant will destroy the correlation and we shall have r v = V* as we already saw from another point of view was the case in a hierarchical battery, in Section 4 of Chapter I. 3. Effect on communalities. The formula is thus a very useful formula, including partial correlation as a special case. If the original variances are each taken as unity, the numerator R% qty for i =*= j gives the new covariances, while pf and pf are the new variances. It also includes as a special case the formula known as the Otis-Kelley formula, which is applicable when two variates have both shrunk to the same extent (a restriction not always recognized). If we put q { = # ; and therefore p i = p. it becomes * = ij - <f = <j - P* p*(I - r<,) = 1 - Rq 1 "~ R v = p* = -^ = ~t the Otis-Kelley formula. 2 - 2 It has a still further application (Thomson, 19386, 456), for if a matrix of correlations in the wider population has been analysed by Thurstone's process, this same formula gives the new communalities (with one exception) to be expected in the sample, if we put i = j and understand by R ii9 the communality in the wider population, by r u , the 176 THE FACTORIAL ANALYSIS OF HUMAN ABILITY communality in the sample (and not a reliability coefficient, which is the usual meaning of this symbol). Writing the usual symbol h* for communality we have the formula in the form Pi* = 3j The exception is the new communality of the trait or quality which has been directly selected, in our example No. 1 the Stanford -Binet scores. For the directly selected trait the new communality is given by (Thomson, 19386, 455 ; and see also Ledermami, 19386). With these formulae we can see what is likely to happen to a whole factorial analysis when the persons who are the subjects of the tests are only a sample of the wider popula- tion in which the analysis was first made. 4. Hierarchical numerical example. We shall take, in the first place, the perfectly hierarchical example of our Chapters I and II. But to save space in the tables we shall consider only the first four tests. Their matrix of correlations, with the one common factor and the four specifics added, and with communalities inserted in the diagonal cells, was as follows : 1 2 3 4 8 *1 *2 % 1 (81) 72 63 54 90 -44 . 2 72 (-64) 56 48 80 . -60 . 3 63 56 (-49) 42 70 71 4 54 48 42 (36) 60 g 90 80 70 60 1-00 . s l 44 . . . 1-00 5 2 . 60 . . 1-00 . *3 t f 71 . . * 1-00 s . t . 80 . . 80 1-00 The bottom right-hand quadrant shows, by its zero entries, that the factors are all uncorrelated with one another, that is, orthogonal. The ttests expressed as linear functions of thfc factoid are THE INFLUENCE OF UNIVARIATE SELECTION 177 2 2 = -8g + -600$ 2 *s = -7g + -714*3 S 4 = -6g + -800,94 These equations are only another way of expressing the same facts as are shown in the north-east, or the south- west, quadrant of the matrix (where only two places of decimals are used for the specific loadings, to keep the printing regular). Let us now suppose that this matrix and these equations refer to a wide and defined population, e.g. all Massa- chusetts eleven-year-olds, and let us ask what will be the most likely matrix of correlations between these tests and factors to be found in a sample chosen by their scores in Test 1 so as to be more homogeneous. The variance of Test 1 in the wider population being taken as unity, let us take that in the more homogeneous select sample as being p^ = -36. We then have, using q i = qiRu, and treating g and the specifics just like tests, the following table : 1 2 3 4 g *i *2 *3 *4 <7 80 576 504 432 720 349 . f . P 60 817 864 902 694 937 1 1 1 p 2 (variance) 36 668 746 813 482 878 1 1 1 For the correlations and communalities, using our formula we get (again printing only two decimal places) : 1 2 3 4 ! s, s* s* 1 (61) 53 44 36 78 28 t 2 53 (46) 38 31 68 -26 73 3 44 38 (32) 26 56 -22 . 4 36 31 26 (21) 46 -18 g 78 68 56 46 1-00 -39 , s i 28 - -26 22 -18 39 1-00 . S 2 . 73 . . . 1-00 S j? 83 -89 83 89 1-00 1-00 178 THE FACTORIAL ANALYSIS OF HUMAN ABILITY In the more homogeneous sample, therefore, the correlations and the communalities of all the tests have sunk. The g column shows what the new correlations of g are with the tests ; and on examination of the matrix we see that these, when cross-multiplied with one another, still give the rest of the matrix. Thus 78 X -46 = -36 (r 14 ) .68 2 -= -46 (A 2 2 ) The test matrix is still of rank 1 (Thomson, 19386, 453), and these g-column entries can become the diminished loadings of the single common factor required by Rank 1. The columns for the specifics s 2 , s 3 (and later specifics also) still show only one entry. In the bottom right-hand quadrant, zero entries show that these specifics are still uncorrelated with one another and with g, that is, g, s Z9 s 39 and $ 4 are still orthogonal. But something has happened to the specific $ t . It has become correlated with g, and with all the tests. It has become an oblique factor, orthogonal still to the other specifics, but inclined to g and the tests. It leans further away from Test 1 than it formerly did, and makes obtuse angles (negative correlation) with the other tests and with g, to which it was originally orthogonal. But since, as we have already pointed out, the test matrix with the reduced communalities is still of rank 1, it is clear that a fresh analysis could be made of the tests into one common factor and specifics, thus a/ == -778g' + -628V * 2 ' = -679g' + -734s 2 + -827*3 In these equations the factors g' 9 $i' 9 S 29 $3, and $ 4 are again orthogonal (uncorrelated), and the loadings shown give the correlations and give unit variances. This is the analysis which an experimenter would make who began with the sample and knew nothing about any test measure- ments in the whole population, The reader, comparing the loadings in these equations THE INFLUENCE OF UNIVARIATE SELECTION 179 with the correlations in the matrix of the sample, will rightly conclude that the specifics from s a onward have not changed. In the matrix it is clear that they are still orthogonal, and their correlations with the tests, in the matrix, are the same as their loadings in the equations. The tests are, in the sample, more heavily loaded with these specifics than they were in the population, but the specifics are the same in themselves. The new specific */ the reader will readily agree to be different from $i* The latter became oblique in the sample, whereas Si' is orthogonal. What now is to be said about the common factors g (in the population) and g r (in the sample) ? From the fact that the loadings of g\ in the sample equations, are identical with the correlations of the original g with the tests, in the sample matrix, one is tempted to imagine g' and g to be identical in nature. But that is not so certain. If we go back to the equations of the tests in the popu- lation, we can rewrite them in the following form Zl = -467' + -800" + -877*! ' Z 2 = -555g' + -576g" + -600s 2 2 3 = -485g' + -504g" + -714*3 z, = - with two common factors g' and g" instead of one common factor g. These equations still give the same correlations. Por example r u = -467 X -417 + -800 X -432 = -540 as before. In these equations the specifics $ a s 3 , $ 4 are the same, and the communalities of Tests 2, 3, and 4 are the same. All that we have done in these three tests is to divide the common factor g into two components. The ratio of the loading of g" to the loading of g' is the same in each of them. The loadings of g* we have made identical with the shrinkages q in the table on page 177. In Test 1 also we have made the loading of g" equal to the shrinkage q t = '8. But in this test g* cannot be looked upon merely as a component of g. To give the correct s, the loading of g' has to be *467 as shown, atod 180 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the communality of Test 1 has been raised from its former value (-81) to 467 2 + -800 2 = -858 while the loading of the specific has correspondingly sunk. The factors g', g" 9 and Si are a totally new analysis of Test 1 in the population. Part of the former specific has been incorporated in the common factors. Now let the factor g" be abolished, i.e. held constant, so that the tests (now of less than unit variance, so we write them with x instead of z) are Variances Xi = -467g' + -877$!' -360 -668 -746 XL = -417g' + -800s 4 -813 The reduced variances are the sum of the squares of the surviving loadings, e.g. 467 2 + '377* = -360 The variances, it will be seen, are the p* 's of our tests as measured in the sample. If each of the last set of equations is divided through by the square root of its variance, we arrive at the equations -827*3 which is the analysis already given as that of an experi- menter who knew only the sample. As to the nature of g' 9 we can say in Tests 2, 3, and 4 that it is possible to regard it as a component of the g of the population. But we cannot do so with assurance in Test 1. There its nature is more dubious. At all events, it is not the same common factor as in the population, and at best we can say that it is one of its components. 5. A sample all alike in Test 1. These phenomena are still more striking if we consider a case where the sample is composed pf persons who are all alike in Test 1. It wb\ild be an Excellent dxe&isfe foV the re'adfe? to, ' THE INFLUENCE OF UNIVARIATE SELECTION 181 the resulting matrix of correlations for tests and population factors in this case. The tests act in this case as though their original equations in the population had been i = g* = -305g' + -630g" + -714*3 + -540g" + -800*4 and then g" had become zero, i.e. a constant with no variance. It perhaps helps to a further understanding of what is happening to the factors during selection if we realize that holding the score of Test 1 constant does not hold its factors g and Si constant. They can vary in the sample from man to man, but since *i = *g + '486*! remains constant, a man in the sample who has a high g must have a low $j that is, these factors are negatively correlated in the sample. And because they are thus negatively correlated, those members of the sample who have high g's, and who will therefore tend to do well in Tests 2, 3, and 4, will tend to have values below average (negative values) for their s l9 which will be therefore negatively correlated with these tests, in this sample. So far in our examples we have assumed the sample to be more homogeneous than the population. But a sample can be selected to be less homogeneous. In such a case the same formulae will serve, if we simply make the capital letters refer to the sample and the small to the population. In fact, the same tables, with their rdles reversed, can illustrate this case. In practical life we usually know which of two groups we would call the sample, and which the population. But mathematically there is no distinction, the one is a distortion of the other, and which is the " true " state of affairs is a question without meaning. It must also throughout be remembered that all these formulae and statements refer, not to consequences which are certain to follow, but to consequences which are to be expected. If actual samples were made the values experi- 182 THE FACTORIAL ANALYSIS OF HUMAN ABILITY mentally found in them for correlations, communalities, loadings, etc., would oscillate about those given by our formula 1 , violently in the case of small samples, only slightly in the case of large samples. 6. An example of rank 2. The above example has only one common factor. We turn next to consider an example with two. Again it is, we suppose, the first test according to which the sample is deliberately selected, and again we suppose the " shrinkage " q l to be -8. The matrices of correlations and communalities, in the population and in the sample, are then as follows, the two factors/! and/ 2 and the specifics being treated in the calculation exactly as if they were tests. To economize room on the page, we omit the later specifics : Correlations in the Population 1 2 3 4 5 /i /2 1 i 1 (65) 46 59 36 41 70 40 59 2 46 (37) 36 26 23 60 10 79 3 59 36 (61) 32 45 50 60 . . 4 36 26 32 (20) 22 40 20 . 5 41 23 45 22 (34) 30 50 /i 70 60 50 40 30 (1-00) 9 , f /2 40 10 60 20 50 . (1-00) . $1 59 . . . . . (1-00) . S 2 . 79 . . . . . (1-00) Correlations in the Sample 1 2 3 4 5 /i /2 *i *2 1 (-40) 30 40 23 26 51 25 40 2 30 (27) 23 17 12 51 -02 -21 85 3 40 23 50 22 35 32 54 -29 . 4 23 17 22 (-13) 14 30 12 -16 . 5 26 12 35 14 (26) 15 44 19 /i 51 51 32 30 15 (1-00) -23 -36 / 25 -02 54 12 44 -23 (1-00) -18 . *i 40 21 -29 -16 -19 -36 -18 (1-00) . *i . 85 . . . . . . (1-00) We see here a new phenomenon. The two common factors fi and / 2 in the population were orthogonal to one another, as is shown by the zero correlation between them. THE INFLUENCE OF UNIVARIATE SELECTION 183 But in the sample they are negatively correlated (- -228) ; that is, they are oblique. We begin to see a generalization which can be algebraically proved, that all the factors, common and specific, which are concerned with the directly selected test(s) become oblique to each other and to all the tests, but the specifics of the indirectly selected tests remain orthogonal to everything, except each to its own test. But the matrix of the tests themselves is still of rank 2, and an experimenter working only with the sample would find this out, although he would know nothing about the population matrix. He would therefore set to work to analyse it into two common factors, orthogonal to one another. A Thurstone analysis comes out in two common factors exactly, and can be rotated until all the loadings are positive. For example : Test Factor// Factor / 2 ' 12345 570 -521 -436 -332 -238 276 . -555 -130 -452 These factors /', however, are clearly a different pair from the factors / in the original population. In the sample, those original factors (/) are oblique ; these (/') are orthogonal. Again the whole phenomenon is reversible. The second matrix (with the orthogonal factors/') might refer to the population, and a sample picked with a suitable increased scatter of Variate 1. All our formulae could be worked backwards, and we should arrive at the matrix beginning (65), referring now to the sample. The/' factors would have become oblique, and a new analysis, suitably rotated, would give us the other factors/. It becomes evident that the factors we obtain by the analysis of tests depend upon the subpopulation we have tested. They are not realities in any physical sense of the word ; they vary and change as we pass from one body of men to another. It is possible, and this is a hope hinted at in Thurstone 5 s book The Vectors of Mind, that if we could somehow identify a set of factors throughout all their changes from sample to sample (in most of which 184 THE FACTORIAL ANALYSIS OF HUMAN ABILITY they would be oblique) as being in some way unique, we might arrive at factors having some measure of reality and fixity. Thurstone, in his latest book Multiple Factor Analysis, believes that he has achieved this, and that his Simple Structure is invariant. His claim is considered below in Section 9 of our Chapter XVIII on Oblique Factors. 7. A simple geometrical picture of selection. The geometrical picture of correlation between tests and factors which was described in Chapter IV is of some help in seeing exactly what happens to factors under selection in some test or trait. In Figure 23, x^ repre- sents the vector of the test or trait which is to be directly selected for, and g and ^ are the axes of the common factor and of its specific taking the case of one common factor only. The circle indicates the circular nature of the crowd of points which represent the population. It is a line of equal density of that crowd, which is densest at the origin and thins off equally in all directions. One-quarter of that Figure 23. Figure 24. crowd are above average in both g and s l9 and another quarter are below average in both. The correlation between g and s is zero. But in the selected sample (Figure 23) the scatter of the persons along the test vector x^ has been reduced. Persons have been removed from the whole crowd to leave the sample, but they have not been removed equally over the whole crowd. The line of equal density has become an ellipse, which is shorter along the line of the test vector x l than at right angles to that line. If we now compare the figure with Figure 18 in Chapter V (page 67), we see that it represents a state of negative correlation between g and $ x . THE INFLUENCE OF UNIVARIATE SELECTION 185 Less than one-quarter are now above average in both g and SD less than one-quarter below average in both. A majority are cases of being above average in the one factor and below in the other. An experimenter coming first to the sample and knowing nothing about the population will naturally standardize each of his tests. He can do, indeed, nothing else. That is to say, he treats the crowd as again symmetrical and our ellipse as a circle (Figure 24). In his space, therefore, the lines g and Si will be at an obtuse angle, just as the axes in Section 2 of Chapter V became acute. He knows nothing about these lines, but chooses new axes for himself. If these are at right angles, one of them may be one of the old axes, but they cannot both coincide with the old axes. 8. Random selection. These considerations, in Sections 1-7, deal with the results to be expected when a sample is deliberately selected so that the variance of one test is changed to some desired extent. The new variances and the changed correlations of the other tests given by our formula _K- are n ot the certain result of our action in selecting for Test 1 . If we selected a large number of samples of the same size, all with the same reduced variance in Test 1, they would not all be alike in the resulting correlations. On the con- trary, they would all be different. But most of them would be like the expected set, few would depart widely from that ; and the departures would be in both directions, some samples lying on the one side, others on the other side, of our expectation. If now, instead of selecting samples which are all alike in the variance of one nominated test, we take a large number of random samples of the same size, what would we find ? Among them would be a number which were alike in the variance of Test 1, and these in the other part of the correlation matrix would have values which varied round about those given by our formula. We could also pick out, instead of a set all alike in the variance of Test 1 ? 186 THE FACTORIAL ANALYSIS OF HUMAN ABILITY a different set all alike in the variance of Test 4, say ; and these would have values in the remainder of the matrix oscillating about our formula, in which Test 4 would replace Test 1. In short, a complex family of random samples would show a structure among themselves such that if we fix any one variance the average of that array of samples obeys our formula.* Random sampling will not merely add an " error specific " to existing factors, it will make complex changes in the common factors. 9. Oblique factors. This chapter, and the next, and the original articles on which they are based, were written be- fore Thurstone's work with oblique factors had been pub- lished. In those days it was assumed that analyses were into orthogonal or uncorrelated factors, and that only such factors could be looked upon as separate entities with any degree of reality inherent in them. When oblique factors are permitted, however, and methods for determining them devised (as is now the case in 1947), it becomes conceivable that the same factors might persist despite selection, and be discoverable, though their correlations with each other would change. Thurstone to-day urges that this is indeed so, and his arguments to this effect are discussed at the end of our Chapter XVIII, after his Oblique Simple Struc- ture has been explained. * On the author's suggestion, Dr. W. Ledermann has since proved this conjecture analytically (Biometrika, 1939, XXX, 295- 304) His results cover also the case of multivariate selection (see next chapter). CHAPTER XII THE INFLUENCE OF MULTIVARIATE SELECTION * 1. Altering two variances and the covariance. In the pre- ceding chapter we have discussed the changes which occur in the variances and correlations of a set of tests, and in their factors, when the sample of persons tested is chosen according to their performance in one of the tests : we are next going to see the results of picking our sample by their performances in more than one of the tests, first of all in two of them. Take again, the perfectly hierarchical example of the last chapter and of Chapters I and II. We must this time go as far as six tests in order to see all the consequences. The matrix of correlations of these tests and their factors will be simply an extension of that printed on page 176. Now let us imagine a sample picked so that the variance of Test 1 and also that of Test 2 is intentionally altered, and further, their covariance (and hence their correlation) changed to some predetermined value. It is at once clear that in these two directly selected tests the factorial composition will in general be changed can indeed be changed to anything which is not incom- patible with common sense and the laws of logic. What, however, will be the resulting sympathetic changes in the variances and co variances of the other tests of the battery ? In Chapter XI we altered the variance of Test 1 from unity to -36. The consequent diminution in variance to be expected in Test 2 was, as is shown on page 177, from unity to '668, and the consequent change in correlation from *72 to -53. Here, however, let us pick our sample so that the variance of the second test is also diminished to 36, and so that the correlation between them, instead of falling, rises to -833. We have, that is to say, chosen * Thomson, 1937 ; Thomson and Ledermann, 1988. 187 188 THE FACTORIAL ANALYSIS OF HUMAN ABILITY people for our sample who tend to be rather more alike than usual in these two test scores, as well as being closely grouped in each, an unusual but not an inconceivable sample. Natural selection (which includes selection by the other sex in mating) has no doubt often preferred indi- viduals in whom two organs tended to go together, as long legs with long arms, and the same sort of thing might occur in mental traits. In terms of variance and covarian ce we have changed the matrix : to the matrix : 1 2 1 2 1-00 -72 72 1-00 36 30 30 36 pp for ---- 30 - = = -833, the new correlation. Notice V(-36 X -36) 6 that the diagonal entries here (unities in R pp and -36, -36 in V pp ) are the variances, not the communalities. 2. Aiiken's multivariate selection formula. We shall symbolically represent the whole original matrix of vari- ances and covariances by : R Rn, where the subscript p refers to the directly selected or picked tests, and the subscript q to all the other tests and the factors. R pq (and also R qp ) means the matrix of Co- variances of the picked tests with all the others, including the factors. R qq means the matrix of variances and Co- variances of the latter among themselves. Since at the outset the tests and factors are all assumed to be stan- THE INFLUENCE OF MULTIVARIATE SELECTION 189 dardized, the variances in this whole R matrix are all unity, and the co variances are simply coefficients of correlation. In our case the R matrix is : Analysis in the Population 1 2 3 4 5 6 g *1 *2 *3 s< s, s 6 1 1-00 72 63 54 45 36 90 -44 . 2 72 1-00 56 48 40 32 80 . -60 . . 3 63 56 1*00 42 35 28 70 . . -71 4 54 48 42 1-00 30 24 60 ... 80 . 5 45 40 35 30 1-00 20 50 ... . -87 . 6 36 32 28 24 20 1-00 40 ... . -92 g 90 80 70 60 50 40 1-00 . . o 44 . 1-00 . *i 60 . . . . . LOO . ... *3 . . 71 . . . 1-00 ... S 4 . . 80 . 1-00 . $ 5 . . . 87 .1-00 . 5 . . . . 92 .... . 1-00 The R pp matrix is the square 2x2 matrix, the R qq matrix the square 11 X 11 matrix, while R pq has two rows and eleven columns, R qp being the same transposed. Our object is to find what may be expected to happen to the rest of the matrix when R pp is changed to V pp . Formulae for this purpose were first found by Karl Pearson, and were put into the matrix form in which we are about to quote them by A. C. Aitken (Ait ken, 1934). The matrix changes to : and in order to explain the meaning of these formulae we shall carry out the calculation for, a part of the above matrix only (the first four tests), with a strong recommendation to the reader to perform the whole calculation systematically. If we confine ourselves to the first four tests we have R PP = 00 -72 72 1-00. 00 -42 42 1-00 190 THE FACTORIAL ANALYSIS OF HUMAN ABILITY ^.= R, P = 63 56 63 54 54 48 56 48 3. The calculation of a reciprocal matrix. The most tiresome part of the calculation, if the number of directly selected tests is large, is to find R^ 1 the reciprocal of the matrix R pp . By the reciprocal of a matrix is meant another matrix such that the product R -i pp - P ~ L = / where I is the so-called " unit matrix " which has unit entries in the diagonal and zero entries everywhere else. Such a reciprocal matrix can be found by means of Ait ken's method of pivotal condensation as follows (Ait ken, 1937a). Write the given matrix with the unit matrix below it and minus the unit matrix on its right, thus : Check Column 1-00 -72 1-0000 . 72 72 1-00 . -1-0000 72 1-00 , . 1-00 1-00 1-00 4816 7200 1-0000 201G 1-0000 1-4950 -2-0764 4180 - -7200 1-0000 , 2800 1-0000 - 1-0000 2-0704 -1-4950 5814 -1-4950 2-0764 p 5814 As before, we -divide the first row of each slab through by its first member, writing the result in a row left blank for that purpose. Each pivot is thus unity, the whole calculation is made easier, and the process continues until the left-hand column no longer has any contents, when the numbers in the middle column are the reciprocal matrix. For a larger example of this automatic form of calculation see the Addendum on pages 350-1. That the matrix is THE INFLUENCE OF MULTIVARIATE SELECTION 191 indeed the reciprocal we can check by direct calculation. We have 1-00 -721 [~ 2-0764 -1-49501 _ Tl .1 72 1-OOj L~ 1-4950 2 -0764 J ~~ [ 1 J Matrix multiplication is carried out by obtaining the inner products (see footnote, page 31) of the rows of the first matrix with the columns of the second. Thus 1 X 2-0764 -72 X 1-4950 = 1 1 X 1-4950 + -72 X 2-0764 =0 are the two upper entries in the product matrix. When the reciprocal matrix R rp ~ l has thus been calculated, the best way of proceeding is to find and D = R P qq -R qp C In the case of our example these are r f 2 ' 0764 1-49501 r -68 -541 _ T-4709 -40371 C ~ L-l-4950 2-0764JL-56 -48 J ~~ [-2209 -1894 J Tl-00 -421 T-63 -561 T -4709 -40371 D ~ [_ '42 1-OOJ L' 54 '48 JL '2209 -1894J 00 -421 _ T-4204 -36041 42 1-OOJ L-3604 -3089 J 5796 -05961 0596 -6911J subtraction of matrices being carried out by subtracting each element from the corresponding one. We next need V r- P 36 '301 T -4709 -40371 _ T-2358 -20221 y pp^ ~ L.SO -36J L-2209 -1894J ~" L* 2208 -1893J which gives us the new covariances of the directly selected tests with those indirectly selected. For V M we need still C'(V pp C) where the prime indicates that the matrix is transposed (rows becoming columns) , T-4709 -22091 T-2358 -20221 _ [-1598 -13701 MrapC) 1^.4037 .I894j[_-2208 -1893J "~ L1370 -1175J and then f- |_- c>v c ~ ' 596 ] + r 1598 ' I87o i C V *^ ~ L-0596 -6911 J T L-1370 -1175 J T-7394 -19661 -8086J 192 THE FACTORIAL ANALYSIS OF HUMAN ABILITY We now can write down the whole new 4x4 matrix of variances and covariances. In the same way, had we included the other tests and the factors, we would have arrived at the whole new 13 X 13 matrix for all the variances and covariances which we now print.* The values calculated above for the first four tests will be recognized in its top left-hand corner. (The diagonal entries are variances, not communalities.) Covariances in the Sample 1 2 3 4 5 6 g *i 2 ^3 ^4 ^5 ^ 1 36 30 24 20 17 14 34 13 05 2 30 36 22 19 16 13 32 04 18 . 3 24 22 74 20 16 13 33 -14 -07 71 ... 4 20 19 20 81 14 11 28 -12 -06 80 . 5 17 16 16 14 87 09 23 -10 -05 . -87 6 14 13 13 11 09 92 19 -08 -04 92 g 34 32 33 28 23 19 47 -19 -10 . *i 13 04 -14 -12 -10 -08 -19 70 32 . <? 05 18 07 06 05 04 10 32 43 2 3 71 1-00 . *4 . . . 80 . . . . . 1-00 . G . . . . 87 . . . 1-00 *e . . . . . 92 . . 1-00 4. Features of the sample covariances. Examination of this matrix shows the following features : (1) The specifics of the indirectly selected tests have remained unchanged. They are still orthogonal to each other and all the other tests and factors (except each to its own test), are still of unit variance, and have still the same covariances with their own tests, though these will become larger correlations when the tests are restan- dardized ; (2) The specifics of the directly selected tests have become oblique common factors, correlated with everything except the other specifics ; * In such calculations on a larger scale, the methods of Aitken's (1937a) paper are extremely economical. Triple products of matrices of the form XY~ 1 Z can thus be obtained in one pivotal operation (see Appendix, paragraph 12 ; and Chapter VIII, Section 7, page 139) THE INFLUENCE OF MULTIVARIATE SELECTION 193 (3) The matrix of the indirectly selected tests is still of the same rank (here rank 1) ; (4) The variances of the factors g, s 1? and s z have been reduced to -47, -70, and -43. An experimenter beginning with this sample, and knowing nothing about the factors in the wider population, would have no means of knowing these relative variances, and would no doubt standardize all his tests. He certainly would not think of using factors with other than unit variance. And even if he were by a miracle to arrive at an analysis corresponding to the last table, with three oblique general factors, he would reject it (a) because of the negative correlations of some of the factors, and (b) because he can reach an analysis with only two common factors, and those orthogonal. It is therefore practically certain that he will not reach the population factors, at least as far as the directly selected tests are concerned. His data and his analysis will be as follows. The variances are all made unity and the covariances converted into correlations. The analysis into factors is a new one, not derived from the last table. Analysis in the Sample 1 2 3 4 5 6 g' h ,9/ * a ' S 3 5 4 5 6 5 6 1 1-00 83 46 38 30 24 82 45 -35 2 83 1-00 43 35 28 22 77 45 . -46 . 3 46 43 1-00 26 21 16 56 . -83 ... 4 38 35 261 00 17 13 46 . -89 . 5 30 28 21 17 1-00 11 37 -93 . 6 24 22 16 13 11 1-00 29 -96 g' 82 77 56 46 37 291 00 . h 45 45 . . . . . 1 00 s i' 35 . . . . . 1-00 S 2 ' 46 . . . 1-00 .... s 3 . . 83 . . 1-00 . s 4 . . 89 . . . 1-00 . S 5 . . . . 93 . . 1-00 . *6 . . . . . 96 . 1-00 5. Appearance of a new factor. The most noticeable change in this sample analysis, as compared with the A. 7 194 THE FACTORIAL ANALYSIS OF HUMAN ABILITY population analysis on page 189, is the appearance of a new " factor " h linking the directly selected tests, a factor which is clearly due entirely to that selection. What degree of reality ought to be attributed to it ? Does it differ from the other factors really, or have they also been produced by selection, even in the population, which is only in its turn a sample chosen by natural selection from past generations ? Otherwise the analysis is still into one common factor and specifics. The loadings of the common factor are less than they were in the population, and this, as our table of variances and covariances shows, is due to a real diminution in the variance of the common factor. The new common factor g' is a component of the old one. The loadings of Si and s 2 have also sunk, because they have been in part turned into a new common factor. The loadings of the other specifics have risen. But this is entirely because the variance of the tests has sunk due to the shrinkage in g, and is not due to any new specifics being added. All these considerations make it very doubtful indeed whether any factors, and any loadings of factors, have absolute meaning. They appear to be entirely dependent upon the population in which they are measured, and for their definition there would be required not only a given set of tests and a given technical procedure in analysis, but also a given population of persons. In our example, the covariance of Tests 1 and 2 in the new matrix V pp was made larger than would naturally follow from the changed variances of Tests 1 and 2, so that the correlation increased. In consequence the new factor h is one with positive loadings in both tests. We might equally well, however, have decreased the covariance in F^,, for example making r-36 .041 pp L-04 -36 J and in that case (the reader is strongly recommended to carry out the calculations as an exercise) the new factor h will be an interference factor, with negative loading in one THE INFLUENCE OF MULTIVARIATE SELECTION 195 of the two tests. In this case the experimenter, with a dislike for such negative loadings, would probably " ro- tate " his factors away from any position which had any simple relation to the factors of the population. Again, the formulae, moreover, can all be worked backward, the sample treated as the population and the population as the sample ; though as we said before, sam- ples in real life are certainly, as a rule, more homogeneous in nearly every quality than the complete population. NOTES, 1945. In Professor Thurstone's coming new edition, or new book, part of which I have been privileged to see in manu- script, he gives what he mildly calls " a less pessimistic inter- pretation than Godfrey Thomson's of the factorial results of selec- tion." His newer work on oblique factors certainly entitles him, I think, to hope that invariance of underlying factorial structure may be shown to persist underneath the changes, and we shall await further results with interest. 1948. Since the above was written the new book referred to has appeared (Multiple Factor Analysis). There in his Chapter XIX will be found Thurstone's above-mentioned interpretation, which is further described and discussed on our pages 292 et seq. below. 1945. It has sometimes been thought that Karl Pearson's selection formulae used in these chapters are only applicable when the vari- ables concerned are normally distributed in both population and sample. They have, however, a much wider application (Lawley, 1943) and in particular are still applicable when the sample has been made by cutting off a tail from a normal distribution, as happens when children above a certain intelligence quotient are selected for academic secondary education (Thomson, lQ43a and 1944). PART IV CORRELATIONS BETWEEN PERSONS CHAPTER XIII REVERSING THE ROLES* 1. Exchanging the rdles of persons and tests. In all the previous chapters the correlations considered have been correlations between tests, and the experiments envisaged were experiments in which comparatively few tests were administered to a large number of persons. For each test there would, therefore, be a long list of marks. The whole set of marks would make an oblong matrix, with a few rows for the tests, and a very large number of columns for the persons we will choose that way of writing it, of the two possibilities. From such a sot of marks we then calculated the correlation coefficients for each pair of tests, and our analysis of the tests into factors was based upon these. In the process of calculating a correlation coefficient we do such things to the row of marks in each test as finding its average, and finding its standard deviation. We quite naturally assume that we can legitimately carry out these operations. We assume, that is, that in the row of marks for one test these marks are comparable magnitudes which at any rate rise and fall with some mental quality even if they do not strictly speaking measure it in units, like feet or ounces. The question we are going to ask in this part of this book is whether, in the above procedure, the r61es of persons and of tests can be exchanged (Thomson, 1935ft, 75, Equation 17), and if so what light this throws upon * The first explicit references to correlations between persons in connexion with factor technique seem to have been made inde- pendently and almost simultaneously by Thomson (19356, July) and Stephenson (1935a, August), the former being pessimistic, the latter optimistic. But such correlations had actually been used much earlier by Hurt and by Thomson, and almost certainly by others. See Burt and Davies, Journ. Exper. Pedag., 1912, 1, 251. 199 Persons X X X X X X X X X X X X X X X X X X X X X 200 THE FACTORIAL ANALYSIS OF HUMAN ABILITY factorial analysis. Instead of comparatively few tests (perhaps two or three dozen ; fifty-seven is the largest battery reported up to date) and a very large number of persons, suppose we have comparatively few persons, and a large number of tests, and find the correlations between the persons. In that case our matrix of marks would be oblong in the other direction, with a large number of rows for the tests, and a small number of columns for the persons, and each correlation, instead of being as before between two rows, would be between two columns. Taking only small numbers for purposes of an explanatory table, we would have in the ordinary kind of correlations a table of marks like this : Tests while for correlations between persons we would have a table of marks like this : Persons Tests But we meet at once with a serious difficulty as soon as we attempt to calculate a correlation coefficient between two persons from the second kind of matrix. To do so, we must find the average of each column, just as previously we found the average of each row for the other kind of correlation. But to find the average of each column (by adding all the marks in that column together and dividing by their number) is to assume that these marks are in some sense commensurable up arid down the column, although each entry is a mark for a different test, on a scoring system which is wholly arbitrary in each test (Thomson, 19356, 75-6). X X X X X X X X X X X X X X X X X X X X X REVERSING THE ROLES 201 To make this difficulty more obvious, let us suppose that the first four tests are : 1. A form-board test ; 2. A dotting test ; 3. An absurdities test ; 4. An analogies test. In each of these the experimenter has devised some kind of scoring system. Perhaps in the form-board test he gives a maximum of 20 points, and in the dotting test the score may be the number of dots made in half a minute. But to find the average of such different things as this is palpably absurd, and the whole operation can be entirely altered by an arbitrary change like taking the number of seconds to solve the form board instead of giving points. 2. Ranking pictures, essays, or moods. This is a very fundamental difficulty which will probably make correla- tions between persons in the general case impossible to calculate. In certain situations, however, it does not arise, namely where each person can put the " tests " in an order of preference according to some criterion or judg- ment (Stephenson, 19356), and it is with cases of this kind that we shall deal in the first place. Usually the " tests " here are not really different tests like those named above, but are perhaps a number of children's essays which have to be placed in order of merit, or a number of pictures in order of aesthetic preference, or a number of moods which the subject has to number, indicating the frequency of their occurrence in himself. Indeed, the subject might not only give an order of preference to, say, the essays, but might give them actual marks, and there would be no absurdity in averaging the column of such marks, or in correlating two such columns, made by different persons. Such a correlation coefficient would show the degree of resemblance between the two lists of marks given to the children, or given to a set of pictures according to their aesthetic value. It would indicate, therefore, a resemblance between the minds of the two persons who marked the essays or judged the pictures. A matrix of correlations between several such persons might look exactly like the matrices of correlations between tests which occur in F.A. 7* 202 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Parts I and II, and could be analysed in any of the same ways. What would the " factors " which resulted from such an analysis mean when the correlations were between persons ? Take an imaginary hierarchical case first. 3. The two sets of equations. In test analysis the common factor found was taken to be something called into play by each test, the different tests being differently loaded with it. The test was represented by an equation such as * 4 = -Kg + -8*4 For each of the numerous persons who formed the sub- jects of the testing, an estimate was made of his g, and another estimate could be made of his s 4 . The different tests were combined into a weighted battery for this purpose of estimating a man's amount of g. His score in Test 4 would then be made up of his g and s 4 inserted in the above specification equation. *4-9 = -6g9 + -854., would be the score of the ninth person in Test 4. By analogy, when we analyse a matrix consisting of correlations between persons, we arrive at a set of equations describing the persons in terms of common and specific factors. Corresponding to a hierarchical battery of tests, we could conceivably have a hierarchical team of persons, from which we would exclude any person too similar to one already included. Each person in the hierarchical team would then be made up of a factor he shared with everyone else in the team, and a specific factor which was his own idiosyncrasy. An equation like would now specify the composition of the ninth person. g' is something all the persons have, s 9 ' is peculiar to Person 9. The loadings now describe the person, and the amount of g' " possessed " or demanded by each test can be estimated by exactly the same techniques employed in Part I. The score which Test 4 would elicit from Person 9 would be obtained by inserting the g' and s 9 ' " possessed " REVERSING THE ROLES 203 by that test into the specification equation of Person 9, giving This equation is to be compared with the former equation Both equations ultimately describe the same score, but 2 9 . 4 is not identical with s 4 . 9 . The raw score X is the same, but the one standardized z is measured from a different zero, and in different units, from the other. Disregarding this for the moment, we see that with the exchange of roles of tests and persons, the loadings and the factors have also changed rdles. Formerly, persons possessed different amounts of g, and tests were differently loaded with it. Now, tests possess different amounts of g', and persons are differently loaded with it. We feel impelled to inquire further into the relationships of these complementary factors and loadings. The test which is most highly saturated with g is that one which, in terms of Spearman's imagery, requires most expenditure of general mental energy, and is least depen- dent upon specific neural engines. It correlates more with its fellow-members of the hierarchical battery than any other test among them does. It represents best what is common to them all. The man, in a hierarchical team of men, who is most highly saturated with g' is that man who is most like all the others. His correlations with them are higher than is the case for any other man in the team. He is the indi- vidual who best represents the type. But a nearer ap- proach to the type can be made by a weighted team of men, just as formerly we weighted a battery of tests to estimate their common factor. 4. Weighting examiners like a Spearman battery. Corre- lations of this kind between persons were used long before any idea of what Stephenson has called " inverted factorial analysis " was present. The author and a colleague found in the winter of 1924-5 a number of correlations between experienced teachers who marked the essays written by 204 THE FACTORIAL ANALYSIS OF HUMAN ABILITY fifty schoolboys upon " Ships " (Thomson and Bailes, 1926). One table or matrix of such correlations, between the class teacher and six experienced head masters who marked the essays independently of one another, was as follows : Te A B C D E F Te . 60 69 56 69 63 67 A 60 9 53 50 54 55 68 B 69 53 . 60 65 66 64 C 56 50 60 67 67 65 D 69 54 65 67 54 69 E 63 55 66 67 54 f 69 F 67 68 64 65 69 69 . In the article in question, these different markers were compared by correlating each with the pool of all the rest. These correlations are shown in the first row of the table below. Purely as an illustrative example, let us make also an approximate analysis of this matrix, and take out at any rate its chief common factor. On the assumption that it is roughly hierarchical, we can use Spearman's formula Saturation - [T - 2 More easily we can insert its largest correlation coefficient as an approximate communality for each test, and find Thurstone's approximate first-factor loadings (see Chapter II, page 24). We get for the saturations or loadings the second and third rows of this table : Correlation with pool of rest Spearman saturations Thurstonc method Te A B D E F 77 -67 -76 -73 -76 -75 -82 814 -704 -796 -766 -798 -788 -861 81 -73 -80 -78 -80 -80 -85 We see that F is the most " typical " examiner of these essays, in the sense that he is more highly saturated with what is common to all of them ; while A conforms least to the herd. With the same formula which in Part I we used to esti- * See Chapter IX, page 154. REVERSING THE ROLES 205 mate a man's g from his test-scores, we could here estimate an essay's g' from its examiner scores. That is to say, the marks given by the different examiners would be weighted in proportion to the quantities Saturation with g' I saturation 2 where g' is that quality of an essay which makes a common appeal to all these examiners. Their marks (after being standardized) would therefore be weighted in the propor- tions -814/(1 -814 2 ), etc., that is : Te A B C D E F 2-41 1-40 2-17 1-85 2-20 2-08 3-33 or -72 -42 -65 -56 -66 -63 1-00 to make global marks for the essays, which could then be reduced to any convenient scale. If this were done, the result would be the " best " estimate * of that aspect or set of aspects of the essay which all these examiners are taking into account, disregarding all that can possibly be regarded as idiosyncrasies of individual examiners. Whether we think it the best estimate in other senses is a matter of subjective opinion. We may wish the " idiosyn- crasies " (the specific, that is) of a certain examiner to be given great weight. It clearly would not do, for example, to exclude Examiner A from the above team merely because he is the most different from the common opinion of the team, without some further knowledge of the men and the purpose of the examination. The " different " member in a team might, for example, be the only artist on a com- mittee judging pictures, or the only Democrat in a court judging legal issues, or the only woman on a jury trying an accused girl. But in non -controversial matters, if all are of about equal experience, it is probable that this system of weighting, restricting itself to what is certainly common to all, will be most generally acceptable as fairest. * Best whether we adopt the regression principle or Bartlett's. For if only one " common factor " is estimated, the difference is one of unit only, and the weighting in the text is the " best " on both systems. 206 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 5. Example from " The Marks of Examiners." This form of weighting examiners' marks has probably never yet been used in practice. But it has been employed, by Cyril Burt, in an inquiry into the marks given by examiners (Burt, 1936). As an example, we take the marks given independently by six examiners to the answer papers of fifteen candidates aged about 16, in an examination in Latin. (The example is somewhat unusual, inasmuch as these candidates were a specially selected lot who had all been adjudged equal by a previous examiner, but it will serve as an illustration if the reader will disregard that fact.) The marks were (op. cit., 20) : F Examiners Cand. A B C D E F I 39 43 52 37 43 40 2 39 44 50 43 43 46 3 44 51 55 47 46 46 4 37 46 43 44 40 43 5 38 47 55 35 43 45 6 45 50 54 45 45 49 7 42 52 51 45 44 46 8 43 49 53 47 46 46 9 32 42 49 34 36 38 10 37 40 48 37 39 42 11 38 42 47 39 36 39 12 40 44 50 41 36 42 13 38 43 50 36 34 41 14 35 45 49 37 40 40 15 32 38 41 28 34 34 The correlations between the examiners calculated from this table are (the examiner with the highest total correla- tion leading) : F A B E D C F . 86 84 82 84 71 A 86 . 80 74 85 71 B 84 80 . 80 81 67 E 82 74 80 , 72 69 D 84 85 81 72 . 48 C 71 71 67 69 48 . If, assuming this table to be hierarchical, we find each examiner's saturation with the common factor by Spear- REVERSING THE ROLES 207 man's formula, we obtain (with Professor Burt, op. cit. 9 294): F A B E D C 95 -92 -91 -87 -84 -72 In the sense, therefore, of being most typical, F is here the best examiner. The proportionate weights to be given to each examiner, in making up that global mark for the candidate which will best agree with the common factor of the team of examiners, are, as before Saturation 1 saturation 2 provided the marks have first been standardized. The resulting weights, giving F the weight unity, are : F A B E D C 1-00 -61 -54 -37 -29 -15 (If the weights are to be applied to the raw or unstan- dardized marks, they must each be divided by that examiner's standard deviation.) The marks thus obtained are only an estimate of the 46 true " common-factor mark for each child, just as was the case in estimating Spearman's g ; and the correlation of these estimates with the " true " (but otherwise undis- coverable) mark will be, as there (Chapter VII, page 106) S -Vr + S where S is the sum of all the six quantities Saturation 2 1 saturation 2 In our case this gives r m = -98 The best examiner's marking itself correlated with the hypothetical " true " mark to the amount *95, so that the improvement is not worth the trouble of weighting, especially as the simple average of the team of examiners gives *97. But in some circumstances the additional labour might be worth while, and there is an interest in 208 THE FACTORIAL ANALYSIS OF HUMAN ABILITY knowing which examiners conform least and which most to the team, and having a measure of this. After the saturation of each examiner with the hypothet- ical common factor has been found, the correlations due to that factor can be removed from the table exactly as in analysing tests in Chapter II, pages 27 and 28, or in Chapter IX, page 155. The residues, as there, may show the presence of other factors ; and " specific " resem- blances or antagonisms between pairs of examiners, or minor factors running through groups of examiners, may be detected and estimated. In short, all the methods of Parts I and II of this book there used on correlations between tests may be employed on correlations between examiners. The tests have come alive and are called examiners, that is all. But since the child's performance, judged by the different examiners differently, is here nevertheless the same identical per- formance, our interpretation of the results is different. The two cases throw light on one another. A Spearman hierarchical battery of tests may estimate each child's general intelligence, which is there something in common among the tests. The examiners may have been instructed to mark exclusively for what they think is general intelli- gence. In that case their weighted team will estimate for each child a general intelligence, which is something in common among the somewhat discrepant ideas the examiners hold on this matter. 6. Preferences for school subjects. In the previous sec- tions we have discussed correlations between examiners who all mark the same examination papers. The purpose of their marking these papers is to award prizes, distinc- tions, passes, and failures to the candidates. The exam- iners are a means to this end ; the reason for employing several of them is to obtain a list of successes and failures in which we can have greater confidence. The technique described is one which enables us to combine their marks, on certain assumptions, to greatest advantage. But it can, as in the inquiries described in The Marks of Examiners, be turned to compare individual examiners, and to evaluate the whole process of examining. REVERSING THE ROLES 209 It is only a step to another, very similar, experiment in which objects evaluated by the " examiners " are not the works of candidates in an examination, but are objects chosen for the express purpose of gaining an insight into the minds of those asked to judge them. Thus we might ask several persons each to evaluate on some scale the aesthetic appeal of forty or fifty works of art (Stephenson, 19366, 353), or ask a number of school pupils each to place in order of interest a list of school subjects. Stephenson (1936a) asked forty boys and forty girls attending a higher school in Surrey, England, thus to place in order of their preference twelve school subjects represented by sixty examination papers, and calculated for about half these pupils the correlation coefficients between them. To explain the kind of outcome that may be expected from such an experiment it will be sufficient for us to quote his data for a smaller number of pupils, say eight girls, avoiding anomalous cases for simplicity in a first consideration. The correlations between them were as follows (op. cit., 50) : Girl 3 4 5 7 17 18 19 20 3 59 31 26 02 -16 38 35 4 59 , 75 42 23 -01 -66 -- -03 5 31 75 65 29 -02 -18 -08 7 26 42 65 50 15 -54 17 17 02 -23 29 50 60 52 72 18 -16 -01 02 -15 60 , 09 79 19 -38 -66 -18 54 52 09 . 40 20 -35 03 -08 -17 72 79 40 . This table at once suggests that these girls fall into two types. Girls 3, 4, 5, and 7 correlate positively among themselves ; they have somewhat similar preferences among school subjects. Girls 17, 18, 19, and 20 correlate positively among themselves. But the two groups correlate negatively with one another. The two types were different in their order of preference, Type I tending, for example, to put English and French higher, and Physics and Chemistry lower, than Type II (though both were agreed that Latin was about the least lovable of their studies !). 210 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 7. A parallel with a previous experiment This experi- ment, it will be seen, forms a parallel to that inquiry (also by Stephenson) described in Chapter I, Section 9, where tests fell into two types, verbal and pictorial, with correla- tions falling there as here into four quadrants. If we call the two types of school pupil here the linguistic (L) and the scientific (S), and again use C for the cross-correlations, the diagram corresponding to that on page 16 of Chapter I is : The chief difference between the two cases is that there the cross-correlations, though smaller than hierarchical order in the whole table would demand, were nevertheless positive. Here, however, the cross-correlations are actually negative. It is true that the signs of all the correlations in the C quadrants can in either case be reversed, by reversing the order of the lists either of all the earlier or all the later variables (there tests, here pupils). But that is not really permissible in either case. We have no doubt which is the top and which the bottom end of a list of marks, whether in a verbal test or a pictorial test ; and to reverse the order of preference given by either the linguistic or the scientific pupils would be simply to stultify the inquiry. There is, therefore, a real difference between the cases. In the present set of correlations something is acting as an " interference factor." In Chapter I we explained the correlations and their tetrad-differences by the hypothesis of three uncorrelated factors g, v 9 and p required in various proportions by the tests, and possessed in various amounts by the children. The loadings which indicated the proportions of the factors in each test we tacitly assumed to be all positive. Thur- REVERSING THE ROLES 211 stone expressly says that it is contrary to psychological expectation to have more than occasional negative loadings. 8. Negative loadings. Let us endeavour to make at least a qualitative scheme of factors to express the correlations between the pupils, factors possessed in various amounts by the subjects of the school curriculum, and demanded in various proportions by each pupil before he will call the subject interesting. One type of pupil weights heavily the linguistic factor in a subject in evaluating its interest to him. The other type weights heavily the scientific factor in a subject in judging its attraction for him. But to explain actual negative correlations between pupils we must assume that some of the loadings are negative, assume, that is, that some of the children are actively repelled by factors which attract others. Common sense does not think thus. Common sense says that two children may put the subjects in opposite orders, even though they both like them all, provided they don't like them equally well. But then common sense is not anxious to analyse the children into uncorrelated additive factors. If each child is thus expressed as the weighted sum of various factors, two children can correlate negatively only if some of the loadings are negative in the one child and positive in the other, for the correlation is the inner product of the loadings. Since Stephenson has found numerous nega- tive correlations between persons, and since few negative correlations are reported between tests, we seem here to have an experimental difference between the two kinds of correlation, and if ever correlations between persons come to be analysed as minutely and painstakingly as correla- tions between tests, it would seem that the free admission of negative loadings would be necessary.* The present matrix can in fact be roughly analysed into two general factors, one of which has positive loadings in all pupils, while the other is positively loaded in the one type, negatively loaded in the other. 9. An analysis of moods. A still more ingenious appli- cation by Stephenson of correlations between persons is in an experiment in which for each person a " population " * See Stephenson, 1936&, 349. 212 THE FACTORIAL ANALYSIS OF HUMAN ABILITY of thirty moods, such as " irascible," " cheerful," " sunny," were rated for their prevalence and intensity for each of ten patients in a mental hospital, and for six normal persons (Stephenson, 1936c, 363). This time the correla- tion table indicated three types, corresponding to the manic-depressives, the schizophrenes, and the normal persons, each type correlating positively within itself, but negatively or very little with the other types. These experiments were only illustrative, and it remains to be seen whether factors which will prove acceptable psycho- logically will be isolated in persons in the same manner as g, and the verbal factor, have been isolated in tests. The parallel between the two kinds of correlation and analysis is, however, certainly likely to throw light on the nature of factors of both kinds. CHAPTER XIV THE RELATION BETWEEN TEST FACTORS AND PERSON FACTORS 1. Burfs example, centred both by rows and by columns. In the examples we have just considered, there is no doubt that correlations between persons can be calculated without absurdity. In the matrix of marks given by a number of ex- aminers (marking the same paper) to a number of candidates, either two candidates can be correlated, or two examiners. The heterogeneity of marks referred to in Chapter XIII, Section 1, does not enter as a difficulty. Still keeping to such material, let us ask ourselves what the relation is between factors found in the one way, and factors found in the other. Qualitatively, we have already suggested that factors and loadings change roles in some manner. The most determined attempt to find an exact relationship has been that made by Cyril Burt, who concludes that, if the initial units have been suitably chosen, the factors of the one kind of analysis are identical with the loadings of the other, and vice versa (Burt, 19376). The present writer, while agreeing that this is so in the very special circum- stances assumed by Burt, is of opinion that his is a very narrow case, and that the factors considered by Burt are not typical of those in actual use in experimental psycho- logy. Theoretically, however, Burt's paper is of very great interest. It can be presented to the general reader best by using Burt's own small numerical example, based on a matrix of marks for four persons in three tests : Persons abed 1 Tests 2 3 6 204 3 113 33 11 213 214 THE FACTORIAL ANALYSIS OP HUMAN ABILITY It will be noticed that this matrix of marks is already centred both ways. The rows add up to zero, and so do the columns. The test scores have been measured from their means, and then thereafter the columns of personal scores have been measured from their means ; or it can be done persons first, tests second, the end result being the same. Burt does not give the matrix of raw scores from which the above matrix comes. If we take the doubly centred matrix as he gives it, the matrices of variances and covarianccs formed from it are : Test Covariances 123 1 j 56 28 28 2 j 28 20 8 3 ! 28 8 20 Person Covariances a b c d a 54 18 36 b 18 14 4 8 c 4 2 2 d 36 8 2 26 Notice that in both these matrices the columns add to zero, just as they do in the matrices of residues in the " centroid " process. 2. Analysis of the Covariances. Burt next proceeds to analyse each of these by Hotelling's method. It seems clear that there will exist some relation between the two analyses, since the primary origin of each matrix is the same table of raw marks, and to show that relation most clearly Burt analyses the covariances direct, and not the correlations which could be made from each table (by dividing each covariance by the square root of the product of the two variances concerned). For the two Hotelling analyses he obtains (and the Thurstone factors before rotation would here be the same) : RELATION OF PERSONAL TO TEST FACTORS 215 Analysis of the Tests x 1 = 2 Vl4 YI # 2 = A/14 Yi + V6 Y 2 #3 ~Vl4 YI V6 Y 2 Analysis of the Persons d= 2V6/!- V2/2 In both cases two factors are sufficient (there will always be fewer Hotelling or Thurstone factors than tests with a doubly centred matrix of marks, for a mathematical reason). The reader can check that the inner products give the co variances, e.g. co variance (bd) = V 6 X 2 V 6 2 V 2 X V 2 = 12 "~ 4 = 8 The method of finding Hotelling loadings was described in Chapter V, and the reader can readily check that the coefficients of YU for example, do act as required by that method. For if we use numbers proportional to 2y^l4, \/14, and \/14, namely 1, |, J, as Hotelling multipliers we get : 56 28 28 28 20 8 28 8 20 56 28 28 1410 4 14 410 84 42 42 proportional to 1 J \ as required. The largest total (84) is the first "latent root," and the multipliers 1, i, ^, have to be divided, according to Chapter V, by the square root of the sum of their squares, and multiplied by the square root of 84, giving 2V14 V 14 -~V 14 21G THE FACTORIAL ANALYSIS OF HUMAN ABILITY 3. Factors possessed by each person and by each test. Burt then goes on to " estimate," by " regression equa- tions," the amount of the factors y possessed by the persons, and the amount of the factors / possessed by the tests. There is a misuse of terms here, for with Hotelling factors there is no need to " estimate " ; they can be accurately calculated : but that is a small point. The first three equations can be solved for the y's there is indeed one equation too many, but it is consistent. And the four equations of the second group can be solved for the /'s again they are consistent. Since the equations are con- sistent, we can choose the easiest pair in each case to solve for the two unknowns. Choosing the two equations for x v and # 2 we obtain Yl = 2714 *' __ x, + x 12 - ~ For the other set of factors we naturally choose the equations in a and c, and have f Jl Now, since we are very liable to confusion in this dis- cussion, let us remind ourselves what these factors y and these factors / are. The factors y are factors into which each test has been analysed. They do not vary in amount from test to test, but each test is differently loaded with them. They vary in amount from person to person. The factors /are factors into which each person has been analysed. These do not vary in amount from person to person, but from test to test. Each person is differently loaded with them, that is, made up of them in different proportions. The y's are uncorrelated fictitious tests : the /'s are uncorrelated fictitious persons. RELATION OF PERSONAL TO TEST FACTORS 217 Now, from the equations I Yl = I we can find the amount of each factor y x and y 2 possessed by each person, by inserting his scores x l and x 2 in these equations, scores which are given in the matrix : a bed - 6 2 4 3 i _ i _ 3 33 11 I 2 3 Thus the first person possesses y x in an amount 6/2 A/14, because his x l is 6. For the four persons and the two factors wo find the amounts of these factors possessed by each person to be : Factors y, y 2 d 3 A/ 6 1 A/6 2 A/14 4. Reciprocity of loadings and factors. These are the amounts of the factors y possessed by the four persons. If now the reader will compare them with the loadings of the factors /in the second set of equations on page 215, he will see a resemblance. The signs are the same, and the zeros are in the same places. Moreover, the resemblance becomes identity if we destandardize the factors f l and / 2 , measuring the former in units \/84< times as large, and the latter in units yl2 times as large, 84 and 12 being the non-zero latent roots of both matrices, In these units let us 218 THE FACTORIAL ANALYSIS OF HUMAN ABILITY use fa and <f> 2 for them. The equations on page 215 giving the analysis of the persons then become 3V6 , r 3 , - b = (^M/O + ( Via/.) = ~ It will be seen that the loadings of fa and < 2 are identical with the amounts of YI and y 2 in the table on page 217. A similar calculation could be made comparing the amounts of fi and / 2 possessed by the tests with the loadings of YI and y 2 (suitably destandardized) in the analysis of the tests. As we said at the outset, if suitable units are chosen for the marks and the factors, the loadings of the personal equations are the factors of the test equations, and the factors of the personal equations are the loadings of the test equations. But only for doubly centred matrices of marks. It would be wrong to conclude in general that loadings and factors are reciprocal in persons and tests. Indeed, even for doubly centred matrices of marks, this simple reciprocity holds only for the analysis of the covariances and not for analyses of the matrices of corre- lations. Except by pure accident (and as it happens, Burt's example is in the case of test correlations such an accident), the saturations of the correlation analysis will not be any simple function of the loadings of the covariance analysis. 5. Special features of a doubly centred matrix. But in any case, a matrix of marks which has been centred both ways is one in which only a very special kind of residual association between the variables is present. Most of what we commonly call the association or resemblance between either tests or persons, the amount of which we gauge by the correlation coefficient, is due to something over and above this. We can write down an infinity of possible raw RELATION OF PERSONAL TO TEST FACTORS 219 matrices from which Burt's doubly centred matrix might have come. To the rows of the latter matrix we can add any quantities we like without in the slightest altering the correlations between the tests, but making enormous changes in the correlations between the persons. Let us, for example, add 10 to the top row, 13 to the middle row, and 16 to the bottom row. There results the matrix : abed 1 4 12 10 14 2 16 14 12 10 (A) 3 19 13 17 15 This gives as correlations between the persons : a bed a b c d 1-00 -75 -84 -14 75 1-00 -28 -76 84 -28 1-00 -42 - -14 -76 -42 1-00 Next, without changing this matrix of correlations between persons in the slightest, wo can add any quantities we like to the columns of the matrix of marks, and produce an infinity of different matrices of correlations between tests. If, for example, we add 5, 2, 8, and 9 to the four columns, we have a matrix of raw marks : abed 1 9 14 18 23 2 21 16 20 19 3 24 15 25 24 This has the same correlations between persons, but the correlations between tests are now : 123 1 2 3 1-00 -16 -16 1-00 23 -92 24 92 1-00 Or instead, by adding suitable numbers to the columns and to the rows, we might have arrived at the matrix : 220 THE FACTORIAL ANALYSIS OF HUMAN ABILITY a b c d 1 44 48 18 10 2 63 57 27 13 (C) 3 58 48 24 10 or equally well at : d 1 35 45 37 43 2 34 34 26 26 (D) 3 34 30 28 28 The order of merit of the persons in each test is quite different in each of these matrices. The order of difficulty of the tests for each person is quite different in each. If we consider the ordinary correlation between Tests 1 and 2, we find that it is negative in (B) 9 zero in (D), and positive in (C), yet all of these matrices reduce to Burt's matrix when centred both ways. It is clear that they contain factors of correlation which are absent in the doubly centred matrix. The averages of the rows and the columns of (C) are as follows : a b c d Average 1 44 48 18 10 30 2 63 57 27 13 40 3 58 48 24 10 35 Average 55 51 23 11 The correlation between two tests is clearly influenced very much by the fact that here the person a is so much cleverer than the person d. Similarly, the correlation between two persons is influenced by the fact that Test 1 is more difficult than Test 2. As soon as the matrix is centred both ways, all the correlation due to these and similar influences is almost extinguished. Centred by rows, (C) becomes : 14 18 12 20 23 17 13 27 23 13 11 25 RELATION OF PERSONAL TO TEST FACTORS 221 and all the tests are equally difficult on the average. Centred by columns as well, it becomes : 6 2 4 3 1 1 3 33 1 - 1 and not only are all the tests equally difficult on the average, but all the persons are equally clever on the average. It is to the covariances still remaining that Burt's theorem about the reciprocity of factors and loadings applies. It does not apply to the full covariances of the matrix centred only one way, in the manner usually meant when we speak of covariances or of correlations. 6. An actual experiment. Since the first edition of this book, Burt's The Factors of the Mind has appeared (London, 1940). In Part I Burt discusses with keen penetration the logical and metaphysical status of factors, concluding " that factors as such are only statistical abstractions, not concrete entities." Part II discusses the connexion between the different methods of factor analy- sis ; and appendices give worked examples of Burt's special methods of calculation. His principle of reci- procity of tests and persons is seen in an actual illustrative experiment, in his Part III on the distribution of tempera- mental types. This experiment was on twelve women students, selected because the temperamental assessments made by various judges on them were more unanimous than in the case of the other students. Each, therefore, was a well- marked temperamental type. They were assessed for the eleven traits seen in the table below. The assessments over each trait were standardised, i.e. measured in such units and from such an origin that their sum was zero and the sum of their squares twelve, the number of persons, so that the group was (artificially) made equal in an average of sociability, sex, etc. The correlations between the traits were then calculated and centroid factors taken out, the first two of which I shall call by the Roman letters u and v. These two are possessed in some amount by 222 THE FACTORIAL ANALYSIS OF HUMAN ABILITY each of the persons, and required, in degrees indicated by the saturation coefficients, by each of the traits. These saturation coefficients have been found by analysis of the correlations between the traits. Now according to the reciprocity principle, if we analyse instead the correlations between the persons, find factors which we may indicate by Greek letters, and measure the amounts of these possessed by the eleven traits, these amounts ought to be the same ^s the saturation coefficients of the Roman factors u, v 9 etc. Burt therefore further standardizes the assessments, by persons this time, and finds the total scores on each trait, which are, by a property of centroid factors (see page 100) proportional to the amounts of a centroid Greek factor possessed by the eleven traits ; and the test of the reciprocity hypothesis is to see whether these totals are similar to the saturations of a Roman factor. The figures (from Burt's page 405) are given in the table below : Saturations of the Amounts of the Roman factors Greek factor u V oc 671 508 587 878 213 489 827 483 378 951 233 297 824 241 280 780 -268 001 898 - -159 - -089 259 - -104 - -337 564 - -667 - -447 830 - -490 - -489 412 - -685 - -525 Sociability Sex. Assertiveness Joy. Anger Curiosity . Fear Sorrow Tenderness Disgust . Submissiveness Clearly the amounts of a do not correspond to the saturations of u ; nor should they, for a general factor has already been eliminated by the double standardization. They do, however, agree reasonably well with the satura- tions of the second Roman factor v 9 and confirm Burt's prediction that, even in this sample, and with factors which are not exactly principal components, the reci- procity principle would still hold approximately. PART V THE INTERPRETATION OF FACTORS CHAPTER XV THE DEFINITION OF g 1. Any three tests define a " g." This concluding part will be devoted to an attempt to answer the questions: " What are factors ? What is their psychological and physio- logical interpretation ? On what principles are we to decide between the different possible analyses of tests (and persons) ? " It may seem strange to have deferred these considerations so long, and to have discussed methods of analysing tests, and of estimating factors, before asking explicitly what they mean. But that is how "factors " have arisen. Whatever else they are, they certainly are not things which can be identified with clearness first, and discussed and measured afterwards. Their definition and interpretation arise out of the attempt to measure them. We shall begin by discussing, in the present chapter, the definition and nature of g. It will be remembered that the idea of g arose out of Professor Spearman's acute observation that correlation coefficients between tests tend to show hierarchical order : that is, that their tetrad-differences tend to be zero or small ; or in more technical terms still, that the rank to which a matrix of correlation coefficients can be " reduced " by suitable diagonal elements tends towards rank one. This fundamental fact is at the basis of all those methods of factorial analysis which magnify specific factors, and a reason for it, based on the idea that it is a mathematical result of the laws of probability, will be advanced in Chapter XX. In consequence of this fundamental fact, correlation coefficients between a number of variables can be adequately accounted for by a few common factors. To be adequately described by one only a g the " reduced " rank of the correlation matrix has to be one 9 within the limits of sampling error. This trouble of sampling error is very liable to obscure 225 226 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the issue, and we will remove it during most of the present chapter, as we did in Parts I and II, by supposing that we have defined our population (say all adult Scots, or all men, for that matter) and have tested every one of them. Suppose now that we have three tests and have, in this whole population, measured their correlation coefficients : 1 2 3 1 1 r lz ?*13 2 7*12 1 7-23 3 7*13 7*23 1 If, as is usually the case, these coefficients are all positive, and if each of them is at least as large as the product of the other two, we can explain them by assuming one g and three specifics s l9 s 2 , and s 3 . There are many other ways of explaining them, but let us adopt this one. We have thereby defined a factor g mathematically (Thomson, 1935a, 260). It is then for the psychologist to say, from a consideration of the three tests which define it, what name this factor shall bear and what its psychological description is. The psychologist may think, after studying the tests, that they do not seem to him to have anything in common, or anything worth naming and treating as a factor. That is for him to say. Let us suppose that at any rate he does not reject the possibility, but that he would like an oppor- tunity of studying other tests which (mathematically speaking) contain this factor, and have nothing else in common, before finally deciding. In that case the experimenter must search for a fourth test which, when added to these three, gives tetrad- differences which are zero ; and then for a fifth and further tests, each of which makes zero tetrad-differences with the tests of the pre-existing battery. This extended battery the experimenter woulci lay before the psychological judge, to obtain a ruling whether the single common factor, of which it is the now extended but otherwise unaltered definition, is worthy of being named as a psychological factor. 2. The extended or purified hierarchical battery. Mathe- matically, any three tests with which the experimenter THE DEFINITION OF g 227 cared to begin would define " a " g, if we except temporarily the case, to which we shall later return, of three correlation coefficients, one of which is less than the product of the other two. The experimental tester, however, might in some cases have great difficulty in finding further tests, to add to the original three, which would give zero tetrad- differences. Unless he could do so, it is unlikely that the psychological judge would accept the factor as worthy of a name and separate existence in his thoughts. It is, for example, an experimental fact that starting with three tests which a general consensus of psychological opinion would admit to have only " intelligence " as a common requirement, it has proved possible to extend the battery to comprise about a score of tests without giving any tetrad-differences which cannot be regarded as zero. Even that has not been accomplished without difficulty, and without certain blemishes in the hierarchy having to be removed by mathematical treatment. But the fact that with these reservations it is possible, and that psychological judgment endorses the opinion that each test of this battery requires " intelligence," is the main evidence behind the actual " existence " of such a factor as " g, general intelli- gence." It must be noted that the word " existence " here does not mean that any physical entity exists which can be identified with this g. It does mean, however, that, as far as the experimental evidence goes, there is some aspect of the causal background which acts "as if " it were a single unitary factor in these tests. The process of making such a battery of tests to define general intelligence (see Brown and Stephenson, 1933) has not in fact taken the form of choosing three tests as the basal definition and then extending the battery. Instead, a number of tests which, it was thought from previous experience, would act in the desired way have been taken, and the battery thus formed has then been purified by the removal of any tests which broke the hierarchy. The removal of such tests does not, of course, mean that they do not contain g, but it means that g is not their only link with the other tests of the battery, and that therefore they are unsuitable members of a set of tests intended to define g. 228 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Further, the actual making of such a hierarchical battery has not been accomplished under the ideal conditions which we have been assuming, namely, that the whole population has been accurately tested. There always re- mains some doubt, therefore, whether, without the blurring effect of sampling error, the hierarchy would continue to be near enough to perfection. But these details should not be allowed to obscure the simplicity of the main argument. The important point to note is that the experimenter has produced a battery of tests which is, he claims, hierarchical; that the mathematician assures him that such a battery acts " as if" it had only one factor in common (though it can also be explained in many other ways), and that the psychologist, who may be the same person as the experi- menter, agrees that psychologically the existence of such a factor as the sole link in this battery seems a reasonable hypothesis. 3. Different hierarchies with two tests in common. Now, it must be remembered that, starting with three other tests, which may contain two of the former set, it may very well be possible to build up a different hierarchy. Only experiment could show whether this were possible in each case, there is no mathematical difficulty in the way. Such a hierarchy would also define " a " g, but this would be usually a different factor from the former g. If there were three tests common to the two hierarchies, then the two 's could be identified with one another (sampling errors apart), and the three tests would be found to have the same saturations with the one g as with the other. But if only two tests were common to the two batteries this would not in general be the case, and the different satura- tions of these tests with the two g's would show that the latter were different (Thomson, 1935a, 261-2). Under such circumstances the psychologist has to choose. He cannot have both these g's. Both are mathematically of equal standing, it is a psychological decision which has to be made. When one g is accepted, the other, as a factor, must then be rejected and a more complicated factorial analysis of the second hierarchy has to be built up which is consistent with this. A simple artificial example will THE DEFINITION OF g 229 illustrate this. Suppose that four tests give a perfect hierarchy of correlations thus : 1 2 3 4 1 1-00 72 63 54 2 72 1-00 56 48 3 63 56 1-00 42 4 54 48 42 1-00 On the principle that the smallest possible number of common factors must be chosen, the analysis of these tests would be Zl == -9g -\ 23 = -7g + Zt = -6g + Suppose now that Tests 2 and 4 are brigaded with two other tests, 5 and 6, in a new experiment, and that the correlations found are : 2 4 5 6 6 1-00 -48 -42 -54 48 1-00 -56 -72 42 -56 1-00 -63 54 -72 -63 1-00 This is also a perfect hierarchy, and the principle of parsimony in common factors leads to the analysis (B) But this analysis is inconsistent with the former, for the saturations of z z and # 4 with their common link have changed. If the factor g has been accepted as a psycho- logical entity, then the factor g' cannot be. To be con- sistent we must begin our equations for z % and * 4 in the 230 THE FACTORIAL ANALYSIS OF HUMAN ABILITY same manner as before, and although we may split up their specifics to link them with the new tests, the only link between them themselves must be g. We can then complete the analysis in various ways,* of which one is + -529150A + A/'36Z 4 + -463006/i + V'51J 5 + -595294A + V'19* 6 4. A test measuring "pure g." Although the hierarchical battery defines a g, it does not enable it to be measured exactly (but only to be estimated) unless either it contains an infinite number of tests, or a test can be found which conforms to the hierarchy and has a g saturation of unity .f In the latter case this test which is " pure g " is such that when it is considered along with any other two tests of its hierarchy, its correlations with them, multiplied together, give the intercorrelation of those two with one another : if k is the " pure " test, then its g saturation being i r ik r jk __ -^ r H No such '' pure " test of the g which is defined by the Brown -Stephenson hierarchy of nineteen tests has yet been found. Such a pure test, with full g saturation, must not be confused with tests which are sometimes called tests of pure g because they do not contain certain other factors, in particular the verbal factor. Thus the " S.V.PJ 99 * Four tests are insufficient as a defining battery for two common factors. f It is understood, of course, that even such a test would give different measures of a man's g from day to day, if the man's per- formance in it varied (as it undoubtedly would) from day to day. By measuring with exactness is meant, in this part of the text, measurement free from the uncertainty due to the factors out- numbering the tests. The reader is reminded that we are assuming sampling errors to be nil, the whole population having been tested, THE DEFINITION OF g 231 (Spearman Visual Perception) tests are referred to by Dr. Alexander (1935, 48) as a " pure measure of g " ; but their saturations with g are given by him (page 107) as 757, -701, and -736 respectively, so that in each case only about half the variance is " g." A possible alternative to the plan of first defining g and then seeking to improve its estimate would be to begin with three tests satisfying the relation r ik r jk ~ r ij which were reasonably acceptable as a definition of general intelligence, and give greater content to the psychological significance of this g by discovering tests which were hierarchical with these three. The lack of an exact measure of what is at present called g is a serious practical defect. Another possible way of remedying this will be referred to below in connexion with what are there called " singly conforming " tests. First, however, let us con- sider the case where three tests are such that 5. The Heywood case. In such a case the g saturation of the test k, if we calculate it, is greater than unity, which is impossible. Yet it is possible, in theory at least, to add tests to such a triplet to form an extended hierarchy with zero tetrad-differences. There can be one such case (but only one) in a hierarchy. Wo shall call them Heywood cases, as this possibility was first pointed out by him (Heywood, 1931). As an artificial example consider these correlations : 1 2 3 4 5 1 1-000 945 840 735 630 2 945 1-000 720 630 540 3 840 720 1-000 560 480 4 735 630 560 1-000 420 5 630 540 480 420 1-000 This is a perfect hierarchy, every tetrad-difference being exactly zero. It is, moreover, a perfectly possible set of correlations, and passes the tests required for a matrix of correlations to be possible. For example, the determinant of the matrix is positive (see Chapter IV ? Section 3, page 282 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 58). But when we calculate the g saturations of the tests we find them to be : Test g saturation 1-05 so that a single general factor is an impossible explanation of this hierarchy as far as Test 1 is concerned. The correlations of Test 1 with the other tests are possible, and they give exactly zero tetrad-differences : but yet the test cannot be a " two-factor " test, for the correlations of the first row are too high to be explained in that way. We might well have possessed the hierarchy of Tests 2, 3, 4, and 5 first, before we discovered Test 1. We should then have analysed these four as follows in a two-factor analysis 2 -600*3 Z 5 = -6g + -800*5 We then, let us suppose, discover Test 1, with its impossible g saturation. We want to retain the above analysis for the other tests. Now can we analyse Test 1 to explain its correlations with them ? We can do so in several ways. If we give it arbitrarily the loading -955 for g, we must use the specific of each test to give the additional correlation required. We thus arrive at the following possible but complicated analysis of Test 1 z l = -955g + -196*2 + -127* 3 + -093* 4 + -071* 5 + -141*! Here Test 1 is seen as containing each of the specifics of the four other tests, and only a small specific loading of its own. We have used up nearly all its variance in ex- plaining the correlations. Clearly there must be a limit to this process. If another test were added to the hier- archy, we might entirely exhaust the available variance of Test 1 in explaining its correlations. Or, indeed, the reader might add, we might more than exhaust it, and prove the impossibility of adhering to the pre-existing analysis. But this is not so. Such a test would only prove the impossibility of its own existence, if we may make THE DEFINITION OF g 233 an Irish bull. Suppose, for example, a Test 6 were to turn up with the correlations : 12345 6 -882 -756 -672 -588 -504 Such a test, when brigaded with Tests 2, 3, 4, and 5, would be given the analysis If now we use even the whole specific of this test as a link with Test 1, we cannot explain the correlation *882. We would need for that a loading of -150 for S Q in Test 1, and we have not enough variance left in Test 1 for this. But when this happens, we find that we have allowed the matrix of correlations to become an impossible one. If we add Test 6 to our matrix and calculate its determinant, we find it negative, which cannot occur in practice. The Test 6 could not occur, if the previous five tests already existed. Or vice versa, if Tests 2-6 existed, the Heywood case given would be impossible. The rule governing its possible existence has been given by Ledermann, namely, that the g saturation of the Heywood case cannot exceed + S S where S is the quantity familiar from Spearman's formula 2 S =2 -. r -<f- i - V for the remainder of the hierarchy (i = 2, 3, 4 . . .). If, then, we have a large hierarchy, we shall find it impossible to discover a test which conforms to it and which at the same time has a g saturation greater than unity. If we have a small hierarchy containing a Heywood case, we shall find it impossible to discover many tests to add to it, except indeed by the formal device of adding tests which do not correlate with it at all. All these considerations make it appear likely that if a Heywood test can be found to conform to a hierarchy, the g defined by that hierarchy must be abandoned. The seeker for a test for pure g is thus in a delicate position. He wants to find a test with F.A. 8* 234 THE FACTORIAL ANALYSIS OF HUMAN ABILITY full saturation of unity. But he must just hit the mark. If the saturation exceeds unity, his whole hierarchy must be abandoned as a definition. And even when the exact saturation of unity has been found, there seems to be too narrow a line dividing the perfect from the impossible, and the reality of the g seems to be balanced on a knife edge. In actual practice, of course, sampling errors would make the situation less acute and could for some time be called in to explain a certain amount of excess saturation over unity, 6. Hierarchical order when tests equal persons in number. If a test cannot be found whose saturation with g is unity (" pure g "), the other method of measuring g exactly would seem to be to extend the hierarchy until it comprised so many tests that the multiple correlation with g r - V s + 1 became practically unity. For S increases with the number of tests, being the sum of the positive quantities v i - V There is here a point of some theoretical interest, namely, what happens when we have increased the number of hierarchical tests until they are as numerous as the persons to whom they are given ? This, in view of the difficulty of finding tests to add to a hierarchy, is admittedly not a question likely to trouble experimenters, but its theoretical implications are considerable. It can be shown that whenever we have a matrix of correlations based upon the same number of tests as persons, its determinant is zero. Now the determinant of a hierarchical matrix (with unity in each diagonal cell) can be shown to be of the form (i - iy )(i - v)(i - v)(i - v) + v (* - VXi ~ VX 1 ~ V) + (i - ry) v (i ~ VX 1 ~ V) + (i - ry)(i - v) v (i - v) . . . THE DEFINITION OF g 235 and it is clear that each of these quantities is positive unless we have a case of pure g, or a Hey wood case. A case of pure g will leave one of the rows of the above sum non-zero. To make the whole sum zero, one case must be a Heywood case, giving 1 r ig * negative. It would seem, therefore, that by the time we have added hierarchical tests to make them equal in number to the persons, we will necessarily have added a Heywood hierarchical case (of which there can be only one in a hierarchy). But we have agreed that the discovery of a Heywood case will cause us to abandon the hierarchy as a definition of g I Mathematically this seems to mean that although the quantity S increases with each new test, provided it is not a Heywood case, yet S does not increase indefinitely, and the multiple correlation does not converge to perfect correlation. The case discussed above, where the number of tests is increased to equal the number of persons, may seem to the reader to be an academic case only. But the case of reducing the number of persons until they equal the number of tests is one which could easily be realized in practice, and presents equal theoretical difficulties. This draws at- tention from a new point of view to what has already been emphasized in Part III, the dependence of any definition of factors on the sample of persons tested. If we have a perfect hierarchy of, say, 50 tests, in a popula- tion of, say, 1,000 persons, and we reduce the number of persons by discarding some at random, it is, of course, to be expected that the correlations will change, and the hierarchy become disturbed. It would, however, at first sight appear possible to discard them so skilfully as not to disturb the hierarchy, or at least not disturb it much. But it would seem from the above considerations that try as we might, we could not, as the number of persons decreased towards fifty, prevent the correlations changing so as to give us a Heywood case, if we clung to hierarchical order. Or to put the same point in another way : a 236 THE FACTORIAL ANALYSIS OF HUMAN ABILITY sample of fifty persons from the above thousand, if it gives hierarchical order, will give a Hey wood case, and its g will be impossible. If the g corresponding to the original analysis on the thousand persons were anything real, such as a given quantity of mental energy available in each person, then it ought always to be possible, one might erroneously think, to find fifty persons and fifty tests to give a hierarchy, without a Hey wood case. But that cannot be easily said. It is impossible, from the correlations alone, to distinguish a real g from one imitated by a fortuitous coincidence of specifics. Even if g were a reality, a sample of persons equal in number to the tests could not give a hierarchy without a Heywood case, and their apparent g would be fortuitous. Now the case of a test of pure g is on the border line of the Heywood cases. It is clear then that it will be suspect, as being probably only fortuitous, if the number of persons does not far exceed the number of tests. 7. Singly conforming tests. There remains one other conceivable method of measuring g exactly,* by the use of certain tests which, when they are all present, destroy the hierarchy, although any one of them can enter the battery without marring it " singly conforming " tests (Thomson, 1934c ; and 19350, 253-6). It will be remem- bered from the chapters on estimation that the reason factors cannot be measured exactly, but have to be esti- mated only, is that they outnumber the tests. Every new test which conforms to a hierarchy adds a new specific (unless it is pure g) 9 and thus continues the excess of factors over tests. It can occur, however, that the correlation of two tests with each other breaks a hierarchy, although either of them alone conforms otherwise. Such a case occurs in the Brown-Stephenson battery, for example, one of whose correlation coefficients has to be suppressed before the hierarchy is acceptable. In such a case, if the psychologist is prepared to accept * By " exactly " is meant, with the same exactness as the test scores, without the additional indeterminacy due to an excess of factors over tests. THE DEFINITION OF g 237 either test as a member of the battery, the erring correlation coefficient must be due to these two tests sharing some portion of their specifics with one another. If, as may happen (apart from error which we are supposing absent), their intercorrelation shows that they have only one specific factor between them, and differ only in their saturations, then they enable the estimate of g to be turned into accurate measurement. For example, consider the following matrix of correlations : 1 2 3 4 5 6 1 669 592 458 335 251 2 669 . 566 438 870 240 3 592 566 . 387 283 212 4 458 438 387 , 219 164 5 335 870 283 219 120 6 251 240 212 164 120 . This is a perfect hierarchy except for the correlation r 25 = -870 Every tetrad-difference, which does not contain this correlation, is zero. If either Test 2 or Test 5 is removed from the battery, there remains a perfect hierarchy. If Test 5 is removed, we can calculate from the remaining battery the g saturations : Test 1234 6 837 800 707 548 300 1 837 3 707 4 548 5 400 300 g saturation If we remove Test 2 and restore Test 5, we get the fol- lowing : Test g saturation From either hierarchy we can estimate g. The correla- tion of our estimates with " true g " will be S S + 1 saturation 2 saturation 14 where S = 238 THE FACTORIAL ANALYSIS OF HUMAN ABILITY and we find for the two hierarchies the g correlations of 92 and -90. Suppose now that we had left both Tests 2 and 5 in the battery with which to estimate g, after calculating their g saturations from the two separate hierarchies, what in- fluence would this have had upon the accuracy of our estimate ? It is of some interest actually to carry out this calculation by Aitken's method, using all the tests with the g saturations given above. A calculation keeping three places of decimals gives for the regression coefficients : Test Regression coefficient 005 1-856 -003 -001 -1-213 -002 which suggests (what would actually be the case if more decimals were retained throughout) that all the regression coefficients except those for Tests 2 and 5 vanish. If we calculate the multiple correlation of this battery with g, by finding the inner product of the g saturations with the above regression coefficients, we find that it is exactly unity. The reason for this is that the correlation of Tests 2 and 5 is such as to show that their specifics are identical, the two tests differing only in their loadings. Their equations are If the whole of s 2 is identical with the whole of s 59 their intercorrelation should bo 8 X -4 + V(l T 8 a 7(l~~4 a ) = -870 and this is its experimental value. We could, therefore, have seen at the beginning, if we had tested the above fact, that these two tests would make a perfect battery for measuring g. We have the simul- taneous equations * 2 = *g + -6s z, - -4 + -9175 THE DEFINITION OF g 239 from which we can eliminate s by multiplying by 917 and 600 respectively, numbers which are exactly in the ratio of the regression coefficients found above 1-856 and 1-213. In fact, we could have performed the regression calcula- tion on these two tests alone, when it would have appeared as follows : 1-000 870 800 870 1-000 400 -1-000 -1-000 870 870 1-200 (4 1135) 2431 8700 -1-000 1131 1 -0000 -2960 3-5787 -8000 4-1135 4652 5040 1-8593 1-2176 6417 giving (more exactly) the same regression coefficients as before. We see, therefore, that under certain hypothetical circumstances, a more exact estimate of g can be obtained from two of these " singly conforming " tests than the hierarchy with which they conform individually. Those circumstances are, that their correlation with one another (the correlation which breaks the hierarchy because it is too large) should either equal or should approach this value. It cannot in actual practice be expected to equal it, as in our artificial example. For we have disregarded errors, which are sure in some measure to be present. At what stage will the pair of singly conforming tests cease to be a better measure of g than the better of the two hierarchies made by deleting either the one or the other ? If in our example the correlation -870 of Tests 2 and 5 be imagined to sink little by little, the correlation of their estimate with g will sink from unity. The better of the two hier- archies gives a multiple correlation of '922. When the 240 THE FACTORIAL ANALYSIS OF HUMAN ABILITY correlation r 25 has sunk from '870 to -847, these two singly conforming tests will give the same multiple correlation, 922. If this defect from the full -870 is due entirely to error, then a fall to '847 corresponds to reliabilities of the two tests of the order of magnitude of -98, if they are equally reliable. This is a very high reliability, seldom attained, so that in a case like our example quite a small admixture of error would make the singly conforming tests no better at estimating g than the hierarchy. We are here, however, neglecting the fact that error would also diminish the efficiency of the hierarchy. Nevertheless, the chance of finding a pair of singly conforming tests, highly reliable, and having no specifics except that which they share, seems small, as small as the chance of finding a test of pure g, perhaps. It might possibly turn out, however, that a matrix of several (say t) singly conforming tests would be practicable. Such a set would measure g exactly if among them they added only t 1 new specifics to the hierarchy. Their saturations would be found by placing them one at a time in the hierarchy, and then their regres- sion on g calculated by Aitken's method. The necessity for the hierarchy in the background, in all this, is clear : it is there to assure us that each singly conforming test is compatible with the definition of g, and to enable its g saturation to be calculated. 8. The danger of" reifying "factors. The orthodox view of psychologists trained in the Spearman school is that g is, of all the factors of the mind, the most ubiquitous. " All abilities involve more or less g," Spearman has said, al- though in some the other factors are " so preponderant that, for most purposes, the g factor can be neglected." With this view, the present author has always agreed, provided that g is interpreted as a mathematical entity only, and judgment is suspended as to whether it is any- thing more than that. The suggestion, however, that g is " mental energy," of which there is only a limited amount available, but avail- able in any direction, and that the other factors are the neural machines, is one to be considered with caution. The word energy has a definite physical meaning. " Mental THE DEFINITION OF g 241 energy " may convey the meaning that the energy spoken of is the same as physical energy, though devoted to mental uses. If that meaning is accepted, innumerable difficulties follow, not the least being the insoluble questions of the connexion of body and mind, and of freewill versus determinism. A less obscure difficulty is that there seems to be no easily conceivable way in which the " energy " of the whole brain can be used in any direction indifferently, except by the " neural engines " also all taking part. The energy of a neurone seems to reside in it, and the passage of a nerve impulse along a neurone seems to resemble rather the burning of a very rapid fuse, than the conduction of electricity, say, by a wire. If " mental energy " does not mean physical energy at all, but is only a term coined by analogy to indicate that the mental phenomena take place " as if " there were such a thing as mental energy, these objections largely disappear. Even in physical or biological science, the things which are discussed and which appear to have a very real existence to the scientist, such as " energy," " electron," " neutron," " gene," are recognized by the really capable experimenter as being only manners of speech, easy ways of putting into comparatively concrete terms what are really very abstract ideas. With the bulk of those studying science there exists always the danger that this may be taken too literally, but this danger does not justify us in ceasing to use such terms. In the same way, if terms like " mental energy " prove to be useful, and can be kept in their proper place, they may be justified by their utility. The danger of " reifying " such terms, or such factors as g, v, etc., is, however, very great, as anyone realizes who reads the dissertations produced in such profusion by senior students using these new factorial methods. CHAPTER XVI " ORTHOGONAL SIMPLE STRUCTURE " 1. Simultaneous definition of common factors. In a sense, Thurstone's system of multiple common factors is a generalization of the original Spearman system which had only one. It recognizes that matrices of correlation coefficients are not usually reducible to rank 1, but that they are usually reducible to a low rank, and it replaces the analysis into one common factor and specifics by an analysis into several common factors and specifics, keeping the number of common factors at a minimum. It does not lay the great stress on the ubiquity and domin- ance of g which is found in the Spearman system. Spearman's system, having defined g as well as possible by an extended hierarchy, goes on then to definitions of the next most important factors, by similar means. It looks upon any complex matrix of correlations as being due to lesser hierarchies superimposed upon the g hierarchy. Moving in accordance with a very commonly held belief which almost certainly has some justification, it has sought and found " verbal " and " practical " factors to add to g, and is groping for some kind of character or emotional factor which would complete the main picture. " One at a time " has been its motto. Moving along another route, Thurstone has endeavoured to define several factors by one matrix of correlations. Although the campaign of the Spearman school was presumably the only method open to pioneers, a student must be struck by the fact that the standard definition of g is made by a battery of tests (Brown and Stephenson, 1933) which is not really reducible to rank 1 until a large verbal factor has been removed by mathematical means. Just as a battery to define g has to be purified either by the actual removal of tests or by the mathematical removal of factors before it is suitable as such a definition, so not 242 "ORTHOGONAL SIMPLE STRUCTURE" 243 every battery will define a group of common factors, Thurstone batteries, like Spearman's, have to be composed of selected tests, and purified if the selection is not com- plete ; and batteries which give incompatible analyses are conceivable. 2. Need for rotating the axes. Actually, experimenters have first assembled a number of tests which appeared to them to be likely to contain only, say, r common factors, factors which they have already suspected to exist and have tentatively named. They have then ascertained by using Thurstonc's approximate communalities whether a reduced rank of r can be achieved as a sufficiently close approximation, by examining the residues after r factors have been " taken out." By analogy with Spearman's purification process, they might then remove any tests which were preventing this ; but such purification has not been very usual though it seems just as justifiable here as in a hierarchy. Let us suppose that a battery, assembled because it appeared, psychologically, to contain r common factors, does give a matrix which can be reduced to rank r. As was explained towards the end of Chapter II, the loadings given by the " centroid " process then include a number of negative values, and these the psychologist has difficulty in accepting in any large numbers. For it is hard for him to conceive of psychological factors which help in some tests and hinder in others, except in rare cases. The mathematician ran then " rotate " the factor axes within the common-factor space (Thurstone's principle forbids him to go outside it) in search of a position which will satisfy the psychologist. One way of doing this has already been sketched in Chapter II, Section 8. It has been used with excellent effect by W. P. Alexander (Alexander, 1935), but involves assuming (a) that the com- munality of a certain test is entirely due to one factor ; (b) that the communality of a second test is entirely due to this factor and one other ; (c) and so on for r 1 tests, where r is the number of factors. The criterion of success with this method is to see whether, when these assumptions are made, negative loadings disappear ; and whether the 244 THE FACTORIAL ANALYSIS OF HUMAN ABILITY consequent loadings of those tests about which no assump- tions are made are compatible with the psychologist's psychological analysis of them. It cannot be too emphati- cally pointed out that the first factors which emerge from the " centroid " process and the minimum-rank principle need not have psychological significance as unitary primary traits. It is only after rotation to a suitable position that this can be expected. 3. Agreement of mathematics and psychology. It becomes increasingly clear that the whole process is one by which a definition of the primary factors is arrived at by satisfying simultaneously certain mathematical principles and certain psychological intuitions. When these two sides of the process click into agreement, the worker has a sense of having made a definite step forward. The two support one another. Obviously the goal to be hoped for along this line of advance will be the discovery of some mathematical process which always leads to a unique set of factors mainly acceptable to the psychologist. If such could be dis- covered and found to produce a few factors over and above those recognized as already known by other means, the new factors would stand a good chance of acceptance on the strength of their mathematical descent only. And no doubt the psychologist would be prepared to make a few concessions and changes in his previous ideas to fit in with any mathematical scheme which already gave much satisfaction and was objective and unique in its results. It is here that Thurstone's notion of " simple structure " is offered as a solution ( Vectors, Chapters 6-8). This idea is that the axes are to be rotated until as many as possible of them are at right angles to as many as possible of the original test vectors ; and that the battery is not suitable for defining factors unless such a rotation is uniquely possible, a rotation which will leave every axis at right angles to at least as many tests as there are factors, and every test at right angles to at least one axis. When the vectors of a test and a factor are at right angles, the loading of the factor in that test is zero. Thurstone's " simple structure " is therefore indicated by " ORTHOGONAL SIMPLE STRUCTURE 1 * 245 a large number of zeros in the matrix of loadings, so large that there will be only one position of the axes (if any) which satisfies the requirement. His search, be it repeated, is for a set of conditions which will make the solution unique. We have seen him approaching this goal by stages. Unless the battery is large, so that n > (see Chapter II, Section 9), the communalities are not unique. Even when the battery is large enough, the axes representing factors may be rotated to positions among which there is no one specially marked out. Then comes the demand that there be this large number of zero loadings. Most batteries of tests will not allow this demand to be satisfied, but with some it can just be attained. Only these last, it is Thurstone's conviction, are suitable for defining primary factors, and it is his faith that the factors thus mathematically defined will be found to be acceptable as psychologically separable unitary traits. 4. An example of six tests of rank 3. To make our remarks more definite and concrete, let us suppose that we have a battery of six tests whose matrix of correlations can be reduced to rank 3. In practice, of course, six tests are far too few, and more than three factors quite likely. The matrix of loadings given by the " centroid " system contains at first negative quantities. Thus from the correlations : 1 2 3 4 5 6 1 . 525 000 000 448 000 2 525 . 098 306 349 000 3 000 098 , 133 314 504 4 000 306 133 . 000 000 5 -448 349 314 000 . 307 6 000 000 504 000 307 . with the communalities 674 -634 -558 -415 -490 -493 246 THE FACTORIAL ANALYSIS OP HUMAN ABILITY we get by the " centroid " process the matrix of loadings : 1 2 3 4 5 6 -542 -612 -074 629 -342 -348 529 -492 -191 281 -182 -550 628 -143 -274 429 -424 -359 It is the factor axes indicated by these loadings that Thurstone wishes to rotate until there are no negative loadings and enough zero loadings to make the position uniquely defined. For this last purpose he finds, empiri- cally, that it is necessary to require (a) At least one zero loading in each row ; (b) At least as many zero loadings in each column as there are columns (here three) ; and (c) At least as many XO or OX entries in each pair of columns as there are columns. By an XO entry is meant a loading in the one column opposite a zero in the other. " At least one zero loading in each row." This means that no test may contain all the common factors. In making up the battery, then, the experimenter, with some idea in his mind as to what the factors are, will endeavour to ensure that they are not all present in any one test. This would, for example, exclude from a Thurstone battery any very mixed group test, or a mixed test like the Binet- Simon which is itself a whole battery of varied items. " At least as many zeros in each column as there are columns," that is, as there are common factors. This means that in a Thurstone battery no factor may be general, but must be missing in several tests. The requirement as to the number of XO or OX entries is intended to ensure that the tests are qualitatively distinct from one another. Now, these requirements cannot generally be met by a matrix of loadings. It will in general be impossible to rotate the axes (keeping them orthogonal) until every axis is at right angles to r test vectors. The above "ORTHOGONAL SIMPLE STRUCTURE 24? example has, however, been constructed so that this can be done. The correlations were in fact made from the loadings : ABC 1 2 3 4 5 6 718 -438 702 475 206 644 821 639 546 n and the centroid loadings must therefore be capable of being rotated rigidly into this form, retaining ortho- gonality. 5. Two-by-two rotation. The problem for the experi- menter, however, is to discover this " simple structure," if it exists ; he is not, like us, in the position of knowing that it does exist, and what it is. Thurstone's original method was to make a diagram of two of the centroid factors, and rotate them ; then to make other diagrams of two factors at a time, and rotate them, in each rotation endeavouring to obtain some zero loadings. Let us illustrate by our artificial example, taking first the centroid factors I and II. Using their centroid loadings as co- ordinates, we obtain Figure 25, where each test is represented by a point, and the centroid axes by the co-ordinate axes marked I and II. At once we notice Figure 25. that the test-points 3, 4, and 6 are almost collinear on a radius from the origin, and that if we rotate the axes clockwise through about 42 the new position of I, labelled I t in the diagram, will almost pass through these test-points, while the new axis IIx will almost pass through test-point 1. On these new axes, therefore, Tests 3, 4, and 6 will have hardly any projections on axis Hi ; that is, will have hardly any loadings in a factor along II x . The 248 THE FACT OHIAL ANALYSIS OF HUMAN ABILITY new co-ordinates of the test-points on I x and II i could be measured on the diagram, and the reader is advised, in doing rotations, always to make a diagram and find the approximate new loadings thus, as this furnishes an ex- cellent check on the arithmetical calculation. This arithmetical calculation is based on the fact that if a test-point has the co-ordinates x and y with reference to the original centroid axes I and II, its new co-ordinates on the rotated axes Ii and Hi will be x cos 42 y sin 42 on I x and x sin 42 + y cos 42 on 11^ From tables we find sin 42 = -669, and cos 42 = -743, and the calculation can be done readily on a machine, or with Crelle's multipli- cation tables. We have then : Old loadings I II New loadings 1 542 -612 007 -817 2 629 -342 239 -675 3 529 492 722 012 4 281 -182 331 -053 5 628 -143 371 -526 6 429 424 602 --028 multipliers *743 669 for I t loadings, 669 -743 for II t loadings. At this point a check should be made by seeing that the sum of the squares of the new pair of loadings is identical with the sum of the squares of the old pair, for each test. We have now obtained our desired three zero (or near zero) loadings in factor II j. Accepting the ap- proximations to zero as good enough for the present, we next make Figure 26 from the loadings of I x and III in the same way as we made the former figure. In this, Test 1 falls quite near the Figure 26. origin. Tests 5 and 6 are ap- "ORTHOGONAL SIMPLE STRUCTURE" 249 proximately on one radius, and Tests 2 and 4 on another, and these radii are at right angles to one another. If we rotate the axes I x and III rigidly through a clockwise turn of about 49 they will pass almost through these radial groups and nearly zero projections will result.* Using sin 49 = 755 and cos 49 = -656 we perform a similar calculation to the preceding, using the loadings I and III as starting-point and obtaining loadings on I 2 and lilt (the subscript indicating the number of rotations that axis has undergone). We have finally, putting our results together, the table of loadings FA.f 1 2 3 4 5 6 060 -817 -043 420 -675 --048 329 -012 -670 632 -053 111 037 -526 -460 124 028 -690 Clearly, this is an approximation to the loadings of the factors A 9 B 9 and C which we who are in the secret (as a real experimenter is not) know to have been used in making the correlations : Illi here is A, I 2 here is B 9 and II! is C. The small loadings are not quite zero, and the other load- ings not quite the same, but a further set of rotations would refine the results and bring them nearer to the ABC values. 6. New rotational method.- When this two-by-two rota- tional method is used on a large battery of tests, with perhaps six or seven factors instead of three, it is not only laborious but somewhat difficult to follow. Thur- stone has, however, devised a method of rotation which takes the factors three at a time, and to this we now turn, still using our small artificial example as illustration. In * The rotation might with advantage have been carried a little further. t The matrix symbols, using Thurstone's notation, are given for the convenience of mathematical readers. Others should ignore them. 250 THE FACTORIAL ANALYSIS OF HUMAN ABILITY this example, since there are only three factors, this new method leads to a complete solution at once. With more factors the matter would be more complicated. If the reader will think of the three centroid factors as represented by imaginary lines in the room in which he is sitting (Figure 27), he will be aided in following the explanation of this newer method. Imagine the first m Figure 27 (not to scale). centroid axis to be vertically in the middle of the room, and the other two centroid axes on the carpet, at right angles to the first and to each other. The test-points are in various positions in the room space, if we take their three centroid loadings as co-ordinates and treat the distance from floor to ceiling as unity. Imagine each test-point joined by a line to the origin (in the middle of the carpet, where the axes cross). The lengths of these lines are the square roots of the communalities, and the loadings on the first centroid factor are their projections on to the vertical axis, the height, that is, of each test-point above the floor. Thurstone now imagines each of these lines or com- munality vectors produced until it hits the ceiling, making a pattern of dots on the ceiling. These extended vectors now all have unit projection on the first centroid axis, for we agreed to call the distance from floor to ceiling unity. Their y and z co-ordinates on the ceiling will be Correspondingly larger than their loadings on the second "ORTHOGONAL SIMPLE STRUCTURE 251 and third centroid factors, and can be obtained by dividing each row of the centroid loadings by the first loading. In our case this gives us the following table, obtained in the manner just mentioned from the table on page 246. Ext ended centroid projections I e II e III e 1 1-000 1-129 137 2 99 544 553 3 99 930 361 4 99 648 -1-957 5 99 228 436 6 99 988 837 He The second and third columns are now the co-ordinates of those dots on the ceiling of which we spoke. A diagram of the ceiling, seen from above, is given in Figure 28, and the important point about it is that the dots form m a a triangle. If the reader will now picture this triangle as drawn on the ceiling of his room, and remember that the origin, where the centroid axes crossed, is in the middle of the carpet, he can next imagine an inverted three-cornered pyramid, with the triangle on the ceiling as its base, the origin in the middle of Figure 28. the carpet as its apex and the communality vectors 1, 4, and 6 as its edges. The vector 5 lies on one of the faces of this pyramid ; vector 2 lies on another ; vector 3 lies on the remaining face, all springing from the origin and going up to the ceiling. 7. Finding the new axes. If now we choose for new axes (in place of the centroid axes) three lines at right 252 THE FACTORIAL ANALYSIS OF HUMAN ABILITY angles respectively to the three plane faces of our pyramid, the test projections on these axes will clearly have the zeros we desire. The three vectors 1, 2, and 4 all lie in one face, and will have zero projections on the axis A' at right angles to that face. The vectors 1, 5, and 6 will have zero projections on the line B' at right angles to their face. The vectors 3, 4, and 6 will have zero projections on C" at right angles to their face. The reader should visualize these new axes in his room. It remains to be shown how the other, non-zero, projections are to be calculated, and to inquire whether these new axes are orthogonal, and whether they can be identified with the original A, B, and C, The first step is to obtain the equa- tions of the three sides of the triangle in the diagram. Where there are many tests and the dots are not perfectly collinear, one plan is to draw a line through them by eye, and measure the distances a and b it cuts off on the axes, then using the equation Or we can write down the equations of the lines joining points at the corners, either actual test points, or the places where our lines intersect, using the equation (Iv mu) -\- (m v) y + (u I) z Q when /, m are the co-ordinates of one corner, and u, v of another. We obtain in our case 2-121 + 2-094*/ - l-777a = for line 1, 2, 4 - 1-080 + -700i/ + 2-1172 = 1, 5, 6 2-476 + 2-794y + -340* = 4, 3, 6 where y means the extended II, and z the extended III. Before we go further we have to divide each equation through by the root of the sum of the squares of its coefficients, so that the new coefficients sum to unity when squared this is called normalizing and is necessary in order to keep the communalities right and for other reasons. The equations then are : -611 + -603*/ -5122 = (1) -436 + -2S3y + -854* = (2) 660 + -745t/ + -091s = (3) ORTHOGONAL SIMPLE STRUCTURE" 253 and it is clear, from the way in which they have been reached, that these equations will be satisfied by the ex- tended co-ordinates of certain of the rows in the table on page 251. Consider the first equation and write its co- efficients above the columns of that table, placing '611 over the first column, thus : 611 603 512 Weighted X y z sum 1 1-000 1-129 137 000 2 99 544 -553 000 3 99 -930 361 -1-357 4 99 648 1-957 000 5 99 228 436 697 6 99 -988 837 -1-635 If we multiply each column by the multiplier above it and add the rows we get the quantities shown on the right. The zeros are in the right places for factor A. The other loadings are, however, negative (that can be easily put right by changing all the signs of the multipliers, which we are at liberty to do) and are too large, because, of course, it is the extended loadings which have been used. We must multiply each of them by the original first centroid load- ing of the row, since at an earlier stage we divided by it. If we do so, we find that we get exactly the loadings of column A of the table originally used to make the corre- lations. Thus : 1-357 X -529 = -718 697 X -628 = -438 1-635 X -429 = -701 the difference from -702 being due to rounding off only. Or more simply, we could have applied our multipliers to the centroid loadings themselves, not to the extended projections. The zeros will obviously remain, and for the other loadings we obtain at once those of factor A. Simi- larly, using eqns. (2) and (3) we get the loadings of factors B and C exactly, except for an occasional difference due to 254 THE FACTORIAL ANALYSIS OF HUMAN ABILITY rounding off at the third decimal place. We have, indeed, found the matrix product FA, 542 -612 -074 629 -342 --348 529 492 -191 281 --182 --550 628 -143 -274 429 424 -359 611 -436 -660 603 --283 -745 512 --854 -091 . -821 . -475 -639 718 -206 . . -644 . 438 . -546 702 . except, as has been already said, for occasional dis- crepancies in the third decimal place. The procedure we have described has enabled us to discover this last matrix, with which in fact we began. And by analogy (is the deduction sound ?) an experimenter with experimental data who follows this procedure and reaches simple structure concludes that that is how his correlations were made. Certainly that is how they may have been made. The matrix A beginning with -611 is the rotating matrix which turns the axes I, II, III into the new posi- tions A 9 B, C. Its columns are the direction-cosines of A, B, and C with reference to the orthogonal system I, II, III. Are A 9 B, and C orthogonal ? The cosines of the angles between them can by a well-known rule be found by premultiplying the rotating matrix by its transpose. When we do so we find A'A I, viz. : 611 603 -512 436 283 -854 660 -745 -091 611 -436 -660 -603 --283-745 512 --854 -091 (again allowing for third decimal place discrepancies). That is to say, the angles between A, B, and C have zero cosines, they are right angles. The axes A, B, and C, were drawn at right angles to the three planes which form the pyramid mentioned above, and therefore these three planes are also at right angles to one another. (Our rough sketch in Fig. 27 made the pyramid too acute.) It follows that A, B, and C are actually the edges of the pyramid. In our example (though this need not be the case) they happen to pass each through a "ORTHOGONAL SIMPLE STRUCTURE" 255 test-point in the room, A through Test 6, B through Test 4, and C through Test 1. These tests are not identical with the factors, for each test contains a specific element, not in the common-factor space, but at right angles to it. What we have called a test-point is the end of the test vector projected on to the common-factor space. The complete test-vectors are out in a space of more dimensions, of which the three-dimensional common-factor space is a subspace. 8. Landahl preliminary rotations. When there are more than three centroid factors, the calculations are not so simple. If the common-factor space is, for example, four-dimensional, then the table of extended vectors, in addition to its first column of unities, will have three other columns. The two-dimensional ceiling of our room, in our former analogy, has here become three-dimensional, a hyper-plane at right angles to the first centroid axis. On paper its dimensions can only be graphed two at a time, and no complete triangle will be visible among the dots. But sets of dots will be seen to be collinear, lines can be drawn through them, and a procedure similar to that out- lined above followed. This will become clearer when we work a four-dimensional example. First, however, it is desirable to explain, on our simple three-dimensional example, a device which facilitates the work on higher dimensional problems, called the Landahl rotation. It is unnecessary in the three-dimensional case, and we are using it only to explain it for use with more than three dimensions. A Landahl rotation turns the centroid axes solidly round until each of them is equally inclined to the original first centroid axis. In our imagined room the first cen- troid axis ran vertically from the middle of the floor to the middle of the ceiling, while the other two were drawn on the floor itself. Imagine all three (retaining their orthogonality) to be moved, on the origin as pivot, until they are equally inclined to the vertical. That is a Landahl rotation. The lines through the test-points have not moved. They remain where they were, and still hit the ceiling in the same pattern of dots. The projections of the extended vectors on to the original first centroid 256 THE FACTORIAL ANALYSIS OF HUMAN ABILITY axis all still remain unity. But for the next step in this method we need their projections on to the Landahl axes. We obtain these by post-multiplying the matrix of een- troid extended loadings by a Landahl matrix, an orthogonal matrix with each element in its first row equal to j- 9 . . ^ c where c is the order of the matrix ; that is, its number of rows or columns (Landahl, 1938). We need a Landahl matrix of order 3, for example : 577 -577 -577 816 --408 -408 000 -707 707 The element *577 is the cosine of the angle which each axis makes, after rotation, with the original position of the first centroid axis. When the table of extended vector projections on page 251 is post-multiplied by the above matrix, the following table results, giving the projections of the extended vectors on to the Landahl axes L, M, N : Projections on Landahl axes L M N 1 1-498 213 020 2 1-021 066 746 3 182 1-212 701 4 048 542 2-225 5 760 792 176 6 229 1-572 388 From this table three diagrams LM 9 LN, and MN can be made, and the reader is advised to draw them. Each of them shows a triangular distribution of dots and in this simple three-dimensional example only one of them is needed. But in a multi-dimensional problem several are needed, and usually only one line is used on each diagram "ORTHOGONAL SIMPLE STRUCTURE 5 * 257 employed. Here, from the LN diagram we find the equations of the three sides of the triangle to be : 2-2051 l-450n + 3-332 = 368Z + l-72Tn -586 = 1-837/ -277n + -528 = We want to make these homogeneous in Z, m, and n, and so we add, after each of the numerical terms, the factor 577 (/ + m + n), which equals unity. The equations then are : 282Z + l-923ra + -473n = 030Z -338m + l-389n = 2-142Z + -305m + -028n == After normalizing these become : 141Z + -961m + -236n = 021Z -236m + -971n = - -990Z + -141m + -013/1 Writing the coefficients as columns in a matrix, and premultiplying by Landahl's matrix (since at an earlier stage we post-multiplied by it) we obtain 609 -436 -660 -603 -283 -745 513 853 -090 the same matrix A as we arrived at (page 254) without the use of Landahl's rotation. The advantage of using a Landahl rotation appears only in problems with more than three common factors. The reader can readily make a Landahl matrix of any required order, say 5. Fill the first row with the root reciprocal of 5, '447. Complete the first column by putting in the second place -894, (because -447 2 + -894 2 = 1) and below that zeros. The second row must then be completed with equal elements, all negative, such that the row sums to zero. Then the second column is completed in a similar way, and the third row, and so on. The reader should finish it. There are alternative forms possible, one of which is used below. 258 THE FACTORIAL ANALYSIS OP HUMAN ABILITY An unfinished Landahl matrix : 447 -447 -447 224 224 224 289 289 289 9. A four-dimensional example. The following example of a problem with four common factors is only partly worked out, so that the reader can finish it as an exercise. It also is an artificial example, and orthogonal simple structure can be arrived at. The centroid analysis gave four centroid factors with the loadings shown in this table : 447 447 894 224 000 866 000 000 000 000 Centroid loadings F I II III IV 1 727 517 094 126 2 575 105 553 049 3 810 289 246 246 4 588 417 367 -382 5 524 -583 450 183 6 549 435 398 013 7 624 318 -187 -254 8 594 .KK-1 \J*J JL 239 084 9 626 252 169 562 10 645 307 -357 109 After these have been " extended " (i.e. divided in each row by the first loading) they were post-multiplied by a Landahl matrix, one of the alternative forms, viz. : 5 5 5 5 5 5 5 5 5 5 5 .5 5 5 5 5 and the resulting projections on the Landahl axes were thus found to be : ORTHOGONAL SIMPLE STRUCTURE" 250 L M N P 1 1-007 704 122 166 2 1-115 068 848 030 3 679 678 625 018 4 218 1-492 158 132 5 311 199 453 1-660 6 455 247 1-270 522 7 -107 598 808 701 8 308 235 1-094 833 9 1-015 387 285 883 10 376 1-099 070 454 Six diagrams can be made, and it is advisable to draw them all, though not all are necessary. The LN diagram is shown in Figure 29. We scan it for collinear points (not necessarily radial) which have all or nearly all the other points on one side of their line, and note the line 5, 4, 10, 9. Its equation is readily found to be approximately 738/ + l-327n -371 = 0. We make this homogeneous by substituting for unity, after the numerical term -371, the quantity -5 (I + ra + n + p), for -5 is the cosine of the angle each of the Landahl axes makes with the original first centroid axis. This gives us the equation 5532 -185m + M41n -185^ = 0. Three more equations are needed, and one of them can indeed be obtained from the same diagram, on which points 5, 7, 8, 6 are very nearly collinear. The reader is advised to draw the remaining diagrams and complete the calculations following the steps of our previous example. The above equation refers to a line which makes a fairly big angle with N. It is desirable to look for the remaining three lines making large angles (approaching right angles) with L, M 9 and P. It will be remembered that in our earlier example the sign of one equation had to be changed at the end of the calculation because large negative values were appearing in the final matrix of loadings. This can be obviated 260 THE FACTORIAL ANALYSIS OF HUMAN ABILITY by attending to the following rule. If the other test-points are on the same side of the line as the origin the numerical term must be positive in the N 6 equation ; if they are on the * 8 side remote from the origin 2 the numerical term must be 3 negative. In the adjacent diagram, the origin and the ^ other points are on oppo- x *^ x * L site sides of the line through Vx ^x Q 5, 4, 10. 9 and therefore x.^ y x - v ^ the numerical term must be Figure 29. " negative, as it is (371). Had it been positive all the signs of the equation would have required to be changed. 10. Ledermanrts method of reaching simple structure. Ledormann has pointed out that when simple structure can be attained (whether orthogonal or oblique) then as many r-rowed principal minors of the reduced correlation matrix must vanish as there arc common factors ; and that it follows that the same number of vanishing deter- minants must be discoverable in the table of centroid loadings. Thus, for example, in the table of centroid loadings on page 246 the three determinants composed respectively of rows 1, 2, and 4 ; of rows 1, 5, and 6 ; and of rows 3, 4, and 6 all vanish, and these rows are where the zeros come in the three columns of the simple structure. This gives an alternative method of reaching simple structure. Test every possible r-rowed determinant in the centroid table of r factors. If r of them are discovered to vanish, then simple structure may be and probably is possible. Each of these vanishing determinants will provide a column of the rotating matrix A, for which pur- pose we delete any one of its rows and calculate all the r 1 rowed minors from what is left. The column has then to be normalized. This process works equally well for oblique simple structure (see Chapter XVIII). Its draw- back, when the number of factors is large, is the necessity of calculating so many determinants to discover those that vanish. "ORTHOGONAL SIMPLE STRUCTURE" 261 11. Leading to oblique factors. In this chapter we have kept our factors orthogonal ; that is, independent, un- correlated with one another. It is natural to desire them to be different qualities, and convenient statistically. In describing a man, or an occupation, it would seem to be both confusing and uneconomical to use factors which, as it were, overlapped. Yet in situations where more familiar entities are dealt with, we do not hesitate to use correlated measures in describing a man. For instance, we give a man's height and weight, although these are correlated qualities. Often, moreover, a battery of tests which will not permit simple structure to be reached if orthogonal factors are insisted on will nevertheless do so if the factors are allowed to sag away a little from strict orthogonality. Even as early as in Vectors of Mind, Thurstone expressly permitted this. It can clearly be defended on the ground that even if the factors were uncorrelated in the whole population, they might well be correlated to some extent in the sample of people actually tested. I was at one time under the impression that this comparatively slight de- parture from orthogonality was all that was contemplated by Thurstone. But lately he has assured me in correspond- ence that he arid his fellow-workers now have the courage of their convictions, and permit factors to depart from orthogonality as much as is necessary to attain simple structure, even if they are then found to be quite highly correlated. A chapter on these oblique factors* is there- fore necessary (Chapter XVIII), and out of them arise Thurstone's " second order factors." First, however, there is the chapter on " Limits to the Extent of Factors," retained unchanged from the first edition of this book, which, except in its last paragraph, deals only with orthogonal factors. * It must be clearly understood that this obliquity or correlation of factors is quite a different matter from the correlation of estimates, even of orthogonal factors, due to the excess of factors over tests, described on pages 120 to 129, CHAPTER XVII LIMITS TO THE EXTENT OF FACTORS* 1. Boundary conditions in general. Before we discuss further the question whether a given set of common-factor loadings can be rotated into " simple structure " it is desirable to consider a wider problem, in itself quite unconnected with Thurstone's particular theory of factors ; the problem, namely, of drawing conclusions from correla- tion coefficients as to what there is in common between tests, or other variates. From one correlation coefficient, if it is significant in proportion to its standard error, it is natural to assume that the variates share some causal factor, though that factor may be a very abstract thing. But the circumstance that the correlation is not perfect shows that other causal factors too are at work. These may dilute the correlation in various ways. Some cause may be influencing the variate (1) but not the variate (2). Or vice versa some cause may be influencing (2) but not (1). Or both these things may be happening. Or some cause may be helping the one variate, and hindering the other. In any case, however, if the two variates are expressed as weighted sums of uncorrelated factors z z m)! one at least of the factors a must be identical with one at least of the factors b, in order that any correlation may result. If we next consider three tests and low correlations (up to '5), we find great elasticity in the possible explanations. f Suppose all three correlations equal *5. We have, then, among innumerable possibilities, two extreme forms of * Orthogonal factors, we must now say. f Brown and Thomson, page 142 ; Thomson, 19196, Appendix J. R. Thompson. 262 LIMITS TO THE EXTENT OF FACTORS 263 explanation possible, one with only one general factor, the other with no general factor Zl = -707a + -707$! # 2 = '707a + -707*2 }>one general factor z z = -707 + -707*3 or *. = -7076 + -707c = -707c + -707d = -7076 + -707d no general factor So long as the correlations do not average more than 5,* they can (usually) be imitated without a general factor, although one can be used if desired. That is, they can be imitated either by a three-factor if we may so designate a factor running through three tests or (usually) by two-factors running through only two tests, though in certain cases this may prove impossible, especially if the average correlation is not far below -5. As soon, however, as the average correlation rises above 5,f some use must be made of a three-factor general to all three tests, as the reader can readily convince himself by trial. In the above example, if we wish to increase the correlation of Tests 1 and 2 while using the second form of equations, we see that since we have exhausted all the variance on the factors fc, c, and d, we can do so only by using either b in Test 2, or d in Test 1, and thus making it into a three-factor. 2. The average correlation rule (Thomson, 19366). When we have more tests, say n, then we can usually do without an n-factor (or general factor) so long as the average corre- lation does not exceed (n 2)/(n 1)4 Again, of course, an n-factor may be used if desired, but its use is not usually compulsory, as it certainly is in some measure as soon as * This is an approximate condition. For an exact form, see the Mathematical Appendix, paragraph 20. See also later in this chapter. f See previous footnote. j Approximate condition, see previous footnote, and consult Appendix. 264 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the average correlation rises past this point. Further, if the average correlation is still lower, we can in turn, as a rule, dispense with (n l)-factors as soon as the average sinks below (n 3)/(n 1), and with factors of less extent as it sinks still further. To know approximately what is the least-extensive kind of factors we can manage with, we have to see where the average correlation fits in, in the series of fractions 1 2 3 n ~3 n -2 n 1 n 1 n 1 n 1 n 1 As soon as the average correlation uses past (n p)/(nl) 9 we can no longer have (p 1) zeros in every column of the matrix of loadings. Usually (though not necessarily) we can manage to have (p 1) zeros at or below that point. The reason for this rule can be appreciated if we reflect that the highest possible correlations we can get with a given number of zero loadings will be reached by abolishing all factors of less extent. For example, with two-factors only, the highest possible correlations between five tests will be obtained by a pattern of loadings like this : xxxxoooooo xoooxxxooo oxooxooxxo ooxooxoxox oooxooxoxx If there are to be no specifics, and if we take the case where all the correlations are alike (which is in fact the maximum correlation possible), we see that the square of every loading must be 1/4, or in general l/(n 1). Each correlation will therefore be equal to 1/4 or l/(n 1). In the series of fractions 123 444 the average correlation just reaches the first, which can be tv\ ___ /vj considered as - , n being 5 and p being 4. And p 1 IT 1 LIMITS TO THE EXTENT OF FACTORS 265 or three zeros are just possible in each column of loadings. Again, consider five tests in which we use only three- factors. The maximum correlation is given by a pattern just like the last one, except that the noughts and crosses have to change places. Since there are six loadings, the square of every loading must be 1/6, and the pattern shows that every correlation is three times this, or 1/2. The average correlation, therefore, now reaches the next of the above fractions 123 444 and when n p __ 2 n ^1 ~~ 4 we have p = 3 ; and p 1 or two zeros are just possible per column (represented by the crosses in the former diagram), as we know is true from the way in which we made the correlations. It should be noted that the rule works with certainty only in one direction. What it asserts to be impossible, is impossible. But when it docs not say that a given number of zero loadings per column is impossible, it is not certain to be possible. The rule is necessary, but not sufficient. Usually, however, it is a fairly safe guide, and when it does not say the zeros are impossible, they can generally be nearly if not quite reached, with the greater ease, of course, the more the average correlation falls below the critical value. It should also be re-emphasized that these considerations have, so far, nothing to do with Thurstone's theory. In terms of our geometrical analogy, we are here considering the whole space (not merely a common-factor space) and asking whether orthogonal axes can be found each of which is at right angles to some of the test vectors. We are at liberty to take as many axes as we like, extending the dimensions of our space as we please. As an example, consider the set of correlations used in the last chapter : F.A. 9* 266 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 1 2 3 4 5 6 1 9 525 000 000 448 000 2 525 . 098 306 349 000 3 000 098 . 133 314 504 4 000 306 133 . 000 000 5 448 349 314 000 . 307 6 000 000 504 000 307 , The average correlation is -199, and n, the number of tests, is 6. The series of critical fractions is therefore 1234 5555 and the average correlation falls just short of the first one, for which, since n p = 1, p = 5. This leaves open the possibility that we can use factors which have p 1 or four zeros in each column of loadings, that is, that we can manage with two-factors each linking only two tests. But as '199 is so near to 1/5, and as the correlations are far from being all alike, we may expect to find this difficult or even not quite possible. Trial shows that we can nearly, but not quite, manage with two-factors. The following set of loadings, for example, while not perhaps the nearest approach to success, comes fairly close : Factor Test 1 2 3 4 5 6 II III IV VI VII VIII IX 734 -679 658 . -300 301 607 318 -613 895 278 -606 -682 446 504 . -477 . -387 682 -732 giving correlations : 1 2 1 . 483 000 000 412 000 2 483 . 090 285 309 000 3 000 090 . 124 289 465 4 000 285 124 . 000 000 5 412 309 289 000 t 283 6 000 000 465 000 283 which average '183 instead of -199. LIMITS TO THE EXTENT OF FACTORS 267 3. The latent-root rule (Thompson, 1929 ; Black, 1929 ; Thomson, 19366 ; Ledermann, 1936). A more scientific rule for ascertaining how " extensive " * the factors must be to explain the correlations is based upon the calculation of the largest " latent root " of the matrix of correlations. The exact calculation of the largest latent root is a very troublesome business, but luckily there are approximations. We have already met the term " latent root," in passing, in connexion with Hotelling's process.f If the largest latent root lies between the integers s and (s + 1), then ^-factors are inadequate, and n - s zeros are impossible. Like the previous rule, this one is " neces- sary," but not " sufficient." It assures us that s-t actors are inadequate, but it does not assure us that (s -f~ 1)- factors are adequate, though they usually are if the latent root is not too near 5 + 1. The easiest approximation to the largest latent root is, when the correlations are positive Sum of the whole matrix, including diagonal elements n In the case of the above example the whole matrix, including unities in the diagonal elements, sums to 11*972, so that the approximate largest latent root is 1-995, which leaves it just barely possible that two-factors will suffice. As we know by trial, they just won't. A better approximation is Sum of the squares of the column totals Sum of the whole matrix the diagonal elements being included for both numerator and denominator. (This quantity is, in fact, the sum of the squares of the first-factor loadings in Thurstone's " centroid " process.) * Meaning by an " extensive " factor one which has loadings in many tests. Thus a two-factor is less " extensive " than a three- factor, and so on. t See Chapter V, Section 4. 268 THE FACTORIAL ANALYSIS OF HUMAN ABILITY In our example we have : 1-000 525 000 000 448 000 525 1-000 098 306 349 000 000 098 1-000 133 315 504 000 306 133 1-000 000 000 448 349 315 000 1-000 308 000 000 504 000 308 1-000 Totals 1-973 2-278 2-050 1-439 2-420 1-812 = 11-972 Squares 3-893 5-189 4-203 2-071 5-856 3-283 = 24-495 24-495 Approximate largest latent root = 2-046 * This time the better approximation definitely cuts out the possibility that two-factors will suffice. 4. Application to the common-factor space. All of the above applies to factors in general, and the calculations are carried out with unity in each diagonal cell. To apply these rules to the problem of the attainability of " simple structure," we have to adapt them to the common-factor space. For this purpose they must be applied either to the matrix with correlations " corrected " for communality (the best plan), or with certain modifications to the matrix with communalities in the diagonal. By "correcting" a correlation coefficient for communality is meant dividing it by the square root of each of the communalities of the two tests concerned. The result is the correlation which would ensue if the specifics were abolished. In the case of our small example, the correlations " corrected " for communality are : 1 2 3 4 5 6 1 1-000 803 000 000 780 000 2 803 1-000 165 597 626 000 3 000 165 1-000 276 601 961 4 000 597 276 1-000 000 000 5 780 626 601 000 1-000 626 6 000 000 961 000 626 1-000 The average of the corrected coefficients is -362. In the * The exact value to three places of decimals calculated by the method given in Aitken, 1937&, 284 jfjf., is 2-086. LIMITS TO THE EXTENT OF FACTORS 269 series of fractions with denominator (n 1) 1234 5555 this value '362 is below 2/5, or (n p)/(n 1) where n 6 tests. We see, therefore, that p = 4, and that the possibility of having^ 1, or three zeros in every column, is not denied. This is in agreement with the analysis (an orthogonal " simple structure ") arrived at in Chapter XVI, page 254 (and see page 247). The first approximation to the largest latent root of the matrix with correlations " corrected " for communality and with unity in each diagonal cell gives Sum of whole matrix . __ ____ = 2'oi.Z n and as this is less than 6 3, three zeros are still possible in each column. The more accurate approximation to the root Sum of the squares of the column totals __ 49*2718 Sum of the whole matrix 16-870 shows by its nearness to 6 3 that three zeros, if they are possible (and we know they are), must just barely be possible.* Instead of applying the latent-root test to the matrix corrected for communality, we can apply it to the matrix of ordinary correlations, with the communalities in the diagonal cells, but with the following change. Instead of comparing the latent root with the series of integers 1, 2, 3 ... we have to compare it with the sum of 1, 2, 3 ... communalities, taking these in their order of magnitude, largest first (Ledermann). We shall illustrate * Exact root is 2-954. It is tempting to surmise that Thurstone's search for unique orthogonal simple structure is really a search for a matrix, corrected for communality, with an integral largest root, equal to n r ; but it must be remembered that the criterion though necessary is not sufficient when the number of factors is restricted to r. 270 THE FACTORIAL ANALYSIS OF HUMAN ABILITY this on the same example. The matrix of ordinary cor- relations, with communalities, is : 674 -525 -000 -000 -448 -000 525 -634 -098 -306 -349 -000 000 -098 -558 -133 -314 -504 000 -306 -133 -415 -000 -000 448 -349 -314 000 -490 -000 000 -000 -504 -000 -000 -493 Sums 1-647 1-912 1-607 -854 1-601 -997 = 8-618 Squares 2-713 3-656 2-582 -729 2-563 -994 = 13-237 1 ^*?^T Approximate largest root * *" = 1 '536 o'ulo The communalities arranged in order of magnitude and summed are : 123456 674 -634 -558 -493 -490 -415 Continued sum -674 1-308 1-866 2-359 2-849 3-264 The latent root 1 -536 is larger than the second of these but less than the third, so the possibility of three zeros per column is left open, in agreement with the former tests and with the known facts. It would seem from the present writer's experience, however, that the test applied to the ordinary matrix in this way does not always agree exactly with that applied to the matrix with correlations corrected for communality, and that the latter is more accurate. 5. A more stringent test. The above tests only refer to the possibility of obtaining the required number of zero loadings with orthogonal factors " orthogonal simple struc- ture/' Even when orthogonal simple structure cannot be reached, it may be possible to attain simple structure with oblique factors. Moreover, the approximations used for the largest latent root above are only valid, in general, when all the correla- tions are positive. In view of the fact, however, that few psychological correlations are negative this is not a great difficulty. Further, while these tests show definitely when ortho- gonal simple structure cannot be attained, it does not LIMITS TO THE EXTENT OF FACTORS 271 follow with certainty that it can actually be reached when the tests are satisfied, though it usually can. An exact criterion has been given (Ledermann, 1936), and is described in the Appendix, pages 377-8, which avoids all the above defects. It requires at present, however, a prohibitive amount of calculation. In general, simple structure will be attainable with a battery of tests only when the battery has been picked with that end in view. There is a certain incompatibility about Thurstone's demands which makes their fulfilment only possible in special circumstances. He wants as few common factors as possible to explain the correlations ; but he wants these common factors to have no loadings in a large number of the tests. This is rather like wanting to run a school with as few teachers as possible, but each teacher to have a large number of free periods. If we begin by reducing the number of common factors to its minimum (as Thurstone does), we will generally find that the second requirement cannot be fulfilled. It can, how- ever, be fulfilled in some cases, and it is exactly these cases which Thurstone relies on to define his primary factors. It is his faith that factors found in this mathema- tical way will turn out to be acceptable to the psychologist as psychological entities. CHAPTEll XVIII OBLIQUE FACTORS, AND CRITICISMS 1. Pattern and structure. So long as the factors are orthogonal, the loadings in the matrix of loadings are also the correlations between the factor and the tests, but this ceases to be the case when the factors are correlated. The word " loading " continues to be used for the coefficients such as /, m, and n in equations like z = Zoc + w$ + wy and the matrix or table of these is called a pattern, while the matrix of correlations between tests and factors is called a structure. Thus of the two matrices on page 182 (Chapter XI, Section 6), the upper one is both a pattern and a structure, for the factors are orthogonal, whereas the lower one is a structure only. From the upper table we can say that using the correlations of the factors with Test 1 as coeffi- cients in a linear equation for that test score. But we cannot say from the lower table that */= -51/t + -25/ 2 + -40* The correlations here cannot serve as coefficients. Moreover, as soon as the factors become oblique, it becomes necessary to distinguish between " reference vectors " and " primary factors." The reference vectors are the positions to which the centroid axes have been rotated so that the test-projections on to them include a number of zeros. Each reference vector is at right angles to a hyperplane containing a number of communality vectors. A hyperplane is a space of one dimension less than the common-factor space. In our first example in Chapter XVI the hyperplanes were ordinary planes, the 272 OBLIQUE FACTORS 273 faces of the three-cornered pyramid there referred to (see page 250) and each reference vector was at right angles to one of those faces. The primary factor corresponding to a given reference vector is the line of intersection of all the other hyper- planes, excluding, that is, the hyper plane at right angles to the reference vector. In our three-dimensional common- factor space the primary factor was the edge of the pyra- mid where those two faces met, excluding that face to which the reference vector was orthogonal. Now, when the reference vectors turn out to be at right angles to each other, as they did in that example, each reference vector is identical with its own primary factor (compare page 254 in Chapter XVI). But not when the reference vectors turn out to be oblique. In Chapter XVI we did not distinguish them, and called their common line the " factor." But in this chapter the distinction must be kept clearly in mind. It is the primary factors Thurstone wants. The reference vectors are only a means to an end. Thurstone's second method of rotation described in Chapter XVI, the method in which the communal ity vectors are " extended," and lines drawn on the diagrams which are not necessarily radial lines, will not keep the axes orthogonal, but seeks for the axes on which a number of projections are zero, regardless of whether the resulting directions are orthogonal or oblique. In general they will be oblique, and the examples worked in Chapter XVI only gave orthogonal simple structure because they had been devised so as to do so. The test of orthogonality is that the matrix of rotation, premultiplied by its transpose, gives the unit matrix (see page 254). Or in other words, that the inner products of the columns of the rotating matrix are all zero. They are the cosines of the angles between the reference vectors, and the cosine of 90 is zero. 2. Three oblique factors. To illustrate Thurstone's method when the resulting factors are oblique we shall next work an example devised to give three oblique common factors. Consider this matrix of correlations : 274 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 1 2 3 4 5 6 7 1 728 167 372 153 105 126 2 728 696 583 651 347 638 3 167 696 857 775 709 740 4 372 583 857 543 797 473 5 153 651 775 543 504 828 6 105 347 709 797 504 433 7 126 638 740 473 828 433 which, with guessed communalities, gives these centroid loadings : F 1 449 682 165 2 825 478 129 3 906 336 020 4 846 133 457 5 808 208 412 6 697 336 335 7 767 173 468 When these projections on the eentroid axes are " ex- tended," that is, when each row is divided by the first loading in that row, we obtain this table : in. I 1-000 -1-519 367 ,2 99 579 156 3 99 371 022 4 99 157 540 5 99 257 510 6 99 482 481 7 99 226 610 The columns II e and III e in this table represent the co- ordinates of the " dots on the ceiling " in our analogy of Chapter XVI, p. 250. When we make a diagram of them we OBLIQUE FACTORS 275 obtain Figure 30. We see that a triangular formation is present, and we draw the dotted lines shown. It is not essential, it may be remarked in passing, that there be no points else- where than on the lines, provided they are addi- tional to those required to fix the simple structure. Had it not been for the desirability of keeping the example small we would have increased the number of tests, and not only arranged for further points to fall on these lines, but also included some whose dots fell in- side the triangle, repre- senting tests which involve all three factors. We find the equations of these lines to be approximately 475 -f -50j/ + -95s = (line 1, 2, 7) 1-113 + -183y - 2-119^ == (line 1, 4, 6) 403 - l-091t/ + -256* = (line 7, 5, 3, 6) The coefficients of each equation have to be " norma- lized," that is, reduced proportionately so that the sum of their squares is unity (for they are to be direction cosines). These normalized coefficients are then written as columns in a matrix as follows : Figure 30. 405 -464 -338 426 -076 916 809 883 -215 = A The table of centroid loadings on page 274 must now be post-multiplied by this rotating matrix to obtain the projections of the tests on the three reference vectors which are at right angles to the planes defined by the dotted lines in our diagram. We obtain this table : 276 THE FACTORIAL ANALYSIS OF HUMAN ABILITY V-FA (Simple) Structure on the Reference Vectors L' B f D' 1 025 Oil 812 2 026 460 689 3 526 428 003 4 769 001 262 5 083 755 006 6 696 053 000 7 006 782 000 We have labelled the columns L', B', and Z)' for a reason which will become apparent later, when wo explain how the correlations were, in fact, made. This table is a simple structure, formed by the projections on the reference vectors. It has a zero (or near-zero) in each row, and three or more in each column, in the positions to be anticipated from Figure 30 ; for example, tests 3, 5, 6, and 7, which are collinear in the figure, have zeros in column D'. Now let us test the angles between the reference vectors. To do this we premultiply the rotating matrix by its transpose A'A = C 405 -426 -809 404 -076 -883 338 --916 -215 405 -4G4 -338 426 -076 --91 6 SCO -883 -215 1 - -494 - -079 -.494 i --103 079 -103 1 This gives the cosines of the angles between the reference vectors and we see that they are obtuse. The angles are approximately : 120 95 120 t 96 95 96 OBLIQUE FACTORS 277 As soon as we know that the reference vectors are not orthogonal, we have to take account of the fact that the primary factors are not identical with them. Each prim- ary factor is the line in which the hyperplanes intersect, excluding that hyperplane to which the corresponding reference vector is orthogonal. In a three-dimensional common-factor space like ours the primary factors lie along the edges of the pyramid which the extended vectors form. Let us return to our mental picture, which the reader can place in the room in which he is sitting. The origin, immediately below the point in Figure 30, is in the middle of the carpet. Figure 30 itself is on the ceiling, seen from above as though translucent. The radial lines with arrowheads are the projections of the primary factors on to the ceiling. The projections of the reference vectors are not drawn, to avoid confusion in the figure. They are near, but not identical with, the primary factors. The reader should not be misled by the fact that two of the primary factors lie along the same lines as Tests 1 and 7. It was necessary to allow this in devising an ex- ample with very few tests in it (to avoid much calculation and printing large tables). But with a large number of tests the lines of the triangle could have been defined without any test being actually at a corner. 3. Primary factors and reference vectors. At about this stage a disturbing thought may have occurred to the reader. We have sought for, and obtained, simple structure on the reference vectors. That is to say, we have found three vectors, three imaginary tests, which are uncorrolated each with a group of the actual tests, namely where there are zeros in the table on page 276. The entries in that table are the projections of the actual tests on the reference vectors. But the primary factors are different from the reference vectors. The projections of the tests on to the primary factors will be different and will not show these zeros. Those projections are, in fact, given in this table (never mind for the moment how it is arrived at) : 278 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Structure on the Primary Factors LED 1 2 3 4 5 6 160 -162 -832 408 -666 -793 866 -809 -176 934 -495 -401 541 -927 -152 842 -472 -132 468 -915 -150 These numbers are the correlations between the primary factors and the tests, and none is zero. The primary factor structure is not " simple," it is the reference vector structure that is simple. Why then not use the reference vectors as our factors ? A two-fold answer can be given to this, one general, the other particular to this example. The latter will become clear when we divulge how the example was made. The former requires us to return to the distinction between structure and pattern. A structure is a table of correlations, a pattern is a table of coefficients in a " specification " equation specifying how a test score is made up by factors. The entries in a pattern are loadings or saturations of the tests with the factors, but not correlations. Pattern and structure are only identical when the reference vectors are orthogonal and coincide with the primary factors. When the reference vectors are oblique (usually at obtuse angles) the primary factors are different and are themselves usually at acute angles. When the primary factors and reference vectors thus separate, the structure of the reference vectors and the pattern of the primary factors are identical except for a coefficient multiplying each column ; and vice versa the structure of the primary factors is identical (except for similar coefficients) with the pattern of the reference vectors. In particular, where there are zeros in the reference vector structure there will also be zeros in the primary factor pattern. The general theorem of the reciprocity of reference vectors and primary OBLIQUE FACTORS 279 factors (to use our present terms), that is, the reciprocity of (a) a set of lines orthogonal to hyperplanes, and (6) another set of lines which are the intersections in each case of the remaining hyperplanes, is an instance of the reci- procity which runs through the whole of n-dimensional geometry between hyperplanes of k dimensions and of (n k) dimensions. It occurs in several other places in the geometry of factorial analysis : for instance, tests, persons, and factors are all in one sense reciprocal and exchangeable. The particular fact about the zeros in the primary factor pattern can be seen readily from the geometrical analogy. For a test vector which lies in a hyperplane can be com- pletely defined as a weighted resultant of the primary factors which are also in that hyperplane, without any assistance from other primary factors. In our drawing of the reader's study, for example, on page 250, the vector of the Test 2, which is the line 02, lies upon the plane face 014 of the pyramid, and can be completely described by a weighted sum of the primary factors along the edges 01 and 04, without bringing in the edge 06 at all. The primary factor which lies along that edge will therefore have a zero weight in the row of the pattern which speci- fies Test 2. This pattern on the primary factors will be very similar to the structure on the reference vectors already given for our example in the table on page 276. It can, in fact, be calculated from that table by multiplying the first column by 1-163, the second column by 1-166, and the third by 1-017, giving the following : FAD' 1 (Simple) Pattern on the Primary Factors LED 1 2 4 5 6 7 029 -013 -826 030 -536 -701 612 -499 -003 895 -001 -266 096 -880 -006 809 -062 -000 008 -912 -000 2$0 THE FACTORIAL ANALYSIS OP HUMAN ABILITY Thus although the primary factors differ from the reference vectors (the angles between the primary factors and their corresponding reference vectors are, in fact, 31, 31, and 11), yet if the structure on the reference vectors is " simple," the pattern on the primary factors will be " simple." The entries in the above table can be used as coefficients in specification equations, and if for clearness we omit the near-zero coefficients entirely, we have found that the test-scores can be considered as made up thus : Score in Test 1 -826d + Specific 2 - -5366 + -701d + 3 - -612Z + -4996 + 4 = -8951 + -266d + 5 - -8806 + 6 - -809Z + 99 99 99 7 = '9120 -f- ,, 4. Behind the scenes. It is now time to divulge what these " tests " really are and how the " scores " were made whose correlations we have been analysing, and to compare our analysis with the reality. The example is a simpler and shorter variety of a device used by Thurstone and published in April 1940 in the Psychological Bulletin. The measurements behind the correlations were not made on a number of persons, but were made on a number of boxes only eight boxes, to keep down the amount of calculation and printing. These boxes were of the follow- ing dimensions : Length Breadth Dep 1 2 2 1 2 3 2 3 3 3 2 2 4 6 3 2 5 4 4 2 6 5 3 1 7 5 4 3 8 4 4 2 Sum 32 24 16 Mean 4 3 2 OBLIQUE FACTORS 281 The " tests " were seven functions of these dimensions, and are shown in the next table, which also shows the score each box (or " person ") would achieve in that test. It is as though someone was unable for some reason to measure the primary length, breadth, and depth of these boxes (as we are unable to measure the primary factors of the mind directly) but was able to measure these more complex quantities like LB, or \/(L 2 ~\- D 2 ) (as we are able to measure scores in complex tests) : Boxes = Persons Test Formula 1 2 3 4 5 6 7 8 Sum Mean 1 D 2 1 9 4 4 4 1 9 4 36 4-500 2 ED 2 6 4 6 8 3 12 8 49 6-125 3 LB 4 6 6 18 16 15 20 16 101 12-625 4 V(& + D*) 2-244-243 61 6-32 4-47 5-10 5-83 4-47 36-28 4-535 5 L-\-B* 6 7 7 15 20 14 21 20 110 13-750 6 L 2 + D 5 12 11 38 18 26 28 18 156 19-500 7 B 2 2 2 3 4 3 4 4 24 3-000 With these scores the sums of squares and products of deviations from the mean are : 12 345 67 1 66 50-5 22 5 10-2 25 29 3 2 50 5 72-9 98 4 16-8 112 3 100 5 16 3 22 5 98-4 273 9 47-9 259 2 398 -5 36 4 10 2 16-8 47 9 11-4 37 91 3 4-7 5 25 112-3 259 2 37-0 283 5 288 41 6 29 100-5 398 5 91-3 288 800 36 7 3 16 36 4-7 41 36 6 From these the correlations could be calculated by dividing each row and column by the square root of the diagonal cell entry. But that would make no allowance for specific factors, which in all actual psychological tests play a considerable part. In the example devised by Thurstone on which this is modelled there are no specific factors, but it was decided to introduce them here into tests 5, 6, and 7, by increasing their sums of squares. In addition, by an arithmetical slip, a small group factor was added to these three tests, and this was not discovered for some time. It was decided to leave it, for in a way it makes the example 282 THE FACTORIAL ANALYSIS OF HUMAN ABILITY more realistic, and may be taken to represent an experi- mental error of some sort running through these three tests. With these changes, the correlations are found, and are* those with which we began this chapter and which we have already analysed into three oblique factors L, B 9 and D. Let us now compare that analysis with the formulae which we now know to represent the tests. The pattern on page 2 79, for example, shows that Test 2 depends only on factors B and D : and that is correct, for it was, in fact, their product J?D, and L did not enter into it. The analysis gives the test score as a linear function of B and D, 536& + -701d whereas it was really a product. But the analysis was correct in omitting L. Similarly, the analyses into the other factors can be compared with the actual formulae, and in almost every case the factorial analysis, except for being linear, is in agreement with the actual facts. Tests 5 and 6, true, appear in the analysis to omit factors L and D respectively, although these dimensions figured in their formulae. But it would appear that they were swamped by reason of the other dimension in the formulae being squared ; and also possibly the specific and error factors we added did something towards obscuring smaller details. Also the process of " guessing " communalities, though innocuous in a battery of many tests, is a source of con- siderable inaccuracy when, as here, the tests are few. 5. Box dimensions as factors. We can now explain the particular reason for selecting the primary factors, and not the reference vectors, as our fundamental entities. The fundamental entities in the present example can reason- ably be said to be the length, breadth, and depth of the boxes, given in the table on page 280. Now, the columns of that table are correlated with one another, as the reader can readily check, the correlation coefficients being L with B, -589 L D, -144 B D, -204 These correlations are due to the fact that a long box naturally tends to be large in all its dimensions. It could, OBLIQUE FACTORS 288 of course, be very, very shallow, but usually it is deep and broad, The reference vectors were, it is true, correlated, but negatively. They were at obtuse angles with one another (see page 276) and obtuse angles have negative cosines corresponding to negative correlations. So the reference vectors do not correspond to the fundamental dimensions length, breadth, and depth. What, then, are the angles and hence the correlations between the primary factors ? We shall find that they are acute angles, and their cosines agree reasonably well with the above correlations between the length, breadth, and depth. The algebraic method of finding these angles is given in the mathematical appendix, but it is perhaps desirable to give a less technical account of it here. We need the direction-cosines of the primary factors, that is, the cosines of the angles they make with the orthogonal centroid axes. Each primary factor is the intersection of n 1 hyperplanes in our simple case is the intersection of two planes. In n-dimensional geometry a linear equation defines a hyperplane of n 1 dimensions. For example, in a plane of two dimensions a linear equation is a line (of one dimen- sion) hence the name linear. But in a space of three dimensions a " linear " equation like ax + by + cz = d is a plane. Two such equations define the line which is the intersection of two planes. Now, the equations of the three planes which form the triangular pyramid of which we have previously spoken are just those equations we have already obtained and used in our example, viz. : 405o? + -426i/- + -809^ = 4640 + -076*/ -8882 = 338o? -916i/ + -215* = These equations taken two at a time define the three edges of the pyramid, which are our primary factors, and if we express each pair in the form a 284 THE FACTORIAL ANALYSIS OF HUMAN ABILITY then the direction cosines are proportional to a, b, and c, which only require normalizing to be the direction cosines. When the direction cosines are found in this way, and written in columns to form a matrix, they prove to have the values 797 400 453 -835 -503 -187 843 517 -192 This is the rotating matrix to obtain the projections, i.e. the structure, on the primary factors, and if the centroid loadings on page 274 arc post-multiplied by this there results the table we have already quoted on page 278. The above matrix, premultiplied by its transpose, gives the cosines of the angles between the primary factors. We obtain 1 506 150 506 1 164 150 164 1 - DC-'D Compare these with the correlations between the columns of dimensions of the boxes, viz. : 1 589 144 589 1 204 144 204 1 The resemblance is quite good, and shows that it is the primary factors, and not the reference vectors, which represent those fundamental although correlated dimen- sions of length, breadth, and depth in the boxes. 6. Criticisms of simple structure. Thurstone's argument is then, of course, that as this process of analysis leads to fundamental real entities in the case of the boxes (and also in his " trapezium " example, Thurstone, 1944, p. 84, with four oblique factors), it may be presumed to give us fundamental entities when it is applied to mental measurements. And I confess that the argument is very strong. CRITICISMS 285 My fears or doubts arise from the possibility that the argument cannot legitimately be reversed in this way. There is no doubt that if artificial test scores are made up with a certain number of common factors, simple structure (oblique if necessary) can be reached and the factors identified. But are there other ways in which the test scores could have been made ? Spearman's argument was a similar reversal. If test scores are made with only one common factor, then zero tetrad-differences result. But zero tetrad-differences can be approached as closely as we like by samples of a large number of small factors, with very few indeed common to all the tests. However, Thurstone's simple structure is a much more complex phenomenon than Spearman's hierarchical order, and yet he seems to have had no great difficulty in finding batteries of tests which give simple structure to a reason- able approximation. I am not sceptical, merely cautious, and admittedly much impressed by Thurstone's ability both in the mathematical treatment and in the devising of experiments. Moreover, his idea of " second-order factors," to which we turn in the next chapter, promises a reconciliation of Spearman's idea of g, Thurstone's primary factors, and (so he tells us in a recent article) my own idea of sampling the " bonds " of the mind. Thurstone might, I think, put his case in this way. He assembles a battery of tests which to his psychological intuition appear to contain such and such psychological factors, some being memory tests, some numerical, etc., etc.. no test, however, containing (to his mind) all these expected factors. He then submits their correlations to his calculations, reaches oblique simple structure, and compares this analysis with his psychological expectation. If there is agreement, he feels confirmed both in his psy- chology and in the efficacy of his method of finding factors mathematically. Usually there will not be complete agreement, and he is led to modify his psychological ideas somewhat, in a certain direction. To test the truth of these further ideas he again makes and analyses a battery. Especially he looks to see if the same factors turn up in various batteries. He uses his analyses as guides to 286 THE FACTORIAL ANALYSIS OF HUMAN ABILITY modifications of his psychological hypotheses, or as con- firmation of them. In Great Britain Thurstone's hypo- thesis of simple structure has been, I think it is correct to say, rather ignored than criticized. The preoccupation of most British psychologists since 1939 with the tasks the war has brought is partly to blame for this neglect, and partly also the fact that most of them have imbibed during their education a belief in and a partiality for u Spear- man's g," a factor apparently abolished by Thurstone. Since his work on second-order factors rehabilitates g, this objection should disappear, and his method be at least accepted as a device a very powerful device for arriving if desired at g and factors orthogonal to it, indirectly. He himself thinks that the oblique first-order factors are more real and more invariant. An early form of response to his work was to show that his batteries could also be analysed after Spearman's fashion. Holzinger and Harman (1938), using the Bifactor method, reanalysed the data of Thurstone's Primary Mental Abilities and found an important general factor due, as they truly say, " to our hypothesis of its existence and the essentially positive correlations throughout." Spear- man (1939) in a paper entitled Thurstone's Work Reworked reached much the same analysis, and raised certain practical or experimental objections, claiming that his g had merely been submerged in a sea of error. But there is more in it than that. As I said in my contribution to the Reading University Symposium (1939) Thurstone could correct all the blemishes pointed out by Spearman and would still be able to attain simple structure. I said on that occasion that however juries in America and in Britain might differ at present, the larger jury of the future would decide by noting whether Spearman's or Thurstone's system had proved most useful in the hands of the prac- tising psychologist. I now think that they will certainly also consider which set of factors has proved most invariant and most real. Very likely the two criteria may lead to the same verdict. But for the present the two rival claims are in the position described by the Scottish legal phrase, 44 taken ad avizandum " ; and perhaps before the judge CRITICISMS 287 returns to the bench the matter may have been settled out of court by Thurstone's reconciliation via " second-order " factors. 7. Reyburn and Taylor's method. These South African psychologists have proposed to let psychological insight alone guide the rotations to which axes are subjected, and have criticized simple structure (see especially their paper 1943a), for lack of objectivity, failure to produce in variance under change of tests, and on the grounds that at best it yields the factors that were put in. They agree that harmony, within the limits of error, between psycho- logical hypothesis and the mathematical simple structure to a certain extent confirms both, but they criticize especi- ally those who assume that even in complete previous ignorance of the factors we are entitled to assign objectivity and psychological meaning to those indicated by simple structure. And anyhow, they urge, simple structure is now, when obliquity is permitted, all too easily reached to prove anything. They themselves do not necessarily insist on a g (see their 1941#, pages 253, 254, 258, etc.). Their own plan is to choose a group of tests which their psychological knowledge, and a study of all that is pre- viously known, leads them to consider to be clustered round a factor. They therefore cause one of their axes to pass through the centroid of this cluster, keeping all axes orthogonal. This factor axis they do not subse- quently move. They then formulate a hypothesis about a second factor and select a second group of tests, through whose centroid (retaining orthogonality) they pass their second factor axis. And so on. There is some affinity between this and Alexander's method of rotation (see page 36). The arithmetical details of their method are as follows. They first obtain a table of centroid loadings in the usual way. Then, having chosen a group of tests which they think form, psychologically, a cluster, they add together the rows of the centroid table which refer to those tests, thus obtaining numbers proportional to the loadings of their centroid. These, after being normalized, form the first column of their rotating matrix. For example, 288 THE FACTORIAL ANALYSIS OF HUMAN ABILITY consider this (imaginary and invented) table of loadings Loadings 1 II III A 2 1 4 3 1 -26 2 5 .g 6 70 3 6 -3 - 3 54 4 5 2 1 30 5 4 -4 2 36 6 5 .4 2 45 7 5 2 1 30 8 7 -4 1 66 9 7 .0 3 62 10 6 .4 4 68 Reyburn and Taylor now decide, let us suppose, that Tests 9 and 10 are, in their psychological view, very strongly impregnated with a verbal factor, and they decide to rotate their original factors until one of them passes through the centroid of these two tests. They extract their rows, add them together, and normalize the three totals thus : (9) -7 (10) -6 2 4 3 4 1-3 816 6 376 7 439 Sum of squares 2-54 = 1-594 2 obtained by dividing by 1*594. If the columns of the original table are multiplied by these three numbers and the rows added, the result is the first column of the rotated factor loadings in the table below. To get the other two columns we must complete the rotating matrix in such a manner that the axes remain orthogonal. How this is done will be explained separately later. Meanwhile, consider the matrix 816 -399 -417 376 183 -909 439 898 CRITICISMS 280 Its first column is composed of the above numbers. It is orthogonal, for the sum of the squares of any row or column is unity, and the inner product of any two is zero. When the original table of loadings is post-multiplied by this we get the rotated table : 1 2 3 4 5 6 7 8 9 10 Rotated Loadings 258 -015 -440 257 -793 064 47] -564 -022 37? -073 -390 088 -266 -530 646 -093 155 289 -253 -390 465 -116 -656 778 -047 -110 816 047 -113 260 699 540 300 359 450 300 660 620 681 At this point two checks must be made. The sum of the squares (h z ) of each row must still be the same : and the inner product of each pair of rows must still be the same (it is sufficient to test consecutive rows only). For example, in the original matrix the inner product of rows 7 and 8 was 5 X -7 + -2 X -4 -1 X -1 = -42 and in the rotated matrix it was 289 X -465 + -253 X -116 + -390 X -656 = -420. The first factor now goes through the centroid of Tests 9 and 10, and we scan the loadings it has in the other tests to see if these are consistent with their psychological nature. For instance, Test 5 has practically no loading on this verbal factor is this consistent with our psychological opinion of this test ? If this scrutiny is satisfactory, the psychologist using this method then proceeds to consider where he will place his second factor ; for the second and third columns of the above loadings have still no necessary psychological mean- ing as they stand. Exactly the same procedure is carried F.A S 10 !20a THE FACTORIAL ANALYSIS OF HUMAN ABILITY out with them, the first column being left unaltered. Suppose the psychologist decided on Tests 5, 7, 8 as being a cluster round (say) a numerical factor. He adds their rows (5) -266 -530 (7) -253 -390 (8) -116 -656 635 1-576 374 -928 when normalized and uses their normalized totals as the first column of a matrix to rotate these last two columns. The matrix must be orthogonal, and it is in fact r 374 -928 928 -374 When the second and third columns are rotated by post- multiplication by this, the final result is : Final Rotated Loadings 1 -258 -414 151 2 -257 -237 -760 3 -471 -191 -532 4 -377 -389 078 5 -088 -591 -049 6 -646 109 -144 7 -289 -457 -089 8 -465 -652 138 9 -778 -120 -002 10 -816 -122 -001 (The same checks must now be repeated.) The psycho- logist now scans column two to see if the loadings of his numerical factor agree reasonably with his idea of each test, and is rather sorry to see two negative loadings, but consoles himself by thinking that they are small. He must finally try to name his third factor, present to an CRITICISMS 291 appreciable extent only in tests 2 and 3. If he thinks he recognizes it, he is content. 8. Special orthogonal matrices. To carry out the above process the reader needs to have at his disposal orthogonal matrices of various sizes, such that he can give the first column any desired values. The following will serve his purpose. Except for the first one, they are not unique, and alternatives can be made. Order 2 Order 3 u v V U mq mp Iq lp p ~q + v 2 = 1 m = 1 It was from this formula that the matrix used in the last section, with first column of '816, 376, -439, was made. For if we set p = -439 we have q = -898 and from mq = -816 we have m = -909 and thence I = -417 Order 4. a b c -d b a d c c d a b d f\ -b a This one was used by Reyburn and Taylor in their 1939 article (page 159). Similar matrices of higher order can be made by a recipe given by them, viz. multiplying together two or more of the above, suitably extended by ones and zeros. For example, a matrix, orthogonal and with arbitrary first column, of order 5, can be made by multiplying together : mq -Iq P mp Ip ~-q X I m 292 THE FACTORIAL ANALYSIS OF HUMAN ABILITY fa c where I* + m 2 = p* + q* = X 2 + ^ = ic 2 + cp 2 = 1. 9. Identity of oblique factors after univariate selection. Thurstone, in his recent book Multiple Factor Analysis (1947), discusses in Chapter XIX the effects of selection, and shows by examples that if a battery of tests yields simple structure with oblique factors (including, of course, the orthogonal case), then after univariate selection the same factors are identified by the new structure, which is still simple. If, for example, the battery which gives the correlations on our page 245, and yields Figure 28 on page 251, has the standard deviation of Test 2 reduced to one-half, then by the methods described on our pages 171-6 we can calculate that the matrix of correlations and communalities becomes : 1 589 295 -044 -140 366 000 2 295 302 049 159 183 000 3 - -044 049 555 115 304 506 4 -140 159 115 371 -087 000 5 366 183 304 -087 439 322 6 000 000 506 000 322 493 The rank of this matrix is still 3 as it was before selection, and three centroid factors are found to have loadings I II III 1 409 647 058 2 379 244 -315 3 569 -444 184 4 160 -271 - -522 5 585 174 257 6 506 -350 337 CRITICISMS 293 When these are " extended " in the manner of our page 251 and a diagram like Figure 28 made, we obtain Figure 81. It is still a triangle, and although its measurements are different, the same tests are found defining each side as before. The cor- ners of the triangle may, with Professor Thurstone, reasonably be claimed to represent the same fac- tors as before selection, although their correla- tions have changed. The plane of Figure 31 is not the same as the plane of Figure 28, being at right angles to a differ- ent first centroid. When adjustment is made for this, as Professor Thur- stone has presumably done in his chapter (though, I protest, without sufficient explanation), then the directly selected test point has not moved, while the other points have moved radially away from or towards it. If the above matrix of centroid loadings is postmulti- plicd by the rotating matrix obtained from the diagram, viz. | -721 -443 -641 -499 -201 -744 480 -874 -190 we obtain the new simple structure on the reference vectors, Figure 31. I 1 2 3 4 5 6 562 459 702 B 394 180 472 732 484 455 294 THE FACTORIAL ANALYSIS OF HUMAN ABILITY If this is compared with the table on page 247 it will be seen that the zeros are in the same places, although the non-zero entries have altered (except in Test 6, which was uncorrelated with the directly selected Test 2, and therefore is unaffected in composition). If the correlations between the factors are calculated by the method of pages 283-4, factor A is found to be still uncorrelated with B and C, but these last two have a correlation coefficient of *3 : that is, they are no longer orthogonal but at an obtuse angle of about 107|. 10. Multivariate Selection and Simple Structure. But though Thurstone must, I think, be granted his claim that univariate selection will not destroy the identity of his oblique factors, but only change their intcrcorrelations, the situation would seem to be very different with multivariate selection. Multivariate selection is not the same thing as repeated univariate selection. The latter will not change the rank of the correlation matrix with suitable communalities, nor will it change the position of zero loadings in simple struc- ture. Repeated univariate selection will, it is true, cause all the correlations to alter, but only indirectly and in such a way as to preserve rank, simple structure, and factor identity. But in multivariate selection it is envisaged that the correlation between two variables may itself be directly selected, and caused to have a value other than that which would naturally follow from the reduction of standard deviation in two selected variables. Selection for correla- tion is just as easily imagined as is selection for scatter. Indeed in natural selection it is possibly even commoner. Once we select for the correlations, however, as well as for scatter, new " factors " emerge, old ones change. In our Chapter XII we suppose a small part R pp of the whole correlation matrix to be changed to V pp9 and found that one new factor is created (page 193) or, indeed, two new oblique factors (page 192). We might have supposed R pp to be a larger portion of R : and there is nothing to prevent us supposing selection to go on for the whole of U, and writing down a brand-new table of coefficients, whose CRITICISMS 295 " factors " would be quite different from those of the origi- nal table. In our example of page 245, for instance, where the three oblique " factors " coincided in direction with the communal parts of Tests 1, 4, and 6, there is nothing to prevent us from writing down, as having been produced by selection, a new set of correlation coeffici- ents whose analysis would identify the " factors " with the communal parts of Tests 2, 3, and 5. In fact, all we would have to do would be to renumber the rows and columns on page 245. Such fundamental changes could be produced by selection : and perhaps they have been, for natural selection has had plenty of time at its disposal. Professor Thurstone (his page 458, footnote, in Multiple Factor Analysis) classes the new factors produced by selection as " incidental factors (which) can be classed with the residual factors, which reflect the conditions of particular experiments." But we can hardly dismiss them thus easily if, as is conceivable, they have become the main or perhaps the only factors remaining, the others having disappeared ! It may be admitted at once, however, that the actual amount of selection from psychological experiment to psychological experiment is not likely to make such alarming changes in factors. For the use to which factors are likely to be put in our age, in our century or more, they are like to be independent enough of such selection as can go on in that time, and in that sense Professor Thurstone is justified in his thesis. Nor am I one to deny " reality " to any quality merely because it has been produced by selection, and may not abide for all time. 11. Parallel proportional profiles. A method which, like Thurstone's simple structure, is meant to enable us to arrive at factors which are real entities, or to check whether our hypotheses about the factor composition of tests are correct, has been put forward by R. B. Cattell (1944&, 1946), and has interesting possibilities which its author will no doubt develop. The essence of his idea is that " if a factor is one which corresponds to a true functional unity, it will be increased or decreased 'as a whole '," and therefore if the same tests are given under 296 THE FACTORIAL ANALYSIS OF HUMAN ABILITY two different sets of circumstance, which favour a certain factor more in one case and less in the other, the loadings of the tests in that factor should all change in the same pro- portion. Experimental trials of this principle may be ex- pected soon from its author. Among " different circum- stances " he mentions different samples of subjects, differ- ing, say, in age or sex, and different methods of scoring, or different associated tests in the battery. But he prefers another kind of change of circumstance ; namely a change " from measures of static, inter-individual differences to measures from other sources of differences in the same variables." He instances, among his examples, inter- correlating changes in scores of individuals with time, or intercorrelating differences of scores in twins. We may thus have two, or several, centroid analyses, and the mathe- matical problem is to find rotations which will leave the profile of loadings of a certain factor similar in all the factor matrices. It may even be that the profiles of several fac- tors could be made similar. These factors would then satisfy CattelPs requirement as corresponding to " true functional unities." The necessary modes of calculation to perform these rotations have not yet been more than adumbrated, however. 12. Estimation of oblique factors. In applying the method of section 2 of Chapter VII (pp. 107-10) to oblique factors, it is important to note that we must use, below the matrix of correlations of the tests, in a calculation like that on page 108, the matrix of correlations of the primary factors with the tests. These are the elements of the structure on the primary factors, F(A')~ 1 D, given at the top of page 278, transposed so that columns become rows and vice versa. It would not do to use the structure on the reference vectors, which is all that most experimenters content themselves with calculating. Ledermann's short cut (section 3 of Chapter VII, pp. 110-12) requires considerable modification in the case of oblique factors. See Thomson (1949) and the later part of section 19 of the Mathematical Appendix, page 378, CHAPTER, XIX SECOND-ORDER FACTORS 1. A second-order general factor. The reason why the factors arrived at in the " box " example were correlated was that large boxes tend to have all their dimensions large. There is a typical shape for a box, often departed from, yet seldom to an extreme degree. Therefore the length, breadth, and depth of a series of boxes are corre- lated, and so also are Thurstone's primary factors in such a case. There is a size factor in boxes, a general factor which does not appear as a first-order factor (those we have been dealing with) in Thurstone's analysis, but causes these primary factors to be correlated. Possibly, therefore, when oblique factors appear in the factorial analysis of psychological tests, there is a hidden general factor causing the obliquity. This factor or factors (for there might be more than one) can be arrived at by analys- ing the first-order factors, into what Thurstone calls second-order factors, factors of the factors. Of course, whether such a procedure could be justified by the reliability of the original experimental data is very doubtful in most psychological experiments. The super- structure of theory and calculation raised upon those data is already, many would urge, perhaps rather top-heavy, and to add a second storey unwise. But we should not, I think, let this practical question deter us from examining what is undoubtedly a very interesting and illuminating suggestion, which may turn out to be the means of recon- ciling and integrating various theories of the structure of the mind. If we take the primary factors of our " box " example of Chapter XVIII, they were correlated as shown in this matrix : P.A.IO* 297 298 THE FACTORIAL ANALYSIS OF HUMAN ABILITY 1 506 150 506 1 164 150 164 1 If we analyse these in their turn into a general factor and specifics we obtain, using the formula g saturation ~ the saturations of the primary factors with a second-order g as 680, -744, and -220 ; and each primary factor will also have a factor specific. We have now replaced the analysis of the original tests into three oblique factors by an analysis into four orthogonal factors, one of them general to the oblique factors and presumably also general to the original tests, though that we have still to inquire into. We must also inquire into the relationship of the specifics of the original tests to these second-order factors, which are no longer in the original three-dimensional common-factor space, but in a new space of four dimen- sions. Are the original test-specifics orthogonal to this new space ? With only three oblique factors, an analysis into one g is always possible (except in the Hey wood case, which will often occur among oblique factors). If there had been four or more oblique factors, we would have had to use more second-order general factors unless the tetrad-differences were zero. Thurstone's " trapezium " example already referred to had four oblique factors, and his article should be consulted by the interested. 2. Its correlations with the tests. Let us turn now to the question what the correlations are between the seven original tests and the above second-order g. To obtain these Thurstone uses an argument equivalent to the fol- lowing : We may first note that each reference vector makes an acute angle with its own primary factor, but is at right angles to every other primary factor, for these are all contained in the hyperplane to which it is orthogonal. SECOND -ORDER FACTORS 299 The cosines of the angles can be obtained by premultiplying the rotation matrix of the reference vectors by the trans- pose of the rotation matrix of the primary factors. Correlations between Primary Factors and Reference Vectors DA' 1 X A =D 797 -400 -453 835 -187 -517 503 -843 -192 405 -464 -338 426 -076 -916 809 --883 -215 860 858 988 These cosines in the diagonal of the matrix D give us the angles 31, 31, and 11 which we have already mentioned on page 280 as the angles between each primary factor and its own reference vector. Each row of the first of the above matrices represents the projections of the primary factor on to the orthogonal centroid axes. These are, in fact, the loadings of the prim- ary factors, thought of as imaginary or possible tests, in the orthogonal centroid factors I, II, and III. Following Thurstone, we add these three rows below the seven rows of our original seven real tests, extending the matrix F in length thus : ra I // III r * 1 449 -682 165 211 2 825 478 129 574 3 906 336 020 787 4 846 133 457 666 wanted 5 808 208 412 719 6 697 336 335 597 7 767 173 468 683. L 797 400 453 6801 B 835 187 -517 744 r known D 503 843 192 220 J This lengthened matrix we want to post-multiply by a column vector (<| in Thurstone's notation) to give the 300 THE FACTORIAL ANALYSIS OF HUMAN ABILITY correlations of the tests, including the imaginary tests L, B, and D 9 with the second-order g. In other words, we want to know by what weights each column must be mul- tiplied so that their weighted sum is the correlation of each row with g. Suppose these weights are u, v, and w. Since we already know from our second- order analysis what r g is for each of the primaries L, B, and D, we have three equations for u, v, and w, the solution of which gives us their values. We have 797u + -400u + -4530) = -680 835w + -1870 -5I7w -744 503^ -843a + -192w = -220 and these equations can be solved in the usual way, if the reader wishes. The values are -798, '198, and -077. A closer examination of them, however, which can be most readily expressed in matrix notation, leads to an easier plan especially desirable if the number of primary factors were greater. In matrix form the above equations are T<l* = r, whence ^ = T l r g and since T is merely a short notation for DA" 1 we have - AD-V, That is to say, the centroid loadings F of the seven tests have to be post-multiplied by this, giving a matrix (a single column) F<]; = FAD'V, But FA we already know. It is (see page 276) the simple structure V on the reference vectors. So we merely have to multiply the columns of V by D~ 1 r g and add the rows to get the correlation of each test with g. These multipliers are, that is to say : 680 -860 = -791 744 -858 = -867 220 -983 = -224 The results are the same as by the former method, except SECOND-ORDER FACTORS 301 for discrepancies due to rounding off decimals, and are given to the right of the preceding table. 3. A g plus an orthogonal simple structure. In his own examples Thurstone has not calculated the loadings of the original tests' with the other orthogonal second-order factors, the factor specifics. This can, however, clearly be done by the same method as above. Since the correlations of the general factor with the three oblique factors are 680, -744, and -220, the correlations of each factor specific with its own oblique factor are -733, -668, and -975. For example, 733 2 = 1 -680*. The second-order analysis therefore is : 680 744 220 733 668 975 E Dividing the rows by the divisors already mentioned, viz. 860, -858, and -983, we obtain the matrix 791 867 224 853 779 992 and when the matrix V is post-multiplied by this we obtain the following analysis of the original seven tests into a general factor plus an orthogonal simple structure of three factors : General Factor plus Simple Structure G - VD^E g X P 8 1 211 021 009 805 2 574 022 358 683 3 787 449 333 006 4 666 656 001 260 5 719 071 588 -006 6 597 593 041 000 7 683 005 609 000 302 THE FACTORIAL ANALYSIS OF HUMAN ABILITY The zero or very small entries in X, (3, and 8 are in the same places as they are for I/, B', and D' in the oblique simple structure V (see page 276). What we have now done is to analyse the box data into four orthogonal factors corresponding to size, and ratios of length, breadth, and depth. In terms of our pyramidal geometrical analogy we have " taken out a general factor " by depress- ing the ceiling of our room, squashing the pyramid down until its three plane sides are at right angles to each other. The above structure, being on orthogonal factors, is also a pattern, so that the inner products of its rows ought to give the correlation coefficients with the same accuracy, if we have kept enough decimal places in our calculations, as do the rows of the centroid analysis F : and so they do. For example, the correlation between Tests 1 and 2 is, from F, 449 X -825 + -682 X -478 -165 X -129 -^ -675 and from G it is 211 X -574 + -021 X -022 + -009 X -358 +-805 X -683 --675 The " experimental " value was -728, the difference of 053 being due to the inaccuracy of the guessed com- munalities, or in an actual experimental set of data to sampling error and to the rank of the matrix not being exactly three. We can see here a distinct step towards a reconciliation between the analyses of the Spearman school and those of Thurstone using oblique factors. But we must not forget that if the oblique factors are not oblique enough, the Heywood embarrassment will occur, and a second- order g be impossible. The orthogonal factors of G are more convenient to work with statistically, but it is possible that the oblique factors of V are more realistic both in our artificial box example and in psychology. They corre- sponded in our case to the actual length, breadth, and depth of the boxes. The factors X, (3, and 8 of matrix G correspond to these dimensions after the boxes have all been equalized in " size." CHAPTER XX THE SAMPLING OF BONDS 1. Brief statement of views. The purpose of this chapter is to give an account of the author's own views as to the meaning of " mental factors." This can perhaps be done most clearly by first expressing them somewhat emphati- cally and crudely, and afterwards adding the details and conditions which a consideration of all the facts demands. In brief, then, the author's attitude is that he does not believe in factors if any degree of real existence is attributed to them ; but that, of course, he recognizes that any set of correlated human abilities can always be described mathematically by a number of variables or " factors," and that in many ways, among which no doubt some will be more useful or more elegant or more sparing of unneces- sary hypotheses. But the mind is very much more com- plex, and also very much more an integrated whole, than any naive interpretation of any one mathematical analysis might lead a reader to suppose. Far from being divided up into " unitary factors," the mind is a rich, comparatively undifferentiated complex of innumerable influences on the physiological side an intricate network of possibilities of intercommunication. Factors are fluid descriptive mathematical coefficients, changing both with the tests used and with the sample of persons, unless we take refuge in sheer definition based upon psychological judg- ment, which definition would have to specify the particular battery of tests, and the sample of persons, as well as the method of analysis, in order to fix any factor. Two experimental observations are at the bottom of all the work on factors, the one that most correlations between human performances are positive, the other that square tables of correlation coefficients in the realm of mental measurement tend to be reducible to a low rank by suitable diagonal elements. The first of these (i.e. the predomi- 303 304 THE FACTORIAL ANALYSIS OF HUMAN ABILITY nance of positive correlations) appears to be partly a mathematical necessity, and partly due to survival value and natural selection. The second (i.e. the tendency to low rank) is a mathematical necessity if the causal back- ground of the abilities which are correlated is comparatively without structure, so that any sample of it can occur in an ability. This enables one to say that the mind works as if it were composed of a smallish number of common faculties and a host of specific abilities ; but the phenomenon really arises from the fact that the mind is, compared with the body, so Protean and plastic, so lacking in separate and specialized organs. 2. Negative and positive correlations.* The great major- ity of correlation coefficients reported in both biometric and psychological work are positive. This almost certainly represents an actual fact, namely that desirable qualities in mankind tend to be positively correlated ; for though reported correlations may be selected by the unconscious prejudices of experimenters, who are usually on the look- out for things which correlate positively, yet as those who have tried know, it is really very difficult to discover negative correlations between mental tests. Besides, even in imagination we cannot make a race of beings with predominantly negative correlations. A number of lists of the same persons in oi*der of merit can be all very like one another, can indeed all be identical, but they cannot all be the opposite of one another. If Lists a and b are the inverse of one another, List c, if it is negatively correlated with a, will be positively correlated with b. Among a number n of variates, it is logically possible to have a square table of correlation coefficients each equal to unity ; that is, an average correlation of unity. But the farthest the average correlation can be pushed in the negative direction is l/(n 1). That is, if n is large, the average correlation can range from + 1 to only very little below zero. Even Mother Nature, then, by. natural selection or by any other means, could not endow man * This section refers to correlations between tests. The greater frequency of negative correlations between persons has already been discussed in Chapter XIII, Section 8. THE SAMPLING OF BONDS 305 with abilities which showed both many and large negative correlations. If they were many, they would have to be very small ; if they were large, they would have to be very few. Natural selection has probably tended, on the whole, to favour positive correlations within the species.* In the case of some physical organs it is obvious that a high positive correlation is essential to survival value for example, between right and left leg, or between legs and arms. In these cases of actual paired organs, however, it is doubtless more than a mere figure of speech to speak of a common factor as the cause. Between organs not simply related to one another, as say eyes and nose, natural selection, if it tended towards negative correlation, would probably split the genus or species into two, one relying mainly on eyesight, the other mainly on smell. Within the one species, since it is mathematically easier to make positive than negative correlations, it seems likely that the former would largely predominate. To say that this was due to * An important kind of natural selection is the selection of one sex by the other in mating. Dr. Bronson Price (1936) has pointed out that positive cross -correlation in parents will produce positive correla- tion in the offspring Price further shows that this positive cross- correlation in the parents will result if the mating is highly homo- gamous for total or average goodness in the traits, a conclusion which, it may be remarked here, can be easily seen by using the pooling square described in our Chapter VI. Price concludes : " The intercorrelations which g has been presumed to illumine are seen primarily as consequences of the social and therefore marital importance which has attached to the abilities concerned." Price in his argument makes use of formulae from Sewall Wright (1921). M. S. Bartlett, in a note on Price's paper (Bartlett, 19376), develops his argument more generally, also using Wright's formulae, and says : " Price contrasts the idea of elementary genetic components with factor theories. ... It should, however, be pointed out that a statistical interpretation of such current theories can be and has been advocated. Thomson has, for example, shown . . .", and here follows a brief outline of the sampling theory. " On the basis of Thomson's theory," Bartlett adds, " I have pointed out (Bartlett, 1937a) that general and specific abilities may naturally be defined hi terms of these components, and that while some statistical interpretation of these major factors seems almost inevitable, this may not in itself render their conception invalid or useless." 300 THE FACTORIAL ANALYSIS OF HUMAN ABILITY a general factor would be to hypostatize a very complex and abstract cause. To use a general factor in giving a description of these variates is legitimate enough, but is, of course, nothing more than another way of saying that the correlations are mainly positive if, as is the case, most people mean by a general factor one which helps in every case, not an interference factor which sometimes helps and sometimes hinders. 3. Low reduced rank. It is, however, on the tendency to a low reduced rank in matrices of mental correlations that the theory of factors is mainly built. It has very much impressed people to find that mental correlations can be so closely imitated by a fairly small number of common factors. Ignoring the host of specific factors to which this view commits them, they have concluded that the agreement was so remarkable that there must be some- thing in it. There is ; but it is almost the opposite of what they think. Instead of showing that the mind has a definite structure, being composed of a few factors which work through innumerable specific machines, the low rank shows that the mind has hardly any structure. If the early belief that the reduced rank was in all cases one had been confirmed, that would indeed have shown that the mind had no structure at all but was completely undiffcr- entiatcd. It is the departures from rank 1 which indicate structure, and it is a significant fact that a general tendency is noticeable in experimental reports to the effect that batteries do not permit of being explained by as small a number of factors in adults as in children, probably because in adults education and vocation have imposed a structure on the mind which is absent in the young.* By saying that the mind has little structure, nothing derogatory is meant. The mind of man, and his brain, too, are marvellous and wonderful. All that is meant by the absence of structure is the absence of any fixed or strong linkages among the elements (if the word may for a moment be used without implications) of the mind, so that any sample whatever of those elements or components can be assembled in the activity called for by a " test," * See also Anastasi, 1936, THE SAMPLING OF BONDS 307 Not that there is any necessity to suppose that the mind is composed of separate and atomic elements. It is pos- sibly a continuum, its elements if any being more like the molecules of a dissolved crystalline substance than like grains of sand. The only reason for using the word " elements " is that it is difficult, if not impossible, to speak of the different parts of the mind without assuming some 66 items " in terms of which to think. For concreteness it is convenient to identify the elements, on the mental side, with something of the nature of Thorndike's " bonds," and on the bodily side with neurone arcs ; in the remainder of this chapter the word " bonds " will be used. But there is no necessity beyond that of convenience and vividness in this. The " bonds " spoken of may be identified by different readers with different entities. All a "bond " means, is some very simple aspect of the causal background. Some of them may be inherited, some may be due to education. There is no implication that the combined action of a number of them is the mere sum of their separate actions. There is no commitment to " mental atomism." If, now, we have a causal background comprising in- numerable bonds, and if any measurement we make can be influenced by any sample of that background, one measurement by this sample and another by that, all samples being possible ; and if we choose a number of different measurements and find their intercorrelations, the matrix of these intercorrelations will tend to be hierarchical, or at least tend to have a low reduced rank. This has nothing to do with the mind : it is simply a mathematical necessity, whatever the material used to illustrate it. 4. A mind with only six bonds. We shall illustrate this fact first by imagining a " mind " which can form only six " bonds," which mind we submit to four " tests " which are of different degrees of richness, the one requiring the joint action of five bonds, the others of four, three, and two respectively (Thomson, 19276). These four tests will (when we give them to a number of such minds) yield correlations with one another. For we shall suppose the 308 THE FACTORIAL ANALYSIS OF HUMAN ABILITY different minds not all to be able to form all six of the possible bonds, some individuals possessing all six, others possessing smaller numbers. We have only specified the richness of each test, but have not said which bonds form each ability. There may, therefore, be different degrees of overlap between them, though some will be more frequent than others if we form all the possible sets of four tests which are of richness five, four, three, and two. If we call the bonds a, 6, c, d, e 9 and /, then one possible pattern of overlap would be the following : esi nun ttd" 1 a b c d e 2 b C d e 3 t t d e 4 9 f C d , If we for further simplicity suppose these bonds to be equally important, and use the formula Correlation = geometrical mean of the two totals we can calculate the correlations which these four tests would give, namely : 1 2 4 A/20 2 A/15 2 A/10 2 4 3 2 4 2 A/20 2 A/1 5 2 A/12 1 A/10 2 A/8 1 V12 2 V8 A/6 and we notice that all three tetrad-differences are zero. However, if we picked our four tests at random (taking THE SAMPLING OF BONDS 309 care only that they were of these degrees of richness) we would not always or often get the above pattern : in point of fact, we would get it only 12 times in 450. Nevertheless, it is one of the most probable patterns. In all, 78 different patterns of the bonds are possible always adhering to our five, four, three, and two the probability of each pattern ranging from 12 in 450 down to 1 in 450. One of the two least-probable patterns is the following : f f Test Bonds 1 a b c d e 2 a b c f t 3 t f t d e 4 . . d e This pattern gives the correlations : 1 2 3 4 This time the tetrads are not zero, but 2 4 1 2 3 4 3 A/20 2 2 3 2 2 V 10 2 A/20 1 VI 5 1 2 A/* 2 V6 A/120 6 A/120 It is possible in this way to calculate the tetrad-differences for each one of the 78 possible patterns of overlap which can occur. When we then multiply each pattern by the expected frequency of its occurrence in 450 random choices of the four tests, we get 450 values for each tetrad- difference, distributed as follows : 310 THE FACTORIAL ANALYSIS OF HUMAN ABILITY Values of F x V 120 Frequency of F, F, F* 2 8 7 4 6 8 14 5 9 2 6 4 27 34 23 3 6 12 30 2 75 72 48 1 61 66 72 99 54 81 ^ 56 78 36 2 67 42 42 3 16 30 60 -4 30 36 18 -5 -6 4 12 18 450 450 450 Although the distribution of each F about zero is slightly irregular, the average value of each F is exactly zero. For F! the variance is c2 = _2,164 = . 04() 120 X 450 We see, then, that in this universe of very primitive- minded men, whose brains can form only six bonds, four tests which demanded respectively five, four, three, and two bonds would give tetrad-differences whose expected value would be zero, the values actually found being grouped around zero with a certain variance. There is no particular mystery about the four " richnesses " five, four, three, and two, by the way. We might have taken any four " richnesses " and got a similar result. If we per- formed the still more laborious calculation of taking all possible kinds of four tests, we should have obtained again a similar result. If there are no linkages among the bonds, the most probable value of a tetrad-difference will always THE SAMPLING OF BONDS Bll be zero ; and if all possible combinations of the bonds are taken, the average of all the tetrad-differences will be zero. With only six bonds in the " mind," however, the scatter on both sides of zero will be considerable, as the above value of the standard deviation of F^ shows, viz. or = ^/-040 = -20 5. A mind with twelve bonds. But as the number of bonds in the mind increases, the tetrad-differences crowd closer and closer to zero. Let us, for example, suppose exactly the same experiment as above conducted in a universe of men whose minds could form twelve bonds (instead of six), the four tests requiring ten, eight, six, and four of these (instead of five, four, three, and two) (Thom- son, 19276). This increase in complexity enormously increases the work of calculating all the possible patterns of overlap, arid the frequency of each. There are now 1,257 different square tables of correlation coefficients and still more patterns of overlap, some of which, however, give the same correlations. When each possibility is taken in its proper relative frequency (ranging from once to 11,520 times) there are no fewer than 1,078,110 instances required to represent the distribution. They have, nevertheless, all been calculated, and the distribution of F l was as follows : V1920 VI 920 VI 920 VI 920 *l Freq. *! Freq. *! Freq. *i Freq. 20 225 7 17,760 3 31,432 - 13 624 18 1,800 6 74,392 4 72,676 - 14 3,792 16 1,755 5 15,744 . PJ 53,808 - 15 4,144 15 4,600 4 52,085 6 49,328 - 16 3,970 14 3,840 3 121,608 7 21,240 - 18 112 12 19,610 2 42,384 - 8 41,951 19 456 11 10,632 1 28,096 9 5,896 - 20 584 10 8,360 122,699 10 29,184 -24 28 9 26,696 1 63,024 11 8,960 8 37,735 -2 81,208 12 15,672 Total 1,078,110 This table again gives an average value of F^ exactly equal to zero. But the separate values of the tetrad- 312 THE FACTORIAL ANALYSIS OF HtJMAN ABILITY difference are grouped more closely round zero than before, with a variance now given by 1,920 X 1,078,110 This is rather less than half the previous variance. Doubling the number of bonds in the imagined mind has halved the variance of the tetrad-differences. If we were to increase the number of potential bonds supposed to exist in the mind to anything like what must be its true figure, we would clearly reach a point where the tetrad- differences would be grouped round zero very closely indeed. The principle illustrated by the above concrete example can be examined by general algebraic means, and the above suggested conclusion fully confirmed (Mackic, 1928a, 1929). It is found that the variance of the tetrad-differ- ences sinks in proportion to lj(N 1), where N is the number of bonds, when N becomes large, and the above example agrees with this even for such small N's as 6 and 12: for 6 1 12 X -040 = -018 as found. In this mathematical treatment, bonds have been spoken of as though they were separate atoms of the mind, and, moreover, were all equally important. It is probably quite unnecessary to make the former assumption, which may or may not agree with the actual facts of the mind, or of the brain. Suitable mathematical treatment could probably be devised to examine the case where the causal background is, as it were, a continuum, different proportions of it forming tests of different degrees of richness. And as for the second assumption, it is in all likelihood merely formal. Let the continuum be divided into parts of equal importance, and then the number of these increased and their extent reduced, keeping their importance equal. What is necessary, to give the result that zero tetrads are so highly probable, is that it be possible to take our tests with equal ease from any part of the causal background ; that THE SAMPLING OF BONDS 313 there be no linkages among the bonds which will disturb the random frequency of the various possible combinations ; in other words, that there be no " faculties " in the mind. And it is also necessary that all possible tests be taken in their probable frequency. In any actual experiment, of course, it is quite imprac- ticable to take all possible tests, which are indeed infinite in number. A sample of tests is taken. If this sample is large and random, then there should, in a mind without separate u faculties," without linkages between its bonds, be an approach to zero tetrads. The fact that this ten- dency attracted Professor Spearman's attention, and was sufficiently strong to make him at first believe that all samples of tests showed it, provided care was taken to avoid tests so alike as to be almost duplicates (which would be " statistical impossibilities " in a random sample), indicates that the mind is indeed very free to use its bonds in any combination, that they are comparatively unlinked. 6. Professor Spearmari s objections to the sampling theory. A theory very similar to that of the sampling theory (but, as will be explained, with an entirely different meaning of sampling) had previously been considered by Professor Spearman (Spearman, 1914, 109 footnote), but had been dismissed by him because it would give a correla- tion between any two columns of the correlation matrix equal to the correlation between the two variates from which the columns derived, both of which correlations (he added) would on this theory average little more than zero (see also Spearman, 1928, Appendices I and II). A further objection raised by him (Abilities, 96) is that the " doctrine of chance," as he calls the sampling theory, would cause every individual to tend to equality with every other individual, than which, as he said, anything more opposed to the known facts could hardly be imagined. These conclusions, however, have been deduced from a form of sampling, if it can be called sampling, which differs from that proposed by the present writer in the sampling theory. In the " doctrine of chance " discussed by Spear- man, .each ability is expressed by an equation containing every one of the elementary components or bonds, each 314 THE FACTORIAL ANALYSIS OF HUMAN ABILITY with a coefficient or loading (see Thomson, 19356, 76; and Mackie, 1929, 30). The different abilities differ only in the loadings of the " bonds," and although some of these may be zero, the number of such zero loadings is insignificant. But the sampling theory assumes that each ability is composed of some but not all of the bonds, and that abilities can differ very markedly in their " richness," some needing very many " bonds," some only few. It further requires some approach to " all-or-none " reaction in the " bonds" ; that is, it supposes that a bond tends either not to come into the pattern at all, or to do so with its full force. This does not seem a very unnatural assumption to make. It would be fulfilled if a " bond " had a threshold below which it did not act, but above which it did act ; and this property is said to characterize neurone arcs and patterns. When this form of sampling is assumed and it is submitted that this is the normal meaning of sampling then neither do the correlations become zero with an infinity of bonds, nor men equal ; but the rank of the correlation matrix tends to be reducible to a small number, if all possible correlations are taken, and finally to be one as the bonds increase without limit. It is important to realize what is meant by the rank tending to rank 1 as more and more of the possible corre- lations are taken. When the rank is 1 the tetrad- differences are zero. But clearly, the reader may say, taking more and more samples of the bonds to form more and more tests will not change in any way the pre-existing tetrad-differences, will not make them zero if they are not zero to start with. That is perfectly true ; but that is not what is meant. As more and more tests are formed by samples of the bonds, the number of zero and very small tetrads will increase and swamp the large tetrads. The sampling theory does not say that all tetrads will be exactly zero, or the rank exactly 1. It says that the tetrads will be distributed about zero (not because each is taken both plus and minus, but when all are given their sign by the same rule) with a scatter which can be reduced without limit, in the sense that with more bonds the pro- THE SAMPLING OF BONDS 315 portion of large tetrads becomes smaller and smaller ; always provided all possible samples are taken, i.e. that the family of correlation coefficients is complete. With a finite number of tests this, of course, is not the case ; but if the tests are a random sample of all possible tests, there will again be the approach to zero tetrads. The same will be true if the tests are sampling not the whole mind, but some portion of it, some sub-pool of our mind's abilities. If we stray from this pool and fish in other waters, we shall break the hierarchy ; but if we sampled the whole pool of a mind, we should again find the tendency to hierarchical order. If the mind is organized into sub- pools (such as the verbal sub-pool, say), then we shall be liable to fish in two or three of them, and get a rank of 2 or 3 in our matrix, i.e. get two or three common factors, in the language of the other theory. 7. Contrast with physical measurements. The tendency for tetrad-differences to be closely grouped around zero appears to be stronger in mental measurements than else- where ; stronger, for example, than in physical measure- ments (Abilities, 142-3). In the comparisons which have been made, there has been some injustice done to the physical distributions ; for diagrams have been published showing all the larger tetrads lumped together on to a small base so as to make the distribution look actually U-shaped. If, however, equal units are used throughout, the tetrad-differences are seen to be distributed here also in a bell-curve centred on zero (Thomson, 1927a),* though with a variance a good deal larger than is found in mental measurements (especially, of course, when the latter have been purified of all tests which give large tetrad-differ- * In the paper quoted (Thomson, 1927a), the author mistakenly took each tetrad-difference with the sign obtained by beginning in every case with the north-west element. It is, however, Professor Spearman's practice to take every tetrad-difference twice, once positive and once negative. If this be done, a histogram like that on page 249 of the paper quoted becomes, of course, perfectly symmetrical. This change could be made throughout the paper without in any way affecting its main argument. The figure on page 249 (Thomson, 1927a) should be compared with that on page 143 of The Abilities of Man, 316 THE FACTORIAL ANALYSIS OF HUMAN ABILITY ences !). In spite of the difficulty of arriving, therefore, at a fair judgment with such evidence, it seems nevertheless likely that physical measurements do indeed show a weaker tendency to zero tetrads. For the tendency to zero tetrads, outlined above, due to the measurements sampling a complex of many bonds, will show itself only when the measurements in a battery are a fairly random sample of all the measurements which might be made. Now, in physical measurements this is not the case. We do not measure a person's body just from anywhere to anywhere. We observe organs and measure them leg, cranium, chest girth, etc. The variatcs are not a random sample of all conceivable variates. In other words, the physical body has an obvious structure which guides our measurements. The background of innumerable causes which produce just this particular body which is before us cannot act in all directions, but only in linked patterns. The tendency to zero tetrad-differences in the mind is due to the fact that the mind has, comparatively speaking, no organs. We can, and do, measure it almost from anywhere to anywhere. No test measures a leg or an arm of the mind ; every test calls upon a group of the mind's bonds which intermingles in most complicated ways with the groups needed for other tests, without being a set pattern immutably linked into an organ. Of all the conceivable combinations of the bonds of the mind we can, without great difficulty, take a random sample, whereas in physical measurements we take only the sample forced on us by the organs of the body. Being free to measure the mind almost from anywhere to anywhere, we can get a set of measure- ments which show " hierarchical order " without overgreat trouble. We can do so because the mind is so compara- tively structureless. Mental measurements tend to show hierarchical order, and to be susceptible of mathematical description in terms of one general factor and innumerable specifics, not because there are specific neural machines through which its energy must show itself, but just exactly because there are no fixed neural machines. The mind is capable of expressing itself in the most plastic and Protean way, especially before education, language, the subjects of THE SAMPLING OF BONDS 317 the school curriculum, the occupation, and the political beliefs of adult life have imposed a habitual structure on it. It is not without significance that the " factor " most widely recognized after Spearman's g is the verbal factor v, the mother-tongue being, as it were, the physical body of the mind, its acquired structure. 8. Interpretation of g and the specifics on the sampling theory. We saw in Chapter III that the fraction express- ing the square of the saturation of a test with g expresses in the sampling theory the fraction of the whole mind, or of the sub-pool of the mind, which that test forms. If the hierarchical battery is composed of extremely varied tests, which cover very different aspects of the mind's activity, this fraction may be taken as being of the whole mind of the whole mind, that is, of an ideal man who can perform all of these tests perfectly, and all others which can extend their hierarchy. When we estimate a person's g, from such a battery, we are deducing a number which expresses how far that person is above or below average in the number of these bonds which his mind can form. This interpretation of g agrees well with an opinion arrived at, from quite another line of approach, by E. L. Thorndike, who on and near page 415 of his Measurement of Intelligence enunciates what has been called by others the Quantity Hypothesis of intelligence that one mind is more intelli- gent than another simply because it possesses more inter- connections out of which it can make patterns. The difference in point of view between the sampling theory and the two-factor theory is that the latter looks upon g as being part of the test, while the former looks upon the test as being part of g. The two-factor theory is therefore compelled to postulate specific factors to account for the remainder of the variance of the test, and has to go on to offer some suggestion as to what specific factors are perhaps neural engines. The sampling theory simply says that the test requires only such and such a fraction of the bonds of the whole mind the same fraction which, on the two-factor theory, g forms of the variance of the test. For it, specific factors are mere figments, which do not arise unless, as can be done, the mathematical 318 THE FACTORIAL ANALYSIS OF HUMAN ABILITY equations which represent the tests are so manipulated that there appears to be only one link connecting them all. The sampling theory does not make this transformation of the equations (see Appendix, paragraph 6). Those who do so, if they adhere to the interpretation that g means all the bonds of the whole mind, have to suppose that the whole mind first takes part in each activity, but that in addition a specific factor is concerned ; which specific factor, since they have already invoked the whole mind, must be for them a second action of part of the mind annulling its former assistance which is absurd. The two-factor equations then do not allow us to consider g as being all the bonds of the mind. They are mathematically equivalent to the sampling equations, but not psychologically or neurologi- cally. To the holder of the sampling theory, the factors of the other view are statistical entities only, g an average (or a total) of all a man's bonds, a specific factor the contrast between performance in any particular test and a person's general ability (Bartlett, 1937, 101-2). As a manner of speaking, the two-factor theory appears to the author to be much more likely to " catch on " with the man in the street, but much more likely to lead to the hypostatization of mere mathematical coefficients. The sampling theory lacks the good selling-points of the other, but is comparatively free from its dangers, and seems much more likely to come into line, in due time, with physio- logical knowledge of the action of the nervous system. 9. Absolute variance of different tests. It will be noted, too, that on the sampling theory the different tests will naturally have different variances, the " richer " tests having a wider scatter. This seems only natural. It is customary, at any rate in theoretical discussions, to reduce all scores in different tests to standard measure, thereby equalizing their variance. This seems inevitable, for there is no means of comparing the scatter of marks in two different tests. But it does not follow that the scatter would be really the same if some means of comparison were available. When the same test is given to two different groups we have no hesitation in ascribing a wider variance to the one or the other group, and it seems con- THE SAMPLING OF BONDS 319 ceivable that a similar distinction might mentally be made between the scores made by one group in two different tests. The writer is completely in accord with M. S. Bart- lett when he says (Bartlett, 1935, 205) : " I think many people would agree . . . that the variation in mathematical ability displayed even in a selected group such as Cam- bridge Tripos candidates cannot be altogether put down to the method of marking adopted by the examiners." We may put these mathematics marks into standard measure, and we may put the marks scored by the same group in, say, a form-board test, also into standard measure. But that does not imply that at bottom the two variances are equal, if only we had some rigorous way of comparing them. Our common sense tells us plainly that they are not equal in the absolute sense, though for many purposes their difference is irrelevant. It seems to be no defect, then, but rather a good quality, of the sampling theory to involve different absolute variances. 10. A distinction between g and other common factors. The writer is inclined, as the earlier sections of this chapter imply, to make a distinction in interpretation between the Spearman general factor g and the various other common factors, mostly if not all of less extent than g, which have been suggested. When properly measured by a wide and varied hierarchical battery, g appears to him to be an index of the span of the whole mind, other common factors to measure only sub-pools, linkages among bonds. The former measures the whole number of bonds ; the latter indicate the degree of structure among them. Some of this " structure " is no doubt innate ; but more* of it is probably due to environment and education and life. Its expression in terms of separate uncorrelated factors suggests what is almost certainly not the case, that the " sub-pools " are separate from one another. The actual organization is likely to be much more complicated than that, and its categories to be interlaced and inter- woven, like the relationships of men in a community, plumbers and Methodists, blonds, bachelors, smokers, conservatives, illiterates, native-born, criminals, and school-teachers, an organization into classes which cut 320 THE FACTORIAL ANALYSIS OF HUMAN ABILITY across one another right and left. No doubt these too could be replaced, and for some purposes replaced with advantage, by a smaller number of uncorrelated common factors and a large number of factors specific to plumbers, smokers, and the rest. But the factors would be pure figments. What the factorist calls the verbal factor, for example, is something very different from what the world recognizes as verbal ability. The latter is a compound, at least of g and v 9 and possibly of other factors. The v of the factorist is something uncorrelated withg, something which the person of low g is just as likely to have as the person with high g. But this is only so long as factors are kept orthogonal. The acceptance of oblique factors, as in Thurstone's system, would change this, and be in accordance with popular usage. Further, it is improbable that the organization of each mind is the same. The phrase " factors of the mind " suggests too strongly that this is so, and that minds differ only in the amount of each factor they possess. It is more than likely that different minds perform any task or test by different means, and indeed that the same mind does so at different times. Yet with all the dangers and imperfections which attend it, it is probable that the factor theory will go on, and will serve to advance the science of psychology. For one thing, it is far too interesting to cease to have students and adherents. There is a strong natural desire in mankind to imagine or create, and to name, forces and powers behind the fa$ade of what is observed, nor can any excep- tion be taken to this if the hypotheses which emerge explain the phenomena as far as they go, and are a guide to further inquiry. That the factor theory has been a guide and a spur to many investigators cannot be denied, and it is probably here that it finds its chief justification. CHAPTER XXI THE MAXIMUM LIKELIHOOD METHOD OF ESTIMATING FACTOR LOADINGS (by D. N. Lawley) 1. Basis of statistical estimation. In recent times attempts have been made to introduce into factorial analysis statis- tical methods developed in other fields of research. In particular the method of statistical estimation put forward by Fisher (1938, Chapter IX), and termed the method of maximum likelihood, has been applied by Lawley (1940, 1941, 1943) to the problem of estimating factor loadings. This method has the property of using the largest amount of available information contained in the data and gives " efficient " estimates, where such exist, of all unknown parameters, i.e. estimates which, roughly speaking, are on the average nearer the true values than those obtained by other, " inefficient," methods of estimation. Before using the maximum likelihood method for esti- mating factor loadings it is necessary to make certain initial assumptions. We assume that both the test scores and the factors, of which they are linear functions, are normally distributed throughout the population of indi- viduals to be tested. This assumption of normality has been the subject of some criticism, but in practice it would appear that departure from strict normality of distribution is not very serious. It is also necessary to make some hypothesis concerning the number of general factors which are present in addition to specifics. We shall later on show how this hypothesis may be tested, and how it may be determined whether the number assumed is, in fact, sufficient to account for the data. 2. A numerical example. In order to illustrate the calcu- lations needed we shall reproduce an example used by Lawley (1943), where eight tests were given to 443 indi- F.A 11 321 322 THE FACTORIAL ANALYSIS OF HUMAN ABILITY viduals. The table below gives the correlations between the eight tests, unities having for convenience been placed in the diagonal cells. In this example the hypothesis made is that two general factors, together with specifics, are sufficient to account for the observed correlations. 1 2 3 4 5 6 7 8 1 1-000 312 405 457 500 350 521 564 2 312 1-000 460 316 279 173 339 288 3 405 460 1-000 394 380 258 433 323 4 457 316 394 1-000 460 222 516 486 5 500 279 380 460 1-000 239 -441 417 6 350 173 258 222 239 1-000 302 262 7 521 339 433 516 441 302 1-000 547 8 564 288 323 486 417 262 547 1-000 The method of estimation about to be described is one of successive approximations. Each successive step in the calculations gives a set of factor loadings which are nearer to the final values than those of the previous set. To start the process it is only necessary to guess or to find by some means (e.g. by a centroid analysis) first approxima- tions to the factor loadings. Any set of figures within reason will serve the purpose, though, of course, the better the approximation the. fewer steps in the calculation will be needed. For illustration we shall take as first approxi- mations to the factor loadings the set of values given below: Tests JLllttl loading in 1 2 3 4 5 6 7 8 Factor I 73 50 66 66 62 40 73 70 Factor II 17 - -27 47 08 06 02 10 29 Specific variance -4382 -6771 -3435 -5580 -6120 -8396 -4571 -4259 Under the loadings are written the corresponding first approximations to the specific variances (the total variance of each test being taken to be unity). They are as usual found by subtracting from unity the sums of squares of the loadings for each test. The calculations necessary for obtaining second approxi- MAXIMUM LIKELIHOOD METHOD 323 mations to the loadings in factor I may now be set out as follows : (a) 1-666 -738 1-921 1-183 1-013 -476 1-597 1-644 (b) 5-647 3-895 5-132 5-129 4-830 3-100 5-647 5-412 (c) 4-917 3-395 4-472 4-469 4-210 2-700 4-917 4-712 hi = 45-724 1/&! = 0-14789 (d) -727 -502 -661 -661 -623 -399 -727 -697 The first row of figures, row (a), is found by dividing the trial loadings in factor I by the corresponding specific variances. The figures in row (b) are then given by the inner products (see footnote, page 31) of row (a) with the successive rows (or columns) of the correlation table printed above, and row (c) is obtained by subtracting from the figures in row (b) the corresponding loadings in factor I. The quantity hi is given by the inner product of rows (a) and (c), and hence, taking the square root of the reciprocal of this quantity, we find I /hi. Finally, row (d) is obtained by multiplying the figures in row (c) by l/h i9 or -14789. The resulting numbers are then second approximations to the loadings of the tests in factor I. The most direct way of obtaining second approximations to the loadings in factor II is to find the residual matrix which results from removing the effect of factor I, and to treat it in the same way as the original matrix, using this time the trial loadings in factor II. A less direct but con- siderably shorter method may, however, be obtained by using once more the original matrix and modifying the process slightly. The necessary calculations are as shown below : (e) -388 -399 1-368 -143 -098 -024 -219 -681 (/) -330 --560 --980 -150 -113 -038 -190 -580 Pl = -0234 (g) '177 278 495 -085 -068 -027 -107 -306 k\ = 1-1080 1/&! = -9500 (h) -168 --264 --470 -081 -065 -026 -102 -291 Row (e) is found by dividing the trial loadings in factor II by the corresponding specific variances, while the numbers in row (/) are given by the inner products of row (e) with the rows of the correlation table. The step by which row (g) is obtained from row (/) is a little more complicated than the corresponding step in 324 THE FACTORIAL ANALYSIS OF HUMAN ABILITY the calculations for the first- factor loadings. From each number in row (/) we subtract not only the corresponding trial loading in factor II, but also a correction which eliminates the effect of factor I ; this correction consists of the corresponding number in row (d) multiplied by 0234, the inner product of rows (e) and (d). Thus, for example, the number -177 in row (g) is equal to -330 -170 -727 X ( -0234). In general, where more than two factors are assumed to be present and where further approximations are being calcu- lated for the loadings in the rth factor, there will be (r 1) such corrections to be subtracted, one for each of the preceding factors. Having found row (g) the quantity k\ is now given by the inner product of rows (e) and (g), from which, taking the square root of the reciprocal, we derive l/A^. Row (h) is then obtained by multiplying the figures in row (g) by 1/fex, or -9500. We have thus found second approximations to the loadings in factor II. The whole cycle of calculations may now be repeated over and over again until the required degree of accuracy is reached. In practice, provided that the initial trial loadings are not too far out, one repetition of the process will usually be found sufficient. In our example the final estimates (with possible slight errors in the last decimal place) were as follows : Tests Loading in 12 3 4 5678 Factor I -725 -503 -664 -661 -623 -399 -726 -694 Factor II -172 261 --468 -087 -069 -027 -106 -291 Specific variance -445 -679 -340 -556 -607 -840 -462 -434 Having obtained these figures, there is, of course, no objection to rotating the factors as desired in order to reach a psychologically acceptable position. 3. Testing significance. A difficulty in most systems of factorial analysis is to know how many factors it is worth- while to " take out," and to decide how many of them may be considered significant. From a statistical point of MAXIMUM LIKELIHOOD METHOD 325 view objections can be raised against the majority of methods at present in use for this purpose. When, how- ever, the number of individuals tested is fairly large, the maximum likelihood method provides a satisfactory means of testing whether the factors fitted can be considered sufficient to account for the data. To illustrate this let us return to the example of the previous section. It is first of all necessary to calculate the matrix of residuals obtained when the effect of both factors is removed from the original correlation matrix. For this purpose we use the final estimates of the loadings as already given. The residual matrix, with the specific variances inserted in the diagonal cells, is as follows : 12345678 (.445) 008 -004 -037 -036 -056 024 -Oil 008 (-679) -004 -006 --016 021 -001 -015 004 -004 (-340) -004 --001 -006 -001 --002 037 -000 -004 (-556) -042 -044 -027 -002 036 -016 -001 -042 (-607) Oil -019 035 056 -021 -006 -044 -Oil (-840) -009 023 024 -001 -001 -027 019 -009 (-462) -012 Oil -015 -002 -002 -035 -023 -012 (-434) We are now able to calculate a criterion, which we shall denote by to, for deciding whether the hypothesis that only two general factors are present should be accepted or rejected. Each of the above residuals is squared and divided by the product of the numbers in the corresponding diagonal cells. Thus, for example, the residual for Tests 4 and 7 is squared and divided by the product of the fourth and seventh diagonal elements, giving the result == - 002838 - There are altogether 28 such terms, one for each residual, and w is obtained by forming the sum of these terms and multiplying it by 443, the number in the sample. The result is found to be 20-1. When the number in the sample is fairly large w is distributed approximately as ^ 2 with degrees of freedom given by J|(n m) 2 n w}, 326 THE FACTORIAL ANALYSIS OF HUMAN ABILITY where n is the number of tests and m is the assumed num- ber of factors. To test whether the above value of w is significant we now use a ^ 2 table such as is given by Fisher and Yates (1943, page 31). In our case, putting n = 8 and m = 2, the number of degrees of freedom is 13. Entering the ^ 2 table with 13 degrees of freedom, we find that the 1 per cent, significance level is 27-7. This means that if our hypothesis that only two general factors are present is correct, then the chance of getting a value of w greater than 27-7 is only 1 in 100. If, therefore, we had obtained a value of w greater than 27-7 wo should have been justified in rejecting the above hypothesis and in assuming the existence of more than two general factors. In our case, however, the value of w is only 20-1, well below the 1 per cent, significance level. We have thus no grounds for rejection, and although we cannot state that only two general factors are present, we have no reason to assume the existence of more than two. It must be emphasized that the method described above is not applicable if other, inefficient, estimates of the loadings are substituted for the maximum likelihood estimates. For the value of ^ 2 would in that case be greatly exaggerated, causing us to over-estimate its significance. For this reason we cannot, for example, use the method for testing the significance of the re- siduals left when factors have been fitted by the centroid method. 4. The standard errors of individual residuals. A method has now* been developed for finding the standard errors of individual residuals. This should be useful when a few of the residuals are very large, while the rest are small. In such a case one or more of the residuals may be highly significant, when tested individually, even though the value of x a does not attain significance. The method ignores errors of estimation of the specific variances, which are not, however, likely to be very large provided that the number of tests in the battery is not too small. Let us denote by I i9 m i the estimated loadings of the i th test in the first and second factors respectively (assuming * Lawley in the Proc. Roy. Soc. Edin., 1949. MAXIMUM LIKELIHOOD METHOD 327 the existence of only two factors). Let v i be the specific variance of the i fh test, and let us write * Then the standard error of the residual for the i th and j i} tests (i = j) is given by v where , and This formula may, of course, be easily extended to take into account any number of factors. Let us illustrate the use of the above formula with the same numerical example as before. If we wish to test the significance of the residual for the first and fourth tests after removing two factors, we have /! = -725 Z 4 - -661 h = 6-7185 Hence e = -33845 n m l = -172 v l m 4 -087 4 k = 1-0528 e u = -48329 e u = -44479 = -55551 =- -08554 and = . 196 Thus the residual in question has a value of 037 with a standard error of -020. It is clearly not significant. 5. The standard errors of factor loadings. When maxi- mum likelihood estimation has been used, we are able to find the standard errors of not only the residuals but also the estimated factor loadings. Using the same notation as in the preceding section, the sampling variance of I i9 the 328 THE FACTORIAL ANALYSIS OF HUMAN ABILITY loading of the i (h test in the first factor is (assuming the test to be standardised) and the standard error is the square root of this. The covariance between any two first factor loadings l { and / ; is given by The formulae for the variances and covariances of the subsequent factor loadings are more complex. Thus the variance of m iy the loading of the i Ut test in the second factor, is while the covariance between m l and MJ is The results for the general case, where more than two factors have been assumed present, may be written down without difficulty. Each factor will give rise to one more term within the curly brackets than the preceding factor. It should be noted that the last of such terms, and that alone, is multiplied by \. One interesting property of maximum likelihood esti- mates is that the loadings in any one factor are uncorre- lated with those in any other factor. Thus any one of I l9 Z 2 , / 3 , ..... is uncorrelated with any one of m l9 m 2 , It must be stressed that all the above results are applic- able only to the unrotated loadings. In our numerical example, we find 1+7 = 1-14884 h 1 +7 = 1-9498 k MAXIMUM LIKELIHOOD METHOD 329 Hence the variance of 1^, for example, is 1>14884 X 1-14884 X -725 a = '001810 ]l | X 1-14884 X -725 a [ [ J 443 [ while that of m^ is 1-9498 f 1 1-14884 X -725 2 J X 1-9498 X -172 2 X -001617 Thus the loading of test 1 in the first factor is -725 with a standard error of V -001 8 10 = -043 and its loading in the second factor is -172 with a standard error of V-001617 = -040 6. Advantages and disadvantages. To sum up : the chief advantage of the maximum likelihood method of estimating factor loadings is that it does lead to efficient estimates and does provide a means of deciding how many factors may be considered necessary. It unfortunately takes, however, much longer to perform than a ccntroid analysis, particularly when the battery of tests is a large one and when several factors are to be fitted. The chief labour of the process lies in the calculation of the various inner products ; although in this respect it does not differ greatly from Hotelling's method of finding " principal components." The maximum likelihood method is thus likoly to be most useful in cases where accurate estimation is desirable and where it is proposed to make a test of significance. The method also possesses the advantage of being independent of the units in which the test scores are measured. The same system of factors is therefore obtained whether the correlation or the covariance matrix is analysed. The loadings in the one case are directly proportional to those in the other. Postscript. For a more detailed exposition of Lawley's method, with checks, see Emmett (1949). P.A. 11* CHAPTER xxit SOME FUNDAMENTAL QUESTIONS IT seems advisable to conclude with a brief discussion of some of the fundamental theoretical questions needing an answer. Among these are the following, of which (1) and (3) are rather liable to be forgotten by those actually engaged in making factorial analyses : (1) What metric or system of units is to be used in factorial analysis ? (2) On what principle are we to decide where to stop the rotation of our factor-axes or how to choose them so that rotation is unnecessary ? (3) Is the principle of minimizing the number of common factors, i.e. of analysing only the communal variance, to be retained ? (4) Are oblique, i.e. correlated, factors to be permitted ? 1. Metric. Most of the work done in factorial analysis has assumed the scores of the tests to be standardized ; that is to say, in each test the unit of measure has been the actual standard deviation found in the distribution. This is in a sense a confession of ignorance. The accidental standard deviation which happens to result from the par- ticular form of scoring used in a test means, of course, nothing more. Yet there is undoubtedly something to be said for the probability of real differences of standard deviation existing between tests (see Chapter XX, Section 9). In that case, if we knew these real standard deviations, we would use variances and covariances and analyse them, not correlations (compare Hotelling, 1933, 421-2 and 509-10). Burt has urged the use of variances and covariances, which are indeed necessary to him to enable his relation between trait factors and person factors to hold (see Chap- ter XIV, page 214). But the variances and covariances he actually uses are simply the arbitrary ones which arise 330 SOME FUNDAMENTAL QUESTIONS * 881 from the raw scores, and depend entirely upon the scoring system used in each test. It would seem necessary to have some system of rational, not arbitrary, units. Hotelling has already suggested one such, based upon the idea of the principal components of all possible tests, but it would seem to be unattainable in practice (Hotel- ling, 1933, 510). Another can be based on the ideas of the sampling theory and has already been foreshadowed in Chapter XX, Section 9. Tests quite naturally have different variances on that theory, since they comprise larger or smaller samples of the " bonds " of the mind (see Thomson, 1935&, 87). In a hierarchical battery these natural variances are measured by the " coefficient of richness " (Chapter III, Section 2, page 45). The " richness " of Test k is given by r ik r jk the same quantity as the square of Spearman's " satura- tion with g." It is, on the sampling theory, the fraction which the test forms of the pool of bonds which is being sampled, and is the natural variance of the test in compari- son with other tests from that pool. The " saturation with g " of Spearman's theory is the " natural standard deviation " of the sampling theory. Even in a battery which is not hierarchical, the formula (Chapter IX, Section 5, page 154) 2 J^t T -2A will give a rough estimate of the natural standard deviation of each test. The general principle is that tests which show the most total correlation have the largest natural variance. 2. Rotation. Our views on the rotation of factors will depend on what we want them to do. Burt looks upon them as merely a convenient form of classification and is content to take the principal axes of the ellipsoids of density, or that approximation to them given by a good centroid analysis, as they stand, without any rotation. He " takes out " the first centroid factor, either by calculation or 332 THE FACTORIAL ANALYSIS OF HUMAN ABILITY by selecting a very special group of persons each of whom has in a battery of tests an average score equal to the population average, each of the tests also having the same average as every other test in tKe battery over this sub- group of persons (Burt, 1938). He concentrates attention on the remaining factors, which are " bipolar," having both positive and negative weights in the tests. When, as in the article referred to, he is analysing temperaments, this fits in well with common names for emotional charac- teristics, for those names too are usually bipolar, as brave-cowardly, extravagant-stingy, extravert-introvert, and so on. Thurstone, on the other hand, emphatically insists on the need for rotation if the factors are to have psycho- logical meaning (Thurstone, 1938a, 90). The centroid factors are mere averages of the tests which happen to form the battery, and change as tests are added or taken away, whereas he wants factors which are invariant from battery to battery. I think he would put invariance before psychological meaning, and say that if a certain factor keeps turning up in battery after battery we must ask ourselves what its psychological meaning is. His own opinion, backed up by a great deal of experimental work of a pioneering and exploratory nature, is that his principle of rotating to " simple structure " gives us also psychologically meaningful and invariant factors. The problems of rotation and metric are not unconnected, and one piece of evidence in favour of rotating to simple structure is that the latter is independent of the units used in the tests. If instead of analysing correlations we analyse co variances, with whatever standard deviations we care to assign to the tests, we get a centroid analysis quite different from the centroid analysis of correlations. But if we rotate each to simple structure the tables are identical, except, of course, that in the covariance structure each row is multiplied by the standard deviation of the test. For example, if we take the six tests of Chapter XVI, Section 4 (page 246) and ascribe arbitrary standard deviations of 1, 2, 3, 4, 5, and 6 to them, we can replace the SOME FUNDAMENTAL QUESTIONS 333 correlations and communalities by covariances and vari- ance-communalities, and perform a centroid analysis. Since we know the proper communalities* it comes out exactly in three factors with no residues, and gives the centroid structure : I II III 1 372 567 462 2 948 1-278 -060 3 1-969 1-016 -337 4 1-002 1-072 2-118 5 2-992 593 1-716 6 3-379 2-493 337 When this is rotated to simple structure, by post- multiplication by the matrix 802 -389 -453 592 -416 -691 080 822 -564 the resulting table is : 1 2 3 4 5 6 A 2-154 2-187 4-213 B 950 619 2-577 820 1-278 2-732 This is identical with the simple structure found from the correlations, if the rows here are divided by 1, 2, 3, 4, 5, and 6, the standard deviations. It is definitely a point in favour of simple structure that it is thus independent * If we have to guess communalities, our two simple structures will differ slightly because the highest covariance in a column may not correspond to the highest correlation. But with a battery of many tests this difference will be unimportant, and could be annulled by iteration. 334 THE FACTORIAL ANALYSIS OF HUMAN ABILITY of the system of units employed. Spearman's analysis of a hierarchical matrix into one g and specifics also has this property of independence of the metric. If the tetrad- differences of a matrix of correlations are zero, and we analyse into one general factor and specifics, it is immaterial whether we analyse correlations or co variances. The loadings obtained in the latter case are exactly the same except, of course, that each is multiplied by the appropriate standard deviation. At this point one is reminded of Lawley's loadings* found by the method of maximum likelihood, for these possess the property that the unrotated loadings obtained from correlations are already the same as the unrotated loadings obtained from covariances, if the latter are divided by the standard deviations. Centroid analyses, or principal component analyses, do not possess this property. The loadings obtained by these means from covariances cannot be simply divided by the standard deviations to give the loadings derived from correlations, though the one can be rotated into the other. Lawley's loadings need no such rotation. They are, as it were, at once of the same shape whether from covariances or from correlations and only need an adjustment of units, such as one makes in changing, say, from yards to feet. A field which is 50 yards broad and 20 polos long has the same shape as one which is 150 feet broad and 330 feet long. (American readers may need I do not know to be told that a pole equals 5| yards.) Now, as we have seen, this property of equivalence of covariance and correlation loadings is also possessed by simple structure. It would thus not be unnatural to hope that Lawley's method might lead straight to simple * In accordance with our definition on page 272, the term " load- ing " means a coefficient in a specification equation, an entry in a 44 pattern." In the present chapter it is used throughout and is strictly correct when the axes referred to are orthogonal. If the axes are oblique, then much of what is said really refers to the items in a structure, not in a pattern ; but the word " loading " is still used to avoid circumlocutions, and because the structure of the reference vectors is, except for a diagonal matrix multiplier, identical with the pattern of the factors. SOME FUNDAMENTAL QUESTIONS 835 structure, without any rotation. But this does not seem to be the case. If we take the known simple structure loadings of a set of correlations as trial loadings in Lawley's method, they ought to come out unchanged from his calculations, if they are the goal towards which those calculations converge ; but they don't. Clearly, then, simple structure is not the only position of the axes where the loadings are independent of the units of measurement employed. Indeed, any subsequent post-multiplication of both the simple structure tables both that from corre- lations and that from covariances by the same rotating matrix will leave their equivalence with regard to units unharmed. Simple structure is only one of an infinite number of positions which possess this property. But it is an easily identifiable one. It is difficult to keep one's mind clear as to the meaning of this. Let me recapitulate. There are some processes of analysis which, while they give a perfect analysis in the sense of one which reproduces the correlations (or the Co- variances) exactly, do not give the same analysis for the correlations as for the covariances. The factors they arrive at depend upon the units of measurement employed in the tests. Such, for example, are the principal compon- ents process and the centroid process. Such processes cannot be relied on to give, straight away and without rotation, factors which can be called objective and scien- tific. Some processes, on the other hand, do give analyses which are independent of the units. One such is Lawley's, based on maximum likelihood. Another is Thurstone's simple-structure process, which, though it begins by using a centroid analysis, follows this by rotation of a certain kind. But the principle of independence of units does not distinguish between these processes, which both satisfy it. Still less does it distinguish between systems of factors. For any one of the infinite number of such systems which can be got from either simple structure or Lawley's factors by rotation equally satisfies the principle. Indeed, there can really be no talk of a system of factors satisfying the principle. Any table of loadings whatever, obtained from 336 THE FACTORIAL ANALYSIS OF HUMAN ABILITY correlations, has, of course, corresponding to it a system differing only in that the rows are multiplied by coefficients, a system which would correspond with covariances. The fact that no one has discovered a process which gives both is irrelevant. The argument is rather as follows. If a worker believes that he has found a process which gives the true psychological factors, then that process must be independent of the metric, and simple structure and maximum likelihood are both thus independent, though they do not, alas, agree. Nor must it be forgotten that analyses from correlations are in no way superior to those from covariances. Indeed, correlations are covariances, dependent upon as arbitrary a choice of units namely standard deviations as any other. But centroid axes in themselves, or principal components, without rotation, are clearly inadmissible, for they change with the units used. The chance that such axes are the true ones is infinitesimal, being dependent on the chance composition of the battery, and the system of units which chances to be used. Independence of metric is not sufficient to validate a process, but it is necessary. Its absence does not prove a system of factors to be wrong, bat it makes it certain that the process by which they have been arrived at does not in general give the true factors. 3. Specifics. These form a fundamental problem in factorial analysis, and yet they are practically never heard of in discussions of an analysis. They are considered in Chapter VIII, where it is pointed out that although it is reasonable enough to think that a test may require some trick of the intellect peculiar to itself, yet it is not obvious that these specific factors must be made as large and important as possible ; and that is what the plan of minimizing the rank of a matrix does. The excess of factors over tests, which inevitably, of course, results from postulating a specific in every test, means that the factors cannot be estimated with any great accuracy. Usually the accuracy is very low indeed. The determinate and the indeterminate parts of each of Thurstone's factors in Primary Mental Abilities can be found by post-multiplying Table 7 on his page 98 by Table 3 on his page 96. We find : SOME FUNDAMENTAL QUESTIONS 337 Variance of the Variance of the * actor Estimated Part Indeterminate Part S . . . -611 -389 P . . . -616 -384 N . . . -825 -175 V . . . -662 -338 M . . . -431 -569 W . . . -439 -561 I . . . -397 -603 R . . . -600 -400 D . . . -519 -481 In three cases less than half of the factor variance has been estimated. The average for the nine factors is 56| per cent, of the variance estimated. In other words, the factor estimates have large probable errors, in some cases as large as the estimates themselves. This has serious consequences, not to be overcome by more reliable tests. To anyone pondering this, it seems somewhat para- doxical to be told that, if the battery of tests is large, the guess we make for the communalities does not much matter. No great change will result in the centroid loadings if unity is employed as communality, i.e. if no specifics are assumed to exist. It would seem, then, that their presence or absence is immaterial ! But is that so ? It is true that the larger the battery, the less the inde- terminacy. But Thurstono's battery above was as large as any is ever likely to be. Using unity for every diagonal element in the matrix of such a battery will give factors (supposing the same number of them to be taken out) which will not imitate the correlations quite so well, but which can be estimated accurately. In fact, whether Hotelling's process or the centroid process is used, with unit communalities, each factor can be calculated exactly for a man, given his scores. By exactly we mean that they are as accurate as his scores are. Of course, in any psychological experiment the scores may not be accurate in the sense that they can be exactly reproduced by a repetition of the experiment. Apart from sheer blunders and clerical errors, there is the fact that a man's performance fluctuates from day to day. But 338 THE FACTORIAL ANALYSIS OF HUMAN ABILITY these errors are common to any process of calculation which may be used on the scores. These are not the errors for which we are criticizing estimates of a man's factors. The point we are making is that factors based on com- munalities less than unity have a further, and large, error of estimation, whereas factors based on unit communalities (even if only one or two or a few are taken out) have no such further error of estimation (see page 79). If a few such factors taken out with unit communalities are then rotated (keeping them in the same space, i.e. not changing their number) they still remain susceptible of exact estimation in a man. But the reader may protest that he does not want these factors, for they have no psychological meaning. Better inexact estimates of the proper factors than exact estimates of the wrong ones. It may be retorted that there is no unanimity about which are the proper ones. It is for each to say what he thinks most desirable. If he is content with factors which are in the same space as the tests, he can estimate them exactly, and he can rotate them too if he keeps them in that space. But if he rotates them out of it, to make them more real, let him remember that he loses on the roundabouts exactly what he gains upon the swings, for he can only measure their projections on the test space. Such factors are like trees and houses to a man who is doomed only to see the ground. He can measure the length of a tree's shadow, but not its height. He must remain content with shadows. And so with factors. We must cither remain content with factors which are within the test space, or must reconcile ourselves to measuring only the shadows of factors outside that space. If with twenty tests we postulate twenty-five factors (five general and the twenty specifics), then we are operating in a space five of whose dimensions we know nothing about, and our factor estimates suffer accordingly. The use of communalities enables us to imitate correlations rather better, but dooms us to estimate factors badly. Such is the argument against communalities. For them is the hope that some day, despite their drawbacks, the factors they lead to may prove to be something real, SOME FUNDAMENTAL QUESTIONS 339 perhaps have some physiological basis. Their defender may plead that the estimates of these factors are as good as the estimates we find useful, in predicting educational or occupational efficiency. In the table given above, of the variances of the estimated parts of Thurstone's factors, it is the square roots of those variances which give the correlations of estimates with " fact " (though " fact " here is an elusive ghost of a thing). Those correlations do not look so bad. That for N is -908, and the worst is 630. 4. Oblique factors. I think it is pretty certain that Thurstone took to oblique factors because he wants simple structure at all costs. Certainly oblique factors make it much easier to reach simple structure too easy, Reyburn and Taylor say. It will be found far more often than it really exists, they add. On the other hand, Thurstone can point to his box example and his trapezium example and say with truth that simple structure enabled him to find " realities," can say that the oblique simple struc- ture is something more real, in the ordinary common-sense everyday use of the word, than the orthogonal second- order factors which are an alternative. Other workers, not at all wedded to the ideas of simple structure, have also declared their belief in oblique factors, e.g. Raymond Cattell, and, I think, many who feel inclined to work in terms of " clusters." In ordinary life, weight and height are both measures of something real, although they are correlated. We could analyse them into two uncorrelated factors a and &, or into three for that matter, but certainly no one would use these in ordinary life. It is, however, just conceivable that some pair of hormones (say) might be found which corresponded, not one of them to height and one to weight, but one to orthogonal factor a and another to orthogonal factor b. It is far too early to state anything more than a preference for orthogonal or oblique factors. Opinion is turning, I think, toward the acceptance of the latter. ADDENDA ADDENDA ADDENDUM (1945) TO PAGE 19 Bif actor Analysis THE following example will illustrate some of the points of this method. Consider these correlations, which to save space are printed without their decimal points : 1 2 3 4 5 6 7 8 9 10 11 12 1 57 40 45 63 63 20 28 74 52 45 34 2 57 34 25 53 39 17 44 68 43 39 56 3 40 34 18 57 27 59 16 44 70 73 20 4 45 25 18 27 51 09 12 32 22 20 15 5 63 53 57 27 42 40 26 68 67 63 31 6 63 39 27 51 42 13 18 50 34 30 23 7 20 17 59 09 40 13 08 22 60 64 10 8 28 44 16 12 26 18 08 35 21 18 43 9 74 68 44 32 68 50 22 35 56 50 44 10 52 43 70 22 67 34 60 21 56 78 25 11 45 39 73 20 63 30 64 18 50 78 23 12 34 56 20 15 31 23 10 43 44 25 23 There are two stages in a bif actor analysis. The first problem is to decide how to group the tests so that those are brought together which share a second or group factor. Then the best method of calculating is needed to find the loadings. The grouping can partly be done subjectively by con- sidering the nature of each test and putting together memory tests, or tests involving number, and so on. Holzinger uses a " coefficient of belonging," B, to determine the coherence of a group. B is equal to the average of the intercorrelations of the group divided by their average correlation with the other tests in the battery. The higher B is, the more the group is distinguishable as a group. He begins with a pair of tests which correlate highly with 343 344 ADDENDA one another, and finds their B. Then he adds a third test and finds the B of the three. Then another and another, until B drops too low. There is no fixed threshold for B, but a rather sudden drop would indicate the end of a group. Another plan is to make a graph or profile of each row of correlations and compare these (Tryon, 1939), grouping together those tests with similar profiles. I find it easier to consider only the peaks of each row and compare the rows with regard to these. If we mark, in each row of the above, the five highest correlations in that row, and also the diagonal cell, we get the following set of peaks : 8 9 10 11 12 X X XX X XXX X XXX X XXX XX X X X ? XXX XXX XX X We then see that, in the rows, (a) Tests 3, 7, 10, 11 have identical peaks, (b) 2, 8, 12 (c) ,, 4, 6 ,, ,, and we take these as nuclei for three groups. There re- main Tests 1, 5, and 9. Their average correlations with each of the above nuclei are : 1 2 3 4 5 6 7 1 X X X X 2 X X X 3 X X X 4 X X X X X 5 X X X 6 X X X X X 7 X X X 8 X X X 9 X X X ? 10 X X X 11 X X X 12 X X X a b c 1 39 40 54 5 57 37 35 9 43 49 41 ADDENDA 845 We therefore add Test 1 to group c, Test 5 to group a, and (less certainly) Test 9 to group b. We then rewrite our matrix of correlations with the tests thus grouped : 357 10 11 289 12 146 3 5 7 10 11 2 8 9 12 57 59 70 73 34 16 44 20 1-14 40 18 27 85 57 40 67 63 53 26 68 31 1-78 63 27 42 1-32 59 40 60 64 17 08 22 10 57 20 09 13 42 70 67 60 78 43 21 56 25 1-45 52 22 34 1-08 73 63 64 78 39 18 50 23 1-30 45 20 30 95 6-24 4-62 34 53 17 43 39 1-86 44 68 56 57 25 39 1-21 16 26 08 21 18 89 44 35 43 28 12 18 58 44 68 22 56 50 2-40 68 35 44 74 32 50 1-56 20 31 10 25 23 1-09 56 43 44 34 15 23 72 6-24 4-07 40 63 20 52 45 2-20 57 28 74 34 1-93 45 63 18 27 09 22 20 96 25 12 32 15 84 45 51 27 42 13 34 30 1-46 39 18 50 23 1-30 63 51 4-62 4-07 It will be seen that certain additions have been made in readiness for the various methods of calculation of the g loadings which are then possible. If we symbolize the above table as A D E D B F E F C all methods depend on using only the correlations in the rectangles D, E, and F, since the suspected group factors which increase the correlations in A, in B, and in C do not influence D, E, and F. Each correlation in the latter rectangles is therefore the product of two ^-saturations (see page 9). Thus : r<11 = -40 = l^ r ia -57 = -40 x -34 *' " ' 49 846 ADDENDA where it should be noted that the three correlations come from E, D, and F respectively. But this value for the loading of Test 3 depends upon three correlations only and would, in a real experimental set of data, vary somewhat with our choice of the three. A method of using all the possible correlations in these three rectangles is needed. One such is given by Holzinger in his Manual (1987a). If all possible ways of choosing the two other tests are taken, and the fraction 3t 3j formed in each case ; and if r v the numerators of these fractions are added together to form a global numerator, and their denominators to form a global denominator ; it will then be found that the fraction thus formed is equal to 1-14 X -85 and this time all available correlations have been used. The rule is to multiply the two totals in the row of the test (1*14 X -85) and divide by the grand total of the block formed by the other tests concerned (1, 4, and 6 with 2, 8, 9, and 12, i.e. 4-07). For Test 2 this rule gives 1-86 x 1-21 = ~^4^r = ' 49 > /2 = - 70 - This Holzinger method is not difficult to extend to four or more groups. If we symbolize a four-group matrix by A D E G D B F H E F C K G H K L and consider the first test, then its g-loading I is given by /a _ <k + dg + eg 1 ~ F + H + K where d, e, and g are the sums of its row in D, E, and G. ADDENDA 347 Another method is given by Burt (1940, 478). For the numerator of each g loading he takes the sum of the side totals which Holzinger multiplied. Thus the numerators are : for Test 3, 1-14 + -85 = 1-99 5, 1-78 + 1-32 = 3-10 2, 1-86 + 1-21 = 3-07 12, 1-09 + -72 '= 1-81 6, 1-46 + 1*30 = 2-76. The denominators differ in group a, group fe, and group c, but all are formed from the three quantities 6-24, 4-62, and 4-07. For group a the denominator is : 6-24 It will be seen that the two quantities within the curly brackets are the totals of D and E, the two rectangles from which the numerators of group a come. By analogy the reader can write down the denominators of group b and group c they come to 4-40 and 5-01. Dividing the numerators by the appropriate denominators, we get for the g loadings : Test 357 10 11 289 12 146 g Loading -49 -76 -24 -62 -55 '70 -33 '90 -41 -82 -36 -55 The proof of Burt's formula is surprisingly easy. If the reader will write down, in place of the correlations in D, E, and F, the literal symbols lj, k (for r ik ) since our hypothesis is that only g is concerned in these correlations and will write out the sums, etc., of the above calculation literally, he will find that Burt's formula simplifies almost immediately to one 1 9 that of the test in question. Burt only gives his formula for three groups. It can be extended to the case of more groups, but becomes cumbersome and rather unwieldy, 348 ADDENDA Now comes the test of whether our grouping is correct, and our hypothesis valid that groups a, 6, and c have nothing in common but the factor g. Using the loadings we have found, form all the products lj, k and subtract them from the experimental correlations. All the corre- lations in D, E, and F should then vanish or, in a real set of data (ours are artificial) become insignificant. There should, however, remain residues in A, B, and C due to the second factors running through groups a, b, and c respect- ively. In our example the subtraction of the quantities l % l k gives the following : 357 10 11 289 12 146 g Loadings 49 76 24 62 55 70 33 90 41 82 36 55 3 49 20 47 40 46 5 76 20 22 20 21 7 24 47 22 45 51 10 62 40 20 45 44 11 55 46 21 51 44 2 70 23 63 29 8 33 23 30 14 9 90 63 30 37 12 41 29 14 37 1 82 15 18 4 36 15 31 6 55 18 31 The correlations left in A, if they are due to only one other factor (now that g has been removed) ought to show zero or very small tetrads ; and so they do. Those in B are also hierarchical. Those in C are too few to form a tetrad. The second factor in each of these submatrices can now be found in the same way as g is found from a matrix with no other factor : see page 9 and, later in this book, pages 153 to 155. The reader should complete the Calculation, and will find these loadings ; ADDENDA Factors Test g u v w 3 49 65 , 5 76 30 . 7 24 72 . 10 62 62 . 11 55 71 . 2 70 44 . 8 33 47 . 9 90 11 , 12 41 62 . 1 82 . . 29 4 36 . , 50 6 55 . . 62 349 An actual set of data will not give so perfect a hollow staircase, but at this stage the strict bifactor hypothesis can be departed from and additional small loadings or further factors added to perfect the analysis. Where a bifactor pattern exists, a simple method of extracting correlated or oblique factors has been given by Holzinger (1944) " based on the idea that the centroid pattern coefficients for the sections of approximately unit rank may be interpreted as structure values for the entire matrix." Cluster analysis is connected with the bifactor method, which is possible when clusters do not overlap. But it is by no means rare to find two or three variables entering into several distinct clusters. Raymond CattelPs article (1944a) describes four methods of determining clusters, and gives references which will lead the interested reader back to much of the previous work. See also the later part of our Chapter XVIII, where Reyburn and Taylor's method is described, and see also Tryon's work Cluster Analysis, 1939. 350 ADDENDA ADDENDUM (1945) TO PAGE 92 Variances and Covariances of Regression Coefficients A METHOD of calculating regression coefficients is described on pages 89 to 93. A somewhat longer method has two advantages : it permits the easy calculation of regression coefficients for any criterion (or many) when once the main part of the computation is completed, and, what is of great importance, it enables the probable errors of the coefficients, and of their differences, to be found quickly. Before describing it, a note about page 93 may be useful. The '720 in slab B is the weight for Test 1 when it alone is used ; the weights -611 and -158 in slab C are for Tests 1 and 2 when they alone form the battery ; -582, 153, and -066 are for a battery of Tests 1, 2, and 3 ; and finally the bottom row gives the weights for all four tests. The method referred to in the first paragraph above is to find first of all the reciprocal of the matrix of correla- tions of the tests. The way to do this is described on page 190 and will be better understood from this present example. There is only the one kind of computation throughout, viz. the evaluation of tetrad-differences involving the pivot, which is the number in the top left- hand corner of each slab. The reciprocal matrix appears at the bottom, (and smaller ones on the way down). The check column is sometimes not properly used. The check consists in seeing that the sum of the row is identical with the tetrad. Thus -177 is the sum of its row, and it is also the tetrad 1 X 1-26 - -69 X 1-57 = -177. The reader will see that space could be saved in the calculation opposite by omitting the rows containing ones only ; and also that nearly half the numbers can be written down from symmetry. After the reciprocal matrix has bee>n found it should be checked to see that its product with the original matrix gives the unit matrix (see page 191). The regression coefficients for any criterion are then obtained by multiplying the rows of the reciprocal by the ADDENDA 351 1 -69 -49 39 -1 . . 1-57 69 1 -38 19 , -1 1-26 49 -38 1 27 , -1 1-14 39 -19 -27 1 , -1 85 1 . * . 1 1 t . * . 1 1 t . . * i 1 . 1 . 1 Tetrad 524 -042 -079 690 -1 177 177 1-000 -080 -151 1317 -1-908 338 338 042 -760 079 490 1 371 371 -079 -079 848 390 1 238 238 -690 --490 -390 1 . . -570 -570 1 . , , , 1 1 . . . i 1 1 . 1 757 085 435 080 -1 357 357 1-000 112 575 106 -1-321 472 472 085 836 494 -151 . -1 264 265 -435 -494 1-909 -1-317 -337 -837 -080 151 -1-317 1-908 662 662 1 . , . 1 1 . 1 826 445 --160 -112 -1 223 224 1-000 539 -194 -136-1-211 270 270 -445 2-159 -1-271 --575 -132 -132 160 -1-271 1-916 --106 699 700 -112 -575 -106 1-321 528 528 1 1 2-399 -1-357 --514 --539 -1-357 1-947 --128 -194 --515 -128 1-336 --136 -539 194 --136 1-211 criterion correlations and then adding the columns. In the example of page 91-3 we multiply the first row of the reciprocal by -72, the second by -58, and so on. The addition of the columns then gives the same regression coefficients as were found on page 93. 352 ADDENDA The most important advantage of this method is that whatever the criterion, the variances and covariances of the regression coefficients are proportional to the cells of the above reciprocal matrix (Thomson, 1940, 16 ; Fisher, 1925, 15 and 1922, 611). Their absolute values for any given criterion are obtained by multiplying by 1 - r 2 m , the defect of the square of the multiple correlation from unity, and dividing by the number of " degrees of free- dom " which is for full correlations N p 1 where N is the number of persons tested, and p the number of tests. For partial correlations the degrees of freedom are reduced by the number of variables " partialled out." Thus in our example, where p = 4, if N had been 105, N _ p __ i would be 100. The multiple correlation was 83, and 1 - r z m = -312 (see page 94). The variances and covariances of our four regression coefficients are in this case equal to the reciprocal matrix multiplied by -00312. 0075 -0042 0016 -0017 0042 -0061 0004 -0006 0016 0004 -0042 0004 0017 -0006 0004 -0038 The standard errors of the regression coefficients are the square roots of the diagonal elements : Regression coefficients -390 -222 -018 -431 Standard errors -087 -078 -065 -062 Significant? Yes ? No Yes The correlations of the regression coefficients will be got by dividing each row and column by the square root of the diagonal element. We obtain : 1-00 62 28 31 62 1-00 79 -12 28 -79 1-00 -10 31 -12 -10 1-00 We can now calculate the standard error of the difference between any pair of the regression coefficients and see whether they differ significantly. Take, for example, those ADDENDA 353 for Test 1 (-390) and Test 4 (-431). The difference is -041. Its standard error is the square root of -0075 + -0038 + 2 X -31 X -087 X -062 == -01 4G. .*. standard error of '041 is -121. The difference is therefore not significant when N = 105. Had N been larger it might have been. ADDENDUM TO CHAPTER IV ON THE GEOMETRICAL PICTURE SHORTLY after the publication of the first edition, Mr. Babington Smith pointed out to me a difficulty which a reader might experience in understanding this chapter, if he had previously read about an apparently different space, in which there are as many orthogonal axes as there are persons. In this space I may call it Wilson's a test is repre- sented by a point whose co-ordinates are the scores of the persons in that test. If the scores have been normalized (see footnote, page 6), then the distance from the origin to the test-point will be unity. The test-points will, in fact, be exactly the same points as those spoken of on page 63, where unit distance was measured along each test vector in the positive direction. The space of Chapter IV is the same as the space defined by these points, a subspace of the larger space. The latter has as many dimensions as there are persons, the former as there are tests. The lines joining the origin to the test-points are the same lines as the test vectors of Chapter IV, and the cosines of the angles between them represent here, as there, the correlations between the tests : for this follows from the " cosine law " of solid or multidimensional geometry, since the normalized scores of the tests are direction cosines with regard to the axes equal in number to the persons. In this subspace I introduced the idea of each point representing a person; whose scores are the projections of F.A. 12 354 ADDENDA that point on to the test vectors. Apparently this was an unfamiliar idea to some, in such a space, although I thought it was in common use. It had been used by Hotelling in his space (see Chapter V), where the test lines are ortho- gonal ; and he contemplates that space being squeezed and stretched into the space of Chapter IV (Hotelling, 1933, 428) and refers to Thurstone's previous use of this subspace. Moreover, since writing the above I have no- ticed that Maxwell Garnett in 1919 used the representation of persons by points in the test space (J5.J.JP., 9, 348). And anyhow the ordinary scatter-diagram does so, the diagram commonly made by drawing the test lines at right angles and using his scores as a person's co-ordinates. If in such an ordinary scatter-diagram the test axes are then moved towards one another until the cosine of the angle represents the correlation, the elliptical crowd of dots will become circular if standard scores have been used. (It is to be noted that although the axes have ceased to be otho- gonal, it is still the vertical projections on to each line which represent a person's scores, not the oblique axes). For Sheppard showed in 1898 (Phil. Trans. Roy. Soc., A 192, page 101) that r = cos {bir/(a + b)\ where the scatter-diagram, with its axes drawn through the means, has the quadrant-frequencies b It is not my experience that unsophisticated persons find difficulty with page 55 ; and as for the sophisticated, well, they oughtn't to. ADDENDUM (1948) TO PAGE 162 FRANCIS F. MEDLAND (Pmka 1947, 12, 101-10) has tried nine methods of estimating communality, on a correlation matrix with 63 variables. A method entitled Centroid No. 1 method seemed to be best. A sub-group is chosen of from three to five tests which correlate most highly with the test whose communality is wanted. The highest cor- ADDENDA 355 relation t in each column of the sub-group is inserted in the diagonal cell, and the columns summed. The grand total is also found. Then the estimate of h\ is (Zri+ t,Y Zr + Zt where the numerator is the square of the column total, and the denominator is the grand total. Thus if the cor- relations of the sub-group were (72) -72 -63 -24 72 (-72) -47 -59 63 -47 (-63) -41 24 -59 -41 (-59) 2-31 2-50 2-14 1-83 = 8-78 the estimate of }i{ would be 2-31 2 - -608 8-78 Clearly the same sub-group will usually serve for more than one of its members. Thus from the above example h\ can be estimated to be -712. A graphical method, for which the reader is referred to Medland's article, was about equally accurate but rather more laborious. BURT ROSNER (Pmka 1948, 13, 181-4) has given an alge- braic solution for the comnumalities depending upon the Cayley-IIamilton theorem that any square matrix satisfies its own characteristic equation, but adds that the method " is not at all suited for practical purposes. The com- putational labour is prohibitive." It is however interest- ing theoretically and may suggest new advances. MATHEMATICAL APPENDIX MATHEMATICAL APPENDIX PARAGRAPHS 1. Textbooks on matrix algebra. 2. Matrix notation. 3. Spearman's Theory of Two Factors. 4. Multiple common factors. 5. Orthogonal rotations. 6. Orthogonal transforma- tion from the two -factor equations to the sampling equations. 7. Hotelling's "principal components." 8. The pooling square. 9. The regression equation. 9a. Relations between two sets of variates. 10. Regression estimates of factors. 11. Direct and indirect vocational advice. 12. Computation methods. 13. Bartlett's estimates of factors. 14. Indeterminacy. 15. Finding g saturations from an imperfectly hierarchial battery. 16. Sampling errors of tetrad-differences. 17. Selection from a multivariate normal population. 17 a. Maximum likelihood estimation (by D. N. Lawlcy). 18. Reciprocity of loadings and factors in persons and traits. 39. Oblique factors. Structure and pattern. 19a. Second-order factors. 20. Boundary con- ditions. 21. The sampling of bonds. 1. Textbooks on matrix algebra. Some knowledge of matrix algebra is assumed, such as can be gained from the mathematical introduction to L. L. Thurstone's The Vectors of Mind (Chicago, 1935) ; Turnbull and Aitken's Theory of Canonical Matrices, Chapter I (London and Glasgow, 1932) ; H. W, Turnbull's The Theory of Determinants, Matrices, and Invariants, Chapters I-V (London and Glasgow, 1929) ; and M. BScher's Introduction to Higher Algebra, Chapters II, V, and VI (New York, 1936). I have adopted Thurstone's notation in sections 19 and 19a of the mathematical appendix, and in Chapters XVIII and XIX in describing his work. But I have not made the change elsewhere because readers would then be incommoded in consulting my own former papers. The chief differences are as follows : My M is Thurstone's F, for centroid factors, my Z is Thurstone's S -f- ^/N, and my F is Thurstone's P ~ yTST. 359 360 MATHEMATICAL APPENDIX 2. Matrix notation. Let X be the matrix of raw scores of p persons in n tests, with n rows and p columns ; and when normalized by rows, let it be denoted by Z. The letters z and Z in the text of this book mean standardized scores, which are used in practical work, but in this appendix they mean normalized scores, so that ZZ' = R . . . (1) the matrix of correlations between n tests. For many purposes it is convenient to think of solid matrices like Z as column (or row) vectors of which each element represents a row (or column). Thus Z can be thought of as a column vector z, of which each element represents in a collapsed form a row of test scores. Thus with three tests and four persons z = = Z. (2) In the theory of mental factors each score is represented as a loaded sum of the normalized factors /, the loadings being different for each test, i.e. z = Mf (specification equations) (3) where M is the matrix of loadings, and f the vector of v factors, collapsed into a column from F, the full matrix, of dimensions v X p. We note that p number of persons, n = number of tests, v = number of factors. The dimensions of M are n x v. Equation (3) represents n simultaneous equations, and the form Z MF represents np simultaneous equations. We now have R = ZZ' = (MF)(MF)' = MFF'M . (4) If the factors are orthogonal, we have FF' = I (5) the unit matrix, and therefore R = MM' (6) The resemblance in shape between this and (1) MATHEMATICAL APPENDIX 361 leads to a parallelism between formulae concerning persons and factors (Thomson, 19356, 75 ; Mackie, 1928a, 74, and 1929, 34). 3. Spearman's Theory of Two Factors assumes that M is of the special form M = L . ra, (7) and therefore R = U' + MS .... (8) where M l is the diagonal matrix which forms the right- hand end of M, and I is the first column of M. In this form it is clear that R is of rank 1 except for its principal diagonal. Its component II is the " reduced correlational matrix " of the Spearman case, and is entirely of rank 1. The elements Z^, Z 2 2 , . . . Z M 2 , which form the principal diagonal of ZZ', are called " commonalities." 4. Multiple common factors. When more than one common factor is present, M takes the form M = (M,\M 1 ) .... (9) where M Q is the matrix of loadings of the common factors, represented in the Spearman case by the simple column Z. We have then R = MM 1 = MM* + MS . . . (10) where the " reduced correlation matrix " M M ' is of rank r, the number of common factors, and is identical with R except for having " communalities " in its principal diagonal. 5. Orthogonal rotations. -If we express the v factors / in terms of w new factors 9 by the equation where A is a matrix of v rows and w columns, we have z = Mf=:MA<p . . . (12) an expression of the tests z as linear loaded sums of a different set of factors, with a matrix of loadings MA. 362 MATHEMATICAL APPENDIX If AA' = 1 (13) the new factors 9 are orthogonal like the old ones. They can be as numerous as we like, but not less than the number of tests unless the matrix R is singular. (12) represents a rigid rotation of the orthogonal axes / into new positions, with dimensions added or abolished. 6. The sampling theory. The following transformation is of interest as showing the connexion between the Theory of Two Factors and the Sampling Theory (Thom- son, 19356, 85). We shall write it out for three tests only, but it is quite general. Consider the orthogonal matrix : III mil Iml Urn mml mlm Imm mmm mil Iml Urn Ill mml mlm mml III Imm mlm Imm III Iml llm mmm mil mmm llm mmm mil Iml Imm mlm mml mml mlm Imm Iml mil mmm llm mmm - mil mmm llm Iml Ul Imm mlm Imm III mml mlm mml Ul llm Iml mil mmm Imm mlm mml llm Iml mil -III (14) wherein the omitted subscripts 1, 2, and 3 are to be understood as existing always in that order, so that mil means m^a* If we take for A in Equation (12) the first four rows of this orthogonal matrix, and for M the Spearman form (7) with three tests, the result is to transfer to eight new factors, yielding : 4- ~s = lA<Pi -I- Wj/a^a 4- ZiWia^s 4- Each z is here in normalized units. If, however, we change to new units by multiplying the three equations by Zj, Z 2 , and Z 3 respectively, we have : l i z i = ZiVa^i 4- liMzkVs 4- hh m 'A9i 4- /iW 2 w 3 <p 7 I 2 z 2 = IJIJ.^ 4- m 1 ; 2 f 3 97 2 4- liltfnjpi 4- m^m^ . . (10) *3*3 = W*<Pi 4- mAh<Pz ,4- Ii>n z l 3 (p 3 4- ^m^^ and the variates Z^, Z 2 s 2 , and Z 3 2 3 are now susceptible of the explanation that each is composed of l^N small equal MATHEMATICAL APPENDIX 363 components drawn at random from a pool of N such components, all-or-none in nature. In that case 1^1 2 2 1 9 2 N components would probably appear in all three drawings (?i) ; l-fl^m^N components would probably appear in the first two drawings, but not in the third (94) ; and so on down to m^m^m^ components, which would not appear at all (93, which is missing from the equations). The transformation can, of course, be reversed, and the sampling theory equations converted into the two-factor equations. 7. Hotelling s " principal components " are the principal axes of the ellipsoids of equal density z'R" l z - constant .... (17) ^vhcn the test vectors are orthogonal axes (Hotelling, 1933). To find the principal axes involves finding the latent roots of R~ l . The Hotelling process consists of (a) a rotation of the axes from the orthogonal test axes to the directions of the principal axes ; and (b) a set of strains and stresses along these new axes to standardize the factors, making the ellipsoid spherical and the original axes oblique. The transformation from the tests to the Hotelling factors y being from Equation (3) z = My (M square) the ellipsoids (17) become constant - z'R *z = y / (M / JB" 1 M)Y = y'y . (18) since they become spheres. Therefore we must have M'R~ 1 M = I . . (19) The locus of the mid points of chords of z'R~ l z whose direction cosines are h' is the plane h'R~ l z = 0, and if this is a principal plane it is at right angles to the chords it bisects, i.e. h'R~ l = W which has non-trivial solutions only for | fi-i - X/ | = the roots X of which are the " latent roots " of J?" 1 , while each h' is a " latent vector." 364 MATHEMATICAL APPENDIX Now, if H is the matrix of normalized latent vectors of J?~ l , we have H'R~ 1 H = A where A is the diagonal matrix of the latent roots of .B" 1 ; so that a solution for M corresponding to rotation to the principal axes and subsequent change of units to give a sphere is seen to be M = //A~* . . (20) The latent vectors of R are the same as those of R~ l , or of any power of JK, and Hotelling's process described in the text (Chapter V) finds the latent roots (forming the diagonal matrix D) and the latent vectors (forming H) of R. We then have M == HD* . . . (21) For the convergence of the process, see Hotelling's paper of 1933, pages 14 and 15. Since in Hotelling analyses M is square, we can write y = M~ l z = (HD*)- l z - D-*H~ l z - D~ l (IflH')z - D~ l M'z . (22) Each factor y, that is, can be found from a column of the matrix M, divided by the corresponding latent root, used as loadings of the test scores z. 8. The pooling square. If the matrix of correlations of a + b variates is : ^~ (23) and if the standardized variates a are multiplied by weights u, the standardized variates b by weights w, and each set of scores summed to make two composite scores, the resulting variances and covariances are : u'R aa u 'aa" u'R^w - . . (24) w'R bb w as can be seen by writing out the latter expressions at length. The battery intercorrelation is therefore u'Rdtx>_ we w'RfrU <\/(u'R aa u x w'R bb w) MATHEMATICAL APPENDIX 365 If weights are applied to raw scores, each applied weight must be multiplied by each pre-existing standard deviation, in (25). If there is only one variate in the a team, (25) becomes where r ba represents a whole column of correlation coeffi- cients. The values of w for which this reaches its maximum value will satisfy the equation 8 w f r ba that is w = a scalar X ^w>~V &a . . (28) consistent with the ordinary method of deducing regression coefficients. 9. The regression equation. If z is the one variate in the a team, and z are the b team, and if *.='*. . . . (29) we wish to make S(z z ) 2 a minimum, that is 8w ^ ' ~~ Szz' = w'Szz' Z Q ^ rll'R b bb ~ l z . . . (30) If R is the matrix of correlations of all the tests including z , the regression estimate of any one of the tests from a weighted sum of the others is given by determinant R z = (31) where R z is R with the row corresponding to the variate to be estimated replaced by the row of variates. 9a. Relations between two sets of variates. (Hotelling 1935a, 1936, M. S. Bartlett 1948). If two sets of variates have correlation coefficients Rab A or R 'ba B and if the variates of the B team are fitted with weights b, F.A. 13 366 MATHEMATICAL APPENDIX then the correlations of the B team, thus weighted, with the separate tests of the A team are given by C'b and the square of the correlation coefficient between the two teams is then The maximum intercorrelation, and other points of in- flexion in A, will be given by dX/db = i.e. (CA~*C' - XB)b = . . . (31.3) a set of homogeneous equations in b. We must therefore have | CA-*C' - XB | = . . . (31.4) an equation for A with as many non-zero roots as the num- ber of variates in the smaller team. For any one of these roots A, the weights b are proportional to the co-factors of any row of (CA~*C' XB). The corresponding weights a for the A team are then found by condensing the team B (using weights b) to a single variate and carrying out an ordinary regression calculation. The result is to " factorize " each team into as many orthogonal axes as there are variates. These axes are re- lated to one another in pairs corresponding to the roots A. Each axis is orthogonal to all the others except its own opposite number in the space of the other team, arising from the same root A as it does, to which axis it is inclined at an angle arccos V^T. Where one team has m more variates than the other, m of the roots will be zeros and the corresponding axes will be at right angles to the whole space of the other team. This form of factorizing has been called by M. S. Bartlett (1948) external factorizing, since the position of the " factors " or orthogonal axes in each team, in each space, is dictated by the other team. The weightings corresponding to the largest root give the closest possible correlation of the two weighted teams. MATHEMATICAL APPENDIX 367 If the two teams are duplicate forms of the same tests, this is the maximum attainable battery or team reliability (Thomson 19406, 1947, 1948). In this case Peel (Nature, 1947) has shown that a simpler equation than 31 4 gives the required roots. If A = (Ji 2 Peel's equation is | C \LA | = . . . (31.5) 10. Regression estimates of factors. When in the speci- fications z = Mf . . . . (3) the factors outnumber the tests, they cannot be measured but only estimated. To all men with the same set of scores z will be attributed the same set of estimated factors /, though their u true " factors may be different. The regression method of estimation minimizes the squares of the discrepancies between f and /, summed over the men. The regression equation (31) will be for one factor^ K* p I = (32) I w t # I v ' where m t is a column of M. Expanding, we have /. = m/JR- 1 * and in general /-M'JT 1 * . . . (33) or, separating the common factors and the specifics f = M Q 'R- l z . . . (34) /! = MJt^z . . . (35) the latter of which shows that we know the proportionate weights for each specific (the rows of R~ l ) even before we know whether that specific exists (Wilson, 19346, 194). The matrix of covariances of the estimated factors is M' Tt~ l M MK M- ,,_, MJt^Mj. (36) a square idempotent matrix of order equal to the number of factors, but trace only equal to the number of tests. For one common factor, (34) reduces to Spearman's estimate 368 MATHEMATICAL APPENDIX I - rb 1 ^' < 81 > where S = S, ** i - V while # = M Q 'R- 1 M Q in (36) reduces to S/(l + S), the variance of g. 10a. Ledermann's short cut (1938a, 19396). The above requires the calculation of the reciprocal of the large square matrix R. Ledermann's short cut only requires the reci- procal of a matrix of order equal to the number of common factors. We have B = M M ' + M^ . . . (10) and the identity M 'Mr*(M M ' + Mi>) = (M 'Mr 2 M + I)M ' = (J + I)M ' say. Premultiplying by (/ + J)~ ] and postmultiplying by R~ l we reach (I + J)- 1 M^M^ = M^R' 1 . . (36-1) and the left-hand quantity can then be used in equation (34). 11. Direct and indirect vocational advice. If Z Q is an occupation and z a battery of tests, the estimate of a candidate's occupational ability is = rSB^z .... (37) where the r are the correlations of the occupation with the tests. If z can be specified in terms of the common factors of z, and a specific s independent of z, then an indirect estimate of via the estimated / is possible. We have *o = Wo'/o + S ( 88 ) where m ' is a row of occupation loadings for the common factors / of z, and also / - M,'R- l z Substitution in (38), assuming an average s (=Q) gives z = rao'Mo'tf' 1 * . . (39) But mo'Mo' = r ' .... (40) MATHEMATICAL APPENDIX 369 and (39) is identical with (37) (Thomson, 1936&). If, how- ever, s is not independent of the specifics s of the battery, (40) will not hold, and the estimate (39) made via an estima- tion of the factors will not agree with the correct estimate (37). 12. Computation methods. The " Doolittlc " method of computing regression coefficients is widely used in America (Holzinger, 1937&, 32). Aitken's method, used and explained in the text, is in the present author's opinion superior (Aitken, 1937a and b, with earlier references). Regression calculations and many others are all special cases of the evaluation of a triple matrix product XY~ 1 Z 9 where Y is square and non-singular, and X and Z may be rectangular. The Aitken method writes these matrices down in the form Y X Z and applies pivotal condensation until all entries to the left of the vertical line are cleared off. All pivots must originate from elements of Y. By giving X and Z special values (including the unit matrix /) the most varied operations can be brought under tho one scheme (see Chapter VIII, Section 7). 13. Bartletfs estimates of factors. We have z = M f + Mjfi, where / and / x are column vectors of the common and specific factors respectively and M is a diagonal matrix. Bartlett now makes the estimates / such as will minimize the sum of the squares of each person's specifics over the battery of tests, i.e. 8 gjftfi'/i) = >- (- Afr'M.y (Mr 'a - Mr l M/o) - o Af.'Mr'z = M 'Mr 2 M / = J/o, say, / =--. J- l M 'M t - 9 z . (41) (Bartlett, 1937, 100.) 370 MATHEMATICAL APPENDIX One could also find the estimated specifics as /! = (/ - Mr'MoJ-'M o'MrWr 1 * . . (42) Substituting z = [M ! M x ] we get for the relation between /and/ / i r/ ! j-'Af.'Mr 1 1 [/] 4f = ' and for the co variances of /we get The error variances and covariances of the common factors are 1 = J- 1 . (45) (Bartlett, 1937, 100.) When there is only one common factor, J becomes the familiar quantity (Bartlett, 1935, 200.) As was first noted by Ledermann * / + J^ 1 = (M 'R- l M o r l - K l . (46) (quoted by Thomson, 1938&) ; and using this we see that the back estimates of the original scores from the regression estimates j^o are identical with the insertion of Bartlett's estimates / in the common-factor part of the specification equations, viz. M Q K- l M Q 'R- l z - MJ^Mo'M^z . . (47) (Thomson, 1938a.) Bartlett has pointed out that, using the same identity, in the form K = J(I K) 9 it is easy to establish the rever- sible relation between his estimates and regression esti- mates (Bartlett, 1938.) * Letter of October 23, 1937, to Thomson. MATHEMATICAL APPENDIX . 371 and he summarizes their different interpretation and prop- erties by the formulae E{f Q } ^E{f Q \ =0, JB{(/.-/)(/.-/o)'| =I-K (49) - K~\I - K) . (50) where E denotes averaging over all persons, EI over all possible sets of tests (comparable with the given set in regard to the amount of information on the group factors /o). 14. Indeterminacy. The fact that estimated factors, if the factors outnumber the tests, necessarily have less than unit variance has sometimes been expressed in the case of one common factor by postulating an indeterminate vector i whose variance completes unity. This i may be regarded as the usual error of estimation, and is a function of the specific abilities (Thomson, 19346). The fact that M'R~ 'M in Eqn. (36) is of rank less than its order also expresses the indeterminacy, and allows the factors to be rotated to different positions which nevertheless fulfil all the required conditions. In the hierarchical case the transformation which effects this is (Thomson, 1935a) /=B 9 .... (51) where B means the required number of rows of B^I-2qq'/q'q . . . (52) in which q. = l i \m i (see Equation 7) . (53) as far as there exist tests, after which q is arbitrary. For z == Mf = MJ?9 = M9 since MB = M ..... (54) and z is thus expressed by identical specification equations in terms of new factors <p. For such transformations in the case of multiple factors see Thomson, 1936a, 140 ; and Ledermann, 1938c. If the matrix M is divided into the part M due to common factors and the part M l due to specifics, as in 372 MATHEMATICAL APPENDIX equation (9), then Ledermann shows that if U is any orthogonal matrix of order equal to the number of com- mon factors,' the matrix wherein Q = Mrw J ' will satisfy the equation MB = M Indeterminacy is entirely due to the excess of factors over tests, i.e. to the fact that the matrix of loadings M is not square. It can be in theory abolished by adding a new test which contains no new factor, not even a new specific ; or a set of new tests which add fewer factors than their number, so that M becomes square (Thomson, 1934c ; 1935a, 253). In the case of a hierarchy each of these tests singly will conform to the hierarchy, so that their saturations / can bo found ; but jointly they break the hierarchy. If they add no new factors, g can then be found without any indeterminacy. 15. Finding g saturations from an imperfectly hierarchical battery. The Spearman formula given in Chapter IX, Section 5, is the most usual method. A discussion of other methods will be found in Burt, 1936, 283-7. See also Thomson, 1934a, 370, for an iterative process modified from Hotelling. 16. Sampling errors of tetrad-differences. The formulae (16) and (16A) given in the text are both approximations, but appear to be very good approximations. The primary papers are Spearman and Holzinger, 1924 and 1925. Critical examination of the formulae have been made by Pearson and Moul (1927), and Pearson, Jeffery, and Elder- ton (1929). Wishart (1928) has considered a quantity P which is equal to P'N*/(N l)(N 2), where P' is the tetrad-difference of the covariances a instead of the correla- tions, and obtained an exact expression for the standard deviation cr of P (N - 2)v* - -- Z> 12 D 34 - D + 3J9 13 Z> 31 (55) MATHEMATICAL, where the D's are determinants of the following matrix and its quadrants : a 21 #13 #33 a 44 But approximate assumptions are necessary when the standard deviation of the ordinary tetrad-difference of the correlations is deduced from that of JP. The result for the variance of the tetrad-difference is N + I 2) (1 - )(! - r 34 2 ) - R (56) where R is the 4x4 determinant of the correlations. 17. Selection from a multivariate normal population.- The primary papers are those of Karl Pearson (1902 and 1912). The matrix form given in the text (Chapter XII, Section 2) is due to Aitken (1934), who employed Soper's device of the moment -gen era ting function, and made a free use of the notation and methods of matrices. A variant of it which is sometimes useful has been given by Ledermann (Thomson and Ledermann, 1988) as follows. If the original matrix is subdivided in any symmetrical manner : R P , R pl R qi R 9t R si R, t 1? I? lt to lt tf and R pp is changed by selection to V pp , then each resulting sub-matrix, including V pp itself, is given by the formula where- ^ E pp = B^ - R^V^j ' ^ (57) IT a. Maximum likelihood estimation. The maximum likelihood equations for estimating factor loadings (Lawley, P.A. 13* 374 MATHEMATICAL APPENDIX 1940, 1941, 1943) may be expressed fairly simply in the notation of previous sections. It is necessary, however, to distinguish between the matrix of observed correla- tions, which we shall denote by J? , and the matrix R = M M ' + MS, which represents that part of R which is " explained " by the factors. The equations may then be written Mo' - Mo'/r^Ro . . . (58) These are not very suitable for computational work. It may, however, be shown that Mo'JfT 1 = (I - #)M 'Mr* = (I + Jr'Mo'Mr 2 (59) where, as before, K = Mo'R- l M< J = M 'Mr 2 M . Hence our equations may be transformed into the form Mo' = (I + J^Mc/MrX (60) or alternatively, M ' = J- a (M 'Mr 2 #o - Mo') . (61) When there are two or more general factors the above equations will have an infinite number of solutions corre- sponding to all the possible rotations of the factor axes. A unique solution may, however, be found such as to make J a diagonal matrix. Finally, if we put L = Mo'Mr^o - M ', V = LMr 2 M , then, from the last set of equations V = JM 'Mr 2 M = J 2 . Hence we have MO' == F-*L .... (62) These equations have been found the most convenient in practice, since they can be solved by an iterative process. When first approximations to M and Mj have been ob- MATHEMATICAL APPENDIX 375 tained, they can be used to provide second approximations by substitution in the right-hand side. 18. Reciprocity of loadings and factors in persons and traits (Burt, 19376). Let W be a matrix of scores centred both by rows and columns. Its dimensions are traits X persons (t . p) 9 and its rank is r where r is smaller than both t and p in consequence of the double centring. The two matrices of co variances are WW for traits and W'W for persons, and by a theorem first enunciated by Sylvester in 1883 (independently discovered by Burt), their non-zero latent roots are the same. If their dimensions differ, i.e. t 4= p, the larger one will have additional zero roots. Let the non-zero roots form the diagonal matrix D. Then the principal axes analyses are : W = H t D*F l9 dimensions (t . r)(r . r)(r . p) and W'=^ Htt*F 29 dimensions (p . r)(r . r)(r . t) where H j and H 2 are the latent vectors of WW and W'W 9 while F l is the matrix of factors possessed by persons, F z that of factors possessed by traits. From the analysis of W we have, taking the transpose W' = JfrYZW, dimensions (p . r)(r . r)(r . t) and comparison of this with the former expression for W makes the reciprocity of // 2 and F/, F 2 and ///, evident. 19. Oblique factors. Structure and pattern. In Thur- stone's notation, which we shall follow in this paragraph, the matrix M of our equation (3), when it refers to centroid factors, is called F. Our equation (3) becomes in his notation s =Fp. Since centroid factors are orthogonal, F is both a pattern and a structure. The structure is the matrix of correla- tions between tests and factors, i.e. : Structure = sp' = ( F P)P* = F (PP') = F1 = F = Pattern. When the factors are oblique, however, this is not the case. In that case, Structure = Pattern X matrix of correlations between the factors. Thurstone turns the centroid factors to a new set of 376 MATHEMATICAL APPENDIX positionsj[( still within the common-factor space, and in general oblique to one another) called reference vectors. The rotating matrix is A, and V = FA . . . (63) is the structure on the reference vectors. The cosines of the angles between the reference vectors #re given by A' A. V is not a pattern. Its rows cannot be used as coefficients in equations specifying a man's scores in the tests, given his scores in the reference vectors. The pattern on the reference vectors would not have those zeros which are found in V. The primary factors are the lines of intersection of the hyperplanes which are at right angles to the reference vectors, taken (r - 1) at a time where r is the number of common factors, the number of dimensions in the common- factor space. They are defined, therefore, by the equations of the hyperplanes, taken (r 1) at a time. These equations are A'a? = O . . . (64) where a? is a column vector of co-ordinates along the centroid axes. The direction cosines of the intersections of these hyperplanes taken (r 1) at a time are therefore proportional to the elements in the columns of (A')~ 1 9 and to make them into direction cosines this has to have its columns normalized by post-multiplication by a diagonal matrix D, giving for the structure on the primary factors F(A / )~ 1 D . . . (65) D is also the matrix of correlations between the reference vectors and the primary factors, for A'(A')" a J5 = D . . . (66) Each primary factor is therefore correlated with its own reference vector but orthogonal to all the others, as can also be easily seen geometrically. The matrix of intercorrelations of the primary factors is DA~ l (A'r l D from equation (65). If W is the pattern on the primary factors p, so that test scores s = Wp, MATHEMATICAL APPENDIX 377 then the structure on the primary factors is also sp' = Wpp' where pp ' is the matrix of correlations between the primary factors, and therefore primary factor structure = WDA~ l (A')~ l D . . (67) Also, this structure = F(A')~ 1 D from (65). Equating these we have : WDA~ l = F whence W = FAD" 1 . . . (68) = VD~ l . . . (69) We have, therefore, Structure Pattern Reference vectors . . FA F(A')~ l ] . . , . Primary factors . . ^(A')" 1 ^ FAD" 1 ) ( ' where the reference -vector pattern has been entered by analogy but could easily be independently found. It will be seen that the structure and pattern of the primary factors are identical with the pattern and struc- ture of the reference vectors except for the diagonal matrix Z>. The structure of the one is the pattern of the other multiplied by D. This theorem is not confined to the case of simple structure, but is more general, and applies to any two sets of oblique axes with the same origin O, of which the axes of the one set are intersections of " primes " taken r 1 at a time in the space of r dimensions, and the axes of the other set are lines perpendicular to those primes. By prime is meant a space of one dimension less than the whole, i.e. Thurstone's hyperplane. The projections of any point P on to the one set of axes are identical with the projections thereon of its oblique co-ordinates on the other set, which sentence is equivalent to the matrix identities (see 70) FA = FAD- 1 x D, and F(A f )- 1 D --= JF(A')' 1 X D, Structure | ___ Pattern on) (Cosines to project it on one set} ~~ other set j \ on to the first set. 378 MATHEMATICAL APPENDIX A diagram makes this obvious in the two-dimensional case and gives the key to the situation. A perspective diagram of the three-dimensional case is not very difficult to make and is still more illuminating. The vector (or test) OP is the " resultant " of its oblique co-ordinates (the pattern), but not of its projections (the structure). It is of interest to notice that, either on the reference vectors or on the primary factors Pattern x Transpose of Structure Test-correlations. This serves as a useful check on calculations. It is geo- metrically immediately obvious. For consider a space defined by n oblique axes, with origin 0, and any two points P and Q each at unit distance from 0. The direc- tions OP and OQ may be taken as vectors corresponding to two tests, and cos POQ to the test correlation. Consider the pattern, on these axes, of OP, and the structure, on the same axes, of OQ. The former is com- posed of the oblique co-ordinates of the point P, the latter of the projections on the axes of the point Q, which pro- jections (OQ being unity) are cosines. Then the inner product of those oblique co-ordinates of P with these cosines obviously adds up to the projection of OP on OQ, that is to cos POQ, or the correlation coefficient. In estimating oblique factors by regression, since the correlations between factors and tests must be used, the relevant equation is . / ={F (A')- 1 D}'/?- 1 ^ (70-1) Ledermann's short cut (section 1 0a above) requires consider- able modification for oblique factors. We no longer have R = M M ' + M) 2 . . . (10) but Pattern x transpose of structure + M^ 2 R 9 i.e. in Thurstone's notation (JVUJ-'HW)-^} ' + FI* = R, - (70-2) and using this (Thomson 1949), we reach the equation / = (/ + Jr^A')- 1 ^} '*\- 2 *, . (70-3) where now J = (F (A')- 1 0}'* l r'WAD- 1 ), (70-4) MATHEMATICAL APPENDIX 379 in place of Ledermann's J = M 'M 1 "" 2 M . Only reciprocals of matrices of order equal to the number of common factors are now required, but the calculation, like all concerning oblique factors, is still one of considerable labour. I9a. Second-order factors. The above primary factors can themselves in their turn be factorized into one, two, or more second-order factors, and a factor-specific for each primary. If the rank of the matrix of intercorreiations of the primaries can be reduced by diagonal entries to say two, then the r primaries will be replaced by r -f 2 second- order factors which will no longer be in the original common-factor space. The correlations of the primaries with these second-order factors will form an oblong matrix with its first two columns filled, but each succeeding column will have only one entry corresponding to a factor- specific, thus : r r r r r r r r r r - E (say), where subscripts must be supplied to indicate the primary (the row) and the second-order factor (the column). The primary factors can be thought of as added to the actual tests, their direction cosines being added as rows below jP, which thus becomes : F DA- 1 Imagine this matrix post-multiplied by a rotating matrix VF, with r rows and r + 2 columns, which will give the correlations with the r + 2 second-order factors. The lower part of the resulting matrix will be E, which we already know. That is = E ... (71) - AD~ 1 E . . . (72) 380 MATHEMATICAL APPENDIX and the correlations of the original tests with the second- order factors are then : G = F*F = FAD~ 1 E = VD~ 1 E . (73) G is both a structure and a pattern, with continuous columns equal in number to the general second-order factors, followed by a number of columns equal to the number of primaries, this second part forming an orthog- onal simple structure. 20. Boundary conditions. These refer to the conditions under which a matrix of correlation coefficients can be explained by orthogonal factors which run each through only a given number of tests. The problem was first raised by Thomson (19196) and a beginning made with its solution (J. R. Thompson, Appendix to Thomson's paper). Various papers by J. R. Thompson culminated in that of 1929, and sec also Black, 19"29. Thomson returned to the problem in connexion with rotations in the common-factor space (Thomson, 19366), and Ledermann gave rigorous proofs of the theorems enunciated by Thomson and Thompson and extended them (Ledermann, 1936). A necessary condition is that if the largest latent root of the matrix of correlations exceeds the integer s, then factors which run through s tests only and have zero loadings in the other tests are certainly inadequate. This rule has not been proved to be sufficient, and when applied to the common-factor space only it is certainly not suf- ficient, though it seems to be a good guide. Ledermann (1936, 170-4) has given a stringent condition as follows. If we define the nullity of a square matrix as order minus rank, then if it is to be possible to factorize orthogonally a matrix R of rank r in such a way that the matrix of load- ings contains at least r zeros in each of its columns, the sum of the nullities of all the r-rowed principal minors of R must at least be equal to r. 21. The sampling of bonds. The root idea is that of the complete family of variates that can be made by all possible additive combinations of bonds from a given pool, and the complete family of correlation coefficients between pairs of these, Thomson (19276) mooted the idea and MATHEMATICAL APPENDIX 381 worked out the example quoted in Chapter XX. He had earlier (1927a) showed that with all-or-none bonds the most probable value of a correlation coefficient is VCPiPa)* where the p's are fractions of the whole pool forming the variates, and the most probable value of a tetrad-difference F, zero. Mackie (1928a) showed that the mean tetrad- difference is zero, and its variance, for JF\ PlP^P* + PlPtP* + PzP'tP*) + 2(N 2) where N is the number of bonds in the whole pool. He found for the mean value of r 12 the value VCjPiPzJj and for its variance 2 _ (lj- PiXi_7" P?) ^12^ ~N I~~ This is not the variance of all possible correlation coefficients, but of those formed by taking fractions p^ and j> 2 of the pool. The whole family of correlation coefficients will be widely scattered by reason of the different values of p, " rich " tests having high correlations, and those with low p, low correlations. Mackie (1929) next extended these formulas to variable coefficients (i.e. bonds which no longer were all-or-none). He again found the mean value of F to be zero, and for its variance 2 The presence of - in this is due to Mackie's limitation to TU positive loadings of the bonds. Thomson (19356, 72) removed this limitation and found 382 MATHEMATICAL APPENDIX Similarly, Mackie found for variable positive loadings (1929) = -(i-(-Yl r N t w / and for all loadings Thomson found (1935&) * = N Thomson suggested without proof that in general, when limits are set to the variability of the loadings of the bonds, resulting in a family of correlation coefficients averaging r, these correlations will form a distribution with variance - 2 - /I _ yZ\ r ~~ AT and will give tetrad-differences averaging zero with a variance 2) Summing up, Thomson says (1935fc, 77-8) : " The sam- pling principle taken alone gives correlations of all values . . . and zero tetrad -differences if N be large. Fitting the sampled elements with weights ... if the weights may be any weights . . . destroys correlation when N is infinite. This means that on the Sampling Theory a certain approxi- mation to ' all-or-none-ness ' is a necessary assumption not to explain zero tetrad-differences, but to explain the existence of correlations of ... large size. . . . The most important point in all this appears to me to be the fact that on all these hypotheses the tetrad-differences tend to vanish. This tendency appears to be a natural one among correlation coefficients . ' J A tendency for tetrad-differences to vanish means, of course a still stronger tendency for large minors of the correlational matrix to vanish. In more general terms, therefore, Thomson's theorem is that in a complete family of correlation coefficients the rank of the correlation matrix tends towards unity, and that a random sample of variates from this family will (in less strong measure) show the same tendency. REFERENCES Tins list is not a bibliography, and makes no pretensions to com- pleteness. It has, on the contrary, been kept as short as possible, and in any case contains hardly any mention of experimental articles. Other references will be found in the works here listed. References to this list in the text are given thus (Mackie, 1929, 17), or, where more than one article by the same author comes in the same year (Burt, 1937&, 84). Throughout the text, however, the two important books by Spearman and by Thurstone are referred to by the short titles Abilities and Vectors respectively, and Thurstone 's later book, Multiple Factor Analysis, by the abbrevia- tion M. F. Anal. Other abbreviations are : A.J.P. = American Journal of Psychology. B.J.P. = British Journal of Psychology, General Section. B.J. P. Statist. = British Journal of Psychology , Statistical Section B.J.E.P. = British Journal of Educational Psychology. J.E.P. Journal of Educational Psychology. Pmka. = Psychometrika. AITKEN, A. C., 1934, " Note on Selection from a Multi-variate Normal Population," Proc. Edinburgh Math. Soc., 4, 106-10. 1937a, " The Evaluation of a Certain Triple-product Matrix," Proc. Roy. Soc. Edinburgh, 57, 172-81. 1937&, " The Evaluation of the Latent Roots and Vectors of a Matrix," ibid., 57, 269-304. ALEXANDER, W. P., 1935, " Intelligence, Concrete and Abstract," B.J.P. Monograph Supplement 19. ANASTASI, Anne, 1936, " The Influence of Specific Experience upon Mental Organization," Genetic Psychol. Monographs 8, 245-355. and Garrett (see under Garrett). BAILES, S., and Thomson (see under Thomson). BARTLETT, M. S., 1935, " The Statistical Estimation of G," B.J.P., 26, 199-206. 1937a, " The Statistical Conception of Mental Factors," ibid., 28, 97-104. 19376, " The Development of Correlations among Genetic Com- ponents of Ability," Annals of Eugenics, 7, 299-302. 1938, " Methods of estimating Mental Factors," Nature, 141, 609-10. 1948, " Internal and External Factor Analysis," B.J. P. Statist., 1, 73-81. 383 384 REFERENCES BLACK, T. P., 1929, "The Probable Error of Some Boundary Conditions in diagnosing the Presence of Group and General Factors," Proc. Hoy. Soc. Edinburgh, 49, 72-7. BLAKEY, R., 1940, " Re-analysis of a Test of the Theory of Two Factors," Pmka., 5, 121-36. BBOWN, W., 1910, " The Correlation of Mental Abilities," B.J.P., 3, 296-322. and Stephenson, W., 1933, "A Test of the Theory of Two Factors," ibid., 23, 352-70. 1911, and with Thomson, G. H., 1921, 1925, and 1940, The Essentials of Mental Measurement (Cambridge). BUBT, C., 1917, The Distribution and Relations of Educational Abili- ties (London). 1936, " The Analysis of Examination Marks," a memorandum in The Marks of Examiners (London), by Hartog and Rhodes. 1937a, " Methods of Factor Analysis with and without Successive Approximation," B.J.E.P., 7, 172-95. 19376, " Correlations between Persons," B.J.P., 28,, 59-96. 1938a, " The Analysis of Temperament," B. J. Medical P., 17, 158-88. 19386, " The Unit Hierarchy," Pmka., 3, 151-68. 1939, " Factorial Analysis. Lines of Possible Reconcilement," B.J.P., 30, 84-93. 1940, The Factors of the Mind (London). and Stephenson, W., " Alternative Views on Correlations between Persons," Pmka., 4, 269-82. CATTELL, R. B., 19440, "Cluster Search Methods," Pmka., 9,169-84. 19446, " Parallel Proportional Profiles," Pmka., 9, '267-83. 1945, " The Principal Trait Clusters for Describing Personality," Psychol. Bull, 42, 129-61. 1946, Description and Measurement of Personality (New York). 1948, " Personality Factors in Women," B.J. P. Statist., 1, 114- 130. COOMBS, C. II., 1941, " Criterion for Significant Common Factor Variance," Pmka., 6, 267-72. DAVEY, Constance M., 1926, " A Comparison of Group, Verbal, and Pictorial Intelligence Tests," B.J.P., 17, 27-48. DAVIS, F. B., 1945, " Reliability of Component Scores," Pmka., 10, 57-60. DODD, S. C., 1928, " The Theory of Factors," Psychol. Rev., 35, 211-34 and 261-79. 1929, " The Sampling Theory of Intelligence," B.J.P., 19, 306-27. EMMETT, W. G., 1936, " Sampling Error and the Two-factor Theory," B.J.P., 26, 362-87. 1949, " Factor Analysis by Lawley's Method of Maximum Likeli- hood," B.J.P.Statist., II, (2), 90-7. FERGUSON, G. A., 1941, "The Factorial Interpretation of Test Difficulty," Pmka. 9 6, 323-29. REFERENCES 385 FISHER, R. A., 1922, " Goodness of Fit of Regression Formulae," Journ. Roy. Stat. Soc., 85, 597-612. 1925, " Applications of 4 Student's ' Distribution," Metron, 5 (3), 3-17. 1925 and later editions, Statistical Methods for Research Workers (Edinburgh). 1935 and later editions, The Design of Experiments (Edinburgh), and Yates, F., 1938 and later editions, Statistical Tables (Edinburgh). GAKNKTT, J. C. M., 1919a, " On Certain Independent Factors in Mental Measurement," Proc. Roy. Soc., A, 96, 91-111. 19196, " General Ability, Cleverness, and Purpose," B.J.P., 9, 345-66. GARRETT, II. E., and Anastasi, Anne, 1932, " The Tetrad-difference Criterion and the Measurement of Mental Traits," Annals New York Acad. Sciences, 33, 233-82. HEYWOOD, H. B., 1931, " On Finite Sequences of Real Numbers," Proc. Roy. Soc., A, 134, 486-501. HOLZINGER, K. J., 1935, Preliminary Reports on Spearman-Hoi- zinger Unitary Trait Study, No. 5 ; Introduction to Bif actor Theory (Chicago). 1937, Student Manual of Factor Analysis (Chicago) (assisted by Frances Swineford and Harry Harman). 1940, " A Synthetic Approach to Factor Analysis," Pmka., 5, 235-50. 1944, " A Simple Method of Factor Analysis," Pmka., 9, 257-62. and Harman, H. II., 19376, "Relationships between Factors obtained from Certain Analyses," J.E.P., 28, 321-45. and Harman, H. H., 1938, " Comparison of Two Factorial Analyses," Pmka., 3, 45-60. and Harman, H. II., 1941, Factor Analysis (Chicago), and Spearman (see under Spearman). HORST, P., 1941 , " A Non-graphical Method for Transforming into Simple Structure," Pmka., 6, 79-100. HOTELLING, H., 1933. "Analysis of a Complex of Statistical Variables into Principal Components," J.E.P., 24, 417-41 and 498-520. 1935a, " The Most Predictable Criterion," J.E.P., 26, 139-42. 19356, " Simplified Calculation of Principal Components," Pmka., 1, 27-35. 1936, " Relations between Two Sets of Variates," Biometrika, 28, 321-77. IRWIN, J. O., 1932, " On the Uniqueness of the Factor g for General Intelligence," B.J.P., 22, 359-63. 1933, " A Critical Discussion of the Single-factor Theory," ibid., 23, 371-81. KELLEY, T. L., 1923, Statistical Method (New York). 1928, Crossroads in the Mind of Man (Stanford and Oxford). 1935, Essential Traits of Mental Life (Harvard). 386 REFERENCES LANDAHL, H. D., 1938, u Ccntroid Orthogonal Transformations," Pmka., 3, 219-23. LAWLEY, D. N., 1940, " The Estimation of Factor Loadings by the Method of Maximum Likelihood," Proc. Roy. Soc. Edin., 60, 64-82. 1941, " Further Investigations in Factor Estimation," ibid., 16, 176-85. 1943a, " On Problems Connected with Item Selection and Test Construction," ibid., 61, 273-87. 19436, " The Application of the Maximum Likelihood Method to Factor Analysis," B.J.P., 33, 172-175. 1943e, " A Note on Karl Pearson's Selection Formulae," Proc. Roy. Soc., Edin., 62, 28-30. 1944, " The Factorial Analysis of Multiple Item Tests," ibid., 62, 74-82. 1949, " Problems in Factor Analysis," ibid., 62, Part IV, (No. 41). LEDERMANN, W., 1936, " Mathematical Remarks concerning Bound- ary Conditions in Factorial Analysis," Pmka., 1, 165-74. 1937a, " On the Rank of the Reduced Correlational Matrix in Multiple-factor Analysis," ibid., 2, 85-93. 19376, " On an Upper Limit for the Latent Roots of a Certain Class of Matrices," J. Land. Math. Soc., 12, 14-18. 1938a, " A Shortened Method of Estimation of Mental Factors by Regression," Nature, 141, 650. 19386, " Note on Professor Godfrey Thomson's Article on the Influence of Univariate Selection on Factorial Analysis," B.J.P., 29, 69-73. 1938c, " The Orthogonal Transformations of a Factorial Matrix into Itself," Pmka., 3, 181-87. 1939, " Sampling Distribution and Selection in a Normal Popu- lation," Biometrika, 30, 295-304. 19396, " A Shortened Method of Estimation of Mental Factors by Regression," Pmka., 4, 109-16. 1940, " A Problem Concerning Matrices with Variable Diagonal Elements," Proc. Roy. Soc. Edin., 60, 1-17. and Thomson (see under Thomson). MACKIE, J., 1928a, " The Probable Value of the Tetrad-difference on the Sampling Theory," B.J.P., 19, 65-76. 19286, " The Sampling Theory as a Variant of the Two-factor Theory," J.E.P., 19, 614-21. 1929, " Mathematical Consequences of Certain Theories of Mental Ability," Proc. Roy. Soc. Edinburgh, 49, 16-37. MCNEMAR, Q., 1941, " On the Sampling Errors of Factor Loadings," Pmka., 6, 141-52. 1942, " On the Number of Factors," ibid., 7, 9-18. MKDLAND, F. F., 1947, "An empirical comparison of Methods of Communality Estimation," Pmka., 12, 101-10. REFERENCES 387 MOSIER, C. I., 1939, " Influence of Chance Error on Simple Struc- ture," Pmka., 4, 33-44. PEARSON, K., 1902, " On the Influence of Natural Selection on the Variability and Correlation of Organs," Phil. Trans. Roy. Soc. London, 200, A, 1-66. 1912, " On the General Theory of the Influence of Selection on Correlation and Variation," Biometrika, 8, 437-43. and Filon, L. N. G., 1898, " On the Probable Errors of Frequency Constants and on the Influence of Random Selection on Varia- tion and Correlation," Phil. Trans. Roy. Soc. London, 191, A, 229-311. Jeffery, G. B., and Elderton, E. M., 1929, " On the Distribution of the First Product-moment Coefficient, etc.," Biometrika, 21, 191-2. and Moul, M., 1927, " The Sampling Errors in the Theory of a Generalized Factor," ibid., 19, 246-91. PEEL, E. A., 1947, " A short method for calculating Maximum Battery Reliability," Nature, 159, 816. 1948, " Prediction of a Complex Criterion and Battery Re- liability," B.J.P.Statist., 1, 84-94. PIAGGIO, H. T. H., 1933, " Three Sets of Conditions Necessary for the Existence of a g that is Real," B.J.P., 24, 88-105. PRICE, B,, 1936, " Homogamy and the Intercorrelation of Capacity Traits," Annals of Eugenics, 7, 22-7. REYBURN, H. A., and Taylor, J. G., 1939, " Some Factors of Personality," B.J.P., 30, 151-65. 1941a, " Some Factors of Intelligence," ibid., 31, 249-61. 19416, " Factors in Introversion and Extra version," ibid., 31, 335-40. 1943a, " On the Interpretation of Common Factors : a Criticism and a Statement," Pmka., 8, 53-64. 19436, " Some Factors of Temperament : a Re-examination," ibid., 8, 91-104. SPEARMAN, C., 1904, " General Intelligence objectively Determined and Measured," A.J.P., 15, 201-93. 1913, " Correlation of Sums or Differences," B.J.P., 5, 417-26. 1914, " The Theory of Two Factors," Psychol. Rev., 21, 101-15. 1926 and 1932, The Abilities of Man (London). 1928, " The Substructure of the Mind," B.J.P., 18, 249-61. 1931, " Sampling Error of Tetrad -differences," J.E.P., 22, 388. 1939a, " Thurstone's Work Reworked," J.E.P., 30, 1-16. 19396, " Determination of Factors," B.J.P., 30, 78-83. and Hart, B., 1912, " General Ability, its Existence and Nature," B.J.P., 5, 51-84. and Holzinger, K. J., 1924, " The Sampling Error in the Theory of Two Factors," B.J.P., 15, 17-19. and Holzinger, K. J., 1925, " Note on the Sampling Error of Tetrad-differences," ibid., 16, 86-8. 388 REFERENCES SPEARMAN, C., and Holzinger, K. J., 1929, " Average Value for the Probable Error of Tetrad-differences/' ibid., 20, 368-70. STEPHENSON, W., 1931, " Tetrad-differences for Verbal Sub-tests relative to Non-verbal Sub-tests," J.E.P., 22, 334-50. 1935a, " The Technique of Factor Analysis," Nature, 136, 297. 19356, " Correlating Persons instead of Tests," Character and Personality, 4, 17-24. 19360, " A new Application of Correlation to Averages," B.J.E.P., 6, 43-57. 19366, " The Inverted-factor Technique," B.J.P., 26, 344-61. 1936c, " Introduction to In verted -factor Analysis, with some Applications to Studies in Orexis," J.E.P., 27, 353-67. 1936d, " The Foundations of Psychometry : Four Factor Sys- tems," Pmka., 1, 195-209. 1939, " Abilities Defined as Non-fractional Factors," B.J.P., 30, 94-104. and Brown (sec under Brown). and Burt, C. (see under Burt). SWINEFOBD, FRANCES, 1941, " Comparisons of the Multiple -factor and Bifactor Methods of Analysis," Pmka., 6, 375-82 (see also Holzinger, 19370). THOMPSON, J. R. (see under Thomson, 19196). 1929, " The General Expression for Boundary Conditions and the Limits of Correlation," Proc. Roy. Soc. Edinburgh, 49, 65-71. THOMSON, G. H., 1916, " A Hierarchy without a General Factor," B.J.P., 8, 271-81. 19190, " On the Cause of Hierarchical Order among Correlation Coefficients," Proc. Roy. Soc., A, 95, 400-8. 19196, " The Proof or Disproof of the Existence of General Ability" (with Appendix by J. K. Thompson), B.J.P., 9, 321-36. 19270, " The Tetrad-difference Criterion," B.J.P., 17, 235-55. 19276, " A Worked-out Example of the Possible Linkages of Four Correlated Variables on the Sampling Theory," ibid., 18, 68-76. 19340, " Hotelling's Method modified to give Spearman's g," J.E.P., 25, 366-74. 19346, " The Meaning of i in the Estimate of g," B.J.P., 25, 92-9. 1934c, " On measuring g and s by Tests which break the ^-hierarchy," ibid., 25, 204-40. 19350, " The Definition and Measurement of g (General Intelli- gence)," J.E.P., 26, 241-62. 19356, " On Complete Families of Correlation Coefficients and their Tendency to Zero Tetrad-differences : including a State- ment of the Sampling Theory of Abilities," B.J.P., 26, 63-92. 19360, " Some Points of Mathematical Technique in the Factorial Analysis of Ability," J^.P., 27, 37-54. REFERENCES 889 THOMSON, G. H., 19366, "Boundary Conditions in the Common-factor Space, in the Factorial Analysis of Ability," Pmka., 1, 155-63. 1937, " Selection and Mental Factors," Nature, 140, 934. 1938a, " Methods of estimating Mental Factors," ibid., 141, 246. 19386, " The Influence of Univariate Selection on the Factorial Analysis of Ability," B. J.P., 28, 451-9. 1938c, " On Maximizing the Specific Factors in the Analysis of Ability," B.J.E.P., 8, 255-63. 1938d, " The Application of Quantitative Methods in Psychology," see Proc. Roy. Soc. B., 125, 415-34. 19380, " Recent Developments of Statistical Methods in Psy- chology," Occupational Psychology, 12, 319-25. 1939a " Factorial Analysis. The Present Position and the Problems Confronting Us," B.J.P., 30, 71-77. "Agreement and Disagreement in Factor Analysis. A Summing Up," ibid., 1058. 19396, " Natural Variances of Mental Tests, and the Symmetry Criterion," Nature, 144, 516. 19400, An Analysis of Performance Test Scores of a Representative Group of Scottish Children (London). 19406, "Weighting for Battery Reliability and Prediction," B.J.P., 30, 357-66. 1941, "The Speed Factor in Performance Tests," B.J.P., 32, 131-5. 1943a, " A Note on Karl Pearson's Selection Formulae," Math. Gazette, December, 197-8. 1944, " The Applicability of Karl Pearson's Selection Formula? in Follow-up Experiments," B.J.P., 34, 105. and Bailes, S., 1926, " The Reliability of Essay Marks," The Forum of Education, 4, 85-91. and Brown, W. (see under Brown), and Ledermann, W., 1938, " The Influence of Multi-variate Selection on the Factorial Analysis of Ability," B.J.P., 29, 288-306. 1947, " Maximum Correlation of Two Weighted Batteries," B. J.P. Statist., 1, (1), 27-34. 1948, " Relations of Two Weighted Batteries," B.J.P. Statist., 1, (2), S2-3. 1949, " On Estimating Oblique Factors," B.J.P. Statist., 2, (l), 1-2. THORNDIKE, E. L., 1925, The Measurement of Intelligence (New York). THURSTONE, L. L., 1932, The Theory of Multiple Factors (Chicago). 1933, A Simplified Multiple-factor Method (Chicago). 1935, The Vectors of Mind (Chicago). 1938a, " Primary Mental Abilities," Psychometric Monograph No. 1 (Chicago). 19386, " The Perceptual Factor," Pmka., 3, 1-18. 1938c, " A New Rotational Method," ibid., 3, 199-218. 390 REFERENCES THUBSTONE, L. L., 1940a, " An Experimental Study of Simple Structure," ibid., 5, 153-68. 19406, " Current Issues in Factor Analysis," PsychoL Bull, 37, 189-236. 19440, " Second-order Factors," Pmka., 9, 71-100. 19446, A Factorial Study of Perception (Chicago), and Thurstone, T. G., 1941, " Factorial Studies of Intelligence," Psychometric Monograph, 2 (Chicago). 1947, Multiple Factor Analysis (Chicago). THURSTONE, T. G., 1941, " Primary Mental Abilities of Children," Educ. and PsychoL Meas., 1, 105-16. TRYON, R. C., 1932a, " Multiple Factors vs Two Factors as Deter- miners of Abilities," PsychoL Rev., 39, 324-51. 19326, " So-called Group Factors as Determiners of Ability," ibid., 39, 403-39. 1935, " A Theory of Psychological Components an Alternative to Mathematical Factors," ibid., 42, 425-54. 1939, Cluster Analysis (Berkeley, Cal.). TUCKER, L. R., 1940, " The Role of Correlated Factors in Factor Analysis," Pmka., 5, 141-52. 1944, " A Semianalytical Method of Rotation to Simple Struc- ture," Pmka., 9, 43-68. WILSON, E. B., 1928a, " On Hierarchical Correlation Systems," Proc. Nat. Acad. Sc., 14, 283-91. 19286, " Review of The Abilities of Man," Science, 67, 244-8. 1933a, " On the In variance of General Intelligence," Proc. Nat. Acad. Sc., 19, 768-72. 19336, " Transformations preserving the Tetrad Equations," ibid., 19, 882-4. 1933c, " On Overlap," ibid., 19, 1039-44. 1934, " On Resolution into Generals and Specifics," ibid., 20, 193-6. and Worcester, Jane, 1934, " The Resolution of Four Tests," ibid., 20, 189-92. and Worcester, Jane, 1939, " Note on Factor Analysis," Pmka., 4, 133-48. WISHART, J., 1928, " Sampling Errors in the Theory of Two Factors ," B.J.P., 19, 180-7. WORCESTER, Jane, and Wilson (see under Wilson). WRIGHT, S., 1921, " Systems of Mating," Genetics, 6, 111-78. YATES, (see Fisher And Yates). YOUNG, G., 1939, " Factor Analysis and the Index of Clustering," Pmka., 4, 201-8. 1941, " Maximum Likelihood Estimation and Factor Analysis," ibid., 6, 49-53. and Householder, A. S., 1940, " Factorial Invariance and Signi- ficance," ibid., 5, 47-56, INDEX The numbers refer to pages. Terms which occur repeatedly through the book are only indexed on their first occurrence or where they are denned. Most chapter and section headings are indexed. The names Spearman and Thurstone (which occur so frequently) are not indexed, nor is the author's name. Acceleration by powering, 74. Aitken, 22, 76, 89 ff., 103, 107, 188 (selection formula), 369, 378. Alexander, 36, 51, 91, 114, 243. Anarchic doctrine, 42, 47. Anastasi, 306. Attenuation, 79, 148. Axes, 60 (orthogonal), 61 (Spear- man), 68 and 363 (principal). Babington Smith, 353. Bailes, 204. Bartlett, M. S., 51, 100, 134 ff., and 369 (Bartletfs estimates), 138 (a numerical calculation of), 205, 305, 318-19, 365, 366 (external factorial analysis). Bifactors, 19, 286, 343. Binet, 4, 51-2, 91. Bipolar factors, 332. Black, 267, 380. Blakey, 169. Bonds of the mind, 45 ff., 51. Boundary conditions, 262 ff., 380. " Box " correlations, 280, 297. Brown, 7, 18, 42, 148, 227, 236, 242. Burt, 25, 75, 169, 199, 206 ff. (marks of examiners), 213 ff., 221 (temperament), 289 (covari- ances), 330, 332, 347, 372, 375. Cattell, 295, 339, 349. Centring a matrix, 214, 218 (special features of double centring). " Centroid " method, 23, 98 (and pooling square), 164 (with guessed communalities), 354. Circumflex mark (as x) indicates an estimate, see 83 ff . Cluster analysis, 349. Coefficient of richness, 45. Common-factor space, 63 ff . Communalities, 23, 38 (unique), 161 (approximate), 361. Coombs, 169. Correlation coefficient, 5 (product- moment formula), 57, 64 (as cosine), 83 (as estimation coeffi- cient), 168 (of correlation coeffi- cients), 173 (partial), 199 ff. (between persons). Cosine as correlation coefficient 57, 64. Co variance, 11, 214 (analysis of), 330, 352 (of regression coeffi- cients), 372. Criterion, 85. Davis, 170. Degrees of freedom, 148. Direction cosines, 283. 391 892 Dodd, 43. Doolittle method, 369. INDEX Elderton, 151, 372. Ellipsoids of density, 67, 363. Emmett, 151, 152, 329. Error specifics, 132, 167. Error, standard, see standard deviation. Estimates, 118 ff. (correlated), 124 (calculation of variances and covariances), 134 ff. and 369 (BartletVs). Estimation, 83 ft ., 95 (geomet- rical picture of), 102 (of fac- tors by regression), 110 and 367 (of specifics), 115 (direct or via factors), 367. Etherington, 91. Extended vectors, 250, 274. Factors, 3 (tests), 4 (fictitious), 14 (group), 15 (verbal), 102 (estimation by regression), 168 (limits to number of), 186 (oblique), 193 (creation of new), 240 (danger of reifying), 262 ff. (limits to extent of), 273 (prim- ary), 297 ff. (second order). Filon, 167. Fisher, 150, 321, 352. G, 8 (saturations), 49 (mental energy), 225 (definition of), 230 (pure g), 286. Garnett, 20, 31, 57, 354. Geometrical picture of correla- tion, 55, 66, 353. Geometrical picture of estima- tion, 95. Group factors, 14. Harman, 286. Hart, 7. Heywood, 131, 231, 298. Hierarchical order, 5, 234 (when persons and tests equally nu- merous). Histogram, 13, 152. Hollow staircase pattern, 19. Holzinger, 19, 151, 152, 286, 343, 346, 369, 372. Hotelling, 24, 53, 60-1, 66 ff., 86, 100, 133, 170, 215, 330, 354, 364, 365. Independence of units, 332 ff. Indeterminacy, 336, 371. Inequality of men, 53, 313. Inner product, 31. Invariance of simple structure, 184, 292, 294. Jcffery, 151, 372. Kelley, 74, 99, 157, 175. Landahl, 255. - Latent root, 72-3, 75-6, 215, 267, 363. Lawley, 169, 195, 321, 334, 373. Ledermann, 40, 110, 186, 187, 260, 267, 270, 371, 372, 373, 378, 380. Loadings, 25, 76 (properties of Hotelling). Mackie, 53, 312, 314, 381, 382. McNemar, 167, 169. Matrix, 8, 190 (calculation of re- ciprocal), 350. Maximum likelihood method, 169, 321, 373. Medland, 162, 352. Mental energy, 49, 241. Metric, 328. Minor determinant, 21. Monarchic doctrine, 42, 47. Moods, 211, INDEX 393 Moul, 151, 372. Multiple correlation, 85 ff., 94-5 (calculation of), 98. Multiple -factor analysis, 20 ff. 161 ff. Multivariate selection, 187, 294. Natural units, 331. Negative loadings, 33, 77, 211, 304. Neural machines, 49. Normal curve, 145. Normalized scores, 6, 360. Normalizing coefficients, 252, 275. Oblique factors, 186, 192, 195, 261, 272 ff., 292, 339, 375. Oligarchic doctrine, 42, 47. Order of a determinant, 21. Orthogonal axes, 60. Orthogonal matrices, 291. Otis-Kelley formula, 175. Oval diagrams, 11. Parallel proportional profiles, 295. Parsimony, 15. Pattern and structure, 272, 375. Pearson, K., 5, 151, 167, 171, 373. Peel, E. A., 100, 367. Physical measurements and hier- archical order, 315. Piaggio, 107. Pivotal condensation, 22, 89. Pooling square, 85 ff., 98 (and "centroid" method), 364. Price, 305. Primary factors, 273, 277. Principal components, 53, 60, 66 ff., 69 (advantages and dis- advantages), 71 (calculation of loadings), 78 [calculation of a man's), 170, 363. Product-moment formula, 5. Purification of batteries, 20, 44, 129, 226. Rank of a matrix, 20, 178 (un- changed by selection), 183 (the same), 306 and 382 (low re- duced rank). Reciprocity of loadings and factors, 217 ff., 375. Reference values for detecting specific correlation, 156 ff. Reference vectors, 272, 277. Regression coefficients, 87 ff., 93, 350 (Aitkerfs computation). Regression equation, 92, etc., 365. Reliabilities, 79, 100, 148, 170. Residual matrix, 28, 155. Reyburn, 170, 287 ff., 339, 349. Richness, coefficient of, 45, 331, 381. Rosner, 355 Rotation, Landahl, 255. Rotation of axes, 36, 55, 64, 243, 247 (graphical method), 249 ff. (new method), 331, 361, 379. Sampling error, 143 (two factor s) 9 167, 372 (of tetrad-differences). Sampling theory, 42 ff., 303 ff., 362, 381. Saturations, 8, 17, 153 (Spear- man's formula), 372. Second-order factors, 297 ff., 379. Selection, 171 ff. (univariate), 173 (partial correlation), 184 (geo- metrical picture of), 185 (ran- dom), 187 ff. (multivariate). Sheppard, 354. Sign-changing in " centroid " pro- cess, 28, 30. Significance of factors, 324, 326. Simple structure, 242 If., 245 (numerical example), 294, 332 (independent of units). 394 INDEX Singly conforming tests, 236. Spearman weights, 105-6 (cal- culation of), 372. Specifics, 48 (maximized), 130 (maximized and minimized), 132 (error specifics), 336. Standard deviation, 5, 149 (of variance and of correlation coefficient), 352 (of a regression coefficient). Standardized scores, 6, 10, 360. Stephenson, 18, 199, 201, 209, 211, 212, 227, 236, 242. Subpools of the mind, 50, 319. Swineford, 170. Sylvester, 375. Taylor, 170, 287 ff., 339, 349. Tetrad-differences, 12, 21. Thompson, J. R., 262, 267, 380. Thorndike, 91, 307, 317. Tripod analogy, 57. Tryon, 42, 344, 349. Tucker, 169. Two-factor theory, 3 ff., 361. Unique communalities, 38, 40 (formula). Units, rational, 331. Univariate selection, 171 ff., 292. Variance, 5, 11, 318 (absolute), 331 (natural), 350 (of regres- sion coefficients). Vectors, 56. Verbal factor, 15. Vocational guidance or advice 19, 52, 114, 368. Weighted battery, 10 (Spear- man's weights), 105-6 (cal- culation of), 203 (of examin- ers). Wilson, 41, 57, 110, 168, 367. Wishart, 151, 372. Worcester, 41, 168. Wright, 305. Zero loadings, 65.