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Multiple-factor analysis : its purpose and meaning Hellyer, Sydney 1950

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MULTIPLE-FACTOR ANALYSIS: ITS PURPOSE AND MEANIN( by Sydney Hellyer A Thesis submitted in Partial Fulfilment of The Requirements for the Degree of MASTER OF ARTS in the Department . of PHILOSOPHY AND PSYCHOLOGY The University of Brit i s h Columbia September, 1950 A b s t r a c t The p u r p o s e o f t h e p r e s e n t s t u d y was t h a t o f p r e s e n t i n g t h e f u n d a m e n t a l t h e o r e m s a n d t e c h n i q u e s o f T h u r s t o n e ' s M u l t i p l e - F a c t o r A n a l y s i s i n a manner t h a t w o u l d be u n d e r -s t a n d a b l e t o t h e n o n - m a t h e m a t i c a l l y t r a i n e d s t u d e n t o f p s y -c h o l o g y . The w o r k was i n t r o d u c e d b y a d i s c u s s i o n o f Spearman's " T h e o r y o f Two F a c t o r s " w h i c h i s s o - c a l l e d s i n c e t h e m e t h o d a n a l y s e s e a c h t e s t i n t o a g e n e r a l f a c t o r " g " a n d a s p e c i f i c f a c t o r . I t was f o u n d t h a t s u c h a f a c t o r i z a t i o n c o u l d be p e r -f o r m e d i f t h e c o r r e l a t i o n m a t r i x e x h i b i t e d h i e r a r c h i c a l o r -d e r . The t e t r a d d i f f e r e n c e e q u a t i o n a n d a s t a n d a r d e r r o r f o r m u l a w e r e n e x t d i s c u s s e d a s t e s t s o f t h i s o r d e r . F i n a l l y , m e n t i o n was made o f a f e w o f t h e p r o b l e m s w h i c h i n v e s t i g a t o r s h a v e a t t e m p t e d t o s o l v e b y means o f t h e m e t h o d . Upon t u r n i n g a t t e n t i o n t o t h e m u l t i p l e - f a c t o r t h e o r y , a n i n i t i a l c h a p t e r was d e v o t e d t o t h e p r e s e n t a t i o n o f t h e r e l a -t i o n s h i p b e t w e e n s u c h t h e o r y a n d s c i e n t i f i c m e t h o d i n g e n e r a l . T h i s f a c i l i t a t e d a d i s c u s s i o n o f some o f t h e m a j o r l i m i t a -t i o n s a n d v a l u e s o f t h e t e c h n i q u e . I t was f o u n d t h a t t h e m e t h o d s c o u l d b e s t be d e s i g n a t e d a s c l a s s i f i c a t o r y . F u r t h e r -m o r e , i t was d i s c o v e r e d t h a t f a c t o r a n a l y s i s was n o t o n l y a p p l i c a b l e t o a d i v e r s i t y o f p r o b l e m s i n p s y c h o l o g y b u t a l s o t o t h o s e o f o t h e r s c i e n c e s . F o l l o w i n g t h i s "more g e n e r a l d i s c u s s i o n , c o n s i d e r a t i o n was g i v e n t o t h e d e r i v a t i o n o f t h e f u n d a m e n t a l e q u a t i o n s . I t was p r o v e n t h a t i f t h e o r i g i n a l s c o r e s a r e c o n v e r t e d t o s t a n d a r d f o r m t h e sum o f t h e s q u a r e s o f t h e f a c t o r l o a d i n g s f o r a n y t e s t i s e q u a l t o u n i t y . I t was a l s o shown t h a t e a c h f a c t o r l o a d i n g f o r s t a t i s t i c a l l y i n d e p e n d e n t f a c t o r s i s t h e s q u a r e r o o t o f t h e v a r i a n c e t h a t i s a t t r i b u t a b l e t o t h a t f a c t o r . The f u n d a m e n t a l t h e o r e m o f f a c t o r a n a l y s i s , w h i c h p r o v e s t h a t a n y r e d u c e d c o r r e l a t i o n m a t r i x c a n be f a c t o r i z e d , was t h e n d i s c u s s e d s i n c e i t i s b a s i c t o t h e e n t i r e t h e o r y . The c e n t r o i d m e t h o d o f f a c t o r i n g a r e d u c e d c o r r e l a t i o n m a t r i x was n e x t d e v e l o p e d . A m e t h o d was t h u s o b t a i n e d b y w h i c h t h e c o r r e l a t i o n m a t r i x c o u l d be a n a l y s e d i n t o a common f a c t o r m a t r i x . H o w e v e r , s i n c e T h u r s t o n e d o e s n o t c o n s i d e r t h e s e t o be p s y c h o l o g i c a l l y m e a n i n g f u l f a c t o r s i t was f o u n d t h a t a r o t a t i o n o f r e f e r e n c e f r a m e m u s t t h e n be c a r r i e d o u t . C o n s i d e r a t i o n was g i v e n t o t h e p r o b l e m o f r o t a t i n g a x e s a n d a m e t h o d e x a m i n e d w h i c h e n a b l e s one t o r o t a t e a x e s i n a t w o - d i m e n s i o n a l p l o t . T h i s m e t h o d was f o u n d t o be s u c h t h a t i t c o u l d be g e n e r a l i z e d t o a n r d i m e n s i o n a l f a c t o r p a t t e r n . I t h a d t h e a d v a n t a g e o f f u r n i s h i n g a g r a p h i c r e c o r d o f t h e r e l a t i o n s h i p s among t h e t e s t s . H o w e v e r , i t h a d a d i s a d v a n -t a g e i n t h a t many d i a g r a m s m u s t be p l o t t e d i f t h e number o f d i m e n s i o n s i s l a r g e . Upon e x a m i n i n g t h e p r o b l e m o f a t t a c h i n g p s y c h o l o g i c a l m e a n i n g t o t h e f a c t o r s , i t was n o t e d t h a t s u b j e c t i v i t y m u s t p l a y a l a r g e p a r t a n d s o n o r o t e r u l e s c o u l d be f u r n i s h e d a s a s o l u t i o n t o t h i s p r o b l e m . F o r t h i s r e a s o n s e v e r a l e x a m p l e s ( i ) o f i n t e r p r e t a t i o n s w e r e p r e s e n t e d i n o r d e r t o e n a b l e t h e r e a d e r t o o b t a i n a c l e a r e r i n s i g h t i n t o t h e l o g i c a l s t e p s em p l o y e d . F i n a l l y , t h e l i m i t a t i o n s o f t h i s s t u d y w e r e c o n s i d e r e d a n d a b r i e f d i s c u s s i o n o f t h e p o s s i b l e f u t u r e d e v e l o p m e n t o f f a c t o r a n a l y s i s was p r e s e n t e d . ( i i ) TABLE OF CONTENTS Page INTRODUCTION 1 P u r p o s e o f F a c t o r A n a l y s i s . . . . 1 P u r p o s e o f t h e S t u d y . . . . . . 1 S c o p e o f t h e I n v e s t i g a t i o n . . . . 2 CHAPTER I . SPEARMAN'S METHOD OF FACTORIZATION . 3 P u r p o s e o f t h e C h a p t e r . . . . . 3 T h e o r e t i c a l B a s e s o f Spearman's M e t h o d . . 3 D e s c r i p t i o n o f a M a t r i x . . . . . 5 H i e r a r c h i c a l O r d e r a n d "g" . . . . 7 S p e c i f i c F a c t o r s . . . . . . 9 F a c t o r P a t t e r n s . . . . . . . 10 T e t r a d D i f f e r e n c e E q u a t i o n s . . . . 11 E v a l u a t i o n o f G e n e r a l a n d S p e c i f i c F a c t o r s . 15 E r r o r s i n E m p i r i c a l C o r r e l a t i o n C o e f f i c i e n t s . 20 T e s t f o r H i e r a r c h i c a l O r d e r . . . . 22 E v a l u a t i o n o f F a c t o r s f r o m F a l l i b l e D a t a . . 26 Some U s e s o f t h e M e t h o d . . . . . 29 CHAPTER I I . SCIENCE AND THE FACTOR PROBLEM . . 3 1 P u r p o s e o f t h e C h a p t e r . . . . . 3 1 I m p o r t a n c e o f T h e o r e t i c a l K n o w l e d g e . . . 3 1 P r i m a r y P o s t u l a t e o f S c i e n c e . . . . 32 F a c t o r A n a l y s i s v s F a c u l t y P s y c h o l o g y . . 33 S c i e n t i f i c S i m p l i c i t y . . . . . 35 F a c t o r A n a l y s i s , a C l a s s i f i c a t o r y S y s t e m . . 36 Some L i m i t a t i o n s o f F a c t o r A n a l y s i s . ' . 36 L i n e a r A p p r o x i m a t i o n . . . . . . 39 F a c t o r A n a l y s i s v s S t a t i s t i c s . . . . 40 P u r p o s e o f F a c t o r A n a l y s i s . . . . 4 2 O r i g i n a l S c o r e s and. D e r i v e d F a c t o r P a t t e r n s . 43 CHAPTER I I I . DERIVATION OF THE FUNDAMENTAL EQUATIONS 46 r R e a s o n s f o r G r o u p F a c t o r T h e o r y . . . 46 F u n d a m e n t a l Theorem . . . . . . 4 7 S t a n d a r d S c o r e s a s L i n e a r F u n c t i o n s . . . 48 M a t r i x M u l t i p l i c a t i o n . . . . . 49 E v a l u a t i o n o f F a c t o r L o a d i n g s . . . . 52 T y p e s o f F a c t o r s . . . . . . 56 C o m m u n a l i t y a n d U n i q u e n e s s . . . . 58 D e r i v a t i o n o f F u n d a m e n t a l Theorem . . . 59 M a t r i x R a n k a n d Number o f Common F a c t o r s . . 66 CHAPTER I V . THE CENTROID METHOD . . . . 70 R e c a p i t u l a t i o n o f R e s t r i c t i o n s . . . . TO D e r i v a t i o n o f F i r s t F a c t o r L o a d i n g s . . . 7 1 Page D e r i v a t i o n o f F i r s t R e s i d u a l M a t r i x . . . 78 Need f o r R e v e r s a l o f T e s t S i g n s . . . 80 E v a l u a t i o n o f a S i n g l e Common F a c t o r . . 82 C o m p a r i s o n o f Spearman a n d C e n t r o i d L o a d i n g s . 8 7 E v a l u a t i o n o f Two Common F a c t o r s . . . 88 R e v e r s a l o f S i g n s i n t h e R e s i d u a l M a t r i x . . 92 M e a n i n g o f C e n t r o i d F a c t o r s . . . . 9 7 CHAPTER V. THE ROTATION PROBLEM . . . . 1 0 0 G e o m e t r i c R e p r e s e n t a t i o n o f a F a c t o r M a t r i x . 100 L e n g t h o f a T e s t V e c t o r . . . . . 103 G e o m e t r i c R e p r e s e n t a t i o n o f a C o r r e l a t i o n C o e f f i c i e n t . . . . . . . 104 I l l u s t r a t i o n o f C o r r e l a t i o n R e p r e s e n t a t i o n . 1 08 R e a s o n s f o r R o t a t i o n o f A x e s . . . . 115 M a t h e m a t i c a l B a s e s f o r R o t a t i o n . . . 118 R o t a t i o n i n Two D i m e n s i o n s . . . . 120 C r i t e r i a f o r S e l e c t i n g New A x e s . . . 122 E x t e n s i o n o f R o t a t i o n t o " r " D i m e n s i o n s . . 130 CHAPTER V I . INTERPRETATION OF FACTORS . . . 1 3 3 S u b j e c t i v e N a t u r e o f I n t e r p r e t a t i o n . . . 133 C r i t e r i a f o r S e l e c t i n g E x a m p l e s . . . 135 I n t e r p r e t a t i o n o f a S i m p l e P a t t e r n . . . 136 I n t e r p r e t a t i o n o f a more C o m p l e x P a t t e r n . . 144 E x a m p l e o f I n a d e q u a t e I n t e r p r e t a t i o n . . 152 S u g g e s t i o n f o r F u r t h e r P r a c t i c e . . . 155 CHAPTER V I I . CONCLUSION 1 5 7 L i m i t a t i o n s o f t h e S t u d y . . . . . 157 O t h e r F a c t o r i a l M e t h o d s . . . . . 157 Need f o r C h e c k i n g A s s u m p t i o n s . . . . 158 P o s s i b l e F u t u r e T r e n d s . . . . . 159 BIBLIOGRAPHY 1 6 1 MULTIPLE-FACTOR ANALYSIS; ITS PURPOSE AND MEANING I n t r o d u c t i o n s The v a r i o u s methods of f a c t o r a n a l y s i s were developed as mathematical techniques f o r the i n v e s t i g a t i o n and d e s c r i p t i o n of the i n t e r r e l a t i o n s h i p s e x i s t i n g between p s y c h o l o g i c a l or other v a r i a b l e s . These methods have been and a r e being ap-p l i e d t o the s o l u t i o n of an ever i n c r e a s i n g d i v e r s i t y o f problems i n many f i e l d s o f r e s e a r c h . The scope of such uses i s v ery c l e a r l y i n d i c a t e d by A n a s t a s i and Foley»s ( l ) sum-mary o f types o f i n v e s t i g a t i o n s t h a t have been c a r r i e d out by f a c t o r a n a l y s i s . These authors a l s o d i s c u s s the r e s u l t s o f such p r o j e c t s . Because the i n s t a n c e s when f a c t o r i a l r e s u l t s a r e encoun-t e r e d i n the p s y c h o l o g i c a l l i t e r a t u r e a r e numerous, i t was f e l t t h a t no student o f t h i s s c i e n c e c o u l d be c o n s i d e r e d t o be completely t r a i n e d u n l e s s he possessed a t l e a s t a c u r s o r y knowledge of the methods of f a c t o r a n a l y s i s . A t the same time i t was r e c o g n i z e d t h a t the primary sources of in f o r m a -t i o n r e g a r d i n g these techniques a r e w r i t t e n i n mathematical terminology t h a t i s almost e n t i r e l y incomprehensible t o the average student who has not s p e c i a l i z e d i n the study of more advanced mathematical procedures. The pr e s e n t study, t h e r e -f o r e , was developed w i t h the purpose i n mind of e x p r e s s i n g the formulae and techniques of m u l t i p l e - f a c t o r a n a l y s i s i n a mathematical form t h a t c o u l d be r e a d i l y understood w i t h only a knowledge of elementary a l g e b r a and geometry. Thus i t i s hoped t h a t the reader w i l l be a b l e t o convince h i m s e l f o f the (2) mathematical validity of the methods. At the same time i t i s intended to furnish the student with insight into the meaning of the techniques and the results of fa c t o r i a l studies. Thus the reports of factorial experiments can be read with a more c r i t i c a l understanding both of the manner in which the results were obtained and of the interpretations furnished to explain such results. In the work that follows, because of a necessary re-stri c t i o n as to the scope of the study, one method of anal-ysis has been emphasized. This i s the technique employing Thurstone's Centroid Method. It was the method selected for discussion since i t i s the one most frequently encountered in experimental studies. In conclusion i t cannot be stressed too emphatically that the follwoing presentation of a few of the techniques of multiple-factor analysis i s in no way intended to qualify the reader as a factor analyst. It i s simply intended to make i t possible for the more advanced student to study the literature with a clearer insight into the meaning of the obtained results. Those who wish to become truly proficient i n the application of factor analysis can only do so by studying the primary source material. To do this they must develop sufficient understanding of the more advanced math-ematical procedures employed in such works. For this purpose Thurstone (19) furnishes a mathematical introduction to his text which should be referred to by such students. (3) Chapter I  Spearman's Method of Factorization Before beginning a study of Thurstone's Multiple-Factor Analysis, i t would seem that a brief glance at the basic ideas of the Spearman school w i l l serve a double purpose. In the f i r s t place, since Spearman was the original pioneer in this f i e l d , such a discussion w i l l furnish a histori c a l background against which to examine the results of the past four decades of research. Secondly, since the two factor method of analysis i s simple in structure in comparison with the subject matter that i s to be investigated in the major portion of this work, i t may well serve as a firm foundation upon which to construct our ideas in an orderly manner. During the early years of the twentieth century, many important uses were being discovered for the then new re-search tool, the product moment coefficient of correlation that had been devised by Karl Pearson. It w i l l be remembered that such a coefficient i s a number which indicates the degree of correspondence between any two sets of measures. If the agreement i s perfect the coefficient obtained is 4 - 1 . If the resemblance i s perfect but inverted, such that the largest obtained value i n one set i s compared with the small-est i n the second, the second largest with the second small-est and so for a l l measures in the groups, then the corre-lation i s — 1 . There exist a l l gradations of correlation between these maximum and minimum values. ( 4 ) Spearman, w h i l e working w i t h such c o e f f i c i e n t s , noted two f a c t s concerning them. The f i r s t o f these was t h a t " . . . when any p a i r o f a b i l i t i e s a r e to any ext e n t c o r r e -l a t e d w i t h each o t h e r , t o t h i s e x t e n t they can be regarded as depending upon a common f a c t o r ( e i t h e r simple or complex)." (16, p. i ) . T h i s concept should be r e a d i l y a c c e p t a b l e as an e x p l a n a t i o n f o r the correspondence which a c o r r e l a t i o n c o e f -f i c i e n t demonstrates t o be present i f one but r e c o g n i z e t h a t the simple or complex f a c t o r so evidenced may be due to other than p u r e l y p s y c h o l o g i c a l c a u s a t i o n . The second was t h a t when a s e r i e s of i n t e r c o r r e l a t i o n s between t e s t s were exam-i n e d they always appeared t o e x h i b i t " h i e r a r c h i c a l o r d e r . " T h i s second concept can b e s t be e x p l a i n e d by means of a con-c r e t e example. For i l l u s t r a t i v e purposes l e t us c o n s i d e r the p u r e l y imaginary scores of a group of s u b j e c t s upon f i v e t e s t s of i n t e l l i g e n c e . These scores are then c o r r e l a t e d and the obtained v a l u e s f o r the i n t e r c o r r e l a t i o n s a re l i s t e d i n the v e r y convenient form of a m a t r i x . For the moment, i t w i l l s u f f i c e to c o n s i d e r such a m a t r i x as being merely a square or r e c t a n g u l a r a r r a y of numbers. The mathematical p r o p e r t i e s of such a form are n e i t h e r h e l p f u l nor necessary f o r the present d i s c u s s i o n and so w i l l be l e f t i n abeyance u n t i l such time as a p p l i c a t i o n s r e q u i r e them. The obtained m a t r i x i s shown i n Table I (17, p. 6 ) . ( 5 ) T a b l e I C o r r e l a t i o n M a t r i x 1 2 3 4 5 1 X .72 .56 .40 .48 2 .72 X .63 .45 .54 3 .56 .65 X .35 .42 4 .40 .45 .35 X .30 5 .48 .54 .4.2 .30 X £ 2.16 2.34 1.96 1.50 1.74 L e t u s e x a m i n e t h i s t a b l e more c l o s e l y . The h o r i z o n t a l l i s t s o f numbers a r e known a s r o w s w h i l e t h e v e r t i c a l l i s t s a r e c o l u m n s . They w i l l be s o d e s i g n a t e d f o r t h e r e m a i n d e r o f t h i s s t u d y . Any i n d i v i d u a l n umber, t e r m e d a n e l e m e n t o f t h e m a t r i x , shows t h e amount o f i n t e r c o r r e l a t i o n e x i s t i n g b e t w e e n t h e t e s t whose number i s l i s t e d o n t h e l e f t o f t h a t r o w a n d t h e t e s t whose number i s shown a t t h e t o p o f t h a t c o l u m n . F o r e x a m p l e , t h e number .40 i n t h e f i r s t r o w a n d f o u r t h c o l -umn i s t h e v a l u e o f t h e c o r r e l a t i o n b e t w e e n t e s t one a n d t e s t f o u r , u s u a l l y w r i t t e n r 1 4 . T h e r e i s a s e r i e s o f X's i n t h e s p a c e s c o n s t i t u t i n g w h a t i s g e n e r a l l y d e s i g n a t e d a s t h e p r i n c i p a l d i a g o n a l o f t h e m a t r i x . A c c o r d i n g t o t h e p u r p o s e f o r w h i c h t h e a n a l y s i s i s t o be p e r f o r m e d a n d t h e m e t h o d t o be u s e d i n o b t a i n i n g f a c t o r s , a s e r i e s o f d i f f e r e n t v a l u e s may be i n s e r t e d i n t h e s e s p a c e s . One o f t h e s e v a l u e s w i l l be c o n s i d e r e d l a t e r i n c o n n e c t i o n w i t h t h e m e t h o d t o w h i c h i t s p e c i f i c a l l y a p p l i e s . (6) I f we now examine the t r i a n g u l a r a r r a y of numbers below the p r i n c i p a l d i a g o n a l we see t h a t they e x a c t l y d u p l i c a t e the v a l u e s above t h a t d i a g o n a l . For t h i s r e a s on the m a t r i x i s s a i d t o be symmetric which i s , by d e f i n i t i o n , simply a s i m i l a r i t y of arrangement. When we c o n s i d e r why the m a t r i x e x h i b i t s such an agreement of v a l u e s , we f i n d t h a t t h i s i s a n a t u r a l outcome of the manner i n which the m a t r i x was p r o -duced. In t h i s r e s p e c t c o n s i d e r the t h i r d number i n the f i r s t row, .56, which i s T13 and next note t h a t the c o r r e - -sponding number below the d i a g o n a l i s the f i r s t one i n the -t h i r d row or r 3 1 . T h i s simply means t h a t the c o r r e l a t i o n of t e s t one w i t h t e s t t hree i s e q u a l t o the c o r r e l a t i o n of t e s t t h ree w i t h t e s t one which i s t o be expected. The same w i l l be seen t o h o l d t r u e f o r the other v a l u e s t h a t are e q u a l n u m e r i c a l l y , t h a t i s , r a b = r b a f o r a l l a and b. I f a l l the v a l u e s i n each column are now summed and l i s t e d a t the bottom of the m a t r i x , a c l e a r e r p i c t u r e i s obtained as t o which t e s t c o r r e l a t e s most h i g h l y w i t h a l l the o t h e r s . Examining these v a l u e s i n Table I we observe t h a t the sum of the c o r r e l a t i o n s of t e s t two i s the l a r g e s t and then t e s t s one, t h r e e , f i v e and f o u r f o l l o w i n descending o r d e r . The a r r a y of numbers can be rearranged i n such a man-ner t h a t the t e s t s are i n descending order both as t o rows and columns. T h i s w i l l p l a c e the t e s t s i n the order e x h i b -i t e d i n Table I I . I n t h i s form the o r i g i n a l concept of h i e r -a r c h i c a l order can be more r e a d i l y e x p l a i n e d a s suggested by Thomson (17). (?) Table II Correlation Matrix 2 1 3 5 4 2 X .72 .63 .54 .45 1 .72 X .56 .48 .40 3 .63 .56 X .42 .35 5 .54 .48 .42 X .30 4 .45 .40 .35 .30 X The characteristic which was noted by Spearman was that the coefficients constituting the elements of any two c o l -umns are of such a value that there exists a common ratio throughout the columns. As an example l e t us scrutinize the columns headed 1 and 3 i n Table II. These are 1_ .72 X .56 .48 .40 _3_ .63 .56 X .42 .35 Every element of 3 is exactly seven-eighths of the corre-sponding coefficient i n 1. For example, 8 x .72= .63 . Upon further investigation i t i s clearly seen that every column of the matrix i s some exact proportion of the (8) p receding one. T h i s was the p r o p e r t y which Spearman termed " h i e r a r c h i c a l o r d e r . " H i s e x p l a n a t i o n f o r t h i s phenomenon was t h a t i t was due t o the f a c t t h a t a l l the c o e f f i c i e n t s i n v o l v e d were occasioned by a common f a c t o r and t h a t the p o s i t i o n o f a g i v e n t e s t i n t h i s order was due s o l e l y t o i t s s a t u r a t i o n w i t h t h i s f a c t o r which he c a l l e d "g." The l o g i c u n d e r l y i n g t h i s concept can be r e a d i l y understood i f one remembers t h a t the c o e f f i c i e n t s a r e simply measures of r e l a -t i o n s h i p between the v a r i o u s s e t s o f sco r e s w i t h i n the b a t -t e r y . T h e r e f o r e , i f each o f the c o e f f i c i e n t s between a p a r -t i c u l a r t e s t and the ot h e r s employed i n the b a t t e r y i s a g i v e n f r a c t i o n o f the corresponding c o e f f i c i e n t o b t a i n e d f o r another t e s t , then i t would be a t l e a s t p l a u s i b l e t o hypoth-e s i z e t h a t a common f a c t o r o c c a s i o n e d a p o r t i o n o f bot h these groups of v a l u e s , and t h a t the t e s t w i t h the l a r g e r c o e f f i c i e n t s i s i n f l u e n c e d by t h i s f a c t o r t o a h i g h e r degree. Thus we see t h a t a t e s t i n a b a t t e r y which i n v o l v e d the most "g" would s t a n d a t the head of the h i e r a r c h y . S i m i l a r l y , each o f the other t e s t s would be l o c a t e d i n the order a c c o r d -i n g t o the amount of "g" which i t measured. Concerning t h i s f a c t o r Spearman has t h i s t o say, " . . . t h a t i n t e r p r e t a t i o n of g which - i n the pr e s e n t s t a t e of psychology - appears t o have the g r e a t e s t importance of a l l ; i t i s t h a t which regards t h i s g as measuring a person's "mental energy"." (16, p. 98). I t i s n o t the purpose o f t h i s study t o i n v e s t i g a t e the i m p l i c a t i o n s o f such a hypoth-e s i s . However, i t would be w e l l worthwhile f o r any student O) of f a c t o r i a l a n a l y s i s t o read Spearman's book, The A b i l i t i e s  o f Man, s i n c e i t f u r n i s h e s an e x c e l l e n t r e p o r t concerning a s e r i e s of s c i e n t i f i c experiments c a r r i e d out t o v e r i f y t h i s t h e o r y . I t a l s o s e r v e s as a v e r y i n t e r e s t i n g and v a l u a b l e a n a l y s i s of the o r g a n i z a t i o n of mental p r o c e s s e s . I t might be w e l l f o r the reader to remember a t t h i s p o i n t t h a t the examples employed i n t h i s p o r t i o n of the chap-t e r are p u r e l y f i c t i t i o u s which accounts f o r the p e r f e c t order e x h i b i t e d by the c o r r e l a t i o n a l c o e f f i c i e n t s i n the t a b l e s . For the moment we w i l l continue t o use such f i c t i -t i o u s data and w i l l c o n s i d e r a t the end of the chapter those changes n e c e s s i t a t e d by a c t u a l e x p e r i m e n t a l r e s u l t s . We have now i n v e s t i g a t e d the reasons proposed by Spear-man f o r b e l i e v i n g t h a t a l l mental measurements are dependent t o some degree upon a s i n g l e g e n e r a l f a c t o r . However, s i n c e the t e s t s do not c o r r e l a t e p e r f e c t l y w i t h each o t h e r , t h e r e must be i n each i n d i v i d u a l t e s t a separate s p e c i f i c f a c t o r which i s not c o r r e l a t i n g w i t h "g." T h i s f a c t o r w i l l only i n f l u e n c e the s i z e of the c o e f f i c i e n t s of c o r r e l a t i o n i f a v e r y s i m i l a r t e s t i s i n t r o d u c e d i n t o the b a t t e r y thus con-v e r t i n g the s p e c i f i c f a c t o r i n t o one t h a t i s h e l d i n common by a t l e a s t two of the t e s t s . The Spearman method of f a c t o -r i a l a n a l y s i s i s c a l l e d the "Theory of Two F a c t o r s " s i n c e i t breaks down every c o r r e l a t i o n c o e f f i c i e n t i n t o a s i n g l e gen-e r a l f a c t o r and a s p e c i f i c f a c t o r which i s unique t o each of the t e s t s i n v o l v e d . However, i f two or more v e r y s i m i l a r t e s t s are present i n the battery,' there w i l l be i n c l u d e d one (10) o r more f a c t o r s common t o t h e s e t e s t s . The p a t t e r n o f t h i s breakdown may be more c l e a r l y u n d e r s t o o d by e x a m i n i n g t h e f o l l o w i n g t a b u l a r e x a m p l e s . I n T a b l e I I I t h e c a s e i s i l l u s t r a t e d where no marked s i m i l a r i t y e x i s t s between t h e t e s t s o f t h e g r o u p . T a b l e I I I F a c t o r P a t t e r n f o r T e s t s g s* S4 s 5 1 X X 2 X X 3 X X 4 X X 5 X X I n t h i s i l l u s t r a t i o n t h e X ' s i n d i c a t e t h a t t h e t e s t whose number i s l i s t e d t o t h e l e f t o f t h a t row has a l o a d i n g o r s a t u r a t i o n o f t h e f a c t o r a t t h e head o f t h e c o l u m n . The column h e a d i n g s a r e t h e g e n e r a l f a c t o r " g " a n d s ± t o s 5 w h i c h a r e the s p e c i f i c f a c t o r s i n v o l v e d i n t h e c o e f f i c i e n t s o f t e s t s one t o f i v e i n t h a t o r d e r . Thus we see t h a t i n t h i s c a s e t h e r e i s one g e n e r a l f a c t o r , common t o a l l t e s t s , and f i v e s p e c i f i c f a c t o r s , e a c h one o f w h i c h i s measured by a p a r t i c u l a r t e s t o f t h e b a t t e r y . T a b l e IV f u r n i s h e s a g r a p h i c r e p r e s e n t a t i o n o f t h e f a c -t o r p a t t e r n w h i c h a r i s e s when two o r more o f t h e t e s t s o f a b a t t e r y a r e v e r y s i m i l a r i n c o m p o s i t i o n . ( 1 1 ) Table IV Factor Pattern f o r Tests g a b s 5 1 X X 2 X X S X X 4 X X 5 X X In t h i s table we f i n d that a l l tests again measure Spear-man's general fa c t o r but that now there are two common f a c -t o r s . These are a, which tests one and two both measure and b, which i s measured by tests three and four. Test f i v e , which has nothing i n common with the other tests except i t s measure of "g," s t i l l retains a factor which i s s p e c i f i c to i t s e l f . Spearman has developed a very simple test to be used i n checking that a h i e r a r c h i c a l order e x i s t s between the c o l -umns of a given c o e f f i c i e n t matrix. This method he terms the method of tetrad differences. The general matrix i n Table V, where the r's are to represent the c o e f f i c i e n t s of co r r e l a t i o n , w i l l be used to explain t h i s procedure. (12) Table V C o r r e l a t i o n M a t r i x 1 2 3 4 5 1 X r 1 3 r i+ 2 X 3 X r 3 * r 3 5 4 r 4 1 X 5 r 5 a r 5 3 X The f i r s t s t e p c o n s i s t s i n s e l e c t i n g f o u r c o e f f i c i e n t s such t h a t none of them are members of the p r i n c i p a l d i a g o n a l and such t h a t they a r e a l l f o u r from only two columns and two rows. An example of f o u r elements which s a t i s f y these c o n d i t i o n s are r 1 4 . , r 1 5 , TZ+ and r 2 5 , chosen from the ad-j a c e n t columns 4 and 5 and the n e i g h b o u r i n g rows 1 and 2. Spearman*s t e t r a d d i f f e r e n c e e q u a t i o n i s then w r i t t e n i n the f o l l o w i n g form, (1) I f a l l p o s s i b l e s e l e c t i o n s of f o u r c o e f f i c i e n t s which s a t i s -f y the above r e s t r i c t i o n as to l o c a t i o n a l s o s a t i s f y t h i s e q u a t i o n , then the m a t r i x e x h i b i t s p e r f e c t h i e r a r c h i c a l o r -der. The proof t h a t a m a t r i x which e x h i b i t s h i e r a r c h i c a l or-der n e c e s s a r i l y s a t i s f i e s the t e t r a d e q u a t i o n i s as f o l l o w s , L e t us assume t h a t the m a t r i x of Table V possesses the char-a c t e r i s t i c of h i e r a r c h i c a l order and l e t us f u r t h e r suppose (13 ) t h a t f o r any two p a r t i c u l a r columns, f o r example, 4 and 5 , column 5 i s the p r o p o r t i o n | of 4, where a and b are both p o s i t i v e whole numbers. Then from our study of h i e r a r c h i c a l order we know t h a t t h i s i m p l i e s t h a t t h a t i s , the c o r r e l a t i o n between t e s t s one and f i v e i s the f r a c t i o n % of the c o r r e l a t i o n between t e s t s one and f o u r , b S i m i l a r l y I f these v a l u e s f o r r ^ and TZ5 a r e s u b s t i t u t e d i n t o e q u a t i o n ( l ) , the f o l l o w i n g i s o b t a i n e d . ri4- * (| * * t j - r 2 4 * ( f * = ^14-* rz+~ b1"14"* T z * = 0. T h i s proof can be g e n e r a l i z e d and holds t r u e f o r any f o u r c o e f f i c i e n t s which s a t i s f y the o r i g i n a l r e s t r i c t i o n t h a t they be chosen from on l y two rows and two columns. The converse, t h a t i f a l l the t e t r a d d i f f e r e n c e s are s a t i s f i e d then the g i v e n m a t r i x e x h i b i t s h i e r a r c h i c a l o r d e r , can be proven to h o l d t r u e . The proof i s as f o l l o w s . L e t a l l the c o e f f i c i e n t s be such t h a t they s a t i s f y the t e t r a d e q u a t i o n . Then (14) • - Ai+ X Z 5 — xz+ is • T h i s e q u a t i o n may now be d i v i d e d by r 2 5 - r 1 5 s i n c e n e i t h e r of these f a c t o r s a r e z e r o . L e t the f o u r c o e f f i c i e n t s r 1 4 . , r 1 5 , r 3 + and r3S be con-s i d e r e d now. These s a t i s f y the r e s t r i c t i o n as t o rows and columns and t h e r e f o r e r 1 4 • r 3 5 = r 3 4 - r 1 5 , ' r l s r 3 S * From t h i s and the above v a l u e B±. - Iz±. - T3+ e t c . r i s r 2 5 r 3 S But t h i s i s simply a restatement of the c h a r a c t e r i s t i c of h i e r a r c h i c a l o r d e r , t h a t i s , every c o e f f i c i e n t of one column., i s an exact p r o p o r t i o n of the corresponding, c o e f f i c i e n t of another column. The number of such t e t r a d d i f f e r e n c e s which must be c a l -c u l a t e d f o r any g i v e n m a t r i x i n c r e a s e s r a p i d l y as the number of t e s t s i n the b a t t e r y i s e n l a r g e d . The number of such equations t o be s o l v e d when the number of t e s t s i s 6, amounts t o 45. Whereas, when the number of t e s t s becomes 12, the (15) t o t a l of t e t r a d equations i s 1,485. S i n c e , i n any p r a c t i c a l a p p l i c a t i o n , a r e l a t i v e l y l a r g e b a t t e r y of t e s t s must be employed be f o r e the method may be s a t i s f a c t o r i l y used, i t becomes apparent t h a t c o n s i d e r a b l e computation must be ex-pected i n o b t a i n i n g a Spearman f a c t o r i a l s o l u t i o n . Now t h a t a method has been developed f o r determining whether a g i v e n m a t r i x c o n s t i t u t e s a h i e r a r c h y and t h e r e f o r e whether t h e c o r r e l a t i o n s c a n be c o n s i d e r e d as being composed of two separate f a c t o r s , the problem a r i s e s of o b t a i n i n g a n u m e r i c a l value f o r each t e s t t s s a t u r a t i o n or l o a d i n g w i t h "g" and w i t h i t s s p e c i f i c f a c t o r . The method of i s o l a t i n g the g e n e r a l f a c t o r w i l l be d i s -cussed f i r s t . I f the i n t e r c o r r e l a t i o n between two t e s t s i s due s o l e l y to the f a c t t h a t each of the t e s t s i s a measure of "g," we can-express the c o r r e l a t i o n between such t e s t s as the product of the c o r r e l a t i o n s of each of the t e s t s w i t h "g" i t s e l f . Spearman proves t h i s by a p p l y i n g the formula f o r p a r t i a l c o r r e l a t i o n (8, p. 269), T — r i 2 ~ r i 3 * r&5 » ii2.3 - ',. x ; 1 ==•* y(l-r£ ) ( l - r / 3 ) where r 1 2.3 s i g n i f i e s the p a r t i a l c o r r e l a t i o n between the v a r i a b l e s one and two w i t h v a r i a b l e three h e l d constant or n u l l i f i e d . L e t us now c o n s i d e r the p a r t i a l c o r r e l a t i o n between t e s t s one and two w i t h "g" h e l d c o n s t a n t . The e q u a t i o n be-comes (16) V d - r ^ X l - r - ) But, s i n c e the c o r r e l a t i o n between one and two i s due to the s i n g l e g e n e r a l f a c t o r "g", the v a l u e o f r l 2 £ = 0 . T h e r e f o r e , m u l t i p l y i n g throughout the e q u a t i o n by the denominator of the r i g h t hand s i d e we o b t a i n or vlz = r l g • r Z g . L e t us examine the t e t r a d e q u a t i o n ri4-* TZ5 ~ r24' ri5 = ^* Each of these c o e f f i c i e n t s can be expressed as the product of the i n t e r c o r r e l a t i o n o f the t e s t s i n v o l v e d and "g." The c o e f f i c i e n t s become r i 4 = ri&* r4- g* rz+= r z g - r 4 g and r i 5 = r i s - r s s . Upon s u b s t i t u t i o n the e q u a t i o n becomes r i s - r4g- r2 g • ^ & - r l a- r ^ - TU • r 5 g = 0. F a c t o r i n g out r z & we o b t a i n r2g C ris- *V rsg- r 4 & ' r i^" r s s ) = °-( 1 7 ) We may now d i v i d e by r 2 & s i n c e Tz&£0} i t being the c o r r e l a t i o n of t e s t two w i t h »g." Now m u l t i p l y the e q u a t i o n by r l s=£0, But and t h e r e f o r e the e q u a t i o n becomes Moving the l a s t term t o the r i g h t of the e q u a t i o n we o b t a i n sg r 4 g . = r45 > r 4 g = r!4 = r!5 > D i v i d i n g through by r 4 5 i&. r* 5 r i g : : /r i 4 - r i S _ / r i 4 - ' r i g V* V r 4 5 \ r 4 5 / T h i s v a l u e r e p r e s e n t s the s a t u r a t i o n or l o a d i n g of t e s t one w i t h »g.« The g e n e r a l form of the e q u a t i o n would be 35»e = f : »' (18) where a, b, and c a r e any t e s t s i n the b a t t e r y employed. As an example l e t a, b, c be t e s t one, f o u r and f i v e i n Table I I , page 7 . For these t e s t s our e q u a t i o n becomes L4-5 where and r 1 & -= r 1 4 = .40, r 1 5 = .48 T4S = .50,... (.40)(.48) .SO =.8 . The reader should check f o r h i m s e l f t h a t the l o a d i n g s of the t e s t s a re Table VI "g" Loadings o f the T e s t s T e s t 1 2 3 4 5 Symbol f o r "g" Loading r l g r+g Numerical Value of "g" Loading .8 . 9 . 7 .5 .6 In order t o d e r i v e the formula f o r the v a l u e o f the s p e c i f i c f a c t o r i n each t e s t Spearman a g a i n makes use of the formula f o r p a r t i a l c o r r e l a t i o n . The formula (8, p. 269) i s r i 2 . j — 7 ( 1 - r * ) ( l - r * ) (19) where r 1 Z 3 means the c o r r e l a t i o n between measures one and two w i t h the e f f e c t of a t h i r d v a r i a b l e " t h r e e " p a r t i a l l e d out or h e l d c o n s t a n t . In the present i n s t a n c e the p a r t i a l c o e f f i c i e n t we wish to compute i s the one between any t e s t , say t e s t one of a b a t t e r y , and the s p e c i f i c f a c t o r " s ^ of t h a t t e s t , w i t h the e f f e c t of "g" p a r t i a l l e d out or h e l d c o n s t a n t . T h e r e f o r e the formula becomes r l s g = r* s* ~ rig ' r s ^ . 7(1- r 4 ) ( 1 - r s * a ) But s i n c e S* and "g" are u n c o r r e l a t e d r S l S = 0 and s i n c e r 1 S i . g i s the c o r r e l a t i o n between a t e s t and i t s own s p e c i f i c f a c t o r w i t h the i n f l u e n c e o f "g" h e l d c o n s t a n t , r 1 S i S = 1 . S u b s t i -t u t i n g these v a l u e s i n t o the formula we o b t a i n 1= r*Sft ~ r*« • 0 V ( l - r * ) ( 1 - (0)') 1= r i s * . r 1 c . = yr^ r as x - v - * i g • In g e n e r a l terms f o r any t e s t "a" (3) r a s = Jl-T. z A p p l y i n g t h i s formula we o b t a i n the f o l l o w i n g s p e c i f i c f a c t o r l o a d i n g f o r t e s t one of Table I I , page 7. (20) From Table VI, r l 6 =.8 . / .v±5=Jl- (,8)z = J l - .64 - IM = .60 . The complete l o a d i n g s of a l l the t e s t s of Table I I were computed i n a s i m i l a r manner and a r e l i s t e d i n Table V I I . Table V I I S p e c i f i c F a c t o r Loadings of the T e s t s T e s t g sx s z S3 S4 *5 1 .8 .60 2 .9 .44 3 .7 .71 4 .5 .87 5 .6 .80 Thus the o r i g i n a l m a t r i x , Table I I , has been completely f a c -t o r e d i n t o the g e n e r a l f a c t o r common to a l l t e s t s and the s p e c i f i c f a c t o r s which are each unique to a s i n g l e t e s t of the b a t t e r y . The data we have been s t u d y i n g , however, has been com-p l e t e l y f i c t i t i o u s i n order t h a t the meaning of the methods i n v o l v e d might be more c l e a r l y demonstrated. I t now remains to examine the a c t u a l data d e r i v e d from e x p e r i m e n t a l s t u d i e s . In such r e s u l t s , the c o e f f i c i e n t s of c o r r e l a t i o n a r e not p e r f e c t l y r e l i a b l e due to a v a r i e t y of e x p e r i m e ntal e r r o r s (21) which e x e r t an i n f l u e n c e upon the sc o r e s from which such v a l -ues a r e o b t a i n e d . For t h i s reason no e x p e r i m e n t a l l y obtained m a t r i x w i l l e x h i b i t a b s o l u t e h i e r a r c h i c a l order and, t h e r e -f o r e , none of the t e t r a d d i f f e r e n c e s w i l l be e x a c t l y z e r o . The method t o employ i n a c t u a l p r a c t i c e i s t o ev a l u a t e a l l the t e t r a d d i f f e r e n c e s and then make c e r t a i n t h a t the v a r i -a t i o n s from z e r o t h a t a r i s e a r e only such a s would be due to the u n r e l i a b i l i t y of the measures t h a t were used. For t h i s purpose l e t us examine a t a b l e chosen from Spearman's book (16, p. x i i ) . T able V I I I C o r r e l a t i o n M a t r i x 1 2 3 4 5 1 X .485 .400 .397 .295 2 .485 X .397 .397 .247 3 .400 .397 X .335 .275 4 .397 .397 .335 X .195 5 .295 .247 .275 i l 9 5 X I f the t e t r a d d i f f e r e n c e s a r e computed by formula ( l ) i t w i l l be found t h a t they a r e a l l s l i g h t l y d i f f e r e n t from z e r o . For example, rl3 " r24-~ r23 " ri4-= (.400) (.397)- (.397) (.397) = (.397) [.400-.397] = (.397) (.003) (22) =.001191 =.0012 . Once a l l p o s s i b l e d i f f e r e n c e s have been computed they can be put i n t o the form of a frequency d i s t r i b u t i o n w i t h a mean of z e r o . Spearman has developed the f o l l o w i n g approx-i m a t i o n formula f o r the probable e r r o r of such a d i s t r i b u t i o n (16, p. x i ) . p.e.= ^ | | 9 - [ r * ( l - r ) { t 4 - ( l - R ) s ^ 7 where R = 3 r ( g e | ) - 2 r ^ | E | > N= number of cases i n the sample, n = number of t e s t s i n the b a t t e r y , r=mean of the c o r r e l a t i o n s and s*=mean squared d e v i a t i o n of a l l the r ' s from t h e i r mean. Sin c e probable e r r o r c a l c u l a t i o n s a r e no l o n g e r gener-a l l y employed i n s t a t i s t i c a l s t u d i e s , i t may serve a more u s e f u l purpose i f converted i n t o a standard e r r o r formula which can then be i n t e r p r e t e d i n e x a c t l y the same manner as any other standard e r r o r o f a s t a t i s t i c . The r e l a t i o n s h i p between p.e. and standard e r r o r i s p.e.= .6745 <r or e q u i v a l e n t l y < T ~ .6745 p , e * (23). A p p l y i n g t h i s c o n v e r s i o n f a c t o r t o the above eq u a t i o n we ob-t a i n the formula te) .= f r [rZ^- * F + ( i - R ) s f where aid = the standard e r r o r of a t e t r a d d i f f e r e n c e . For Table V I I I we f i n d the f o l l o w i n g v a l u e s r = . 3 4 2 , s 4=.007, N = 757, n = 5 R = * f i H ) - ^ ( & = t ) = 3 ( . 3 4 2 ) ( | £ | ) - 2 ( . 3 4 2 f ( | 5 | ) = 3 ( . 3 4 2 ) ( i ) - 2 ( . 3 4 2 f (-1) = .342+-. 078 = .420 . From formula (4) ^ J = ^.^z{l-Tf + ( l - R ) s ^ = TIT T [(.342) z ( l . - . 3 4 S ) l 4 - ( l - . 4 2 0 ) . 0 0 7 ] * = 27 ? 5 T [(.H7) (.658)*+ (.580).007^ = 27?5T [0507 4-.004l]^ = .017 . Spearman has the f o l l o w i n g t o say concerning the prob-a b l e e r r o r o f the t e t r a d d i f f e r e n c e computed by means of h i s (24) formula. "Sometimes, t h i s w i l l be a p p r e c i a b l y i n a c c u r a t e . U s u a l l y , however, i t w i l l be near enough f o r the p r e s e n t pur-pose of e s t i m a t i n g the range of sampling e r r o r s , e s p e c i a l l y where (as here) the frequency d i s t r i b u t i o n proves t o be f a i r -l y "normal"." (16, p. x i ) . T h i s statement a p p l i e s e q u a l l y w e l l t o the st a n d a r d e r r o r f ormula. We must now d e r i v e the d i s t r i b u t i o n o f the t e t r a d d i f -f e r e n c e s by formula ( 1 ) . The d i s t r i b u t i o n of the v a l u e s of these e r r o r s i s reco r d e d i n Table IX. Table IX D i s t r i b u t i o n o f T e t r a d D i f f e r e n c e s Step I n t e r v a l f .0251 to .0350 2 .0151 to .0250 1 .0051 t o .0150 1 -.0049 to .0050 4 -.0149 t o -.0050 1 -.0249 t o -.0150 4 -.0549 to -.0250 2 N = = 15 The l a r g e s t d e v i a t i o n from z e r o of any of the t e t r a d d i f f e r e n c e s i s .0346. Knowing t h i s v a l u e and the v a l u e of the standard e r r o r , a c r i t i c a l r a t i o can be computed f o r the data which would be PR- t - L a r g e s t D e v i a t i o n (25) In the preceding example, C R = t =. »^yr 3 -2.04 . We employ t h i s v a l u e as a means of a s s e s s i n g whether our t e t r a d d i f f e r e n c e s arose simply from chance sampling e r r o r s . I f the c r i t i c a l r a t i o i s l e s s than approximately t = 2 . 5 8 f o r N;>50 and a normal d i s t r i b u t i o n , then the d e v i a t i o n l i e s w i t h i n the 1$ l e v e l of con f i d e n c e and so the p o s s i b i l i t y t h a t the d i f f e r e n c e s a r o s e from these e r r o r s i s rea s o n a b l y a s s u r e d . I f the r a t i o i s l a r g e r than t = 2.58 then a f u r t h e r e x p l a -n a t i o n f o r the d e v i a t i o n must be sought. In t h i s r e g a r d , i t must be c o n s t a n t l y borne i n mind t h a t the l e v e l o f confidence t h a t one chooses i n any s t a t i s t i c a l examination of r e s u l t s must always be based upon a l l of the f a c t s known concerning t h a t d a t a . Regarding the use of the probable e r r o r formula, Spear-man (16, p. 140) s a y s , In p r a c t i c e , then, the t e t r a d d i f f e r e n c e s s h o u l d not tend towards z e r o , but i n s t e a d s h o u l d tend t o have j u s t such v a l u e s as would r e s u l t from the sampling e r r o r s a l o n e . The most p e r f e c t comparison between the two, the observed t e t r a d d i f f e r e n c e s and those t o be expected from sampling e r r o r s a l o n e , i s obtained by making a complete f r e -quency d i s t r i b u t i o n of each of these two s e t s o f v a l u e s . A more summary comparison i s got by se e i n g whether or not about h a l f of the observed t e t r a d d i f f e r e n c e s a r e g r e a t e r and h a l f l e s s than t h e i r "probable e r r o r . " Much the same t h i n g i s to see whether the median observed t e t r a d d i f f e r e n c e and the probable e r r o r are about e q u a l . Most summary of a l l i s t o see whether or not the l a r g e s t ob-se r v e d t e t r a d d i f f e r e n c e exceeds about f i v e times the magnitude of t h e i r probable e r r o r . T h i s statement a p p l i e s e q u a l l y w e l l t o the use o f standard (36) e r r o r as our e v a l u a t i n g c r i t e r i o n . The only d i f f e r e n c e l i e s i n the f a c t t h a t the 1$ l e v e l of confidence then becomes t = 2.58. T h i s i s <a more s t r i n g e n t r e s t r i c t i o n than t h a t imposed by Spearman, but even t h i s v a l u e may upon o c c a s i o n be too l e n i e n t . Even though N i s not g r e a t e r than 50 and the d i s t r i -b u t i o n i s not normal, s i n c e the c r i t i c a l r a t i o i s t = 2 . 0 4 i n the f o r e g o i n g example, l e t us assume t h a t the t e t r a d d i f f e r -ences of the t r u e c o r r e l a t i o n c o e f f i c i e n t s are z e r o and t h a t the t e s t s can be decomposed i n t o one g e n e r a l f a c t o r and s p e c i f i c f a c t o r s . In t h i s case some doubt would be p r e s e n t due t o the f a c t t h a t the o b t a i n e d t i s g r e a t e r than t = 1.96, the 5% l e v e l of c o n f i d e n c e , and our d i s t r i b u t i o n i s not t r u l y normal. In a p r a c t i c a l a p p l i c a t i o n of the method the data would have t o be s c r u t i n i z e d more c l o s e l y i n order to d i s p e l or c o n f i r m any doubts. L e t us assume t h a t t h i s process has been performed and t h a t i t has strengthened the b e l i e f t h a t our t e t r a d d i f f e r e n c e s have a r i s e n from chance sampling e r r o r s a l o n e . I t i s now necessary to c o n s i d e r the e v a l u a t i o n of a t e s t ' s s a t u r a t i o n w i t h "g" from f a l l i b l e d ata such as t h a t c o n t a i n e d i n Table V I I I . I f formula ( 2 ) , developed f o r i n -f a l l i b l e d a t a , i s employed, there are s e v e r a l groups of t e s t s which w i l l f u r n i s h t h i s i n f o r m a t i o n . However, s i n c e none of the c o r r e l a t i o n c o e f f i c i e n t s are " t r u e " ones, i t w i l l be found t h a t every such grouping w i l l g i v e a s l i g h t l y d i f f e r e n t v a l u e f o r such a l o a d i n g . Spearman proposes two p o s s i b l e (27) s o l u t i o n s f o r t h i s problem. The f i r s t i s t o d e r i v e a l l p o s s i v l e v a l u e s f o r the "g" l o a d i n g of a p a r t i c u l a r t e s t and then t o accept the mean as the b e s t r e p r e s e n t a t i o n of these v a l u e s . The second method i s t o employ a formula developed by Spearman which bases the v a l u e obtained f o r "g" upon a l l the c o e f f i c i e n t s i n the t a b l e . Spearman b e l i e v e s t h i s l a t t e r method i s p r e f e r a b l e both from the viewpoint o f shortness and r e l i a b i l i t y . The formula (16, p. x v i ) i s (5) r - (A*- A* ) i where ra% = t e s t a's s a t u r a t i o n w i t h "g", A = sum of the c o r r e l a t i o n s between t e s t a and every other t e s t , A1 = sum of the squares of the c o r r e l a t i o n s between t e s t a and every other t e s t , T = t o t a l of a l l c o r r e l a t i o n s i n the m a t r i x . These v a l u e s a r e r e a d i l y o b tained from the m a t r i x . For example, l e t us compute the v a l u e of r ^ i n Table V I I I . In t h i s case A =sum of the f i r s t column i n the m a t r i x , ,'.A= 1.577 . A 1 = t h e sum of the squares of each element of the f i r s t column, .'. A 4=.6599 . T = sum of a l l the c o e f f i c i e n t s i n the m a t r i x , .\T= 6.846 . r„„ = (28) 1.577*- .6399 I i -« 16.846- 2(1.577) 1.847 [5.692 = .707 . The s p e c i f i c f a c t o r l o a d i n g s may s t i l l be computed by formula ( 3 ) . A p p l y i n g t h i s formula we o b t a i n f o r the s p e c i f i c f a c t o r l o a d i n g of t e s t one r l s = 7 1 - ( , 7 0 7 ) " = V I - .4998 = .707 . Table X c o n t a i n s the g e n e r a l and s p e c i f i c f a c t o r load-i n g s f o r the f i v e t e s t s o f Table V I I I . The reader i s ad-v i s e d t o compute these v a l u e s by means of formulas (5.) and ( 3 ) . Table X F a c t o r Loadings of the T e s t s T e s t r s* r s 3 1 .707 .707 2 .672 .740 5 .605 .796 4 .554 .853 5 .398 .917 Thus, any s e t of c o r r e l a t i o n s o b tained between t e s t s of (29) a given battery can be broken down into two factors, pro-vided that the tests are sufficiently different i n compo-sition to avoid any marked overlap between specific factors. Such overlap would be revealed when applying the standard error test to the tetrad differences since i t would destroy the hierarchical order of the matrix and cause tetrad d i f f e r -ences that exceed the maximum value allowable by the theory of probability. The purpose of this chapter has been to explain the methodology of the Spearman school i n the simplest possible mathematical terms and for this reason applications have been omitted. At this point, therefore, l e t us briefly consider a few of the problems to which this factorial method may be applied. Spearman and his colleagues have employed this process to arrive at a highly developed theory of the mental processes in which " g " or "mental energy" i s the main corner-stone. Then, too, other workers in the f i e l d , basing their efforts upon Spearman's definition of intelligence, are attempting to improve mental tests. S t i l l others are direct-ing this analytic approach to the work of selecting from the thousands of available tests a series of tests which w i l l derive an adequate representation of a given individual's potentialities. This would f a c i l i t a t e the work of prediction in both the scholastic and vocational f i e l d s . It i s hoped that such methods w i l l y i e l d much more valid results than those at present employed. In closing, a f i n a l warning must be borne in mind. In (30) a l l the f o r e g o i n g examples a minimum number of t e s t s were d e a l t w i t h i n order t h a t the theory might be more r e a d i l y demonstrated. In any p r a c t i c a l a p p l i c a t i o n o f t h i s theory, however, many more t e s t s than the number used i n these i l l u s -t r a t i o n s must be a d m i n i s t e r e d b e f o r e any v a l i d and meaningful r e s u l t s can be d e r i v e d from the c o l l e c t e d d a t a . (31) Chapter II  Science and the Factor Problem As an introduction to the study of the actual methods and techniques employed i n multiple-factor analysis, i t would seem advisable to consider carefully the principles of scie n t i f i c method in general and the relationship existing between these principles and the processes of fac t o r i a l i n -vestigation. Such a discussion w i l l furnish us with valu-able knowledge in two very important spheres. F i r s t l y , i t _will provide us with an insight into the purpose for which Thurstone developed his analytic methods. Secondly, i t w i l l assist i n clarifying the values and limitations of these pro-cedures . Consideration of this matter w i l l make the importance of such information very apparent. Without such knowledge one would be unable to make a truly valid application of the methods, since i t would not be possible to decide upon a f i e l d of investigation that is,truly suited to factorial study. Furthermore, any results obtained without an understanding of the values and limitations of these techniques could not be interpreted in a meaningful manner. Thus we see that a true knowledge of a sc i e n t i f i c instrument, such as that with which we w i l l deal in the chapters that follow, demands not only an understanding of the formulae which have been developed, but also a clear insight into the theoretical basis upon which these formulae are founded. This i s true since i t i s never ( 3 2 ) possible to remove completely the factor of s c i e n t i f i c i n -sight which must be supplied by a truly proficient experimen-ter. This has been voiced as a shortcoming of f a c t o r i a l analysis, but, in r e a l i t y , this i s not truly so since every s c i e n t i f i c methodology places such a premium upon s c i e n t i f i c acumen. For the above reasons, the major portion of this chap-ter w i l l be devoted to a discussion of the f a c t o r i a l concept against a backgound furnished both by Thurstone's latest book (19) and the fundamental framework upon which s c i e n t i f i c method depends for i t s meaning. Science i s founded upon the primary postulate that a l l phenomena i n the universe constitute an orderly system and that i t i s possible to describe this i n f i n i t y of individual events by means of a f i n i t e number of explanatory concepts. Regarding this postulation Thurstone (19, p. 51) has the following to say, It i s the f a i t h of a l l science that an unlimited number of phenomena can be compre-hended i n terms of a limited number of con-cepts or ideal constructs. Without this f a i t h no science could ever have any motivation. To deny this f a i t h i s to affirm the primary chaos of nature and the consequent f u t i l i t y of sc i e n t i f i c effort. The constructs i n terms of which natural phenomena are comprehended are man-made inventions. To discover a s c i e n t i f i c law i s merely to discover that a man-made scheme serves to unify, and thereby to sim-p l i f y , comprehension of a certain elass of natural phenomena. A s c i e n t i f i c law i s not to . be thought of as having an independent exist-ence which some scientist i s fortunate to stumble upon. i . s c i e n t i f i c law i s not a part of nature. It i s only a way of comprehending nature. ( 3 3 ) The preceding quotation i s w e l l worth further study since i t serves as,an answer to much of the misunderstanding which existed and, to some extent, s t i l l e x i s t s concerning the purpose of the f a c t o r i z a t i o n techniques. The argument raised i s that the factors derived hy means of a given inves-t i g a t i o n are proposed as ac t u a l descriptions of e x i s t i n g en-t i t i e s or f a c u l t i e s w i t h i n the i n t e l l e c t i v e processes of man. The c r i t i c s therefore conclude that t h i s methodology must of necessity lead to a subdivision or compartmentization of the mental function and thus must return us to the formerly held theories of f a c u l t y psychology. The argument continues that t h i s viewpoint, being i n d i r e c t opposition to the theories of h o l i s t i c or c o n f i g u r a t i o n a l i s t psychology, makes the method inapplicable since i t i s at variance with the general theory of psychology. Regarding the difference between the methods and re s u l t s of f a c u l t y psychology and those of f a c t o r i a l a nalysis, Mur-s e l l ( 1 3 , p. 384) states, In one sense i t i s akin to f a c u l t y psy-chology. A psychologist of t h i s school, i f such there s t i l l were, might study some ex-amples of test performance and decide that some of them required memory plus reasoning, others, memory plus imagination, others reasoning plus imagination, and s t i l l others pure memory and pure reasoning. I f he could, by regrouping some test items and adding others, construct a test that would measure nothing but memory, or reasoning, or imagin-a t i o n , he would have carried through the l o g i c Of his position and consider that he had pro-duced a superior instrument. This i s the general l i n e of work followed by factor anal-y s i s . But i t d i f f e r s from the fa c u l t y view-point i n two important respects. F i r s t , i t (34) uses s t a t i s t i c a l techniques i n s t e a d of un-c o n t r o l l e d s p e c u l a t i o n i n determining i t s mental components. Second, i t does not b e g i n w i t h a predetermined s e t of f a c u l t i e s , but seeks t o d i s c o v e r what f a c t o r s or components •are necessary to e x p l a i n the da t a . Thus the f a c t o r s a t which i t a r r i v e s are not thought o f as f a c u l t i e s , but as e x p l a i n i n g concepts based on e m p i r i c a l m a t e r i a l . Thus we see t h a t one major d i f f e r e n c e e x i s t i n g between these two s c h o o l s of thought i s one of b a s i c procedure. For the f a c u l t y p s y c h o l o g i s t , a new f a c u l t y c o u l d be added whenever a new name was s u b j e c t i v e l y d e r i v e d , whereas i n f a c t o r i a l a n a l -y s i s b e f o r e a new f a c t o r can be added t o the d e s c r i p t i v e l i s t i n g i t must be r i g i d l y demonstrated as nece s s a r y i n order t o d e s c r i b e f u l l y the obt a i n e d r e s u l t s . Thus, a l t h o u g h the naming process i s s t i l l s u b j e c t i v e l y dependent upon the psy-c h o l o g i c a l knowledge of the r e s e a r c h worker, the method of f a c t o r a n a l y s i s permits the a c t u a l e x p e r i m e n t a l r e s u l t s t o e x e r t a c o n s t r a i n i n g and d i r e c t i n g i n f l u e n c e upon h i s d e c i -s i o n . , F u r t h e r l i g h t i s thrown upon t h i s s u b j e c t by a c l o s e r examination of the claims made by Thurstone i n h i s l a t e s t book. We f i n d t h a t he makes, no c l a i m f o r the a b s o l u t e e x i s t -ence of f a c t o r s as e n t i t i e s w i t h i n the mind s i n c e no such a b s o l u t i s m i s e i t h e r necessary or f e a s i b l e i n any s c i e n t i f i c c o n s t r u c t . The f a c t o r s , which may be suggested by a.given a n a l y s i s , can never be propounded as. corresponding t o a c t u a l f u n c t i o n a l s u b s t r u c t u r e s w i t h i n the t o t a l mental f u n c t i o n i n g of man, but a r e merely d e s c r i p t i v e terms proposed a s the best a v a i l a b l e means of e x p l a i n i n g a g i v e n f i e l d . The p h y s i c i s t , (35) working i n a s c i e n t i f i c f i e l d whose constructs are more se-curely established than are those of psychology or the s o c i a l sciences, makes no claim that the defined quantities, such as mass, force, etc., i n terms of which he develops his theories, have any absolute existence i n nature. There i s , therefore, no necessity or p o s s i b i l i t y of postulating such absolute existence f o r the t r a i t s or a b i l i t i e s derived by means of a f a c t o r i a l study of a given sphere of i n t e r e s t . The value of a given t h e o r e t i c a l construct i s to be judged against the c r i t e r i o n of s c i e n t i f i c s i m p l i c i t y or par-simony. . This merely means that, the descriptive method has value i f i t successfully subsumes a greater number of seem-ingly i s o l a t e d events or phenomena by means of a construct containing a l e s s e r number of unknown qu a n t i t i e s . I t i s t h i s constant s t r i v i n g f o r greater s i m p l i c i t y , i n the s c i e n t i f i c meaning of the word, which serves as the basis f o r selecting the more valuable of two theories which seem equally plau-s i b l e . There i s , however, no statement, either implied or intended, that either of these constructs describes the a c t u a l s i t u a t i o n e x i s t i n g i n nature. Thus, when Thurstone f a c t p r i a l l y investigated the i n t e r c o r r e l a t i o n s e x i s t i n g be-tween the scores upon a group of i n t e l l i g e n c e t e s t s , he de-r i v e d a pattern of seven primary a b i l i t i e s which he proposed as a means of describing the i n t e l l e c t i v e function. This postulation was based upon the f a c t that these a b i l i t i e s served to describe the differences e x i s t i n g between i n d i v i d -uals i n t h i s function and, at the same time, s a t i s f i e d the (36). c r i t e r i o n of parsimony. As we w i l l see l a t e r i n our study ce r t a i n f a c t o r i a l concepts., such as that of simple structure, f u r n i s h us with the means for v e r i f y i n g the adequacy and s t a b i l i t y of any such construct. By using these methods i t i s possible successively to increase one's confidence i n the v a l i d i t y of any theory even though i t i s never possible to a t t a i n certainty within the s c i e n t i f i c framework. Concerning the l i m i t e d , and as Thurstone expresses i t , humble descrip-t i v e r o l e of factor analysis, Burt ( 2 , p. 97) writes, What distinguishes f a c t o r - a n a l y s i s , therefore, from other ways of discovering how i n d i v i d u a l s and t h e i r numerous a t t r i b u t e s can best be c l a s s i f i e d i s c h i e f l y t h i s : whereas the ancient l o g i c i a n reached h i s d e f i n i t i o n s by examining the meaning of words, the modern f a c t o r i s t reaches his c l a s s i f i c a t i o n s by examining the correlations between the forms of behaviour to which those words very loosely r e f e r . But the u l t e r i o r object i s s t i l l the same; and, whether we are describing persons or t r a i t s , the f a c t o r i a l concepts adopted are simply p r i n c i p l e s of c l a s s i f i c a t i o n . In a l l of the foregoing, the major l i m i t a t i o n of the method becomes -apparent; t h i s i s , that factor analysis f u r -nishes us with no absolute answers to our problems. Its value l i e s i n supplying the means f o r describing or c l a s s -i f y i n g data i n more s c i e n t i f i c a l l y simple terms. However, i n r e a l i t y , t h i s shortcoming i s not as serious as i t may appear ..since,It i s a l i m i t a t i o n which must of necessity permeate a l l s c i e n t i f i c findings by the very nature_of s c i e n t i f i c method. It i s ;this f a c t that i s too poorly understood by the layman and sometimes forgotten by the research worker. However, the very indefiniteness which i s inherent i n a l l s c i e n t i f i c ( 3 7 ) r e s u l t s , w h i l e s e r v i n g as an o b s t r u c t i o n t o u l t i m a t e c e r t a i n -t y , a l s o p r o v i d e s one of the g r e a t e s t s t r e n g t h s of s c i e n c e . T h i s i s true s i n c e t h i s f a c t e n t a i l s t h a t any worker i n the f i e l d of^ s c i e n t i f i c r e s e a r c h must always r e t a i n an open mind and be prepared to a l t e r h i s o p i n i o n i f the weight of e v i -dence so d i c t a t e s . Thus dogmatism and s t a g n a t i o n a r e avoided. Thurstone (19) i n h i s book d i s c u s s e s y e t another c r i t i -cism t h a t has been l e v e l l e d a t the methods he has developed. I t seems w e l l a t t h i s p o i n t to r e c o r d both the c r i t i c i s m and Thurstone's answer i n order t h a t the student may more c l e a r l y see y e t another l i m i t a t i o n upon the r e s u l t s o b t a i n a b l e by means of f a c t o r a n a l y s i s . The argument a g a i n s t f a c t o r a n a l y s i s c o n s i s t s - i n p o i n t -i n g out t h a t any h y p o t h e s i s d e r i v e d by t h i s means f a i l s t o s u c c e s s f u l l y subsume a l l p o s s i b l e d e t a i l s of a c l a s s of events or phenomena and t h a t , f o r t h i s reason, any such s o -l u t i o n s obtained are not capable of g e n e r a l a p p l i c a t i o n . Thurstone, while a d m i t t i n g t h i s l i m i t a t i o n , p o i n t s out t h a t any s c i e n t i f i c c o n s t r u c t s u f f e r s from l o s s i n the degree to which i t corresponds to a l l the i n d i v i d u a l events of e x p e r i -ence. Thus he c o n t i n u e s , even i n the p h y s i c a l s c i e n c e s no t h e o r e t i c a l e q u a t i o n a c t u a l l y d e s c r i b e s a l l p o s s i b l e i n -stances of the e v e n t s which i t i s intended to e x p l a i n . In f a c t , i t might be added t h a t such e q u a t i o n s , which are de-r i v e d f o r an i d e a l i z e d s i t u a t i o n , do not agree e x a c t l y w i t h any s i n g l e event t h a t a c t u a l l y o c c u r s . Thus, he says, con-s i d e r f o r example the i d e a l i z e d e q u a t i o n e x p r e s s i n g -the ( 3 8 ) a c t i o n of a pendulum i n terms of i t s mass, p e r i o d of swing, and l o c u s or path of the c e n t r e of g r a v i t y w i t h r e f e r e n c e to i t s f u l c r u m . The d e s c r i p t i o n f u r n i s h e d by t h i s e q u a t i o n never a c c u r a t e l y p o r t r a y s the swing of an a c t u a l pendulum s i n c e many u n c o n t r o l l e d v a r i a b l e s e n t e r i n t o the p r a c t i c a l s i t u a t i o n . T h i s , however, i n no way serves to i n v a l i d a t e the formula, but merely i n d i c a t e s the l i m i t a t i o n s which must be c o n s t a n t l y borne i n mind i f the theory i s t o be c o r r e c t l y ap-p l i e d . T h i s same l i m i t a t i o n a p p l i e s to any e x p l a n a t i o n de-. r i v e d by s c i e n t i f i c r e s e a r c h employing f a c t o r i a l methods. A . q u o t a t i o n from h i s own work (19, p. 54) may serve t o s t i l l f u r t h e r c l a r i f y t h i s important p o i n t . Every s c i e n t i f i c c o n s t r u c t l i m i t s i t s e l f t o s p e c i f i e d v a r i a b l e s , without any pretense of c o v e r i n g those'aspects of a c l a s s of phenom-ena about which i t has s a i d n o t h i n g . As regards t h i s 1 c h a r a c t e r i s t i c of s c i e n c e , t h e r e i s no d i f f e r e n c e between the s c i e n -t i f i c study o f p h y s i c a l events and the s c i -e n t i f i c study of b i o l o g i c a l . a n d p s y c h o l o g i c a l events. What i s not g e n e r a l l y understood, even by many s c i e n t i s t s , , i s t h a t no s c i e n t i f -i c law i s ever intended to r e p r e s e n t any event p l c t o r i a l l y . The law i s o n l y an ab-s t r a c t i o n from the experimental s i t u a t i o n . No experiment i s ever completely r e p e a t e d . In t h i s r e s p e c t , i t might be. w e l l i f we r e c a l l a f a c t t h a t i s known to every student of experimental psychology. T h i s i s , t h a t i n any experiment the extraneous v a r i a b l e s which are not under o b s e r v a t i o n must be c o n t r o l l e d and h e l d constant to as h i g h a degree as p o s s i b l e . -The e x p e r i m e n t a l r e s u l t s are v a l i d and r e p e a t a b l e i n p r o p o r t i o n t o the e x t e n t t h a t t h i s u l t i m a t e i d e a l s i t u a t i o n i s approximated. In t h i s r e g a r d , any f a c t o r -( 3 9 ) i a l study must be c o n s i d e r e d as an exp e r i m e n t a l s i t u a t i o n which must be c a r e f u l l y a nalysed b e f o r e commencing the i n v e s -t i g a t i o n i n order t h a t as many as p o s s i b l e of the non-exper-i m e n t a l v a r i a b l e s may be i s o l a t e d and c o n t r o l l e d . In t h i s r e s p e c t , the r e s u l t s o f such f a c t o r i a l r e s e a r c h a re dependent upon the knowledge and a b i l i t y of the experimenter t o the same ext e n t t h a t the v a l i d i t y of any ex p e r i m e n t a l r e s u l t s are so dependent. Thus, f o r example, i n a d m i n i s t e r i n g a g i v e n group of t e s t s t o a s p e c i f i c e x p e r i m e n t a l sample, i t i s nec-e s s a r y t h a t a l l p o s s i b l e c o n t r o l s be employed s i n c e the data so obtained a r e to serve as the b a s i s f o r the i n t e r c o r r e l a -t i o n s which are to be a n a l y s e d . The u l t i m a t e r e s u l t s ob-t a i n e d by f a c t o r methods can be no more v a l i d than are the data from which they are d e r i v e d . T h i s , however, i n no way invalidates,, the e x p e r i m e n t a l method i n g e n e r a l , or the method of f a c t o r i a l a n a l y s i s i n p a r t i c u l a r , but merely p l a c e s a p r e -mium upon the experimenter's a b i l i t y . Another p o i n t t h a t must be borne i n mind d u r i n g our l a t e r s t u d i e s i s t h a t , a l t h o u g h the methods a t presen t em-ployed w i l l be found t o express the g i v e n a t t r i b u t e s a s l i n -ear f u n c t i o n s o f a g i v e n number of f a c t o r s , t h e r e i s no nec-e s s a r y h y p o t h e s i s t h a t t h i s i s the u l t i m a t e form t h a t w i l l be found f o r such a r e l a t i o n s h i p . T h i s method i s employed as a c l o s e approximation which i s mathematically sound, but which may a t a l a t e r date, as our knowledge becomes p r o g r e s s i v e l y . more advanced, be r e p l a c e d by a f u n c t i o n which i s f a r more complex i n n a t u r e . (40) The c o n c e p t o f l i n e a r r e l a t i o n s h i p w i l l r e p a y f u r t h e r d i s c u s s i o n . T h i s t e r m i s u s e d t o d e n o t e a f u n c t i o n a l l i n k i n g b e t w e e n v a r i a b l e s , s u c h t h a t a c e r t a i n s e l e o t e d v a r i a b l e may be c o m p l e t e l y d e f i n e d i n v a l u e by a c o m b i n a t i o n o f v a r i o u s o t h e r v a r i a b l e s , a l l o f w h i c h a r e t o t h e f i r s t p o w e r . E a c h o f t h e s e l a t e r v a r i a b l e s h a s a c e r t a i n c o e f f i c i e n t d e t e r m i n e d b y t h e a c t u a l r e l a t i o n s h i p e x i s t i n g . Thus t h e e q u a t i o n , s = a x + b y , i s a l i n e a r r e l a t i o n s h i p . I t i s s o c a l l e d s i n c e , i f t h e l o c u s o r p a t h o f a l l p o i n t s p e r t a i n i n g t o g i v e n f i x e d v a l u e s f o r a , b a n d s a n d a l l p o s s i b l e . . c o m b i n a t i o n s o f v a l u e s f o r x a n d y a r e p l o t t e d , t h e r e s u l t i n g s e r i e s o f p o i n t s w o u l d a l l l i e i n a s t r a i g h t l i n e . S u c h a r e l a t i o n s h i p i s t h e s i m -p l e s t p o s s i b l e f u n c t i o n t o d e r i v e a n d i s t h e one u s e d f o r a p -p r o x i m a t i o n i n t h e f a c t o r i a l m e t h o d s . C e r t a i n p o i n t s c o n -c e r n i n g t h i s l i n e a r e q u a t i o n s e r v e t o i n d i c a t e t h e d i f f e r -e n c e s w h i c h T h u r s t o n e e m p h a s i z e s a s e x i s t i n g b e t w e e n t h e p u r -p o s e o f f a c t o r i a l a n a l y s i s a n d t h e a i m o f p r e d i c t i v e s t a t i s -t i c s . I t i s a l l t h e more i m p o r t a n t t h a t one r e c o g n i z e t h a t a d i f f e r e n c e d o e s e x i s t s i n c e , b e c a u s e many s t a t i s t i c a l f o r m u -l a e a r e e m p l o y e d i n f a c t o r i z a t i o n , t h e r e i s m a r k e d p o s s i b i l -i t y o f c o m p l e t e l y i d e n t i f y i n g t h e s e two f i e l d s w i t h one a n -o t h e r . I n t h e a b o v e l i n e a r e q u a t i o n t h e v a r i a b l e s x a n d y a r e u s u a l l y d e n o t e d a s i n d e p e n d e n t , i n t h a t t h e y a r e t h e v a r i -a b l e s t o w h i c h a l l p o s s i b l e c o m b i n a t i o n s o f v a l u e s a r e a s -c r i b e d . The v a r i a b l e s i s c a l l e d t h e d e p e n d e n t v a r i a b l e s i n c e , f o r g i v e n f i x e d v a l u e s o f a a n d b , when a n y p a r t i c u l a r (41) v a l u e s are-assigned, to x and y, .the-numerical v a l u e of s i s completely determined. The d i f f e r e n c e s between f a c t o r i a l and s t a t i s t i c a l methods l i e i n t h e i r u l t i m a t e o b j e c t i v e s . A f a -m i l i a r type o f s t a t i s t i c a l problem i s one i n which a s e r i e s of v a l u e s are known f o r each member of a g i v e n sample popu-l a t i o n , and the purpose i s t o employ these known q u a n t i t i e s i n such a manner as t o p r e d i c t y e t another unknown v a l u e . In order to s o l v e t h i s problem a l i n e a r r e g r e s s i o n e q u a t i o n i s s e t up i n terms of the known v a r i a b l e s such t h a t , when gi v e n v a l u e s are a s s i g n e d to these v a r i a b l e s , the r e q u i r e d unknown v a l u e i s d e f i n e d . In t h i s e q u a t i o n the v a r i a b l e whose v a l u e i s d e s i r e d i s c l a s s i f i e d as the dependent v a r i -a b l e , w h i l e the other v a r i a b l e s are a r b i t r a r i l y chosen as i n -dependent ones. The c o e f f i c i e n t s of these v a r i a b l e s are c a l l e d the " r e g r e s s i o n c o e f f i c i e n t s . " However, as Thurstone p o i n t s out, i n f a c t o r a n a l y s i s there i s no such d i s t i n c t i o n made between dependent"and independent v a r i a b l e s , s i n c e the whole of the data i s employed and the r e s e a r c h worker i s simply seeking to e x p l o r e a g i v e n domain i n order t o a s c e r -t a i n whether there e x i s t s any u n d e r l y i n g order among the v a -r i a b l e s , and, i f such e x i s t s , what form t h a t order might take. Thus, a l t h o u g h many of the f a m i l i a r s t a t i s t i c a l t o o l s a r e em-ployed, the g o a l t h a t i s i n mind d u r i n g a t r u e f a c t o r i a l study d i f f e r s t o some extent from the more u s u a l s t a t i s t i c a l one, and t h i s d i f f e r e n c e must a t a l l times be kept c l e a r l y i n mind. T h i s d i f f e r e n c e might be s t a t e d as l y i n g 1 i n the f a c t t h a t , whereas one of the b a s i c aims of many s t a t i s t i c a l ( 4 2 ) studies i s that of prediction, the fundamental purpose of factor analysis i s , as i t s name implies, simply one of anal-ysing a given domain to discover any possible order that may e x i s t therein. I t seems advisable at t h i s point to devote closer a t -tention to the purpose f o r which Thurstone developed his multiple-factor a n a l y s i s . In describing the area that i s to be investigated by means of a given f a c t o r i a l study he uses the term domain which he (19, p. 55) defines as "The range of phenomena that i s represented i n any factor analysis . . .." He states that o r i g i n a l l y the methods were developed f o r the purpose of " . . . i d e n t i f y i n g the p r i n c i p a l dimensions or categories of mentality; . . .." but then goes on to indicate that t h i s by no means implies that they are not applicable to yet other domains. In actual f a c t , they may be applied not only i n other psychological spheres but also i n other scien-t i f i c f i e l d s as w e l l . Thurstone foresees the greatest value of t h i s method as l y i n g i n i t s a b i l i t y to deduce hypotheses which may then be tested by means of more d i r e c t laboratory experiments. That i s i t would seem he regards the process as being most valuable when employed i n an exploratory r o l e , and i n t h i s form of f a c t o r i a l experiment no o r i g i n a l hypothesis would be necessary. The data concerning a selected sample from a population would be c o l l e c t e d and analysed f a c t o r i a l l y as to i t s underlying structure i n order to decide f i r s t l y , whether order exists within the domain, and, secondly, what form the order takes i f such i s found to e x i s t . Having (43) completed t h i s work the r e s u l t s might serve as a f r u i t f u l source f o r explanatory concepts and hypotheses. Thus we see that the purpose of f a c t o r i a l s t u d i e s , as so envisaged, i s , f a r from being a t vari a n c e w i t h the general experimental work of a given s c i e n c e , y e t a f u r t h e r a i d to these methods i n the i n i t i a l and c r u c i a l stages. This i s s u e may be f u r t h e r c l a r i -f i e d by Thurstone's own words (19, p. 56). The e x p l o r a t o r y nature of f a c t o r a n a l -y s i s i s o f t e n not understood. F a c t o r a n a l -y s i s has i t s p r i n c i p a l usefulness a t the bor-der l i n e of s c i e n c e . I t i s n a t u r a l l y super-seded by r a t i o n a l formulations i n terms of the science i n v o l v e d . Factor a n a l y s i s i s u s e f u l , e s p e c i a l l y i n those domains where b a s i c and f r u i t f u l concepts are e s s e n t i a l l y l a c k i n g and where c r u c i a l experiments have been d i f f i c u l t to conceive. The new methods have a humble r o l e . They enable us t o make only the crudest f i r s t map of a new domain. But i f we have s c i e n t i f i c i n t u i t i o n and s u f f i c i e n t i n g e n u i t y , the rough f a c t o r i a l map of a new domain w i l l enable us to proceed beyond the e x p l o r a t o r y f a c t o r i a l stage t o the more d i r e c t forms of p s y c h o l o g i c a l experimentation i n the l a b o r a -t o r y . In a n a l y s i n g Thurstone's statement, i t might w e l l be noted that t h i s very r o l e , which he terms humble, i s i n r e a l i t y an immensely important one a t the present stage i n the develop-ment of p s y c h o l o g i c a l knowledge. Before b r i n g i n g t h i s s e c t i o n of our work to a cl o s e i t i s necessary t h a t we consider a problem of fundamental impor-tance to the whole f a c t o r i a l s t r u c t u r e . The f a c t o r s d e r i v e d i n many p s y c h o l o g i c a l s t u d i e s are obtained from the i n t e r -c o r r e l a t i o n s of t e s t scores and th e r e f o r e the problem a r i s e s as to whether changes i n s c o r i n g method w i l l occasion a l t e r -( 4 4 ) a t i o n i n the obtained f a c t o r i a l pattern. This problem i s so basic to a l l the work that follows that i t i s e s s e n t i a l that we give i t very c a r e f u l consideration. The problem i s simply that, i n any t e s t , the scores obtained have, i n actual f a c t , no absolute meaning since they are dependent to a large de-gree upon the s p e c i f i c scoring methods chosen by the test designer. In r e a l i t y , these scores serve only to place the individuals i n rank order i n regard to the a b i l i t y tested. Since the obtained test scores form the basis f o r the i n t e r -correlations which are to be analysed, and since differences i n score w i l l occasion altered c o r r e l a t i o n c o e f f i c i e n t s , i t becomes v i t a l to ascertain whether these changes also e l i c i t differences i n the factor patterns that are derived. I f such were the case, the whole r e s u l t obtained would be dependent upon the whim of the t e s t constructor and so would lose i t s descriptive s i g n i f i c a n c e . I t would be possible to i n v e s t i -gate t h i s problem by t h e o r e t i c a l means but the proof would then be extremely d i f f i c u l t . However, the problem could be most e a s i l y examined by means of an experiment i n which the various methods of recording the scores were employed. The variety of c o r r e l a t i o n matrices that a r i s e could then be factored to ascertain what r e s u l t these changes had brought about i n the basic factor pattern. Such an experiment has now been performed and has shown that the fundamental factor groupings are not distorted by such a l t e r a t i o n s . This s i g -n i f i e s that the methods are " . . . s u f f i c i e n t l y powerful that one can take considerable l i b e r t i e s with the raw scores (45) without s e r i o u s l y a f f e c t i n g the r e s u l t s . " (19, p. 66). I t might he w e l l t o c l o s e t h i s chapter w i t h the f o l l o w -i n g q u a l i f y i n g statement from Conrad (5, p.553). "The method, w h i l e u n i q u e l y v a l u a b l e , i s not a panacea. C e r t a i n l y i t does not r e p l a c e shrewd o b s e r v a t i o n and i n t e r p r e t a t i o n , nor does i t i n any way reduce the need f o r c l e v e r t e s t c o n s t r u c t i o n . " (46) Chapter I I I  D e r i v a t i o n of the Fundamental Eq u a t i o n s In our i n i t i a l d i s c u s s i o n we found t h a t the o r i g i n a l methods of f a c t o r i z a t i o n , d e v i s e d by Spearman, r e s u l t e d i n a f a c t o r p a t t e r n c o n s i s t i n g of a s i n g l e g e n e r a l f a c t o r measured by a l l t e s t s o f the b a t t e r y and s p e c i f i c f a c t o r s each measured by a separate t e s t . In order t o i n s u r e t h a t a m a t r i x c o u l d be so analysed i t was nece s s a r y t o t e s t the v a l u e s of the t e t r a d d i f f e r e n c e s i n order t o make c e r t a i n t h a t they d i f f e r e d from z e r o only by an amount compatible w i t h the theory o f chance e r r o r s . T h i s g e n e r a l theory s u f f i c e d u n t i l i t became apparent, i n the course of numerous f a c t o r i a l s t u d i e s of t h i s type, t h a t , i n a l a r g e percentage o f the i n v e s t i g a t i o n s , t h i s r e s t r i c t i o n d i d not h o l d t r u e . The v a l u e s obtained f o r the t e t r a d d i f f e r e n c e s f o r many of these e x p e r i m e n t a l m a t r i c e s d i f f e r e d from z e r o t o a s i g n i f i c a n t degree. I t thus became apparent t h a t a t h e o r e t i c a l e x p l a n a t i o n must be found f o r t h i s phenomenon or e l s e the tw o - f a c t o r theory would not be completely g e n e r a l i n i t s a p p l i c a b i l i t y . The Spearman 1 igroup, i n s e a r c h i n g f o r a s o l u t i o n t o t h i s dilemma, a r r i v e d a t the c o n c l u s i o n t h a t the h i e r a r c h i c a l order of these m a t r i c e s was being d i s t u r b e d by the f a c t t h a t c e r t a i n t e s t s i n the b a t t e r y were h i g h l y s i m i l a r i n n a t u r e . They f e l t t h a t t h i s f a c t tended t o produce s p u r i o u s common f a c t o r s between t e s t s . In order t o c o r r e c t t h i s s i t u a t i o n they f i r s t d ecided t h a t the t e s t b a t t e r y must, i n such cases, be p u r i f i e d by e l i m i n a t i n g (47) a l l tests which were too similar in nature. Thus the dis-turbers of hierarchical order were removed. Later, however, the overall theory was revised to some extent so as to i n -clude several group or common factors which were evaluated by a judicious recombining of tests within the batteries em-ployed. In this way the two-factor school reconciled their techniques with the observable results of experimentation. These common factors were secondary in nature and did not have the importance of the general or. the specific factors. This solution of the problem did not serve to convince a l l investigators in the factorial f i e l d and a variety of groups began investigating more closely the concept of common factors which were measured by two or more tests in the bat-tery. One series of techniques which developed from this type of investigation are those which we are about to study. The Chicago school became increasingly convinced that i t was these common factors which might well serve as the source of a reference frame for the study of any given domain, and so Thurstone and his co-workers began the process of devel-oping techniques by which common factors could be evaluated. As we w i l l see i n a later section of this chapter, the methods Thurstone has derived are such that they include the Spear-man two-factor theory as a special case. Let us now consider the fundamental theorem of multiple-factor analysis. This theorem states that a correlation matrix, of the form we have already discussed on page 5 of Chapter I, can be factorized into a factor matrix in which (48) the column v a l u e s correspond t o the f a c t o r l o a d i n g s of each t e s t i n the b a t t e r y . T h i s theorem i s so b a s i c t o the e n t i r e theory t h a t i t s proof assumes g r e a t s i g n i f i c a n c e f o r our work. For t h i s reason l e t us t r a c e the steps employed by Thurstone i n developing a.proof of t h i s theorem. In a r r i v i n g a t h i s c o n c l u s i o n Thurstone used the summa-t i o n n o t a t i o n which, f o r l a r g e numbers of terms i n a compu-t a t i o n , tends t o markedly s i m p l i f y the mathematics i n v o l v e d . However, i n order t o make t h i s proof more r e a d i l y under-standable f o r the non-mathematical student, we w i l l a r b i t r a -r i l y r e s t r i c t the number of terms i n the d e r i v a t i o n and so a v o i d the need f o r summation n o t a t i o n . T h i s a r t i f i c e must be borne i n mind throughout the d i s c u s s i o n i n order t h a t the reader r e a l i z e t h a t our r e s u l t s a re p u r e l y i n d i c a t i v e of the f a c t s , and t h a t the number of t e s t s and s u b j e c t s which we w i l l employ i s i n s u f f i c i e n t f o r any a c t u a l f a c t o r i a l study. The student w i l l r e a d i l y r e c o g n i z e t h a t such a l i m i t a t i o n ap-p l i e s t o t h i s p r o o f , but he should a l s o r e a l i z e t h a t such a proof i s l o g i c a l l y a p p l i c a b l e t o a problem of f a r l a r g e r d i -mensions . The f i r s t assumption made by Thurstone i s t h a t i f N sub-j e c t s a r e a d m i n i s t e r e d n t e s t s then i t i s p o s s i b l e t o account f o r the standar d s c o r e s obtained by each i n d i v i d u a l on each t e s t of the b a t t e r y by means of a l i n e a r r e l a t i o n s h i p employ-i n g a s m a l l e r number of f a c t o r s or v a r i a b l e s than there a r e t e s t s i n the g i v e n b a t t e r y . The assumption i s f e a s i b l e as long as a l a r g e number of t e s t s a re employed and the v a r i e t y (49) o f - a b i l i t i e s measured i s r e l a t i v e l y s m a l l e r i n number. At t h i s p o i n t we impose our r e s t r i c t i o n a s t o t h e num-b e r o f t e s t s a n d t h e s i z e o f t h e sample and assume t h a t t h e r e a r e o n l y f o u r t e s t s i n t h e b a t t e r y a n d f i v e s u b j e c t s i n t h e s a m p l e . F u r t h e r m o r e , l e t u s assume t h a t the s t a n d a r d s c o r e s o f our s u b j e c t s on t h e s e f o u r t e s t s can be e x p r e s s e d i n terms o f o n l y t h r e e a b i l i t i e s . T h e r e f o r e t h e s t a n d a r d s c o r e " s " o f any s u b j e c t , f o r e x -ample i n d i v i d u a l o n e , on any t e s t o f t h e b a t t e r y , say t e s t number f o u r , can be e x p r e s s e d i n e q u a t i o n form as f o l l o w s , ( l ) S 4 l = C41 X l l +" C4-2 X £ l •+• C43 X 3 1 • The x ' s a r e s t a n d a r d s c o r e s f o r t h e s u b j e c t upon e a c h o f t h e t h r e e r e f e r e n c e a b i l i t i e s and the c ' s a r e t h e w e i g h t s a s r -s i g n e d t o t h e s e r e f e r e n c e a b i l i t y s c o r e s i n o r d e r t o d e t e r -mine t h e s t a n d a r d s c o r e o f t h e i n d i v i d u a l u p o n t h e e n t i r e t e s t . The c ' s a r e t h e f a c t o r l o a d i n g s w h i c h a r e dependent o n l y upon t h e s p e c i f i c t e s t e m p l o y e d , whereas the x ' s a r e s t a n d a r d s c o r e s w h i c h e x p r e s s t h e i n d i v i d u a l ' s a c t u a l c a -p a c i t y i n r e s p e c t t o t h e t h r e e r e f e r e n c e a b i l i t i e s . F o r o u r p r e s e n t w o r k , t h e r e f e r e n c e a b i l i t i e s w i l l be s e l e c t e d i n s u c h a manner as t o be s t a t i s t i c a l l y i n d e p e n d e n t . By s t a - . t i s t i c a l l y i n d e p e n d e n t "we s i m p l y mean t h a t t h e c o e f f i c i e n t o f c o r r e l a t i o n between any two o f t h e s e a b i l i t i e s i s z e r o f o r t h i s p o p u l a t i o n . A v e r y s i m p l e n o t a t i o n f o r s u c h r e l a t i o n s h i p s as t h e above has been employed by T h u r s t o n e i n w h i c h a l l s u c h (50) e x p r e s s i o n s are rec o r d e d i n the form of a m a t r i x p r o d u c t . In m a t r i x n o t a t i o n a l l such equations as the sample i n equation (1) can be expressed as shown i n Table I. Table I M a t r i x R e p r e s e n t a t i o n of L i n e a r F u n c t i o n s I I I I l l • 1 2 3 4 5 1 2 3 4 5 1 I X1Z X l 5 X14 X 1 ; r 1 s n s« s « s CZ2 I I x z i X22 x 2 J XZ4 X 2 * 2 s 2 2 s « S2* 3 c 3 a C33 I I I X 3 1 X 3 2 X33 X 3 * 3 s 3 3 S 3 5 4 C 4 L C42 C43 4 s 4 1 S 4 t s 4 3 S44. S 4 5 F P S F a c t o r M a t r i x P o p u l a t i o n M a t r i x Score M a t r i x M a t r i x m u l t i p l i c a t i o n i s accomplished by m u l t i p l y i n g each element of any row of F by the corresponding element of any column of P. The sum of a l l v a l u e s d e r i v e d f o r a spe-c i f i c m u l t i p l i c a t i o n of one row times one column d e f i n e s a s i n g l e element of the m a t r i x S. Thus, f o r example, i f we m u l t i p l y the f o u r t h row of m a t r i x F by the f i r s t column of m a t r i x P the r e s u l t i n g e q u a t i o n i s t h a t on the f o r e g o i n g page f o r the standard s c o r e of i n d i v i d u a l one on t e s t f o u r or It e tt i>4a • T h i s process may be f u r t h e r c l a r i f i e d by c o n s i d e r a t i o n of another example. Suppose we wanted the e q u a t i o n f o r the standard s c o r e of s u b j e c t three on t e s t two or " s 2 3 ." T h i s i s o b t ained by m u l t i p l y i n g the second row of m a t r i x F by the t h i r d column of m a t r i x P. The r e s u l t i n g e q u a t i o n i s as (51) f o l l o w s , SZ5 = CZ1 X 1 5 + °ZZ X Z 3 + CZ3 X33 • I n t h i s manner e v e r y e l e m e n t o f t h e s c o r e m a t r i x i s d e f i n e d i n t e r m s o f a l i n e a r f u n c t i o n o f t h e s t a n d a r d s c o r e s " x . " I n m a t r i x F, T a b l e I , t h e r o w s c o n t a i n t h e w e i g h t s o r f a c t o r l o a d i n g s a s s i g n e d t o t h e p a r t i c u l a r t e s t whose number i s a t t h e l e f t o f t h a t r o w , w h i l e t h e c o l u m n s e x h i b i t t h e v a l u e s f o r t h e g i v e n f a c t o r whose number i s a t t h e t o p o f t h a t c o l u m n . T h u s , f o r e x a m p l e , t h e v a l u e s i n r o w 2 a r e t h e f a c t o r l o a d i n g s , o f t e s t two o n e a c h o f t h e t h r e e f a c t o r s , w h i l e t h e amounts i n c o l u m n 3 a r e t h e f a c t o r l o a d i n g s o f a l l f o u r t e s t s o n f a c t o r I I I . M a t r i x P, o n t h e o t h e r h a n d , h a s e a c h r o w c o n t a i n i n g t h e s t a n d a r d s c o r e s o b t a i n e d b y a l l f i v e s u b j e c t s u p o n t h e f a c t o r whose number i s a t t h e l e f t o f t h a t r o w , w h i l e e a c h c o l u m n e x p r e s s e s t h e s t a n d a r d s c o r e u p o n e a c h r e f e r e n c e a b i l i t y o f t h e s u b j e c t whose number i s a t t h e h e a d o f t h a t c o l u m n . Thus t h e s e c o n d r o w o f m a t r i x P c o n t a i n s t h e s t a n d a r d s c o r e s o f a l l s u b j e c t s u p o n f a c t o r I I , w h i l e c o l u m n two c o n s i s t s o f t h e s t a n d a r d s c o r e s o f i n d i v i d u a l two u p o n e a c h o f t h e t h r e e r e f -e r e n c e a b i l i t i e s . F i n a l l y , m a t r i x S, t h e s t a n d a r d s c o r e m a t r i x , i s s o c o n -s t r u c t e d t h a t t h e r o w s i n d i c a t e t h e s t a n d a r d s c o r e s o b t a i n e d by a l l f i v e i n d i v i d u a l s o n t h e t e s t whose number i s t o t h e l e f t o f t h a t r o w , w h i l e t h e c o l u m n s c o n t a i n t h e s t a n d a r d s c o r e s u p o n e a c h o f t h e f o u r t e s t s o b t a i n e d b y t h e s u b j e c t ( 5 2 ) whose number is at the top of that column. Thus the fourth row of matrix S contains the standard scores of a l l five sub-jects on test four and the f i r s t column contains the standard scores of subject one upon a l l four tests. Our f i r s t problem i s to obtain a value for the co e f f i -cients or factor loadings "c" of any particular test of our battery. Let us f i r s t note that, since sn and x,^ are scores con-verted to standard form, i f we sum a l l scores "s" or a l l scores "x" for a given test or a specific a b i l i t y such a sum w i l l be identically zero. Let us test this statement for the scores "s" obtained by each of our five subjects on test one, But any standard score s a i i s obtained from the corresponding raw score y l £ by subtracting the test mean mi and dividing by origin and contracts the standard deviation to unity for our scores. Thus, where yla. i s the raw score obtained by individual one on test one. Therefore the test standard deviation <r . This moves- the mean to the L- l cr ~ o~ a~ 0~ tr m (53) i-Gk -+- y12 4- y 1 3 4- y 1 + 4- y l y - 5m) But, by the d e f i n i t i o n of the average or mean of a set of raw measures "y", t H yn + yiz +y 1 5 +" yi4 +-yi* ) = m -Therefore y 1 ± +- y l z + y 1 3 + y14. + y^ = 5m and upon substitution, f s, • = l _ ( 5 m — 5m) = 0. A s i m i l a r statement holds true for any sum of a l l stand-ard scores f o r any s p e c i f i c test or any given a b i l i t y . Hence <L SJ£ = 0 f o r any test " j " f o r our population ::x 7 ni = 0 f o r any a b i l i t y "m" f o r our group. •I and Since the standard deviation of any d i s t r i b u t i o n of standard scores must equal unity, the variance, which i s sim-ply a~z must a l s o equal t h i s value. Let us consider the f i r s t test of our battery. Since our scores have been converted to standard form the variance of t h i s test i s S l l + S1Z +" S13 +" S14 +" S15 = 1 . Upon substituting the l i n e a r equation values f o r the "s" terms we obtain (54) =i[(Qu x u + C « X z i + C13 X 3 i )* + ( c u x i z + C i * ^ + C « X 3 & )* + ( c 4 1 x 1 3 4- ctz xzi 4- c 1 3 x 3 3 ) - f ( c l t x 1 4 . 4 - c l z x^.4- c 1 3 x 3 4 ) + ( C n X i s " * ~ C 1 2 X f c s ' + " C 1 3 X 3 S ) j . • W h e n we s q u a r e t h e t e r m s i n t h e i n n e r b r a c k e t s t h i s b e c o m e s c r > L r = \ [<2i1x[i+ c ^ x | t 4 - c * , x 3 1 - f 2 c 1 1 c u x I 1 x z l - i - 2 c 1 1 c 1 5 x 1 1 x 3 1 + - 2 c l 2 c 1 3 x 2 1 x 3 1 -+- c ^ x £ 5 + c ^ x | 3 + C j 3 X 3 3 4 - 2 c 1 1 c l i l x t t x 2 3 4 - 2 c 1 1 c 1 3 x 1 3 x 3 3 + 20^0^2^X33 4- c * 1 x | 4 . + c ^ 1 ( x | 4 + c j 3 x 3 + + 2 c 1 1 c l i , x 1 4 x £ + + 2 c 1 1 c 1 3 x w x 3 * + 2 c u c t ^ c 3 ^ + c « x i 5 4 < x £ f f 4 - c ^ x 3 5 + 2 c 1 1 c n x ^ c 4 s 4 2 c l i c 1 3 x 1 J x 3 f f + 2 c w c 1 3 x 2 ? x 3 ^ =1. I f w e e x a m i n e t h e t e r m s i n t h e f o r e g o i n g e x p r e s s i o n , we f i n d t h a t t h e s e m a y b e g r o u p e d i n t o s e r i e s o f f i v e t e r m s a l l o f w h i c h h a v e t h e s a m e c o e f f i c i e n t . T h u s o u r e q u a t i o n b e -c o m e s . 1 4 ( ^ 4 + X " + X £ + X ^ + C ^ X M + X L + * y + < & ( X f 1 + X k + X 3 3 +" X L + X 3 S ) + 2 C H C l t ( X l l X t l + X l i X 2 * . + X 0 3 ^ 3 4- x 1 4 . x Z 4 , 4- x 1 5 x i 5 ) 4 20^ c 1 5 ( x 1 1 x 3 1 4 - x u x 3 i + x 1 3 x 3 3 4 - x 1 4 x ^ + x a ? x 3 j r ) +" 2c 1 2 r C 1 3 ( x 2 1 X 3 1 + X ^ X3Z H~ X Z J X 3 3 + X2,4 x 3 4 . X23" X 3 ^ =1. L e t u s s t u d y t h e q u a n t i t i e s c o n t a i n e d i n t h e i n n e r b r a c k e t s . T h e t e r m w h o s e c o e f f i c i e n t i s c ^ c a n b e e v a l u a t e d a s f o l l o w s , (55) ^04 + x i \ -+-X* 4-x£ + x£. ) = ^ = 1 , s i n c e the x's are standard s c o r e s . T h e r e f o r e X J J ~f- X^ 2 "i~Xjj 4-X^ 4-X15 5 . A s i m i l a r statement h o l d s t r u e f o r the terms whose c o e f -f i c i e n t s are c£z and c ^ . The v a l u e of the term w i t h c o e f f i c i e n t 2c1± c u can be obtained as f o l l o w s , ^ ( x n x z l + x 1 J L x 2 2 + x 1 5 x 2 3 4- x 1 4 x 2 4 + x15. x^ 5 ) = r^^ > s i n c e the x's are standard scores and t h e r e f o r e have a mean e q u a l t o z e r o and a standard d e v i a t i o n e q u a l t o u n i t y . That i s , the above e x p r e s s i o n i s simply the c o e f f i c i e n t of c o r r e -l a t i o n between standa r d scores obtained by our p o p u l a t i o n upon a b i l i t i e s one and two. But the a b i l i t i e s which we have s e l e c t e d as a r e f e r e n c e frame are such as t o be s t a t i s t i c a l l y independent. T h i s means t h a t the c o r r e l a t i o n between any two of these a b i l i t i e s i s i d e n t i c a l l y z e r o . T h e r e f o r e i ( x x i X 2 1 4- X 1 Z Xzz + X 1 3 X 2 3 4- X^ X^ +X15 X2J. )= 0 and x 1 4 x z i + x 1 2 x z z 4 x 1 3 x 2 3 4- x 1 < t xM + x 1 5 = 0. For the same reason the terms whose c o e f f i c i e n t s a r e 2clx c 1 3 and 2c 1 2 c 1 3 a r e i d e n t i c a l l y z e r o . T h e r e f o r e our equ a t i o n r e -duces t o the f o l l o w i n g form. (56) (2) = c* + c* = 1. T h i s can he stated, as an i n t e r m e d i a t e theorem (19, p. 73) which i s " . . . the sum of the squares of the t e s t c o e f f i -c i e n t s of a t e s t f o r a l l o r t h o g o n a l f a c t o r s i s e q u a l t o u n i t y . " For the moment i t i s s u f f i c i e n t t o understand the term or t h o g o n a l as being e q u i v a l e n t t o s t a t i s t i c a l l y i n d e -pendent. We w i l l d i s c u s s the p r o p e r t y of o r t h o g o n a l i t y , as i t r e f e r s t o such f a c t o r s , when we develop the geometric r e p r e s e n t a t i o n of a f a c t o r p a t t e r n . Upon a g a i n r e f e r r i n g to e q u a t i o n (2) we note t h a t the square of each t e s t c o e f f i c i e n t " c " i s t h a t p o r t i o n of the t o t a l v a r i a n c e o f a g i v e n t e s t which i s a t t r i b u t a b l e t o t h a t s p e c i f i c r e f e r e n c e a b i l i t y . Thus i n t h a t e q u a t i o n c l z > i s the square r o o t of t h a t p a r t of the t o t a l v a r i a n c e of t e s t one t h a t i s due to f a c t o r I I . From t h i s we o b t a i n Thurstone's second b a s i c theorem (19, p. 73) t h a t , " . . . each f a c t o r l o a d i n g or t e s t c o e f f i c i e n t c J T n f o r or t h o g o n a l f a c t o r s i s the square r o o t of the v a r i a n c e of t e s t j a t t r i b u t a b l e t o the f a c t o r m." The f a c t o r s i n terms of which the t e s t scores can be i n t e r p r e t e d a re of two g e n e r a l types; common f a c t o r s , which are those measured by two or more t e s t s of a b a t t e r y and unique f a c t o r s , which a r e measured by a s i n g l e t e s t of the group. Since no t e s t can measure a b i l i t y without some degree of e r r o r , t h i s l a t t e r s e t of f a c t o r s can be f u r t h e r s u b d i -v i d e d i n t o s p e c i f i c f a c t o r s , which are a b i l i t i e s i n v o l v e d i n (57) a s i n g l e t e s t , and e r r o r f a c t o r s due t o the i n a c c u r a c y of the instrument. For t h i s reason our complete f a c t o r p a t t e r n must a l l o w f o r a l l such f a c t o r s . However, s i n c e our presen t ob-j e c t i v e i s t o d e r i v e a means of e v a l u a t i n g the common f a c t o r s measured by a b a t t e r y , we w i l l content o u r s e l v e s w i t h denot-i n g both the s p e c i f i c f a c t o r and the e r r o r f a c t o r by u ± . Th i s value i n d i c a t e s the combined e f f e c t o f a l l f a c t o r s i n which a g i v e n t e s t i s unique from a l l other t e s t s of a p a r -t i c u l a r b a t t e r y . We can r e p r e s e n t the f a c t o r i a l breakdown,of our f o u r t e s t s as i n Table I I (19, p. 76). Table I I Complete F a c t o r M a t r i x I I I 1 2 3 4 1 a n « l t 2 a z i a 2 2 3 a 3 1 4 a 42. u + 4. In t h i s t a b l e , the a ! s denote the f a c t o r l o a d i n g s of the fo u r t e s t s upon the common f a c t o r s I and I I , w h i l e the u's denote the sum of a l l the f a c t o r s i n which each t e s t i s unique from a l l the other t e s t s s e l e c t e d . F i n a l l y , each blank space s i g n i f i e s t h a t the t e s t , whose number i s t o the l e f t of t h a t row, has zero l o a d i n g upon the f a c t o r , whose num-ber i s a t the head of the column. (58) By employing the i n f o r m a t i o n d e r i v e d i n e q u a t i o n ( 2 ) , page 56, i t can be s t a t e d t h a t a t\+ a^-h u\t = l=<r{ . The term (a^-H a\t ) i s c a l l e d the communality or common f a c -t o r v a r i a n c e of t e s t one. I t i s so desig n a t e d s i n c e i t r e -presen t s t h a t p o r t i o n of the t o t a l v a r i a n c e t h a t i s due to common f a c t o r s w i t h i n the t e s t . T h i s i s u s u a l l y w r i t t e n (3) a*, 4- = h* where h* denotes the communality of t e s t one. S i m i l a r l y the va l u e i s c a l l e d the uniqueness of t e s t one and i s u s u a l l y w r i t t e n u*. Thus we o b t a i n the f o l l o w i n g e q u a t i o n . (4) h*+-u*=l, or the sum of the communality and uniqueness of any t e s t i s equal t o u n i t y . we a r e now f u l l y prepared t o d e r i v e the fundamental theorem of f a c t o r a n a l y s i s . . In'order t o a v o i d summation no-t a t i o n which may tend t o be c o n f u s i n g f o r the.non-mathemati-c a l student, l e t us a g a i n r e s t r i c t o u r s e l v e s t o a l i m i t e d number of t e s t s , i n d i v i d u a l s and r e f e r e n c e a b i l i t i e s i n order t h a t the f o l l o w i n g proof may be performed as a simple process of a l g e b r a i c m u l t i p l i c a t i o n . The reader i s a g a i n reminded t h a t t h i s r e s t r i c t i o n i s simply an a r t i f i c e i n order t h a t the number of"terms i n the f o l l o w i n g work may be as few as pos-s i b l e . The need f o r t h i s d e v i c e and the v a l u e of m u l t i p l e (59) summation w i l l be impressed on the reader when he notes the l a r g e number of terms w i t h which we s t i l l must d e a l d e s p i t e the r e s t r i c t i o n imposed above. Let us assume t h a t we have a d m i n i s t e r e d f o u r t e s t s t o three s u b j e c t s and t h a t the standard scores o b t a i n e d by these people can be expressed as a l i n e a r f u n c t i o n of two f a c t o r s common to a l l of the t e s t s and a f a c t o r which i s unique to each of the t e s t s i n our s e r i e s . T h i s unique f a c t o r w i l l r e p r e s e n t the combined e f f e c t of the s p e c i f i c f a c t o r of our t e s t and i t s e r r o r f a c t o r . These l i n e a r r e l a t i o n s h i p s may be r e p r e s e n t e d by the m a t r i x product r e c o r d e d i n Table I I I . Table I I I Product f o r a Complete F a c t o r M a t r i x I I I 1 2 1 « n u i 2 a 2 * U 2 3 a 3 1 a 3 2 4 a 4 1 a ^ : F F a c t o r M a t r i x The blank spaces i n m a t r i x F i n d i c a t e t h a t the g i v e n t e s t has zero l o a d i n g on the f a c t o r i n t h a t column. Thus the s t a n d -a r d s c o r e slz of s u b j e c t two on t e s t one can be expressed as u. u. 1 2 3 1 2 3 I x n X 1 2 X13 1 S 1 2 S ! 3 I I X 2 1 X 2 2 X 2 3 2 S 2 3 1 X 3 1 X 3 2 X 3 3 3 S31 S32 S 3 3 2 x 4 1 X4-2 x 4 3 ' 4 S42 S 4 3 3 X 5 1 X 5 2 X 5 3 4 X fc2 X f c 3 P s P o p u l a t i o n M a t r i x Score M a t r i x (60) f o l l o w s , s 1 2, = x l t + a 1 2 x 2 z + u t x 3 2 4- 0 -x 4 2 + 0 - x „ +0-x 6 z = a a i x l i t -+-a12 x 2 2 4 - u 1x 3 j l . Since the number of s u b j e c t s i s three and the scores _"s" are i n standard form, the c o r r e l a t i o n between any two t e s t s , f o r example t e s t s one and two, i s e q u a l t o The l i n e a r f u n c t i o n , which i s e q u a l t o each of these standard s c o r e s , can be ob t a i n e d from the m a t r i x product of Table I I I . These v a l u e s are l i s t e d below. — a i l x l a 4- a 1 2 X Z 1 + U1 X31 • s 2 1 = a 2 l x 1 1 4 - a 2 z X 2 1 • S 1 2 a i l X i z + a 1 2 X 2 Z 4- u.x^ • szz a 2 1 x-iz 4- a 2 2 X 2 Z + U 2 X 4 Z * Sl3 = a i l x 1 3 4- a 1 2 X 2 3 -*-u ax 3 3 • S*3 a Z ! X 1 3 - l ~  azz X 2 5 4-U2X43 • The products needed f o r the above e q u a t i o n are now r e a d i l y o b t ained by m u l t i p l i c a t i o n of each s e t of two rows qf the above v a l u e s . We o b t a i n s n s 2 . i = a i i az.i x i i+ a n a22 x n x 2 i 4 - a i i u « . x n . x 4 i + a i z a 2 i X l 1 i x ^ + a ^ a ^ x ^ +a l t u 2 X g 1 x 4 1 4-Uj^a^ x l 1 L x31 -+• u a a 2 Z x 2 1 x 3 1 + u J u 2 x 3 1 x 4 1 . S l t S 2 2 . ~ a i l a21 X12+ a i l a i l X 1 Z X 2 2 + ^ H ^ X M . X 4 2 + a «. a 21 X12  XZl+&lz a 2 Z , X 2 2 . + a!2 U Z . X Z i X 4 2 + U i a i ! . l X12 X 3 2 + U l & 2 Z X 2 2 X 5 Z " + U 1 U 2 X 3 Z X 4 Z • (61) S13SZ3 — a i l a 2 1 X l 3 " ' " a i l & * 2 X l 3 X Z 3 " l ~ a i l U Z X l 3 X 4 3 + a i Z a 2 1 X 1 3 X Z 3 + a i Z a 2 Z X 2 3 + a a 2 U 2 , X 2 3 X 4 3 " ' " U l ' a 2 1 X 1 3 X 3 3 + U i a z 2 X 2 3 X 3 3 ~^~^i.^-ZX-33 X 4 3 • We can now add these v a l u e s and a l s o c o l l e c t the terms w i t h _ e q u i v a l e n t c o e f f i c i e n t s i n each group. Upon doing t h i s we o b t a i n the f o l l o w i n g , S l l S Z l + S 1 2 . S 2 2 + S 1 3 S 2 3 = , a i l a n ( X l l + X 1 2 + X 1 3 ) " * " a i i a Z . 2 . ( X £ l + X 2 2 + X Z 3 ) 4-a^ a a z ( x n x z i + x i ^ x z i + x i 3 X Z 3 ) + a 4 1 U z ( x w + X l t X 4 2 + X J 3 X 4 3 ) + &1Z a z l ( x i a x z l +x l z x Z 2 4-x13 x 2 3 ) 4 - a 1 2 u 2 ( x ^ x ^ i +x 2 2 x 4 z 4-x 2 3x 4 3 ) 4- a 2 1 u ± ( x 1 1 L x 3 1 +x 1 5 t x 3 2 -f-x 1 3 x 3 5 ) + a 2 2 U l ( X 2 1 X 3 1 + X Z Z X 3 2 " ' ~ X Z 3 X 3 3 ) -+- u 4 u a ( x 3 1 x 4 1 +x 3 2 x 4 2 +-x33 X4 3 ) . T h i s appears a fo r m i d a b l e expansion u n t i l we c o n s i d e r more c l o s e l y the q u a n t i t i e s i n the b r a c k e t s . The f i r s t two terms reduce very markedly when we note the resemblance t o the v a r i a n c e of a standard v a r i a b l e . F or example, the term w i t h a l a a 2 1 a s i t s c o e f f i c i e n t t u r n s out t o be a constant m u l t i p l e of the u n i t v a r i a n c e o f the standard s c o r e s on f a c -t o r I . Thus, ^ 2 = | ( x i i + x a \ + x * 3 ) = l . T h e r e f o r e x* a +x* 2 -+x*3 = 3 . The term w i t h a„„a,, a s a c o e f f i c i e n t reduces i n a s i m i l a r (62) manner s i n c e i t i s three times the v a r i a n c e of scores on the common f a c t o r I I . I f ve now study the other terms of our expansion we . f i n d a c l o s e c o n n e c t i o n between them and the c o r r e l a t i o n ex-i s t i n g between the scores on v a r i o u s p a i r s of f a c t o r s . But the f a c t o r s were chosen i n such a manner as to be s t a t i s -t i c a l l y independent and so the c o r r e l a t i o n between any two must be z e r o . For example, the c o r r e l a t i o n between f a c t o r s I and I I i s r i i r ~~ 3( X11 X 2 l " + " X 1 Z X 2 Z " I " X 1 3 X Z 3 ) = 0« Thus x l t x z i 4- x l z x z z + x 1 5 x „ •• = 0. But t h i s i s the term t h a t has a1± &zz as a c o e f f i c i e n t and t h e r e f o r e t h i s term d i s a p p e a r s from our expansion. A s i m i l a r argument holds t r u e f o r a l l the remaining terms and so our expansion reduces t o S l l SZX + SXl S 2 Z + S a 3 S 2 3 = S ( a i l aiX •+ S 1 Z a 2 2 . ) But we have a l r e a d y noted t h a t r!2 3 ( S H SZ1 S l l S t Z 4 7 S l 3 S23 ) T h e r e f o r e (5) r l z = a u a w + a l t . A s i m i l a r argument would serve to prove t h a t the c o r -r e l a t i o n between any two t e s t s of our b a t t e r y i s produced i n a s i m i l a r manner by the sum of the products of corresponding (63) common f a c t o r l o a d i n g s . Thus the c o r r e l a t i o n between t e s t s two and f o u r c o u l d be expressed as f o l l o w s , r24 = a z i a 4 i 4 _ & z z a 4 Z The uniqueness f a c t o r as would be expected f u r n i s h e s no p a r t of such a c o r r e l a t i o n c o e f f i c i e n t . A simple method of r e c o r d i n g t h i s r e s u l t f o r a l l c o r -r e l a t i o n c o e f f i c i e n t s i s one which employs the m a t r i x n o t a -t i o n . T h i s i s i l l u s t r a t e d i n Table IV. Table IV C o r r e l a t i o n M a t r i x as a Product o f F a c t o r M a t r i c e s I II 1 * u a i z 2 a * i azz 3. a 3 l a « 4 a 4 1 a 4 z . F - 1 2 3 4 1 2 3 4 I a z i a31 a 4 i 1 x t r i z r« r t 4 II a z z *» a 4 Z 2 *z r Z 4 3 x 3 r 3 4 4 r 4 1 r * 3 R S e v e r a l f a c t s are worthy of note i n t h i s m a t r i x product. F i r s t l y , the m a t r i x F 1 i s the transpose of m a t r i x F. That i s every row of F i s w r i t t e n as a column of F 1 . Secondly, i t w i l l be noted t h a t , f o r the moment, we have not e n t e r e d spe-c i f i c c o r r e l a t i o n v a l u e s i n the d i a g o n a l spaces of the c o r -r e l a t i o n m a t r i x R. T h i r d l y , i t can be r e a d i l y shown t h a t the above product i s v a l i d . As a n example, l e t us m u l t i p l y row 1 of m a t r i x F by (64) c o l u m n 2 o f F 1 . T h i s s h o u l d g i v e u s t h e c o r r e l a t i o n b e -t w e e n t e s t s one a n d t w o . Upon c a r r y i n g o u t t h i s m u l t i p l i c a -t i o n we o b t a i n a n a t i + ' a i z ' a 2 2. = r i 2 • We h a v e a l r e a d y p r o v e n t h i s t o be s o i n e q u a t i o n ( 5 ) . I n a s i m i l a r manner i t c a n be shown t h a t e v e r y e l e m e n t o f R i s t h e p r o d u c t o f a g i v e n r o w o f F w i t h a s p e c i f i c c o l u m n o f F 1 . L e t u s now i n v e s t i g a t e t h e a p p r o p r i a t e v a l u e s t o be i n -s e r t e d a s d i a g o n a l e l e m e n t s i n o u r c o r r e l a t i o n m a t r i x . I s a n e x a m p l e , l e t u s c o n s i d e r t h e f i r s t s u c h e l e m e n t i n m a t r i x R. T h i s i s o b t a i n e d by- m u l t i p l y i n g t h e f i r s t r o w o f F by t h e f i r s t c o l u m n o f F 1 . C a r r y i n g o u t t h i s m u l t i p l i c a t i o n we o b -t a i n a i i + " a i z . — x i • B u t we h a v e p r o v e n i n e q u a t i o n ( 5 ) , p age 5 8 , t h a t a i i alz = n i * t h e c o m m u n a l i t y o r common f a c t o r v a r i a n c e o f t e s t o n e . Thus e a c h o f t h e d i a g o n a l e l e m e n t s o f t h e c o r r e l a t i o n m a t r i x m u s t be t h e c o m m u n a l i t y o f t h e g i v e n t e s t . ' T h i s a g a i n i s t o be e x p e c t e d s i n c e t h e c o r r e l a t i o n o f a n y t e s t w i t h i t s e l f , f o r e x a m p l e r x l , i s t h e o n l y e l e m e n t o f a g i v e n r o w o f m a t r i x R w h i c h w i l l be a f f e c t e d b y t h e u n i q u e f a c t o r s . The v a l u e d e -r i v e d f r o m t h e common f a c t o r s a l o n e w i l l u n d e r e s t i m a t e t h i s c o r r e l a t i o n . (65) Upon i n s e r t i n g c o m m u n a l i t i e s i n t h e d i a g o n a l s p a c e s we o b t a i n t h e c o m p l e t e r e d u c e d c o r r e l a t i o n m a t r i x . I t i s c a l l e d r e d u c e d s i n c e e a c h o f t h e d i a g o n a l e l e m e n t s i s a n u n d e r e s t i -m a t i o n o f t h e t r u e c o r r e l a t i o n o f a s p e c i f i c t e s t w i t h i t s e l f . T h i s m a t r i x p r o d u c t i s shown i n T a b l e V ( 1 9 , p . 8 1 ) . T a b l e V R e d u c e d C o r r e l a t i o n M a t r i x a s a M a t r i x P r o d u c t • — I I I 1 2 3 4 1 2 3 4 1 • a n I a 2 1 a 3 l a + 1 1 K ±V r ! 3 r ! 4 2 a z l a zz I I * 1 2 azz a 3 Z a 4 - Z 2 ^Zl r Z 3 r 2 4 3 a 3 i *3Z 3 r 3 1 r 3 Z K r 3 4 4 a 4 1 & 4 Z 4 ± 4 1 ± * 4 Z 1*43 K The a b o v e f i g u r e s y m b o l i z e s t h e f u n d a m e n t a l t h e o r e m o f m u l t i p l e - f a c t o r a n a l y s i s t h e p r o o f o f w h i c h we h a v e b e e n d e m o n s t r a t i n g . T h i s t h e o r e m ( 1 9 , p . 81) i s "The p r o d u c t o f t h e r e d u c e d f a c t o r m a t r i x b y i t s t r a n s p o s e i s t h e r e d u c e d c o r r e l a t i o n m a t r i x . " We h a v e d e v e l o p e d t h i s t h e o r e m i n d e t a i l b e c a u s e i t i s b a s i c t o a l l f u r t h e r w o r k i n t h e f i e l d o f m u l t i p l e - f a c t o r a n a l y s i s . T h i s i s c l e a r s i n c e t h i s k n o w l e d g e f u r n i s h e s p r o o f t h a t a g i v e n r e d u c e d c o r r e l a t i o n m a t r i x w i l l s e r v e a s a s o u r c e f o r t h e common f a c t o r l o a d i n g s o f the. b a t t e r y o f t e s t s . T h ese a r e t h e f a c t o r s w h i c h we a r e s e e k i n g t o e v a l u a t e . We now know t h a t f a c t o r i z a t i o n o f a r e d u c e d c o r r e l a t i o n m a t r i x i s p o s s i b l e a s l o n g a s we r e s t r i c t o u r s e l v e s t o (66) e v a l u a t i n g the common f a c t o r s . That i s , no attempt w i l l be made i n the f o l l o w i n g chapters t o ev a l u a t e the unique f a c t o r s i n a t e s t . The problem has been d i s c u s s e d by Thurstone, how-ever, who suggests the p o s s i b i l i t y of r e d u c i n g t h i s unknown q u a n t i t y by performing a f a c t o r i a l a n a l y s i s i n which the gi v e n t e s t i s j u d i c i o u s l y combined w i t h other t e s t s i n such a manner as to convert i t s s p e c i f i c f a c t o r s i n t o common f a c t o r s f o r t h i s new b a t t e r y . Before b r i n g i n g t h i s chapter t o a c l o s e i t i s necessary t h a t we i n v e s t i g a t e the number of common f a c t o r s t h a t can be d e r i v e d from any g i v e n reduced c o r r e l a t i o n m a t r i x . In theory t h i s number can be r e a d i l y proven t o be e q u a l t o the rank of the c o r r e l a t i o n m a t r i x i t s e l f . The term "rank of a m a t r i x " means the number of rows or columns of t h a t m a t r i x t h a t are not p r o p o r t i o n a l t o or l i n e a r combinations of the other rows or columns. The m a t r i x l i s t e d i n Table VI e x h i b i t s three rows and f o u r columns. Table M a t r i x VI of Rank 1 1 3 5 2 2 6 10 4 4 12 20 8 An i n s p e c t i o n of t h i s m a t r i x r e v e a l s t h a t every column i s a constant m u l t i p l e of any other column. For example, column P 4 i s ~ of column 3. Thus the f o u r rows are a l l l i n e a r l y de-(67) pendent and the rank i s one. In Table V I I there i s r e c o r d e d a m a t r i x which i n s p e c t i o n w i l l prove to be of rank 2, t h a t i s , every column can be proven t o be a l i n e a r combination of the f i r s t two columns. Table V I I M a t r i x of Rank 2 1 3 7 19 2 2 1 4 4 4 5 14 17 I f we c o n s i d e r column 3 i t i s p o s s i b l e to f i n d constant m u l t i p l i e r s c 4 and ot such t h a t i f any element of column 1 i s m u l t i p l i e d by c 4 and the corresponding element of column 2 by cx then the corresponding element of column 3 can be ex-pressed as a l i n e a r f u n c t i o n of these terms. The v a l u e s of Cj^  and cz can be o b t a i n e d as f o l l o w s . For the f i r s t element of column 3 the f o l l o w i n g e q u a t i o n h o l d s t r u e . 7 = 1 ^ + 5cz. For the second element o f column 3, 4 = 2 c 1 - f l c z . I f we s o l v e these two equations s i m u l t a n e o u s l y f o r Cj and cz we f i n d t h a t these constants have the f o l l o w i n g v a l u e s , c t = l (68) L e t u s c h e c k t h a t t h e s e v a l u e s h o l d t r u e f o r t h e t h i r d e l e -ment o f c o l u m n 3, T h a t i s , 1 4 = 4c 1+ 5cz . S u b s t i t u t i n g i n o u r v a l u e s f o r ct a n d cz we o b t a i n 14 = ( 4 ) ( 1 ) + ( 5 ) ( 2 ) = 1 4 . T h e r e f o r e a n y e l e m e n t o f c o l u m n 3 c a n be e x p r e s s e d a s a l i n e a r c o m b i n a t i o n o f c o l u m n s 1 a n d 2. The r e a d e r i s a d v i s e d t o c h e c k t h a t c o l u m n 4 c a n l i k e w i s e be e x p r e s s e d a s a l i n e a r c o m b i n a t i o n o f c o l u m n s 1 a n d 2 w h e r e c o l u m n 1 i s m u l t i p l i e d b y a c o n s t a n t c 3 a n d c o l u m n 2 i s m u l t i p l i e d b y a c o n s t a n t c^« The v a l u e s f o r t h e c o n s t a n t s a r e f o u n d t o be Thus a n y c o l u m n o r r o w o f o u r m a t r i x c a n be d e f i n e d a s a l i n e a r f u n c t i o n o f t h e f i r s t two c o l u m n s o r r o w s a n d s o t h e m a t r i x r a n k i s 2. By a n a l o g y , i f we h a v e a m a t r i x w i t h n r o w s a n d m c o l u m n s a n d i f a n y c o l u m n o f t h i s m a t r i x c a n be e x p r e s s e d a s a l i n e a r f u n c t i o n o f r o f t h e s e c o l u m n s , t h e n o u r m a t r i x h a s r a n k r . (69) S i n c e the number of common f a c t o r s d e r i v a b l e by means of a n a l y s i s i s e q u a l t o the rank o f the c o r r e l a t i o n m a t r i x , i f the rank o f the c o r r e l a t i o n m a t r i x i s r then the number of common f a c t o r s i s a l s o r . As an example, i f the c o r r e l a t i o n m a t r i x has rank 3 then there are three common f a c t o r s which f u l l y determine the c o r r e l a t i o n c o e f f i c i e n t s i t c o n t a i n s . I f we r e c a l l t h a t i n Spearman's t h e o r e t i c a l d e r i v a t i o n of h i s system o f f a c t o r i z a t i o n , the c o r r e l a t i o n m a t r i x had to e x h i b i t h i e r a r c h i c a l order and t h a t t h i s meant t h a t a l l rows of the m a t r i x were p r o p o r t i o n a l , we see t h a t t h i s simply im-p l i e d a m a t r i x o f rank one and thus t h a t one common f a c t o r c o u l d be d e r i v e d . For t h i s reason Spearman's method can be e x p l a i n e d as a s p e c i a l case of the Thurstone method of mul-t i p l e - f a c t o r a n a l y s i s . As we d i s c o v e r e d i n our e a r l i e r work, i n a c t u a l c o r r e l a -t i o n m a t r i c e s d e r i v e d from experimental data t h e r e a r e many extraneous e r r o r f a c t o r s p r e s e n t . T h i s w i l l always make the rank e x h i b i t e d by such a m a t r i x e x a c t l y e q u a l t o the number of t e s t s employed. T h e r e f o r e , i n p r a c t i c e , i t i s never pos-s i b l e t o d e r i v e the t r u e rank t h a t the m a t r i x would possess i f these e r r o r f a c t o r s had not oc c u r r e d s i n c e i t i s never p o s s i b l e t o o b t a i n the t r u e c o r r e l a t i o n c o e f f i c i e n t s . Thus the methods we w i l l study i n succeeding chapters w i l l be such a s t o overcome t h i s d i f f i c u l t y and a v o i d the n e c e s s i t y f o r knowing the tr u e rank of the c o r r e l a t i o n m a t r i x . ( 7 0 ) C h a p t e r I V The C e n t r e - i d M e t h o d I n t h e p r e c e d i n g c h a p t e r we i n v e s t i g a t e d t h e p o s s i b i l i t y o f f a c t o r i n g a c o r r e l a t i o n m a t r i x . T h i s s t u d y f u r n i s h e d u s w i t h a p r o o f t h a t s u c h a n a n a l y s i s c o u l d be p e r f o r m e d i f we r e s t r i c t e d o u r r e s e a r c h i n c e r t a i n w a y s . . To r e c a p i t u l a t e , t h e s e r e s t r i c t i o n s w e r e , f i r s t l y , t h a t t h e c o r r e l a t i o n m a t r i x m u s t be t h e r e d u c e d m a t r i x w i t h t h e v a l u e s f o r c o m m u n a l i t i e s c o n s t i t u t i n g t h e d i a g o n a l e l e m e n t s , a n d s e c o n d l y , t h a t t h e f a c t o r i s a t i o n m u s t be r e s t r i c t e d t o t h e e v a l u a t i o n o f common f a c t o r s o n l y . T h i s l a t t e r l i m i t a t i o n i s r e q u i r e d i n o r d e r t h a t t h e number o f f a c t o r s d e r i v e d be l e s s t h a n n , t h e t o t a l number o f t e s t s i n o u r b a t t e r y . I n a n y p r a c t i c a l p r o b l e m t h e d a t a f u r n i s h u s w i t h t h e c o r r e l a t i o n ^ c o e f f i c i e n t s so t h a t k n o w l e d g e o f t h e c o r r e l a t i o n m a t r i x i s t h e p o i n t a t w h i c h we s t a r t o u r a n a l y s i s . The o b -j e c t i v e i s t o o b t a i n t h e f a c t o r l o a d i n g s o f t h e t e s t s b a s e d u p o n t h e s e e m p i r i c a l c o e f f i c i e n t s . T h e r e a r e many m e t h o d s f o r d e r i v i n g t h e s e v a l u e s . I n t h i s c h a p t e r we w i l l c o n c e n t r a t e on one o f t h e b e t t e r known t e c h n i q u e s , n a m e l y , T h u r s t o n e ' s C e n t r o i d M e t h o d . T h i s p r o -c e d u r e w i l l e v a l u a t e t h e f a c t o r l o a d i n g s i n t e r m s o f s t a t i s -t i c a l l y i n d e p e n d e n t o r , a s t h e y a r e s o m e t i m e s c a l l e d , o r t h o -g o n a l r e f e r e n c e f a c t o r s . Once t h e s e v a l u e s a r e o b t a i n e d , i t t u r n s o u t t h a t , t o make o u r l o a d i n g s p s y c h o l o g i c a l l y m e a n i n g -f u l , we m u s t t h e n p e r f o r m a r e a r r a n g e m e n t o f t h e r e f e r e n c e ( 7 1 ) f a c t o r s . C o n s i d e r a t i o n o f t h i s s e c o n d s t e p w i l l be d e f e r r e d u n t i l t h e f o l l o w i n g c h a p t e r a n d i n t h e r e m a i n i n g p o r t i o n o f t h i s c h a p t e r we w i l l d i s c u s s t h e c e n t r o i d m e t h o d f o r a n a l -y s i n g t h e c o r r e l a t i o n m a t r i x i n t o a f a c t o r m a t r i x w i t h o u t r e f e r e n c e t o t h e r e a r r a n g e m e n t p r o b l e m . I n t h e d e r i v a t i o n s t h a t f o l l o w , we w i l l a d o p t t h e g e n e r -a l p a t t e r n s u g g e s t e d by P e t e r s a n d V a n V o o r h l s ( 1 4 ) a n d d e -v e l o p t h e p r o o f s c o m p l e t e l y i n a l g e b r a i c t e r m s r a t h e r t h a n t h e g e o m e t r i c o n e s e m p l o y e d b y T h u r s t o n e . I t m u s t be b o r n e c l e a r l y i n m i n d t h a t o u r p u r p o s e i s t o e x p l a i n t h e e x i s t i n g c o r r e l a t i o n s i n t e r m s o f t h e s m a l l e s t p o s s i b l e number o f r e f e r e n c e f a c t o r s a n d , t h e r e f o r e , t h a t we w i s h t o a b s t r a c t t h e maximum l o a d i n g s f o r e a c h f a c t o r a s i t i s d e r i v e d . T h i s may be more f u l l y c l a r i f i e d i f we remember t h a t we h a v e a l r e a d y p r o v e n t h a t e a c h s a t u r a t i o n c o e f f i c i e n t f o r a s p e c i f i c t e s t i s t h e s q u a r e r o o t o f t h a t p o r t i o n o f t h e t e s t v a r i a n c e a t t r i b u t a b l e t o t h e f a c t o r . Thus we s e e t h a t t h e a b o v e s t i p u l a t i o n means t h a t we w i s h t o d e r i v e f a c t o r s s u c h t h a t e a c h i n t u r n w i l l a c c o u n t f o r t h e maximum amount o f s u c h t e s t v a r i a n c e . F o r t h e p u r p o s e o f more r e a d y i l l u s t r a t i o n l e t u s c o n -s i d e r a n e x a m p l e i n w h i c h t h r e e t e s t s a r e a d m i n i s t e r e d t o N s u b j e c t s a n d t h e r e s u l t i n g s t a n d a r d s c o r e s a r e t h e n c o r r e l a t e d . We w i l l assume a b a t t e r y o f h y p o t h e t i c a l t e s t s w h i c h c o n t a i n n o e r r o r f a c t o r i n t h e i r s c o r e s . The o b t a i n e d c o r r e l a t i o n c o e f f i c i e n t s w i l l be t r u e o n e s i n s u c h a s i t u a t i o n . L e t u s a l s o assume t h a t we know e a c h t e s t ' s c o m m u n a l i t y a n d s o h a v e ( 7 2 ) a l l t h e b a s i c d a t a f r o m w h i c h t o c o n s t r u c t t h e r e d u c e d c o r -r e l a t i o n m a t r i x . M o r e o v e r , we w i l l assume t h a t t h e i n t e r -c o r r e l a t i o n s i n t h i s f i c t i t i o u s m a t r i x c a n be c o m p l e t e l y de-s c r i b e d i n t e r m s o f two common f a c t o r s . I n C h a p t e r I I I , h o w e v e r , we p r o v e d t h a t s u c h a r e d u c e d c o r r e l a t i o n m a t r i x c o u l d be e x p r e s s e d a s t h e p r o d u c t o f t h e common f a c t o r m a t r i x a n d i t s t r a n s p o s e . L e t u s r e - r e c o r d t h i s f a c t w h i c h we p r e v i o u s l y n o t e d i n T a b l e V, C h a p t e r I I I , T a b l e I R e d u c e d C o r r e l a t i o n M a t r i x a s a M a t r i x P r o d u c t I I I 1 a n 2 ' a z i 3 a 3 1 a 3 2 F 1 2 3 I a i i a z i a 5 , I I a i z a 2 2 a 3 Z 1 2 3 1 K 2 hz r 2 3 3 r 3 1 *3Z R The p r o b l e m t o be s o l v e d i s t h a t o f d e r i v i n g t h e v a l u e s f o r t h e common f a c t o r l o a d i n g s i n F f r o m t h e known v a l u e s i n m a t r i x R. T h a t i s , t h e d a t a f r o m w h i c h we m u s t d e d u c e t h e s e v a l u e s a r e t h o s e r e c o r d e d i n g e n e r a l f o r m i n T a b l e I I . T a b l e I I R e d u c e d C o r r e l a t i o n M a t r i x f o r T h r e e T e s t s 1 2 3 1 *\ riz r ! 3 2 TZl K 3 r 3 1 h i ( 7 5 ) Our p r o b l e m c a n now be s t a t e d i n a more e x p l i c i t f o r m . I t c o n s i s t s i n a b s t r a c t i n g e a c h s u c c e s s i v e f a c t o r l o a d i n g i n s u c h a manner t h a t e a c h w i l l a c c o u n t f o r t h e l a r g e s t p o s s i b l e a m ount o f t h e c o r r e l a t i o n c o e f f i c i e n t s we h a v e o b t a i n e d . L e t u s c o n s i d e r t h e p r o c e d u r e i n a s e r i e s o f s y s t e m a t i c s t e p s . The f i r s t s u c h s t e p c o n s i s t s i n d e r i v i n g t h e v a l u e s f o r t h e maximum l o a d i n g s u p o n t h e f i r s t r e f e r e n c e f a c t o r . F r o m T a b l e I , page 7 2 , we s e e t h a t we c a n w r i t e e a c h o f t h e e l e m e n t s o f t h e f i r s t c o l u m n o f t h e c o r r e l a t i o n m a t r i x o f T a b l e I I a s t h e sum o f t h e p r o d u c t s o f c e r t a i n s a t u r a t i o n c o e f f i c i e n t s . T h e s e e l e m e n t s c a n be e x p r e s s e d a s f o l l o w s , K = a i \ + a i z > T Z 1 = a S L l a H + a Z 2 a i Z a n d r 3 1 = r a 3 1 a l x -+• a 3 2 a 1 J L , w h e r e t h e c o r r e l a t i o n s c a n be f u l l y d e s c r i b e d b y two f a c t o r s . I t m u s t be remembered t h a t t h e v a l u e s on t h e l e f t o f t h e a b o v e e q u a t i o n s a r e t h e known q u a n t i t i e s a n d t h e v a l u e s o n t h e r i g h t a r e t h e unknowns w h i c h we w i s h t o e v a l u a t e . " I f we sum t h e t h r e e e q u a t i o n s t h e r e s u l t i n g e q u a t i o n i s h j - f r 2 1 + r 3 1 = « t l ( a l t +a 2 1 +a 3 1) + a15, ( a 1 4 + azz 4-& 3 Z ) . L e t u s d e n o t e t h e sum o f t h e c o e f f i c i e n t s i n c o l u m n 1 o f t h e r e d u c e d c o r r e l a t i o n m a t r i x b y r± . The a b o v e e q u a t i o n becomes r i = * n (;aii-»- ; & z i + a 3 i )"+ a u ( a u + a z z + a 3 t ) • S i m i l a r l y , f o r t h e s e c o n d c o l u m n we o b t a i n (74) r i Z = a i l a 2 1 ~+  &iz a Z 2 > ^ 2 = a 2 1 + a ! 2 and r 3 2 = a 3 i azi'+" a 3 2 a 22 • Summing the above equations and denoting the sum of the c o e f -f i c i e n t s i n column 2 by TZ the r e s u l t i s -2 "21 ( a l t 4- a 21 + a 3 1 ) "+ a Z Z ( a i z + a 2 z + & 3 Z ) . For the t h i r d column the v a l u e s are r i 3 = a n a 3 i a i z a 3 z t r23 = a 2 1 a 3 l " + a Z Z a 3 2 and h 3 = a ^ + a 3 J l . Upon denoting the sum of column 3 by r 3 we o b t a i n r 3 = a 3 i ( a n + a z i •+ a 3 i ) + a 3 5> ^ l a - 1 "  azz - | - a 3 z ) •• Thus the sums f o r a l l three columns are r t = (a 1 1+ a 2 1 4 - a 3 1 ) + a±z (a l s, 4- a g z + a 3 2 ), . V = « M ( a i l + a Z ! + a 3 1 ) + a 2 Z ( a i 2 . + a22 + S 3 Z > and r 3 = a 3 1 ( a l a 4- a u -+ a 3 1 ) 4- a 3 2 ( a u 4- a t t -+ a 3 Z ) . I f we now sum the above t h r e e e x p r e s s i o n s we o b t a i n r 1 + r 5 L + r 3 =(aw-+ a z i 4 - a 3 1 ) (a 1 4+ a j t l+a J 1 )+(alt+ a 2 z 4 - a ^ ) ( a l t + a^-t-a^). But the sum on the l e f t of the above eq u a t i o n i s the sum of a l l the c o e f f i c i e n t s i n the reduced c o r r e l a t i o n m a t r i x . L e t us denote t h i s t o t a l by r t . We thus o b t a i n the r e s u l t , (75) r t = ( a ± 1 + a t l + a 3 1 )* + ( a l t + a 2 2 + a 3 2 ) z . Our o b j e c t i v e i s t o h a v e t h e f i r s t f a c t o r a c c o u n t f o r t h e maximum p r o p o r t i o n o f t h e t o t a l i n t e r c o r r e l a t i o n s . I n t h e a b o v e e q u a t i o n t h i s means t h a t we w i s h ( a u + a z i + & 3 i )> w h i c h i s t h e sum o f t h e f i r s t f a c t o r l o a d i n g s , t o be a s l a r g e a s p o s s i b l e . I f we r e w r i t e t h e a b o v e e q u a t i o n i n a s l i g h t l y -d i f f e r e n t f o r m i t becomes ( a w + a M + a 3 1 ) z = r t - ( a l t + a M , + a „ f . B u t r t , b e i n g t h e sum o f a l l i n t e r c o r r e l a t i o n s i n t h e m a t r i x , i s a c o n s t a n t f o r a n y p a r t i c u l a r s t u d y . ,Thus t h e q u a n t i t y o n t h e l e f t o f t h e e q u a t i o n w i l l be a maximum when ( a 1 2 -t- a 2 2 + a a ) =0 s i n c e t h e s q u a r e o f t h i s q u a n t i t y i s s u b t r a c t e d f r o m a f i x e d v a l u e . T h i s w i l l r e a d i l y be r e c o g n i z e d a s b e i n g t r u e when one n o t e s t h a t t h e a b o v e q u a n t i t y i s s q u a r e d i n t h e g i v e n e q u a t i o n . T h u s , w h e t h e r i t be e i t h e r a n e g a t i v e o r p o s i t i v e v a l u e , i t w i l l become p o s i t i v e when s q u a r e d a n d w i l l r e d u c e t h e q u a n t i t y on t h e r i g h t . T h i s t e l l s u s t h a t , i n o r d e r t o o b t a i n t h e maximum l o a d i n g s f o r t h e f i r s t f a c t o r , t h e s e c o n d f a c t o r m u s t be s u c h t h a t t h e sum o f i t s l o a d i n g s i s z e r o . E m p l o y i n g t h i s f a c t , o u r f i r s t f a c t o r l o a d i n g s become a m a x i -mum when ( a w + a 2 1 + a 3 1 )*= r t , t h a t i s , when ( l ) a t l + azi_ +- a 3 1 = J% . ( 7 6 ) T h i s c a n be e x p r e s s e d a s f o l l o w s . The f i r s t f a c t o r l o a d i n g s a c c o u n t f o r t h e maximum amount o f t h e i n t e r e o r r e l a -t i o n s when t h e sum o f t h e f i r s t f a c t o r l o a d i n g s f o r a l l t h r e e t e s t s i s e q u a l t o t h e s q u a r e r o o t o f t h e t o t a l o f a l l c o e f f i -c i e n t s i n t h e r e d u c e d c o r r e l a t i o n m a t r i x . We now r e c a l l t h a t o n page 74 we d e r i v e d t h e r e s u l t t h a t r ± , t h e sum o f a l l c o e f f i c i e n t s i n t h e f i r s t c o l u m n o f t h e r e d u c e d c o r r e l a t i o n m a t r i x , i s e q u a l t o t h e f o l l o w i n g , r i =' an ( a n + a z i + a 3 l ) + a i i ( a i i L + a 2 2 . + a « ) -Upon n o t i n g t h a t f o r t h e f i r s t f a c t o r l o a d i n g s t o be a m a x i -mum we h a d t o h a v e a i * + a 2 * + a 3 * = 0 > we g e t t h e f o l l o w i n g r e s u l t r i = a n ( a n ' + a z i + a 3 i ) • S u b s t i t u t i n g t h e v a l u e f o r t h e t e r m i n b r a c k e t s f r o m e q u a t i o n (1) t h i s e x p r e s s i o n becomes r i =  ai.rJ^t • Hence (2) - a i i = v i r T h i s t e l l s u s t h a t t h e maximum s a t u r a t i o n c o e f f i c i e n t f o r o u r f i r s t t e s t u p o n t h e f i r s t r e f e r e n c e f a c t o r i s o b t a i n e d by d i v i d i n g t h e sum o f t h e f i r s t c o l u m n o f o u r r e d u c e d c o r r e l a -(77) t i o n m a t r i x by the square r o o t of the t o t a l f o r a l l c o e f f i -c i e n t s o f t h a t m a t r i x . In order t h a t we may be c e r t a i n t h a t t h i s method of a s s e s s i n g the v a l u e s t o be a s s i g n e d i s completely understood, l e t us now c o n s i d e r the l o a d i n g of t e s t two upon the f i r s t common f a c t o r . On page 74 of t h i s chapter we a l r e a d y have obtained the f o l l o w i n g e q u a t i o n r 2 = a z i ( a l t + a 2 1 + a 3 1 ) + a ^ ( a 1 2 + & 3 Z ) • For the same reason, as b e f o r e ( a 1 2 - i - a 2 2-+ a 3 i l) must be z e r o and t h e r e f o r e VZ = a 2 1 ( a i l +  azi " ^ ^ 3 1 ) • Upon s u b s t i t u t i n g the v a l u e f o r the terms i n the b r a c k e t from e q u a t i o n ( l ) the above e q u a t i o n becomes Tz — az-L^t • T h e r e f o r e T h i s t e l l s us t h a t the maximum f a c t o r l o a d i n g f o r t e s t two on f a c t o r I i s obtained by d i v i d i n g the sum of the c o e f f i c i e n t s i n column 2 of the reduced c o r r e l a t i o n m a t r i x by the square r o o t of the sum of a l l c o e f f i c i e n t s i n the m a t r i x . In a sim-i l a r manner we can prove t h a t The above proof f o r the v a l u e s - o f the f i r s t ..factor l o a d i n g s (78) when onl y two f a c t o r s are present i n the complete f a c t o r ma-t r i x can be g e n e r a l i z e d t o i n c l u d e the case w i t h r f a c t o r s . I t merely e n t a i l s t h a t i n order t o have the maximum l o a d i n g s on the f i r s t f a c t o r , the sums of the l o a d i n g s on each of the other f a c t o r s must equal z e r o . The next step i n the development c o n s i s t s i n o b t a i n i n g the maximum f a c t o r l o a d i n g s f o r the second r e f e r e n c e f a c t o r . I f we c o n s i d e r the f i r s t f a c t o r l o a d i n g s as a s i n g l e columned m a t r i x and m u l t i p l y i t by i t s tra n s p o s e , we o b t a i n the r e s u l t r e c o r d e d i n Table I I I . Table I I I Product of F a c t o r I by I t s Transpose I • 1 2 3 1 ;. 2 3 1 a n I a21 1 aL a n a 2 i a n a 3 i 2 a 2 t 2 ^ 1 a n a L a21 a31 3 3 a 3 i a n a a„ 31 21 a 2 31 M Let us c o n s i d e r the v a l u e s we have a l r e a d y o b t a i n e d f o r the o r i g i n a l c o r r e l a t i o n c o e f f i c i e n t s i n terms of f a c t o r l o a d i n g s . For the elements of column 1 o f the reduced c o r -r e l a t i o n m a t r i x the v a l u e s a r e n i = a i i + a i 2 * r Z l = a 2 i a i l + a 2 2 a i Z and r, „= a„„ a -+- a a . . 3 1 3 1 11 1 31 1 2 • (79) These v a l u e s can be rearranged as f o l l o w s a* — h* — a 2  &zz a i t = r2i~~a2i a n a I l d a 3 i L a i t = r 3 1 - a 3 i a « • A study of the reduced c o r r e l a t i o n m a t r i x , Table I I , page 72 and m a t r i x M i n Table I I I , r e v e a l s t h a t the v a l u e s on the l e f t of the three equations above can be obtained by sub-t r a c t i n g each element of column 1 i n m a t r i x M from the c o r -responding element i n column 1 of the reduced c o r r e l a t i o n m a t r i x R. I t w i l l be noted t h a t the v a l u e s remaining a f t e r t h i s s u b t r a c t i o n are a l l products obtained by combining s e c -ond f a c t o r l o a d i n g s . T h i s f a c t can be r e a d i l y ; proven for- the remaining -elements of m a t r i x R. Thus i f we m u l t i p l y the mat-r i x of f i r s t f a c t o r l o a d i n g s by i t s transpose and s u b t r a c t the o b t a i n e d v a l u e s from the o r i g i n a l reduced c o r r e l a t i o n ma-t r i x we o b t a i n the f o l l o w i n g m a t r i x . T h i s m a t r i x , i n our s p e c i a l case, i s expressed i n terms of the second f a c t o r l o a d i n g s . However, i f the number of f a c t o r s were g r e a t e r than two, these e x p r e s s i o n s would a l l be combinations of l o a d i n g s f o r a l l remaining f a c t o r s . The m a t r i x of Table IV i s c a l l e d the f i r s t r e s i d u a l ma-t r i x s i n c e i t c o n t a i n s the remaining p o r t i o n s of the c o r r e l a -t i o n c o e f f i c i e n t s a f t e r the f i r s t f a c t o r l o a d i n g s have been a b s t r a c t e d . (80) Table IV F i r s t R e s i d u a l M a t r i x 1 2 3 1 a i 2 a t 2 a i 2 a 3 2 2 a 2 2 a i Z a 2 2 a 3 2 3 a 3 2 a ! 2 &3Z a 2 2 a 3 2 At t h i s p o i n t i n the f a c t o r i z a t i o n process we encounter a d i f f i c u l t y s i n c e , i f we apply the preceding technique d i -r e c t l y t o t h i s m a t r i x i n order to e v a l u a t e the second f a c t o r l o a d i n g s , the f o l l o w i n g r e s u l t o c c u r s . To a p p l y the r e s u l t of e q u a t i o n (2) t o o b t a i n the l o a d i n g of the f i r s t t e s t on f a c t o r 11^ a l z , we must f i r s t sum a l l the c o e f f i c i e n t s of column 1 of the r e s i d u a l m a t r i x and then d i v i d e t h i s q u a n t i t y by the square r o o t of the sum of a l l elements of the m a t r i x . Le t the sum of column 1 of the r e s i d u a l m a t r i x be de-noted by r ^ . The r e s u l t of summing t h i s column i s l i s t e d be-low. r i — a i 2 + a z t a i £ ~ * ~ a 3 2 a i 2 k l Z ( a ! 2 + a Z 2 a 3 2 . ) • But the q u a n t i t y i n b r a c k e t s was s e t e q u a l t o z e r o to maxi-mize the f i r s t f a c t o r l o a d i n g s . T h e r e f o r e r^= a,,. 0 = 0 . Let the t o t a l of a l l the elements of t h i s m a t r i x be denoted (81) by r^T. Upon summing t h e s e v a l u e s we o b t a i n T * = a i i + atZ a i t + a 3 Z a i Z 4 " & 1 Z aZZ+ a\z+ a 3 t azZ+ a i i a 3 1 + a 2 Z a 3 S L + a 3 2 « C o l l e c t i n g t h e s e t e r m s i n t o g r o u p s o f t h r e e e l e m e n t s e a c h a n d f a c t o r i n g o u t t h e common f a c t o r i n e a c h s u c h g r o u p we o b t a i n ¥ * l t tel*"1- » M + &3Z ) + a 2 Z ^ 1 2 + a 2 z + • a 3 Z ) + a 3 z ( a 1 Z + a 2 t + « s z ) * B u t f o r t h e r e a s o n a l r e a d y s t a t e d e a c h o f t h e b r a c k e t e d t e r m s i n t h e a b o v e e x p r e s s i o n i s e q u a l t o z e r o . T h e r e f o r e ^ = » l l . O + a t l.O + a s t-0 = 0. A s a n e x a m p l e , t h e f a c t o r l o a d i n g o f t e s t one u p o n r e f e r e n c e f a c t o r I I , t h a t i s , a l z , w o u l d be o b t a i n e d a s f o l l o w s a - ^ - 0. B u t t h i s e x p r e s s i o n i s m a t h e m a t i c a l l y m e a n i n g l e s s a n d s o t h e t e c h n i q u e b y w h i c h we d e r i v e d t h e f i r s t f a c t o r l o a d i n g s c a n -n o t be a p p l i e d t o t h e r e s i d u a l m a t r i x i n i t s p r e s e n t f o r m . T h i s i s s o s i n c e e v e r y s u c h sum w i l l be i d e n t i c a l l y z e r o f o r o u r h y p o t h e t i c a l c a s e i n w h i c h n o e r r o r f a c t o r s a r e i n v o l v e d . F a c e d w i t h t h i s p r o b l e m , T h u r s t o n e h a s d e v e l o p e d a p r o -c e d u r e w h i c h i n v o l v e s a t e m p o r a r y r e v e r s a l o f s i g n f o r s p e -c i f i c a l l y c h o s e n r o w s a n d c o l u m n s o f t h e r e s i d u a l m a t r i x . T h i s r e m o v e s t h e z e r o t o t a l s i n o r d e r t h a t t h e same m e t h o d may be e m p l o y e d t o f i n d t h e s e c o n d f a c t o r l o a d i n g s a s was u s e d t o (82) e v a l u a t e t h e s a t u r a t i o n s f o r t h e f i r s t f a c t o r . S i n c e t h i s p r o c e s s o f r e v e r s i n g s i g n s c a n be more r e a d i l y g r a s p e d when a p p l i e d t o a n a c t u a l r e s i d u a l m a t r i x , we w i l l l e a v e d i s c u s -s i o n o f t h i s p o i n t u n t i l we a r e d e a l i n g w i t h t h e a c t u a l com-p u t a t i o n s i n v o l v e d i n a s p e c i f i c e x a m p l e . I n o r d e r t o c l a r i f y f u r t h e r t h e c o m p u t a t i o n a l m e t h o d w h i c h t h i s f o r m o f s o l u t i o n e m p l o y s , l e t u s now c o n s i d e r s e v -e r a l e x a m p l e s w h i c h i n c r e a s e i n c o m p l e x i t y s o t h a t t h e v a r i -o u s p r o c e d u r e s may be e x p l i c i t l y d e m o n s t r a t e d . As was s t a t e d a t t h e b e g i n n i n g o f t h i s c h a p t e r , we w i l l s i m p l y d e r i v e t h e o r i g i n a l f a c t o r l o a d i n g s a n d w i l l l e a v e T h u r s t o n e ' s r e a r -r a n g e m e n t p r o b l e m u n t i l t h e f o l l o w i n g c h a p t e r . To s e r v e a s a more r e a d i l y u n d e r s t a n d a b l e i l l u s t r a t i o n o f t h e s e p r o c e d u r e s , l e t u s i n v e s t i g a t e t h e s t e p s i n v o l v e d i n f a c t o r i n g a r e d u c e d c o r r e l a t i o n m a t r i x w h i c h c o n s i s t s o f i n -f a l l i b l e d a t a o f r a n k o n e ; t h a t i s , a c o r r e l a t i o n m a t r i x whose r o w s a n d c o l u m n s a r e a l l p r o p o r t i o n a l t o e a c h o t h e r . F o r t h i s p u r p o s e l e t u s e m p l o y , a s o u r i n i t i a l e x a m p l e , t h e m a t r i x w h i c h we f a c t o r i z e d by S p e a r m a n ' s m e t h o d i n C h a p t e r I , page 5 . T h i s m a t r i x i s r e p e a t e d i n T a b l e V w i t h t h e f i c t i - • t i o u s v a l u e s f o r t h e t e s t c o m m u n a l i t i e s i n s e r t e d i n t h e d i a -g o n a l s p a c e s . I t i s n e c e s s a r y t o c a u t i o n t h e r e a d e r a g a i n t h a t t h i s e x a m p l e i s composed o f a s e r i e s o f i n t e r c o r r e l a -t i o n s a n d c o m m u n a l i t i e s o b t a i n e d f o r a g r o u p o f i m a g i n a r y t e s t s w h i c h a r e c o m p l e t e l y r e l i a b l e . ( 8 3 ) T a b l e V R e d u c e d C o r r e l a t i o n M a t r i x 1 2 3 4 5 E 1 .64 .72 .56 .40 .48 2.80 2 .72 .81 .63 .45 .54 3.15 3 .56 .63 .49 .35 .4.2 2.45 4 .4.0 .45 .35 .25 .30 1.75 5 .48 .54 .42 .30 .36 2.10 i j 2.80 3.15 2.45 1.75 2.10 = 12.25 .81 .91 .71 .51 .61 = 3.5 i m = .29 The d a t a w i t h w h i c h we b e g i n o u r a n a l y s i s c o n s i s t s o f t h e c o r r e l a t i o n c o e f f i c i e n t s a n d t h e c o m m u n a l i t i e s . We h a v e now t o a p p l y t h e f o r m u l a e , w h i c h we d e v e l o p e d a t t h e b e g i n -n i n g o f t h i s c h a p t e r , t o d e r i v e t h e f a c t o r p a t t e r n . L e t u s , t h e r e f o r e , f o r g e t t h a t t h i s i s a c a r e f u l l y c o n t r i v e d a r t i -f i c i a l m a t r i x a n d t h a t t h e f o r m o f t h e s o l u t i o n i s a known d e t e r m i n a t e o n e , a n d l e t u s p r o c e e d t o e m p l o y t h e T h u r s t o n e m e t h o d s t o o b t a i n t h e f a c t o r l o a d i n g s . The f i r s t s t e p i n t h e p r o c e d u r e c o n s i s t s i n summing t h e c o l u m n v a l u e s . The r e s u l t s o f t h e s e s u m m a t i o n s a r e l i s t e d i n t h e r o w m a r k e d r , a t t h e b o t t o n o f t h e m a t r i x . I n a n y w o r k i n w h i c h a l a r g e number o f c o m p u t a t i o n s a r e e m p l o y e d i t i s a l w a y s i m p o r t a n t t o c o n t r i v e a s many c h e c k s u p o n t h e o b t a i n e d r e s u l t s a s p o s s i b l e . One s u c h c h e c k a t t h i s p o i n t i s t o sum t h e r o w s o v e r t o t h e r i g h t o f t h e m a t r i x . B e c a u s e o f t h e (84) p a t t e r n i n w h i c h t h e m a t r i x i s c o n s t r u c t e d , t h e s e v a l u e s m u s t a g r e e w i t h t h o s e i n t h e b o t t o m r o w u n l e s s e r r o r s h a v e o c -c u r r e d . The s e c o n d s t e p t o be t a k e n i s t h a t o f summing t h e v a l u e s i n t h e r o w o f c o l u m n sums a t t h e b o t t o m o f t h e m a t r i x . T h i s f u r n i s h e s u s w i t h r t , t h e sum o f a l l c o e f f i c i e n t s i n t h e m a t r i x . The c o l u m n o f sums may a l s o be a d d e d a s a c h e c k u p o n t h i s v a l u e a n d , i f a c a l c u l a t o r i s a v a i l a b l e o r i f t h e number o f c o e f f i c i e n t s i s f e w , a s e c o n d c h e c k i s o b t a i n e d by summing a l l t h e v a l u e s i n t h e m a t r i x . I f n o e r r o r s h a v e o c c u r r e d t h e s e v a l u e s s h o u l d a l l a g r e e a n d t h e r e s u l t i s l i s t e d a t t h e l o w e r r i g h t o f o u r m a t r i x . S i n c e t h e f i r s t f a c t o r l o a d i n g s a r e o b t a i n e d b y d i v i d i n g t h e c o l u m n sums by t h e s q u a r e r o o t o f r t , t h i s v a l u e i s n e x t f o u n d a n d l i s t e d b e l o w r t . I n t h i s r e g a r d T h u r s t o n e a d v i s e s f i n d i n g t h e v a l u e o f a c o n s t a n t m e q u a l t o -r-Lr w h i c h i s more s a t i s f a c t o r y f o r f u r t h e r c o m p u t a t i o n s t h a n i s t h e Jx^ i t s e l f . B o t h o f t h e s e v a l u e s a r e l i s t e d i n t h e s p a c e s i m m e d i a t e l y b e -l o w r t . U p on r e c h e c k i n g t h e f o r m u l a we f i n d t h a t t h e f i r s t f a c -t o r l o a d i n g f o r e a c h t e s t i s o b t a i n e d by d i v i d i n g e a c h c o l u m n sum by A/T~ o r , more e a s i l y , by m u l t i p l y i n g e a c h o f t h e s e sums by t h e e q u i v a l e n t c o n s t a n t m = -7|=. Upon p e r f o r m i n g t h i s o p -e r a t i o n o n e a c h o f t h e c o l u m n sums i n t h e m a t r i x we o b t a i n t h e v a l u e s i n t h e r o w w i t h a j r a t i t s l e f t . T h a t i s , we o b -t a i n t h e f a c t o r l o a d i n g s f o r e a c h o f t h e f i v e t e s t s u p o n f a c -t o r I . A s a n e x a m p l e l e t u s compute t h e f a c t o r l o a d i n g o f t e s t one o n f a c t o r I o r a l t . By f o r m u l a ( 2 ) we h a v e t h a t ( 8 5 ) = ( . 2 9 ) - ( 2 . 8 0 ) = 0.812 = .81 I n a s i m i l a r manner t h e r e m a i n i n g c o e f f i c i e n t ' s a r e e v a l u a t e d . A t t h i s p o i n t a f u r t h e r c h e c k u p o n t h e c a l c u l a t i o n s c a n be e m p l o y e d . L e t u s r e c a l l t h a t we h a v e a l r e a d y d e v e l o p e d a f o r m u l a f o r t h e sum o f t h e f i r s t f a c t o r l o a d i n g s . T h i s i s e q u a t i o n ( l ) , p a ge 7 5 , w h i c h i s r e p e a t e d b e l o w f o r r e a d y r e f e r e n c e . T h i s t e l l s u s t h a t a f u r t h e r c h e c k u p o n o u r c o m p u t a t i o n c a n be o b t a i n e d b y summing t h e f i r s t f a c t o r l o a d i n g s a n d t h a t , e x c e p t f o r m i n o r d i s c r e p a n c i e s due t o t h e m a t h e m a t i c a l a p -p r o x i m a t i o n s made i n r o u n d i n g o f f v a l u e s , t h i s sum s h o u l d e q u a l t h e v a l u e we h a v e a l r e a d y d e r i v e d f o r t h e s q u a r e r o o t o f r t . U pon a p p l y i n g t h i s c h e c k t o o u r f i r s t f a c t o r l o a d i n g s we o b t a i n a n + a 2 1 + a 3 i -+ a 4 i + a 5 1 = . 8 1 + . 9 1 + . 7 1 + .51+'.61 = 3.55 . The m i n o r d i s c r e p a n c y i n t h i s v a l u e i s d i r e c t l y a t t r i b u t a b l e t o t h e f a c t t h a t we h a v e c a r r i e d o u r r o u n d i n g o f f p r o c e s s t o t h e s e c o n d p l a c e o f d e c i m a l s . The c h e c k t h e r e f o r e i n d i c a t e s ( 8 6 ) t h a t i t i s q u i t e s a f e t o assume t h a t n o m a t h e m a t i c a l e r r o r h a s b e e n made i n t h i s s t a g e o f t h e w o r k . S i n c e t h i s i s m e r e l y a n i l l u s t r a t i v e e x a m p l e , l e t u s r o u n d o f f t h e f a c t o r l o a d i n g s t o t h e f i r s t p l a c e o f d e c i m a l s i n o r d e r t o s i m p l i f y t h e r e m a i n i n g a r i t h m e t i c . We h a v e now d e r i v e d t h e maximum f a c t o r l o a d i n g s f o r t h e f i r s t f a c t o r a n d t h e n e x t s t e p c o n s i s t s i n c o m p u t i n g t h e f i r s t r e s i d u a l m a t r i x . I n o r d e r t o do t h i s we m u s t f i r s t c a l c u l a t e t h a t p o r t i o n o f t h e c o r r e l a t i o n c o e f f i c i e n t s w h i c h i s due t o t h e s e f i r s t f a c t o r l o a d i n g s . T h i s , a s we h a v e s e e n b e f o r e , i s m o s t r e a d i l y c a l c u l a t e d b y m u l t i p l y i n g t h e m a t r i x f o r t h e f i r s t f a c t o r l o a d i n g s by i t s t r a n s p o s e . T h i s prod-r u c t i s r e c o r d e d i n T a b l e V I . I 1 • 8 . 2 .9 3 .7 4 .5 5 .6 T a b l e V I P r o d u c t o f t h e F i r s t F a c t o r L o a d i n g s 1 2 3 4 5 , I .8 .9 .7 .5 .6 1 2 3 4 5 1 .64 .72 .56 .40 .48 2 .72 .81 .63 .45 .54 3 .56 .63 .49 .35 .42 4 .40 .45 .35 .25 .30 5 .48 .54 .42 .30 .36 To o b t a i n t h e r e s i d u a l m a t r i x i t i s o n l y n e c e s s a r y t o s u b -t r a c t e a c h e l e m e n t o f t h e a b o v e m a t r i x f r o m t h e c o r r e s p o n d i n g e l e m e n t o f t h e o r i g i n a l r e d u c e d c o r r e l a t i o n m a t r i x . Upon p e r f o r m i n g t h i s o p e r a t i o n we f i n d t h a t e a c h e l e m e n t o f t h e ( 8 7 ) r e s i d u a l m a t r i x i s z e r o . I n o t h e r w o r d s , t h e f i r s t f a c t o r l o a d i n g s a c c o u n t f o r a l l t h e c o e f f i c i e n t s o f c o r r e l a t i o n w h i c h s e r v e d a s o u r i n i t i a l d a t a . I n t h i s c a s e s u c h a r e -s u l t i s t o be e x p e c t e d s i n c e t h e m a t r i x we h a v e f a c t o r e d was s o c o n t r i v e d a s t o h a v e e x a c t l y r a n k o n e . T h a t i s , i t was a r t i f i c i a l l y c o n s t r u c t e d f o r i l l u s t r a t i v e p u r p o s e s i n s u c h a manner a s t o be f a c t o r a b l e i n t o a s i n g l e common f a c t o r . I n t h e f i r s t c h a p t e r we f a c t o r e d t h i s same m a t r i x b y means o f t h e Spear m a n g e n e r a l f a c t o r t e c h n i q u e . I t w i l l be o f i n t e r e s t t o compare t h e f a c t o r l o a d i n g s o b t a i n e d b y t h a t m e t h o d w i t h t h e ones w h i c h we h a v e now d e r i v e d by t h e c e n -t r o i d m e t h o d . To f a c i l i t a t e s u c h a c o m p a r i s o n t h e s e e a r l i e r v a l u e s a r e r e c o r d e d i n T a b l e V I I t o g e t h e r w i t h t h e c o r r e s p o n d i n g v a l u e s t h a t we h a v e now c o m p u t e d . T a b l e V I I F a c t o r L o a d i n g s D e r i v e d b y Spearman a n d C e n t r o i d T e c h n i q u e s T e s t S p e a r m a n L o a d i n g s C e n t r o i d L o a d i n g s 1 .8 .8 2 .9 .9 3 .7 .7 4 .5 .5 5 .6 .6 I t i s r e a d i l y s e e n t h a t t h e s e l a t t e r v a l u e s , o n c e r o u n d e d o f f t o t h e f i r s t d e c i m a l p l a c e , a g r e e e x a c t l y w i t h t h e v a l u e s (88) o b t a i n e d b y Spearman's t e c h n i q u e . I n g e n e r a l , t h e r e f o r e , i t may be s a i d t h a t t h e T h u r s t o n e C e n t r o i d M e t h o d w i l l d e r i v e c o m p a r a b l e v a l u e s t o t h o s e c o m p u t e d b y t h e g e n e r a l f a c t o r t e c h n i q u e i f t h e t e s t s e m p l o y e d a r e o f s u c h a n a t u r e a s t o be e x p l a i n a b l e by means o f a s i n g l e g e n e r a l f a c t o r a n d t h a t , f u r t h e r m o r e , t h i s n e w e r p r o c e d u r e w i l l f a c t o r i a e much more c o m p l i c a t e d s y s t e m s o f i n t e r c o r r e l a t i o n s . H a v i n g s t u d i e d t h e p r o c e s s e s t o be e m p l o y e d i n t e r m s o f t h i s e x t r e m e l y s i m p l e p r o b l e m , l e t u s now p r o c e e d y e t one s t e p f u r t h e r a n d s t u d y a n o t h e r a r t i f i c i a l l y c o n s t r u c t e d ma-t r i x . H o w e v e r , l e t u s now d e v i s e a m a t r i x o f s u c h a n a t u r e a s t o be c o m p l e t e l y d e f i n a b l e i n t e r m s o f two common f a c t o r s . A s t u d y o f s u c h a m a t r i x w i l l f u r n i s h u s w i t h a n o p p o r t u n i t y f o r d i s c u s s i n g t h e p r o b l e m t h a t a r i s e s b e c a u s e t h e sum o f e a c h c o l u m n o f t h e r e s i d u a l m a t r i x s h o u l d i n t h e o r y e q u a l z e r o . I t w i l l a l s o p r o v i d e u s w i t h a n o p p o r t u n i t y f o r s t u d y i n g f u r t h e r c h e c k s t h a t may be a p p l i e d t o t h e m a t h e m a t i -c a l s t e p s i n o r d e r t h a t e r r o r s may be a v o i d e d . A t t h e same t i m e we w i l l o b t a i n f u r t h e r p r a c t i c e i n h a n d l i n g t h e t e c h -n i q u e s w h i c h h a v e b e e n d e v e l o p e d . I n T a b l e V I I I i s l i s t e d a f i c t i t i o u s m a t r i x o f r a n k t w o ; t h a t i s , a m a t r i x w h i c h c a n be a n a l y s e d i n t o two common f a c t o r s . ( 8 9 ) T a b l e V I I I C o r r e l a t i o n M a t r i x o f R a n k Two T e s t 1 2 3 4 H 1 .80 .60 .60 .70 2.70 2 .60 .90 .60 .75 2.85 5 .60 .60 .50 .60 2.30 4 .70 .75 .60 .73 2.78 r j 2.70 2.85 2.30 2.78 r.t = 1 0 . 6 3 Jv; = 3.260 m = .3067 a J i .828 .874 .705 .853 Z a j t = 3.260 The f i r s t f a c t o r l o a d i n g s a r e d e r i v e d i n e x a c t l y t h e same manner a s i n t h e p r e c e d i n g e x a m p l e . H o w e v e r , l e t u s b r i e f l y r e c a p i t u l a t e t h e c o m p u t a t i o n s a n d t h e c h e c k s on e r r o r s t h a t a r e e m p l o y e d . We f i r s t sum e a c h c o l u m n o f t h e m a t r i x a n d o b t a i n t h e r o w o f sums m a r k e d Tj . A c h e c k u p o n t h e s e v a l u e s i s o b t a i n e d by summing e a c h row a n d c o m p a r i n g c o r r e s p o n d i n g sums. The n e x t s t e p c o n s i s t s i n summing t h e r o w r,- a n d t h i s f u r n i s h e s r t , t h e sum o f a l l c o e f f i c i e n t s i n t h e m a t r i x . T h i s v a l u e may a g a i n be c h e c k e d b y summing t h e c o l u m n h e a d e d Z a n d a l s o b y summing a l l c o e f f i c i e n t s i n t h e m a t r i x . We n e x t compute fr^ a n d m w h i c h e q u a l s y p ^ * i s n o w a s i m p l e m a t t e r t o compute e a c h t e s t ' s f i r s t f a c t o r l o a d i n g . T h i s i s done by m u l t i p l y i n g e a c h v a l u e i n t h e r o w r,- by m. E a c h s u c h p r o d u c t i s t h e f i r s t f a c t o r l o a d i n g o f t h e t e s t whose number i s a t ( 9 0 ) t h e h e a d o f t h a t c o l u m n . T h e s e v a l u e s a r e l i s t e d i n t h e r o w a j x . We c a n c h e c k o u r v a l u e s s i n c e t h e sum o f a l l t h e f i r s t f a c t o r l o a d i n g s s h o u l d e q u a l Jr^ . I n p e r f o r m i n g t h i s c h e c k i t i s n e c e s s a r y t h a t one remember t h a t a m i n o r d i f f e r e n c e may be p r e s e n t i n t h i s v a l u e s i n c e we h a v e r o u n d e d o f f e v e r y number t o t h e t h i r d p l a c e o f d e c i m a l s . The n e x t s t e p c o n s i s t s i n c o m p u t i n g t h a t p o r t i o n o f t h e o r i g i n a l c o e f f i c i e n t s w h i c h i s due t o t h e s e f i r s t f a c t o r l o a d i n g s . T h i s may be done m o s t s i m p l y by m u l t i p l y i n g t h e s i n g l e c o l u m n e d f a c t o r m a t r i x b y i t s t r a n s p o s e . The r e -s u l t i n g p r o d u c t i s shown i n T a b l e I X . I 1 .828 2 .874 3 .705 4 .853 Z 3.260 T a b l e I X P r o d u c t o f F i r s t F a c t o r L o a d i n g s ,828 .874 .705 .853 1 2 3 4 1 .686 .724 .584 .706 2 .724 .764 .616 .746 3 .584 .616 .497 .601 4 .706 .746 .601 .728 The r e s i d u a l m a t r i x , w h i c h i s composed o f t h o s e p o r t i o n s o f t h e c o r r e l a t i o n c o e f f i c i e n t s n o t a c c o u n t e d f o r by t h e f i r s t f a c t o r l o a d i n g s , i s f o u n d by s u b t r a c t i n g t h e e l e m e n t s i n t h e m a t r i x o f T a b l e I X f r o m t h e c o r r e s p o n d i n g e l e m e n t s i n t h e o r i g i n a l m a t r i x . A s we h a v e a l r e a d y f o u n d , t h e s e v a l u e s s h o u l d be s u c h t h a t t h e c o l u m n sums a r e z e r o e x c e p t f o r t h o s e ( 9 1 ) e r r o r s o c c a s i o n e d by t h e a p p r o x i m a t i o n s made i n r o u n d i n g o f f o u r r e s u l t s . The r e s i d u a l m a t r i x i s r e c o r d e d i n T a b l e X. T a b l e X F i r s t R e s i d u a l M a t r i x 1 2 3 4 1 .114 - . 1 2 4 .016 - . 0 0 6 2 -.124 .136 -.016 .004 3 .016 -.016 .003 - . 0 0 1 4 -.006 .004 - . 0 0 1 .002 Ch. .001 .001 .002 - . 0 0 1 L .000 .000 .002 - . 0 0 1 B e l o w t h e m a t r i x i n T a b l e X i s l i s t e d a f u r t h e r r o w m a r k e d Ch. T h i s s e r v e s a s a s t i l l f u r t h e r c h e c k u p o n t h e a c c u r a c y o f t h e w o r k . E a c h o f t h e v a l u e s i n t h i s r o w s h o u l d c h e c k v e r y c l o s e l y w i t h t h e c o r r e s p o n d i n g e l e m e n t o f t h e f i n a l r o w w h i c h i s t h e a l g e b r a i c sum o f t h e c o l u m n . The v a l u e s i n t h e c h e c k r o w c a n be o b t a i n e d a s f o l l o w s . L e t u s c o n s i d e r t h e c o m p u t a t i o n o f t h e f i r s t v a l u e . To o b t a i n t h i s v a l u e t a k e t h e sum o f t h e f i r s t c o l u m n i n T a b l e V I I I a n d f r o m t h a t v a l u e s u b t r a c t t h e p r o d u c t o f t h e f i r s t t e s t ' s l o a d i n g o n f a c t o r I w i t h t h e v a l u e f o r t h e J T ^ . Thus t h e f i r s t v a l u e becomes V - f c ^ )(7r7)= 2 . 7 0 - ( . 8 2 8 ) ( 3 . 2 6 ) = 2.70 — 2.699 = .001 (92) L e t us d i g r e s s f o r a moment i n order t o d i s c u s s the v a -l i d i t y o f t h i s check. From page 74 of t h i s chapter we f i n d t h a t the sum of the f i r s t column of the reduced c o r r e l a t i o n m a t r i x can be expressed as f o l l o w s r t + * w a 2 1 + a a i a 3 1 + a\x 4- a n a „ + a u and from e q u a t i o n ( l ) , page 75, the Jx± i s equal t o the f o l -lowing / r ^ = a 1 1 - t - a z l - 4 - a 3 1 . M u l t i p l y i n g / f ^ by the f i r s t t e s t ' s l o a d i n g on f a c t o r I , we o b t a i n a n f** = a?x + ; a l a azi + aai a 3 i • T h e r e f o r e r 4 a 4 1 ^  = < + a 1 J L a « + a i z a 3 a • But the elements on the r i g h t of the above e q u a t i o n c o n s t i t u t e the sum of the f i r s t column of the r e s i d u a l m a t r i x i n Table IV. Thus we see t h a t t h i s check f u r n i s h e s us w i t h a means f o r a s c e r t a i n i n g the accuracy of the elements i n the r e s i d u a l m a t r i x . We are now prepared t o d i s c u s s the problem t h a t was p r e -v i o u s l y mentioned, namely, t h a t a l l the columns of the r e s i d -u a l m a t r i x sum to zer o except f o r minor i n a c c u r a c i e s i n our v a l u e s . T h i s problem was encountered by Thurstone. The method he employed t o overcome t h i s d i f f i c u l t y c o n s i s t e d i n t e m p o r a r i l y changing the s i g n s of certain-columns and rows i n order to o b t a i n as many p o s i t i v e v a l u e s as p o s s i b l e i n the (95) a l t e r e d r e s i d u a l m a t r i x . The c h a n g e s i n s i g n a r e n o t e d a n d h a v i n g o b t a i n e d t h e s e c o n d f a c t o r l o a d i n g s we m u s t t h e n r e -v e r s e t h e s i g n o f t h o s e f a c t o r s t h a t w e r e a f f e c t e d b y t h e o r i g i n a l s i g n c h a n g e . R e g a r d i n g t h e m e a n i n g o f t h e s e c h a n g e s i n s i g n , P e t e r s a n d V a n V o o r h i s ( 1 4 , p . 260) w r i t e , A n y t e s t s c o r e may be e i t h e r p o s i t i v e o r n e g a -t i v e a c c o r d i n g t o t h e way i n w h i c h i t i s o r i e n -t e d . I f , f o r e x a m p l e , a p o s i t i v e s c o r e means " t a c t f u l , " t h e same s c o r e w i t h t h e n e g a t i v e s i g n w o u l d mean " t a c t l e s s . " I t i s , t h e r e f o r e , e n t i r e l y l e g i t i m a t e t o i m a g i n e a l l s c o r e s i n t h e t e s t r e v e r s e d i n s i g n , s o f a r a s t h e r e -m a i n i n g f a c t o r s a r e c o n c e r n e d . The f a c t t h a t we c o u l d n o t i n p r a c t i c e r e v e r s e t h e p a r t o f t h e s c o r e t h a t r e m a i n s a f t e r t a k i n g o u t o f i t f a c t o r 1 n e e d n o t b o t h e r u s , b e c a u s e we a r e m a k i n g t h e c h a n g e m e r e l y c o n c e p t u a l l y a n d s h a l l r e t u r n t o t h e o r i g i n a l s i g n when o u r p u r p o s e h a s b e e n m e t . T h u r s t o n e s u g g e s t s s e v e r a l m e t h o d s t o e m p l o y i n s e l e c t -i n g t h e r o w s i n w h i c h t h e s i g n i s t o be r e v e r s e d . T h e s e m e t h o d s e n a b l e one t o m i n i m i z e t h e number o f n e g a t i v e s i g n s i n t h e m a t r i x . One o f t h e e a s i e s t m e t h o d s t o a p p l y i s t h a t o f n o t i n g w h i c h r o w s o f t h e m a t r i x h a v e t h e m o s t n e g a t i v e s i g n s a n d t h e n o f r e v e r s i n g t h o s e s i g n s f i r s t . I t m u s t be remembered t h a t i n r e v e r s i n g t h e s i g n s i n a n y c o l u m n t h e s i g n s o f t h e c o r r e s p o n d i n g r o w m u s t l i k e w i s e be r e v e r s e d . I n T a b l e X we n o t e t h a t e a c h c o l u m n h a s two n e g a t i v e s i g n s . T h e r e f o r e , l e t u s b e g i n by r e v e r s i n g t h e s i g n i n c o l -umn 1 a n d i n r o w 1. Upon d o i n g t h i s , we o b t a i n t h e m a t r i x r e c o r d e d i n T a b l e X I i n w h i c h t h e t e s t , whose s i g n h a s b e e n r e v e r s e d , i s i n d i c a t e d . ( 9 4 ) T a b l e XI R e s i d u a l M a t r i x w i t h One T e s t R e v e r s e d i n S i g n - 1 2 3 4 - 1 .114 .124 - . 0 1 6 .006 2 .124 .136 - . 0 1 6 .004 S -. 0 1 6 - . 0 1 6 .003 - . 0 0 1 4 .006 ' .004 - . 0 0 1 .002 In t h e a b o v e m a t r i x i t i s now o b s e r v e d t h a t t e s t t h r e e h a s t h e l a r g e s t number o f n e g a t i v e s i g n s . U p o n r e v e r s i n g t h e s i g n s f o r t h i s t e s t we o b t a i n t h e m a t r i x o f T a b l e XII. T a b l e XII R e s i d u a l M a t r i x w i t h Two: T e s t s R e v e r s e d i n S i g n - 1 2 - 3 4 £ ; - 1 .114 .124 .016 .006 .260 2 .124 .136 .016 .004 .280 - 3 .016 .016 .003 .001 .036 4 .006 .004 .001 .002 .013 .260 .280 .036 .013 r t = .589 Jr^= .767 m = 1 . 3 0 4 .339 .365 ^047 .017 Z = .768 - . 3 3 9 .365 - . 0 4 7 .017 S i n c e t h i s m a t r i x c o n t a i n s a minimum o f n e g a t i v e s i g n s we now a p p l y t h e same t e c h n i q u e w h i c h was p r e v i o u s l y u s e d t o ( 9 5 ) d e r i v e t h e f i r s t f a c t o r l o a d i n g s . Thus we o b t a i n t h e s e c o n d f a c t o r l o a d i n g s . I n t h i s c a s e , h o w e v e r , we m u s t remember t h a t t h e s i g n s o f t h e r e s i d u a l s f o r t e s t s one a n d t h r e e h a v e b e e n r e v e r s e d , a n d s o t h e f a c t o r l o a d i n g s w i t h c o r r e c t s i g n s a r e l i s t e d i n t h e l a s t r o w o f t h e m a t r i x . We now compute t h e s e c o n d r e s i d u a l m a t r i x i n e x a c t l y t h e same manner a s t h e f i r s t r e s i d u a l m a t r i x was o b t a i n e d . T h i s t i m e we m u l t i p l y t h e s i n g l e c o l u m n e d m a t r i x composed o f t h e s e c o n d f a c t o r l o a d i n g s b y i t s t r a n s p o s e . T h i s p r o d u c t i s shown i n T a b l e X I I I . I I 1 -.339 2 .365 3 -.047 4 .017 T a b l e X I I I P r o d u c t o f S e c o n d F a c t o r L o a d i n g s 1 2 3 4 I I -.339 .365 -.047 .017 1 2 3 4 1 .115 -.124 .016 -.006 2 -.124 .133 -.017 .006 3 .016 -.017 .002 -.001 4 -;006 .006 -. 0 0 1 .000 To o b t a i n t h e s e c o n d r e s i d u a l m a t r i x we s u b t r a c t t h e v a l u e s i n t h e f i n a l m a t r i x o f T a b l e X I I I f r o m t h e c o r r e s p o n d i n g e l e m e n t s i n t h e f i r s t r e s i d u a l m a t r i x . The r e s u l t i n g v a l u e s a r e l i s t e d i n T a b l e X I V . ( 9 6 ) T a b l e X I V S e c o n d R e s i d u a l M a t r i x 1 2 3 4 1 - . 0 0 1 .000 .000 .000 2 .000 .003 .001 - . 0 0 2 3 .000 .001 .001 .000 4 .000 - . 0 0 2 .000 .002 We n o t e t h a t t h e v a l u e s t h u s o b t a i n e d a r e e x t r e m e l y c l o s e t o z e r o a n d t h a t t h e d i f f e r e n c e s w h i c h e x i s t c o u l d h a v e a r i s e n s i m p l y b e c a u s e e v e r y v a l u e we e m p l o y e d was r o u n d e d o f f t o t h e t h i r d p l a c e o f d e c i m a l s . Thus t h e two f a c t o r s w h i c h we h a v e d e r i v e d a r e s u f f i c i e n t t o e x p l a i n t h e c o r r e l a t i o n s e x i s t i n g b e t w e e n o u r t e s t s . The c o m p l e t e f a c t o r m a t r i x i s r e c o r d e d i n T a b l e XV T a b l e XV C o m p l e t e F a c t o r M a t r i x I I I 1 .828 - . 3 3 9 2 .874 .365 3 .705 - . 0 4 7 4 .853 .017 B e f o r e d i s c u s s i n g t h e r e s u l t we h a v e o b t a i n e d l e t u s c h e c k t o s e e how c l o s e l y t h e s e v a l u e s e x p l a i n o u r o r i g i n a l c o r r e l a t i o n c o e f f i c i e n t s . T h i s we c a n do by m u l t i p l y i n g t h e (97) complete f a c t o r m a t r i x of Table XV by i t s t r a n s p o s e . Table XVI Product of Complete F a c t o r M a t r i x I I I • 1 2 3 4 1 2 3 4 1 .828 -.339 I .828 .874 .705 .853 1 .801 .600 .600 .701 2 .874 .365 I I -.339 .365 -.047 .017 2 .600 .897 .599 .752 3 .705 -.047 3 .600 .599 .499 .601 4 .853 .017 4 .701 .752 .601 .728 A comparison between these v a l u e s and those i n the o r i -g i n a l m a t r i x i n d i c a t e s t h a t our two f a c t o r s a r e s u f f i c i e n t t o e x p l a i n the c o e f f i c i e n t s of t h a t m a t r i x almost e x a c t l y . The s l i g h t d i s c r e p a n c i e s which do e x i s t i n the t h i r d decimal p l a c e a r e very probably due t o the f a q t t h a t we rounded o f f a l l of our o b t a i n e d v a l u e s t o t h a t number of d e c i m a l s . A t t h i s p o i n t i t i s important t o note t h a t , s i n c e the preceding examples were f i c t i t i o u s , i t was p o s s i b l e to f a b -r i c a t e v a l u e s f o r the communalities o f the t e s t s . However, i n any a c t u a l f a c t o r i a l study, these v a l u e s would be unknown and would have to be approximated. As a s o l u t i o n t o t h i s problem, Thurstone (19) suggests t h a t the b e s t approximation f o r these v a l u e s can be obtained by i n s e r t i n g the v a l u e of the l a r g e s t c o r r e l a t i o n c o e f f i c i e n t i n each column to r e p r e -s e n t the g i v e n t e s t ' s communality. Let us now c o n s i d e r what meaning can be a t t a c h e d to the c e n t r o i d f a c t o r l o a d i n g s . I t i s Thurstone.'s c o n t e n t i o n t h a t (98) t h e l o a d i n g s w h i c h w e h a v e o b t a i n e d h a v e n o t r u e p s y c h o l o g i -c a l m e a n i n g . H e i n s i s t s t h a t a r e a r r a n g e m e n t o f t h e s e f a c -t o r s m u s t b e p e r f o r m e d t o o b t a i n a p s y c h o l o g i c a l l y m e a n i n g -f u l r e s u l t a n d t h i s i s t h e b a s i s f o r h i s r o t a t i o n p r o b l e m w h i c h w e w i l l c o n s i d e r i n t h e n e x t c h a p t e r . F o r t h e p r e s e n t , h o w e v e r , l e t u s a n a l y s e t h e m e t h o d b y w h i c h w e a r r i v e d a t t h e f a c t o r l o a d i n g s i n T a b l e X V . T h e f i r s t l o a d i n g i n c o l u m n 1 , w h i c h i s & 1 X , w a s o b t a i n e d b y d i v i d i n g t h e s u m o f t h e f i r s t c o l u m n o f o u r o r i g i n a l c o r r e l a -t i o n m a t r i x b y t h e s q u a r e r o o t o f t h e s u m o f a l l c o e f f i c i e n t s i n t h e m a t r i x . T h a t i s , i t h a s t h e f o l l o w i n g v a l u e , a ± 1 = V±t T h e s u m o f a l l t h e c o e f f i c i e n t s i n c o l u m n 1 o f o u r o r i -g i n a l m a t r i x i s a m e a s u r e o f t h e e x t e n t t o w h i c h t e s t o n e c o r r e l a t e s w i t h a l l o t h e r t e s t s o f t h e b a t t e r y . I f w e h a d d i v i d e d t h i s v a l u e b y r , . , t h e s u m o f a l l c o e f f i c i e n t s , t h e o b t a i n e d v a l u e w o u l d h a v e r e p r e s e n t e d t h a t p r o p o r t i o n o f t h e e n t i r e i n t e r t e s t c o r r e l a t i o n s w h i c h w a s d u e t o t h e c o r r e l a -t i o n b e t w e e n t e s t o n e a n d a l l t h e o t h e r t e s t s . S i n c e w e a c t u a l l y d i v i d e d b y / r ; , t h e f a c t o r l o a d i n g i s n o t e x a c t l y s u c h a p r o p o r t i o n , b u t i s , f o r a n y s p e c i f i c m a t r i x , a c o n -s t a n t m u l t i p l e o f t h a t p r o p o r t i o n . A s i m i l a r a r g u m e n t h o l d s t r u e f o r t h e f a c t o r l o a d i n g s o f a l l t h e r e m a i n i n g t e s t s o n f a c t o r I . T h u s t h e c o l u m n o f f i r s t f a c t o r l o a d i n g s i s a m e a s u r e o f t h e a m o u n t o f t h e t e s t i n t e r c o r r e l a t i o n s w h i c h i s d u e t o a f a c t o r p r e s e n t i n a l l t h e t e s t s s h o w i n g l o a d i n g s o n t h a t f a c t o r . T h e r e f o r e , i t c a n b e s e e n t h a t t h e f a c t o r l o a d -i n g s w h i c h w e h a v e o b t a i n e d a r e m e a s u r e s o f t h e r e l a t i o n s h i p s (99) existing among the tests of the battery. For this reason, they enable us to classify our tests i n a more concise and exact manner on the basis of these interrelationships. In the next chapter we w i l l consider Thurstone's rota-tion problem and a method of performing such a rotation. This w i l l enable us to derive what Thurstone believes are more psychologically meaningful factors. s ( 1 0 0 ) C h a p t e r V  The R o t a t i o n P r o b l e m I n t h e p r e c e d i n g c h a p t e r v e d e v e l o p e d a t e c h n i q u e f o r f a c t o r i n g a r e d u c e d c o r r e l a t i o n m a t r i x i n t o a m a t r i x c o n -s i s t i n g o f common f a c t o r s . H a v i n g s o f a c t o r i z e d t h e m a t r i x , t h e p r o b l e m o f f i n d i n g p s y c h o l o g i c a l m e a n i n g f o r t h o s e f a c -t o r s m u s t t h e n be s o l v e d . I t i s T h u r s t o n e ' s c o n t e n t i o n t h a t t h e f a c t o r s s o d e -r i v e d c a n n o t be d i r e c t l y i n t e r p r e t e d i n a p s y c h o l o g i c a l m a n n e r . He b e l i e v e s t h a t t h e r e m u s t be a f u r t h e r r e a r r a n g e -ment o f t h e f a c t o r s b e f o r e p s y c h o l o g i c a l m e a n i n g c a n be a t -t a c h e d t o them. C o n c e r n i n g t h i s b e l i e f , Thomson ( 1 7 , p.247) h a s t h e f o l l o v i n g t o s a y , " I t c a n n o t be t o o e m p h a t i c a l l y p o i n t e d o u t t h a t t h e f i r s t f a c t o r s v h i c h emerge f r o m t h e " c e n -t r o i d " p r o c e s s a n d t h e m i n i m u m - r a n k p r i n c i p l e n e e d n o t h a v e p s y c h o l o g i c a l s i g n i f i c a n c e a s u n i t a r y p r i m a r y t r a i t s . I t i s o n l y a f t e r r o t a t i o n t o a s u i t a b l e p o s i t i o n t h a t t h i s c a n be e x p e c t e d . " I t i s o u r p u r p o s e , i n t h e p r e s e n t c h a p t e r , t o i n v e s t i g a t e T h u r s t o n e ' s r e a s o n s f o r t h i s b e l i e f a n d a l s o t o s t u d y one o f t h e m e t h o d s t h a t he e m p l o y s i n o b t a i n i n g t h e n e v f a c t o r i a l a r r a n g e m e n t . T h i s c o n s t i t u t e s w h a t T h u r s t o n e d e s i g n a t e s a s t h e r o t a t i o n p r o b l e m . The r e a s o n f o r s o n aming t h i s s t e p i n t h e a n a l y s i s v i l l become c l a r i f i e d a s we p r o c e e d w i t h t h e v o r k t h a t f o l l o w s . B e f o r e d i s c u s s i n g t h e a c t u a l r o t a t i o n a l p r o c e d u r e t h e r e a r e c e r t a i n p r e l i m i n a r y r e s u l t s a n d p r o o f s t h a t m u s t be ( 1 0 1 ) e s t a b l i s h e d a n d u n d e r s t o o d . The f i r s t o f t h e s e i s t h e g e o -m e t r i c i n t e r p r e t a t i o n t h a t c a n be p l a c e d u p o n a f a c t o r i a l a n a l y s i s . S u c h r e p r e s e n t a t i o n i s a n e x t r e m e l y u s e f u l a d j u n c t t o t h e o t h e r t e c h n i q u e s v e h a v e a l r e a d y d e v e l o p e d s i n c e a g r a p h i c m e t h o d o f r e c o r d i n g s u c h a n a n a l y s i s o f t e n c o n v e y s t h e m e a n i n g much more c l e a r l y t h a n w o u l d t h e n u m e r i c a l m a t r i x i t s e l f . T h e r e a r e s e v e r a l d i f f e r e n t g e o m e t r i c a n a l o g u e s t h a t c a n be u s e d t o r e p r e s e n t t h e r e s u l t s we h a v e o b t a i n e d . How-e v e r , s i n c e we a r e t o c o n s i d e r t h e T h u r s t o n e r o t a t i o n p r o b -l e m , i t seems a d v i s a b l e t o l i m i t o u r s e l v e s t o a m e t h o d t h a t he e m p l o y s . T h i s c o n s i s t s i n d e p i c t i n g t h e t e s t b a t t e r y a s a s e r i e s o f v e c t o r s i n a two d i m e n s i o n a l d i a g r a m . A v e c t o r i s s i m p l y a d i r e c t e d l i n e s e gment t h a t i s c h a r a c t e r i z e d by i t s d i r e c t i o n a n d i t s l e n g t h . L e t u s now c o n s i d e r a n a c t u a l f a c -t o r m a t r i x i n o r d e r t h a t t h e a b o v e d e v i c e may be more c l e a r l y u n d e r s t o o d . I n T a b l e I i s l i s t e d a n a r t i f i c i a l l y c o n s t r u c t e d f a c t o r m a t r i x f o r w h i c h we w i l l d e v i s e a g e o m e t r i c a l r e p l i c a . T a b l e I Common F a c t o r M a t r i x I I I -1 .50 .00 2 .70 .30 3 ' .90 .00 4 .00 .80 5 .40 .60 ( 1 0 2 ) I n o r d e r t o r e p r e s e n t t h e t e s t s o f T a b l e I o n a two d i m e n -s i o n a l v e c t o r d i a g r a m we t a k e two a x e s a t r i g h t a n g l e s a n d a r b i t r a r i l y c a l l t hem t h e I a n d I I a x e s . We t h e n p l o t o u r t e s t v e c t o r s u s i n g t h e s e a x e s a s t h e c o o r d i n a t e s y s t e m . The p o i n t o f i n t e r s e c t i o n o f t h e two a x e s i s t a k e n t o be t h e o r i -g i n o f t h e r e f e r e n c e f r a m e . S u c h i s t h e c a s e i n t h e d i a g r a m o f F i g u r e 1. F i g u r e 1 G e o m e t r i c R e p r e s e n t a t i o n o f T e s t s IC 1.O0I 1 1 1 1 I n F i g u r e 1 i s shown t h e v e c t o r r e p r e s e n t a t i o n o f t h e f i v e t e s t s i n T a b l e I . L e t u s c o n s i d e r t h e s t e p s i n v o l v e d i n p l o t t i n g s u c h v e c t o r s i n t e r m s o f a c o n c r e t e e x a m p l e , s a y , t h e p l o t t i n g o f t h e v e c t o r t o r e p r e s e n t t e s t t w o . The r e f e r -e n c e v e c t o r s a r e d r a w n t o be o f a n a r b i t r a r i l y c h o s e n u n i t l e n g t h . T h e r e f o r e , t o p l o t t e s t two we m u s t - m e a s u r e ,70 o f t h e u n i t a l o n g a x i s I a n d .30 o f t h e u n i t a l o n g a x i s - I I . I f ( 1 0 3 ) p e r p e n d i c u l a r s a r e e r e c t e d a t t h e s e two p o i n t s , t h e i r p o i n t o f i n t e r s e c t i o n w o u l d be t h e t e r m i n u s o f t h e g i v e n t e s t v e c -t o r . The r e m a i n i n g t e r m i n i i c a n be p l o t t e d i n a s i m i l a r man-n e r . H a v i n g p l o t t e d t h e t e r m i n i i o f t h e v e c t o r s , we t h e n d raw t h e v e c t o r a r r o w s t h a t r e p r e s e n t t h e d i r e c t i o n a n d mag-n i t u d e ( l e n g t h ) o f t h e g i v e n t e s t v e c t o r s . From t h e manner i n w h i c h t h e v e c t o r s w e r e p l o t t e d , i f we d r o p p e r p e n d i c u l a r s f r o m a g i v e n v e c t o r t e r m i n u s t o e a c h o f t h e a x e s t h e d i s t a n c e s f r o m t h e o r i g i n t o t h e p o i n t s o f i n t e r s e c t i o n r e p r e s e n t t h e v a l u e s o f t h e t e s t ' s l o a d i n g on e a c h f a c t o r . The p o s i t i v e o r n e g a t i v e v a l u e o f s u c h a l o a d -i n g i s o b t a i n e d a s i s t h e s i g n o f a n y d i r e c t e d l i n e s e g m e n t . T h u s , i f t h e d i r e c t i o n o f t h e segment i s t o t h e r i g h t o n a x i s I o r u p w a r d on I I t h e n t h e f a c t o r l o a d i n g i s p o s i t i v e a n d i f t h e d i r e c t i o n i s t o t h e l e f t on f a c t o r I o r downward on I I t h e n t h e f a c t o r l o a d i n g i s n e g a t i v e i n s i g n . The l e n g t h o f t h e t e s t v e c t o r i t s e l f i s a r b i t r a r i l y d e f i n e d t o be p o s i t i v e f o r a r e a s o n we w i l l p r e s e n t l y s e e . Upon c o n -s i d e r i n g t h e d i a g r a m more c l o s e l y we f i n d t h a t two t e s t s h a v e z e r o l o a d i n g on f a c t o r I I a n d s o a r e c o l l i n e a r w i t h f a c t o r I . One t e s t h a s a z e r o l o a d i n g on f a c t o r I a n d s o i s c o l l i n e a r w i t h f a c t o r I I . The r e m a i n i n g t w o , h a v i n g p o s i t i v e l o a d i n g s o n b o t h f a c t o r s , l i e i n t h e f i r s t q u a d r a n t o r p o s i t i v e q u a r -t e r o f t h e c o o r d i n a t e s y s t e m . The l e n g t h o f a n y g i v e n t e s t v e c t o r i n t h e f o r e g o i n g d i a g r a m h a s a n i n t e r e s t i n g a n d i m p o r t a n t v a l u e . T h i s i s r e a d i l y o b t a i n e d i f we remember t h e v a l u e we h a v e a l r e a d y ( 1 0 4 ) d e r i v e d f o r t h e c o m m u n a l i t y o f a t e s t when t h e t e s t b a t t e r y c a n be e x p r e s s e d i n t e r m s o f two common f a c t o r s . T h i s r e -s u l t was r e c o r d e d i n e q u a t i o n ( 3 ) , C h a p t e r I I I , a n d i s r e -p e a t e d b e l o w . w h e r e h* i s t h e c o m m u n a l i t y o f t e s t one a n d a.lx a n d a l z a r e t h e f a c t o r l o a d i n g s o f t h a t t e s t u p o n f a c t o r s I a n d I I . When t h e t e s t i s r e p r e s e n t e d a s a v e c t o r , a l t a n d a 1 5 l a r e t h e l e n g t h s o f t h e p r o j e c t i o n s o f t h a t t e s t v e c t o r u p o n t h e two c o o r d i n a t e a x e s I rand I I . T h i s i s i l l u s t r a t e d d i a g r a m -raatically i n F i g u r e 2. F i g u r e 2 G e o m e t r i c R e p r e s e n t a t i o n o f T e s t C o m m u n a l i t y n x I n F i g u r e 2, t h e f o l l o w i n g e q u a l i t i e s h o l d t r u e . 0B = a 1 1 . . . t h e l o a d i n g o f t e s t one on f a c t o r I , 0 A = a l i t . . . t h e l o a d i n g o f t e s t one o n f a c t o r I I an d f r o m e l e m e n t a r y g e o m e t r y ( 1 0 5 ) 0A= BM . . . o p p o s i t e s i d e s o f a r e c t a n g l e . Upon e m p l o y i n g t h e t h e o r e m o f P y t h a g o r a s we o b t a i n {OB)1 + (BM) 2" = (OM)2- . A f t e r s u b s t i t u t i o n t h i s becomes (OB) 2- + ( O A ) 1 = (OM)2" . B u t we know t h e v a l u e o f t h e l e f t h a n d e l e m e n t s i n t e r m s o f t h e t e s t ' s f a c t o r l o a d i n g s , a n d h e n c e H o w e v e r , (OM) 2 , i s t h e s q u a r e o f t h e l e n g t h o f t h e v e c t o r r e -p r e s e n t i n g t e s t o n e , a n d t h e l e f t h a n d sum, f r o m e q u a t i o n ( 3 ) r e c o r d e d o n p age 1 0 4 , i s e q u a l t o h* t h e c o m m u n a l i t y o f t h e t e s t . T h e r e f o r e , i t c a n be s a i d t h a t t h e l e n g t h o f t h e t e s t v e c t o r i n o u r d i a g r a m i s e q u a l t o t h e p o s i t i v e s q u a r e r o o t o f t h a t t e s t ' s c o m m u n a l i t y . S i n c e t h e s q u a r e r o o t o f t h e c o m m u n a l i t y i s a l w a y s p o s i t i v e , t h e l e n g t h o f t h e v e c t o r m u s t be d e f i n e d t o be p o s i t i v e i n o r d e r t o r e m a i n c o n s i s t e n t w i t h o u r s y s t e m . The f o r e g o i n g r e s u l t s show t h a t we h a v e c o n s t r u c t e d a d i a g r a m w h i c h w i l l f u r n i s h a l l t h e i n f o r m a t i o n t h a t i s c o n -t a i n e d i n t h e f a c t o r m a t r i x . M o r e o v e r , t h e g r a p h i c m e t h o d h a s t h e a d d e d a d v a n t a g e o f e n a b l i n g one t o d i s c o v e r more r e a d i l y t h e p e r t i n e n t i n t e r r e l a t i o n s h i p s b e t w e e n t h e t e s t s . I n a l l t h e f o r e g o i n g w o r k we h a v e d e s i g n a t e d t h e a r r o w ( 1 0 6 ) r e p r e s e n t i n g a t e s t a s a t e s t v e c t o r a n d we n o t e d t h a t a v e c -t o r l i a s n o t o n l y m a g n i t u d e b u t a l s o d i r e c t i o n . H o w e v e r , w h e n e v e r d i r e c t i o n i s c o n s i d e r e d i t m u s t be r e l a t i v e t o some f i x e d f r a m e o f r e f e r e n c e . I n t h e p l o t t i n g o f a f a c t o r m a t r i x , t h i s r e f e r e n c e f r a m e i s f u r n i s h e d b y t h e common f a c t o r a x e s . B u t , i t c a n be shown t h a t a n y g i v e n c o r r e l a t i o n m a t r i x c a n be r e c o n s t r u c t e d f r o m a n i n f i n i t y o f d i f f e r e n t f a c t o r m a t r i c e s . T h i s i s due t o t h e f a c t t h a t t h e c o e f f i c i e n t s o f c o r r e l a t i o n a r e e n t i r e l y i n d e p e n d e n t o f t h e r e f e r e n c e f r a m e . We w i l l now p r o v e t h i s . I n o r d e r t o c a r r y o u t t h e p r o o f we r e q u i r e c e r t a i n b a s i c t r i g o n o m e t r i c f o r m u l a e . L e t u s , t h e r e f o r e , r e v i e w t h e s e f o r m u l a e b e f o r e p r o c e e d i n g w i t h t h e p r o o f i t s e l f . The v a l u e s t h a t we w i l l r e q u i r e a r e t h o s e f o r t h e s i n e a n d c o s i n e o f a n a n g l e a n d a l s o f o r t h e v a l u e o f t h e c o s i n e o f t h e d i f f e r e n c e b e t w e e n two a n g l e s . I n o r d e r t o d e f i n e t h e f i r s t two o f t h e s e v a l u e s l e t u s c o n s i d e r t h e r i g h t - a n g l e d t r i a n g l e o f F i g u r e 3. I n t h i s t r i a n g l e , a , b a n d c a r e t h e l e n g t h s o f t h e s i d e s a n d e i s t h e a n g l e whose s i n e a n d c o s i n e we w i s h t o e v a l u a t e . F i g u r e 3 R i g h t - A n g l e d T r i a n g l e ( 1 0 7 ) By d e f i n i t i o n , t h e s i n e a n d c o s i n e o f t h e a n g l e © a r e t h e r a t i o s b e t w e e n t h e f o l l o w i n g l e n g t h s . s i n e = — a n d c o s e = •§-. c T h e n , t o e v a l u a t e t h e c o s i n e o f t h e d i f f e r e n c e b e t w e e n two a n g l e s , l e t u s c o n s i d e r t h e d i a g r a m o f F i g u r e 4. F i g u r e 4 D i f f e r e n c e B e t w e e n Two A n g l e s I n F i g u r e 4, a n g l e i s t h e a n g l e BOC, w h i l e a n g l e c<. i s AOC. We w i s h t o e x p r e s s t h e v a l u e o f c o s (pi —6). T h i s c a n be e x -p r e s s e d a s f o l l o w s ( 1 5 , p . 1 4 6 ) , ( l ) cos(c< — ) = c o s o C c o s / 3 - f s i n < X s i n / 5 „ E m p l o y i n g t h e f o r e g o i n g v a l u e s , T h u r s t o n e h a s d e r i v e d t h e i m p o r t a n t r e s u l t t h a t t h e c o e f f i c i e n t o f c o r r e l a t i o n i s i n d e -p e n d e n t o f t h e r e f e r e n c e f r a m e . I n F i g u r e 5 a r e d e p i c t e d t h e t e s t v e c t o r s f o r two t e s t s n u m b e r e d one a n d t w o . ( 1 0 8 ) F i g u r e 5 D i a g r a m m a t i c R e p r e s e n t a t i o n o f a C o r r e l a t i o n C o e f f i c i e n t H a i« a 1 4 a M H e r e c< a n d /S a r e t h e a n g l e s b e t w e e n t e s t v e c t o r s one a n d two a n d t h e c o o r d i n a t e a x i s I . A l s o , a n g l e e ^ i s b o t h t h e d i f f e r e n c e (c<-j3) a n d t h e a n g u l a r s e p a r a t i o n o f t h e two t e s t v e c t o r s . The p r o j e c t i o n s o f t h e s e two t e s t v e c t o r s u p o n t h e c o o r d i n a t e a x e s a r e t h e f a c t o r l o a d i n g s o f t h e t e s t s u p o n t h e two common f a c t o r s . T h e i r v a l u e s a r e r e c o r d e d o n t h e f i g u r e . I f we now r e c a l l e q u a t i o n ( 5 ) , C h a p t e r I I I , we n o t e t h a t r u , t h e c o r r e l a t i o n c o e f f i c i e n t b e t w e e n t h e two t e s t s , c a n be e x -p r e s s e d a s f o l l o w s , L e t u s d i v i d e b o t h s i d e s o f t h e a b o v e e q u a t i o n b y t h e p r o d u c t o f t h e s q u a r e r o o t o f e a c h t e s t ' s c o m m u n a l i t y , t h a t i s , by h ^ h . . . T h e s e , we remember, a r e t h e l e n g t h s o f t h e t e s t v e c -t o r s a n d t h e d i v i s i o n may be v a l i d l y p e r f o r m e d s i n c e n e i t h e r o f t h e s e v a l u e s c a n be z e r o . We t h u s o b t a i n ( 1 0 9 ) r i l a i l azi , a i J L a Z Z hx- h± h z h,_ Upon e x a m i n a t i o n o f t h e d i a g r a m a n d c o n s i d e r a t i o n o f t h e d e f i n i t i o n o f t h e s i n e a n d c o s i n e o f ex. a n d 8 we o b t a i n t h e f o l l o w i n g " e q u i v a l e n t v a l u e s ' f o r t h e r a t i o s on t h e r i g h t o f t h e a b o v e e q u a t i o n . a t l h ~ ~ = c o s c < > = c o s 8, a21 h z a42 —• = s i n c < -3, a n d T-——- = s i n /3 . Hence — — I i — = cosc< cos/<5 4- s i n c < s i n j O . h 1 - h z B u t , f r o m e q u a t i o n ( l ) o n p age 1 0 7 , we n o t e t h a t t h e t e r m s on t h e r i g h t o f t h e a b o v e e q u a t i o n a r e t h e e x p a n s i o n o f cos(<X-/S). T h e r e f o r e — — — = cos(f< —8) . * V h z S i n c e (<x—-6) i s e q u a l t o © 1 Z , t h e a n g u l a r s e p a r a t i o n o f t h e two v e c t o r s , r„, _ v \ = c o s e.,. Upon m u l t i p l y i n g b o t h s i d e s o f t h e a b o v e e q u a t i o n by h 4 • h z we o b t a i n (2) r l t = fc^-hji/cos o±z. The i m p o r t a n t f a c t t o n o t e c o n c e r n i n g t h e a b o v e e q u a t i o n i s t h a t t h e c o r r e l a t i o n b e t w e e n t h e two t e s t s , when e x p r e s s e d i n t e r m s o f t h e i r v e c t o r r e p r e s e n t a t i o n , i s t h e p r o d u c t o f t h e ( n o ) l e n g t h s a n d t h e c o s i n e o f t h e a n g u l a r s e p a r a t i o n o f t h e v e c -t o r s . The a b o v e e q u a t i o n c o n t a i n s n o f a c t o r t h a t i s d e p e n -d e n t u p o n t h e s e l e c t e d r e f e r e n c e s y s t e m . T h u s , k n o w l e d g e o f t h e t e s t c o r r e l a t i o n s , t h e d a t a w i t h w h i c h we commence a f a c -t o r p r o b l e m , d e t e r m i n e s t h e l e n g t h o f e a c h v e c t o r a n d t h e r e l a t i o n s h i p b e t w e e n t h e t e s t v e c t o r s b u t d o e s n o t d e t e r m i n e t h e r e f e r e n c e f r a m e we m u s t e m p l o y . The c o o r d i n a t e s y s t e m , w h i c h r e p r e s e n t s t h e common f a c t o r s , m u s t be d e t e r m i n e d by means o f a s u b s i d i a r y c r i t e r i o n . T h i s p o i n t may be f u r t h e r c l a r i f i e d by means o f a s i m p l e e x a m p l e . L i s t e d i n T a b l e I I i s a n a r t i f i c i a l m a t r i x o f i n -t e r c o r r e l a t i o n s a n d c o m m u n a l i t i e s f o r a t h r e e t e s t b a t t e r y . T a b l e I I C o r r e l a t i o n M a t r i x f o r T h r e e T e s t s 1 2 3 1 .49 .30 .00 2 .30 .25 .23 3 .00 .23 .81 I f we w i s h t o r e p r e s e n t t h e a b o v e s y s t e m o f c o r r e l a t i o n c o e f -f i c i e n t s i n t e r m s o f t e s t v e c t o r s we m u s t d e r i v e t h e l e n g t h s o f s u c h v e c t o r s a n d a l s o t h e i r a n g u l a r s e p a r a t i o n . T h i s i n -f o r m a t i o n i s f u l l y d e t e r m i n e d b y t h e v a l u e s r e c o r d e d i n t h e c o r r e l a t i o n m a t r i x . F rom o u r p r e v i o u s r e s u l t we know t h a t t h e l e n g t h o f any s u c h t e s t v e c t o r i s t h e s q u a r e r o o t o f t h e g i v e n t e s t ' s c o m m u n a l i t y . L e t u s d e n o t e t h e l e n g t h s o f t h e ( I l l ) v e c t o r s f o r t e s t s o n e , two a n d t h r e e by L t , Lz a n d L 3 . F r o m t h e m a t r i x o f T a b l e I I t h e s e v a l u e s a r e L 4 =JK=JA9 = .7, L t = y h T = £ 2 5 =.5 a n d L 3 = / h ^ =7781 =.9 . The a n g u l a r s e p a r a t i o n o f t h e s e v e c t o r s c a n be d e t e r m i n e d a s f o l l o w s . L e t u s d e n o t e t h e a n g l e b e t w e e n v e c t o r s 1 a n d 2 b y QIZ, b e t w e e n 2 a n d 3 b y a n d b e t w e e n 1 and. 3 b y e 1 5 . We c a n d e t e r m i n e t h e m a g n i t u d e o f t h e s e a n g l e s by means o f e q u a t i o n ( 2 ) . T h i s e q u a t i o n t e l l s u s t h a t r i z = V n2 c o s e x l . Upon d i v i d i n g b y h^h 4-0 we o b t a i n cos e u = r " E m p l o y i n g t h e a b o v e e q u a t i o n we c a n now d e t e r m i n e t h e v a l u e o f t h e t h r e e a n g l e s a s f o l l o w s . cos e l t . - E i i — , ^ 2 = . 8 6 , " h t . h , ( . 7 ) ( . 5 ) .35 e l t = 3 0 ° . cos e _ r2* _ - 2 5 _ - 2 3 _ 5 i e 2 s = e o ° . r .00 .00 cos e l s = _ J „ J L _ = # 0 0 f h x - h 3 (.7) (.9) .63 ( 1 1 2 ) 6 , 3=90° . We now h a v e d e r i v e d t h e l e n g t h s o f o u r t e s t v e c t o r s a n d t h e a n g l e s s e p a r a t i n g t h e m. H o w e v e r , t h e c o r r e l a t i o n m a t r i x d o e s n o t d e f i n e t h e d i r e c t i o n o f t h e c o n f i g u r a t i o n o f v e c t o r s a s a w h o l e a n d some o t h e r s o u r c e m u s t f u r n i s h t h e r e f e r e n c e f r a m e a g a i n s t w h i c h t h e y a r e t o be p l o t t e d . T h u s , t h e d i a g r a m s o f F i g u r e s 6 a n d 7 b o t h i l l u s t r a t e v e c t o r p a t t e r n s t h a t c o u l d e q u a l l y w e l l r e p r e s e n t t h e g i v e n c o r r e l a t i o n m a t r i x . F i g u r e 6 P o s s i b l e F a c t o r P a t t e r n f o r t h e T h r e e T e s t B a t t e r y 1_.0 •lo 4 ( l i s ) F i g u r e 7 A l t e r n a t e F a c t o r P a t t e r n f o r t h e T h r e e T e s t B a t t e r y n 1.0 .1 -1.0 --8 -.ID -.+ -.2 -Z .4. -t, • j? 1.0 I n o r d e r t o c h e c k t h a t b o t h t h e s e v e c t o r p a t t e r n s f u r -n i s h e q u a l l y a d e q u a t e d e s c r i p t i o n s o f t h e o r i g i n a l c o r r e l a -t i o n m a t r i x , l e t u s d r o p p e r p e n d i c u l a r s f r o m t h e t e r m i n i i o f t h e t e s t v e c t o r s u p o n t h e two s e t s o f r e f e r e n c e a x e s we h a v e c h o s e n . We t h u s o b t a i n t h e t e s t f a c t o r l o a d i n g s f o r e a c h o f t h e a r b i t r a r y r e f e r e n c e s y s t e m s we- h a v e a d o p t e d . . H a v i n g r e -c o r d e d t h e s e f a c t o r l o a d i n g s we c a n t h e n c a l c u l a t e t h e c o r -r e l a t i o n v a l u e s . I n o t h e r w o r d s , i n t h e p r e s e n t c a s e we a r e c o m p u t i n g t h e v a l u e s i n t h e o p p o s i t e d i r e c t i o n t o t h a t n o r -m a l l y a d o p t e d i n o r d e r t o i l l u s t r a t e t h e p o i n t i n q u e s t i o n . The f a c t o r l o a d i n g s a r e m e a s u r e d on F i g u r e s 6 a n d 7 a n d t h e v a l u e s a r e r e c o r d e d i n t h e two t a b l e s , I I I a n d I V . ( 1 1 4 ) T a b l e I I I F a c t o r L o a d i n g s f r o m F i g u r e 6 I I I 1 .70 .00 2 .43 .25 3 .00 .90 T a b l e I V F a c t o r L o a d i n g s f r o m F i g u r e 7 I I I 1 .35 .61 2 .00 .50 3 .78 .45 To c a l c u l a t e t h e c o r r e l a t i o n m a t r i x t h a t g a v e r i s e t o e a c h o f t h e s e s e t s o f f a c t o r l o a d i n g s , we h a v e a l r e a d y s e e n t h a t i t i s s i m p l y n e c e s s a r y t o m u l t i p l y t h e f a c t o r m a t r i x b y i t s t r a n s p o s e . T h i s p r o d u c t i s c o m p u t e d i n T a b l e s V a n d V I f o r e a c h o f t h e m a t r i c e s o f T a b l e s I I I a n d I V . T a b l e V P r o d u c t o f F a c t o r M a t r i x o f T a b l e I I I a n d i t s T r a n s p o s e I I I • 1 2 3 1 2 3 1 .70 .00 I .70 .43 .00 1 .49 .30 .00 2 .43 .25 I I .00 .25 .90 2 .30 .25 .23 3 .00 .90 3 .00 .23 .81 ( 1 1 5 ) T a b l e V I P r o d u c t o f F a c t o r M a t r i x o f T a b l e I V a n d i t s T r a n s p o s e I I I • 1 2 3 1 2 3 1 .55 .61 I .55 .00 T 7 8 1 .49 .30 .00 2 .00 .50 I I .61 .50 .45 2 .30 .25 .23 3 -.78 .45 3 .00 .23 .81 I t i s r e a d i l y s e e n f r o m a n e x a m i n a t i o n o f t h e f o r e g o i n g r e -s u l t s t h a t e i t h e r one o f t h e f a c t o r m a t r i c e s d e s c r i b e s e q u a l l y w e l l t h e o r i g i n a l c o r r e l a t i o n m a t r i x . I n a c t u a l f a c t t h e r e e x i s t s a n i n f i n i t e number o f m a t r i c e s a l l o f w h i c h w o u l d f u r n i s h s u c h a p e r f e c t d e s c r i p t i o n . A c a r e f u l c o m p a r i -s o n o f t h e two v e c t o r d i a g r a m s o f F i g u r e s 6 a n d 7 r e v e a l s t h a t t h e t e s t v e c t o r s a r e i n t h e same r e l a t i o n s h i p t o one a n o t h e r i n b o t h d i a g r a m s , t h e o n l y d i f f e r e n c e c o n s i s t i n g i n t h e l o c a t i o n o f t h e a x e s i n r e l a t i o n t o t h e t e s t v e c t o r c o n -f i g u r a t i o n . I t c o u l d be c o n s i d e r e d t h a t t h e c o o r d i n a t e a x e s h a d b e e n r e v o l v e d o r r o t a t e d i n r e l a t i o n t o t h e t e s t v e c t o r s i n g o i n g f r o m one d i a g r a m t o t h e o t h e r . T h e r e a r e a n i n -f i n i t y o f s u c h p o s i t i o n s t o w h i c h t h e a x e s c a n be r o t a t e d a n d f o r e a c h o f t h e s e p o s i t i o n s a d i f f e r e n t f a c t o r m a t r i x w i l l be d e f i n e d . H o w e v e r , f o r e a c h o f t h e s e p o s i t i o n s , i f t h e t e s t v e c t o r s m a i n t a i n t h e same r e l a t i o n s h i p t o one • a n o t h e r , t h e same c o r r e l a t i o n m a t r i x w i l l be d e f i n e d by t h e v e c t o r s t r u c t u r e . The p r o b l e m , t h e r e f o r e , c o n s i s t s i n a t t e m p t i n g t o e s t a b -l i s h some a u x i l i a r y p r i n c i p l e t h a t w i l l d e f i n e a f a c t o r ( 1 1 6 ) r e f e r e n c e s y s t e m w h i c h c a n be i n t e r p r e t e d p s y c h o l o g i c a l l y . T h i s i s t h e s i t u a t i o n t h a t o c c a s i o n e d T h u r s t o n e ' s r o t a t i o n p r o b l e m . A s we saw i n t h e p r e c e d i n g c h a p t e r , t h e c e n t r o i d m e t h o d o f f a c t o r i z a t i o n d e f i n e s a s e t o f f a c t o r a x e s b u t , a s was m e n t i o n e d a t t h a t t i m e , t h e s e a x e s a r e n o t a c c e p t e d b y T h u r s t o n e a s n e c e s s a r i l y h a v i n g p s y c h o l o g i c a l m e a n i n g . R e -g a r d i n g t h i s m a t t e r T h u r s t o n e ( 1 9 , p . 95) s a y s , I t s h o u l d be c l e a r t h a t a n y i n t e r p r e t a -t i o n o f t h e f a c t o r m a t r i x m u s t be made i n r e l a t i o n t o t h e p r i n c i p l e s t h a t a r e c h o s e n f o r l o c a t i n g t h e c o - o r d i n a t e a x e s i n t h e c o n f i g u r a t i o n d e f i n e d by t h e c o r r e l a t i o n s . I n g e n e r a l , t h e n u m e r i c a l e n t r i e s o f t h e f a c t o r m a t r i x a r e a s a r b i t r a r y a s t h e l o c a -t i o n o f t h e r e f e r e n c e f r a m e . I t i s o n l y t o t h e e x t e n t t h a t t h e r e f e r e n c e f r a m e c a n be i n some s e n s e u n i q u e t h a t t h e n u m e r i c a l e n -t r i e s o f t h e f a c t o r m a t r i x c a n be g i v e n a n y s c i e n t i f i c i n t e r p r e t a t i o n . T h u r s t o n e f u r n i s h e s two c a t e g o r i e s o f c l a s s i f i c a t i o n f o r t h e c r i t e r i a b y w h i c h t h e r e f e r e n c e a x e s c a n be l o c a t e d . The f i r s t o f t h e s e he c o n s i d e r s t o be s t a t i s t i c a l i n n a t u r e i n t h a t t h e c o o r d i n a t e a x e s a r e d e t e r m i n e d by m a x i m i z i n g t h e amount o f t e s t v a r i a n c e t h a t i s a c c o u n t e d f o r by t h e s q u a r e o f e a c h f a c t o r l o a d i n g a s i t i s d e r i v e d . T h i s i s t h e c r i -t e r i o n t h a t was e m p l o y e d when f a c t o r i z i n g t h e c o r r e l a t i o n m a t r i x i n t o t h e i n i t i a l f a c t o r s b y means o f t h e c e n t r o i d m e t h o d . The s e c o n d c l a s s o f c r i t e r i a he d e s c r i b e s a s c o n -f i g u r a t i o n a l i n t h a t t h e e x p e r i m e n t e r makes h i s d e c i s i o n a s t o t h e b e s t r e f e r e n c e f r a m e f r o m a j u d i c i o u s e x a m i n a t i o n o f t h e t e s t v e c t o r c o n f i g u r a t i o n . T h i s i s t h e r e f e r e n c e f r a m e t o w h i c h T h u r s t o n e w o u l d r o t a t e h i s a x e s . ( 1 1 7 ) T h u r s t o n e p r e s e n t s s e v e r a l a r g u m e n t s i n f a v o u r o f t h e r o t a t i o n o f a x e s p r o c e d u r e . H i s m a j o r r e a s o n f o r r o t a t i n g a x e s i s t h a t he b e l i e v e s " s i m p l e s t r u c t u r e " among h i s t e s t v e c t o r s i s t h e m o s t a d e q u a t e a n d t h e s i m p l e s t f o r m o f c l a s s i -f i c a t i o n i n w h i c h t h e r e s u l t s c a n be o b t a i n e d . " S i m p l e s t r u c t u r e " may be d e s c r i b e d a s a c o n f i g u r a t i o n o f t e s t v e c -t o r s s u c h t h a t t h e y a l l h a v e z e r o l o a d i n g s o n one o r more o f t h e r e f e r e n c e a x e s o r p l a n e s . Thus t h e f a c t o r m a t r i x w h i c h i s s y m b o l i z e d by t h e s e v e c t o r s w o u l d h a v e a t l e a s t one z e r o i n e a c h r o w t h a t r e p r e s e n t s a t e s t a n d one o r more z e r o s i n e a c h c o l u m n o f t h e f a c t o r l o a d i n g s . O b v i o u s l y i t i s n o t a l -ways p o s s i b l e t o o b t a i n a c o m p l e t e " s i m p l e s t r u c t u r e " , b u t when s u c h i s o b t a i n e d T h u r s t o n e f e e l s t h a t more c o n f i d e n c e may be p l a c e d i n t h e i n t e r p r e t e d f a c t o r s . T h e n , t o o , he f u r t h e r c o n t e n d s t h a t t h i s c o n f i d e n c e may be a u g m e n t e d by r e p e a t e d e x p e r i m e n t s i n o r d e r t o a s c e r t a i n w h e t h e r s u c h s i m -p l e s t r u c t u r e a l w a y s r e s u l t s f o r t h e b a t t e r y u n d e r c o n s i d e r a r -t i o n . He a l s o s t a t e s t h a t t h e r o t a t i o n m e t h o d p e r m i t s t h e e x p e r i m e n t e r t o e m p l o y b o t h h i s k n o w l e d g e o f t h e v a r i a b l e s i n v o l v e d a n d h i s p r e v i o u s e x p e r i e n c e i n o r d e r t o d e r i v e more m e a n i n g f u l f a c t o r s . T h u r s t o n e makes t h e f i n a l p o i n t t h a t t h e f a c t o r p a t t e r n , w h i c h c o n s i s t s m e r e l y o f t h e p o s i t i v e , z e r o , o r n e g a t i v e s i g n s o f t h e v a r i o u s l o a d i n g s , i s much more v a l u -a b l e a s a s o u r c e o f e x p l a n a t o r y h y p o t h e s e s t h a n a r e t h e n u -m e r i c a l v a l u e s o f t h e l o a d i n g s . I n t h i s r e g a r d , i f we r e t a i n t h e c e n t r o i d f a c t o r s t h e s i g n s t h a t a r e o b t a i n e d a r e mere a r t i f a c t s o f t h e manner i n w h i c h we p l a c e d t h e i n i t i a l a x i s ( 1 1 8 ) i n t h e c e n t r e o f t h e t e s t c o n f i g u r a t i o n . B e c a u s e o f t h e a r -b i t r a r y p l a c e m e n t o f t h a t f a c t o r a l l s u c c e e d i n g f a c t o r s w i l l h a v e t o be b i - p o l a r , t h a t i s , t h e y w i l l h a v e t o c o n s i s t o f f a c t o r l o a d i n g s t h a t a r e b o t h p o s i t i v e a n d n e g a t i v e i n s i g n . The w h o l e p r o c e d u r e o f r o t a t i o n i s one c o n c e r n i n g w h i c h t h e r e i s s t i l l much c o n t e n t i o n among a n a l y s t s . H o w e v e r , s i n c e we a r e s t u d y i n g T h u r s t o n e ' s m e t h o d s a n d t h e r o t a t e d f a c t o r s o f -t e n do f u r n i s h a s i m p l e r a n d more c o n v i n c i n g d e s c r i p t i v e s y s -tem, l e t u s now p r o c e e d t o a n a l y s e t h e s t e p s i n v o l v e d i n t h i s t e c h n i q u e . B e c a u s e , a s we s h a l l s e e l a t e r , a l l f a c t o r p a t t e r n s c a n be r o t a t e d i n a two d i m e n s i o n a l f r a m e o f r e f e r e n c e , we w i l l now d e v e l o p f o r m u l a e b y means o f w h i c h r o t a t i o n i n a p l a n e c a n be p e r f o r m e d . I n o r d e r t h a t we may b e t t e r u n d e r s t a n d t h e m e t h o d i t s e l f , b e f o r e we p r o c e e d w i t h a n a c t u a l p r o b l e m l e t u s e x a m i n e t h e n e c e s s a r y m a t h e m a t i c a l b a s e s . I n F i g u r e 8 i s r e c o r d e d t h e l o c a t i o n o f t h r e e p o i n t s i n r e l a t i o n t o a f i x e d o r t h o g o n a l c o o r d i n a t e s y s t e m XY. F i g u r e 8 R o t a t i o n o f A x e s Y \ \ w s * \ \ * \ \ y > • \ \ \ \ \ * \ \ \ \ \ \ \ X (119) The a x e s a r e t h e n r o t a t e d t h r o u g h a n a n g l e (JP t o a new p o s i -t i o n x'Y'and we w i s h t o f i n d t h e c o o r d i n a t e s o f t h e p o i n t s w i t h r e f e r e n c e t o t h i s new s y s t e m o f a x e s . Upon r e f e r r i n g t o t h e d i a g r a m we n o t e t h a t f o r e a c h p o i n t t h e r e a r e two s e t s o f c o o r d i n a t e s shown. T h e s e a r e ( x , y ) , w h i c h d e n o t e t h e p o i n t ' s c o o r d i n a t e s w i t h r e f e r e n c e t o t h e a x e s XY, a n d ( x ' , y ' ) , t h e p o i n t ' s c o o r d i n a t e s w i t h r e f e r e n c e t o a x e s X 1 Y' . The f o r m u -l a e (15, p . 2 3 6 ) , w h i c h e x p r e s s t h e new c o o r d i n a t e s i n t e r m s o f t h e a n g l e o f r o t a t i o n a n d t h e f o r m e r c o o r d i n a t e s o f t h e p o i n t , a r e a s f o l l o w s . ; (3 ) x'= x c o s u? - f y s i n t p , y'=-x s i n o o + y c o s cp . T h u s , t h e c o o r d i n a t e s o f a n y p o i n t w i t h r e f e r e n c e t o t h e new a x e s c a n be e x p r e s s e d a s a f u n c t i o n o f t h e f o r m e r c o o r d i n a t e s o f t h e p o i n t a n d t h e s i n e a n d c o s i n e o f t h e a n g l e o f r o t a t i o n . As a n e x a m p l e t h e c o o r d i n a t e s o f t h e f i r s t p o i n t become x!^ = x 1 c o s ( p - + - y 1 sina> , y'= - x t s i n u? + y1 c o s u? . T h i s r e l a t i o n s h i p c a n be v e r y c o n c i s e l y r e c o r d e d a s a m a t r i x p r o d u c t . L e t u s w r i t e t h e o r i g i n a l c o o r d i n a t e s a s a two c o l u m n e d m a t r i x a n d t h e t r i g o n o m e t r i c f u n c t i o n s o f t h e a n g l e cp a s a s q u a r e two b y two m a t r i x . The p r o d u c t o f t h e two m a t r i c e s a s r e c o r d e d i n T a b l e V I I w i l l e q u a l t h e new c o o r d i n a t e s o f t h e p o i n t s a f t e r r o t a t i o n h a s t a k e n p l a c e . ( 1 2 0 ) T a b l e V I I New C o o r d i n a t e s a s a M a t r i x P r o d u c t X, • C O S CP - s i n e ? 7z s i n e ? C O S IQ y i x3 y 3 X 3 y; I n o r d e r t o c h e c k t h a t t h e f o r e g o i n g p r o d u c t d o e s i n f a c t f u r n i s h t h e new c o o r d i n a t e v a l u e s we r e q u i r e , l e t u s p e r f o r m t h e m a t r i x m u l t i p l i c a t i o n f o r t h e f i r s t p o i n t . Upon c a r r y i n g o u t t h i s p r o c e s s we o b t a i n x 4 c o s w + y t s i n u? = x [ , - x l s i n c ^ + y x c o s up = y[ . B u t t h e s e a r e t h e v a l u e s we h a v e a l r e a d y r e c o r d e d f o r t h e f i r s t p o i n t ' s new c o o r d i n a t e s . T h u s , t o f i n d t h e two d i m e n -s i o n a l c o o r d i n a t e s o f o u r p o i n t s a f t e r a r o t a t i o n d u r i n g w h i c h t h e a x e s r e m a i n a t r i g h t a n g l e s o r o r t h o g o n a l , we m u s t l o c a t e t h e v a l u e s o f t h e s i n e a n d c o s i n e o f t h e a n g l e o f r o -t a t i o n i n . o r d e r t o e x p r e s s t h e p r e c e d i n g p r o d u c t . T h i s p r o -c e d u r e w i l l be f u r t h e r c l a r i f i e d when we a n a l y s e t h e f o l l o w -i n g e x a m p l e . F o r p u r p o s e s o f i l l u s t r a t i o n we w i l l e m p l o y a f a c t o r m a t r i x w h i c h h a s b e e n u s e d b y T h u r s t o n e f o r t h i s same p r o b l e m . , T h i s w i l l e n a b l e u s t o d i s c u s s t h e v a r i o u s i m p o r -t a n t p o i n t s w h i c h he makes i n r e g a r d t o t h e r o t a t i o n p r o b l e m . The o r i g i n a l f a c t o r m a t r i x ( 1 9 , p. 1 0 7 ) , w h i c h c o n s i s t s o f t h e v a l u e s d e r i v e d i n t h e i n i t i a l s t a g e o f t h e f a c t o r i -( 1 2 1 ) z a t i o n o f a c o r r e l a t i o n m a t r i x , i s l i s t e d i n T a b l e V I I I . T a b l e V I I I O r i g i n a l F a c t o r M a t r i x I I I 1 .762 .000 2 .827 -.355 3 .604 .394 4 .670 .039 5 .460 - . 1 9 7 6 .236 .552 7 .591 .617 8 .315 .736 T h e s e v a l u e s a r e e x p r e s s e d i n t e r m s o f t h e u n r o t a t e d f a c t o r a x e s I a n d I I . The r o w s l i s t t h e t e s t l o a d i n g s o n e a c h o f t h e s e f a c t o r s o r , i n t e r m s o f o u r g e o m e t r i c r e p r e s e n t a t i o n , t h e c o o r d i n a t e s o f t h e t e r m i n a l p o i n t s o f t h e t e s t v e c t o r s i n r e l a t i o n t o t h i s o r i g i n a l r e f e r e n c e s y s t e m . I f we c o n s t r u c t t h e two r e f e r e n c e v e c t o r s a s u n i t v e c t o r s , t h a t i s , a s v e c -t o r s h a v i n g u n i t l e n g t h , we c a n t h e n p l o t t h e t e s t s a s p o i n t s whose c o o r d i n a t e s a r e t h e f a c t o r l o a d i n g s o f t h e t e s t s u p o n t h e two f a c t o r s . The u n i t l e n g t h a t t a c h e d t o e a c h r e f e r e n c e a x i s i s o c c a s i o n e d b y t h e f a c t t h a t t h e s e v e c t o r s a r e c o n -s i d e r e d t o be r e p r e s e n t a t i v e o f p u r e m e a s u r e s o f the- f a c t o r s d e n o t e d a n d s o w o u l d h a v e t h e i r c o m m u n a l i t y e q u a l t o u n i t y . I n F i g u r e 9 we h a v e c o n s t r u c t e d t h e r e f e r e n c e s y s t e m a n d ( 1 2 2 ) p l o t t e d t h e e i g h t p o i n t s r e p r e s e n t a t i v e o f t h e t e s t s o f o u r b a t t e r y . F i g u r e 9 T e s t C o n f i g u r a t i o n w i t h R e f e r e n c e t o O r i g i n a l A x e s TL o -9 0 °® ®* •(, 0 o © •4 0 •2 0 0 -.8 0 -.(. o - 4 0 -2 o o 0 Z 0 -4 0 -6 0 ".g 0 l.«' - 2 0 o © - 4 0 -b 0 - 8 - l o 0 0 ¥e m u s t now d e c i d e u p o n t h e c r i t e r i o n t o be u s e d i n l o -c a t i n g a new a n d more s u i t a b l e r e f e r e n c e f r a m e i n t h e f o r e -g o i n g s t r u c t u r e . T h u r s t o n e s u g g e s t s a number o f p o s s i b l e ( 1 2 5 ) c r i t e r i a t h a t m i g h t be e m p l o y e d . L e t u s l i s t two o f t h e s e . One s u c h c r i t e r i o n w o u l d be t o p l a c e t h e a x e s i n s u c h a man-n e r a s t o e l i m i n a t e o r m i n i m i z e t h e n e g a t i v e p r o j e c t i o n s w h i l e a t t h e same t i m e s e e k i n g t o o b t a i n " s i m p l e s t r u c t u r e . " C o n c e r n i n g t h i s c o n c e p t o f " s i m p l e s t r u c t u r e " a n d T h u r s t o n e ' s u l t i m a t e o b j e c t i v e i n s u g g e s t i n g t h i s c r i t e r i o n , Thomson ( 1 7 , p. 248) s a y s , T h i s i d e a i s t h a t t h e a x e s a r e t o be r o t a t e d u n t i l a s many a s p o s s i b l e o f them a r e a t r i g h t a n g l e s t o a s many a s p o s s i b l e o f t h e o r i g i n a l t e s t v e c t o r s ; a n d t h a t t h e b a t t e r y i s n o t s u i t a b l e f o r d e f i n i n g f a c t o r s u n l e s s s u c h a r o t a t i o n i s u n i q u e l y p o s s i b l e , a r o t a t i o n w h i c h w i l l l e a v e e v e r y f a c t o r a x i s a t r i g h t a n g l e s t o a t l e a s t a s many t e s t s a s t h e r e a r e f a c t o r s , ' a n d e v e r y t e s t a t r i g h t a n g l e s t o a t l e a s t one f a c t o r . When t h e v e c t o r s o f a t e s t a n d a f a c t o r a r e a t r i g h t a n g l e s , t h e l o a d i n g o f t h e f a c -t o r i n t h a t t e s t i s z e r o . T h u r s t o n e ' s " s i m p l e s t r u c t u r e " i s t h e r e f o r e i n d i c a t e d by a l a r g e number o f z e r o s i n t h e m a t r i x o f l o a d i n g s , s o l a r g e t h a t t h e r e w i l l be o n l y one p o s i t i o n o f t h e a x e s ( i f a n y ) w h i c h s a t i s f i e s t h e r e q u i r e -m e n t . H i s s e a r c h , be i t r e p e a t e d , i s f o r a s e t o f c o n d i t i o n s w h i c h w i l l make t h e s o l u t i o n u n i q u e . T h i s c r i t e r i o n i s one t h a t e n t a i l s m i n i m i z i n g w h a t T h u r s t o n e c a l l s t h e " c o m p l e x i t i e s " J o f . t h e t e s t s . T h i s c o n s i s t s i n p l a c i n g t h e a x e s s o t h a t a s many z e r o e n t r i e s a s p o s s i b l e a p p e a r i n t h e t e s t l o a d i n g s a n d , i f p o s s i b l e , a t l e a s t one z e r o t e s t l o a d i n g f o r e a c h t e s t . I f p e r f e c t " s i m p l e s t r u c -t u r e " r e s u l t s , t h e n e v e r y t e s t w i l l h a v e a l e s s e r d e g r e e o f c o m p l e x i t y t h a n t h e b a t t e r y a s a w h o l e . A n o t h e r i m p o r t a n t c o n s i d e r a t i o n i s t h e s c i e n t i f i c k n o w l e d g e a nd acumen o f t h e e x p e r i m e n t e r . The r o t a t i o n p r o c e s s e n a b l e s t h e f a c t o r i s t t o ( 1 2 4 ) e m p l o y a l l t h a t he knows a b o u t t h e p a r t i c u l a r d o m a i n b e i n g i n v e s t i g a t e d . T h i s i n f o r m a t i o n may w e l l s e r v e t o a s s i s t i n m a k i n g a d e c i s i o n a s t o t h e m o s t a d e q u a t e o f a l l p o s s i b l e r o t a t i o n s . I n t h e p r e s e n t i n s t a n c e , s i n c e t h e f a c t o r m a t r i x i s one c o n t r i v e d f o r i l l u s t r a t i v e p u r p o s e s , t h e r e i s no p o s -s i b i l i t y o f e m p l o y i n g s u c h p r i o r k n o w l e d g e c o n c e r n i n g t h e v a r i a b l e s i n v o l v e d . R e t u r n i n g t o t h e s p e c i f i c e x a m p l e o f T a b l e V I I I a n d F i g u r e 9, we e x a m i n e t h e t e s t v e c t o r c o n f i g u r a t i o n i n s e a r c h o f t h e b e s t p o s s i b l e a x i s r o t a t i o n . A t t h i s p o i n t we n o t e t h a t t h e a n g u l a r s p a n o f a l l t h e v e c t o r s a p p e a r s t o be a p -p r o x i m a t e l y 9 0 ° . I f s u c h i s t h e c a s e t h e n we w i l l be a b l e t o r o t a t e t h e a x e s i n s u c h a manner a s t o i n c l u d e a l l t h e t e s t s i n t h e p o s i t i v e f i r s t q u a d r a n t o f t h e r o t a t e d d i a g r a m . We w i l l a l s o be a b l e t o make s e v e r a l o f t h e f a c t o r l o a d i n g s z e r o . L e t u s d i g r e s s f o r a moment i n o r d e r t o d i s c u s s a v e r y s i m p l e t e s t t h a t c a n be e m p l o y e d t o d e t e r m i n e t h e a n g u l a r s e p a r a t i o n o f o u r t e s t v e c t o r s . I n C h a p t e r I I I , e q u a t i o n ( 5 ) , we r e c o r d e d t h e v a l u e o f t h e c o r r e l a t i o n c o e f f i c i e n t b e -t w e e n a n y two t e s t s i n t e r m s o f t h e i r common f a c t o r l o a d i n g s . T h i s was T h i s t e l l s u s t h a t we c a n r e a d i l y e v a l u a t e t h e t e s t i n t e r -c o r r e l a t i o n b e t w e e n a n y two t e s t s i n T a b l e V I I I by t a k i n g t h e p r o d u c t s o f a l l c o r r e s p o n d i n g f a c t o r l o a d i n g s a n d t h e n sum-m i n g a l l s u c h p r o d u c t s . Then,, t o o , e a r l i e r i n t h e p r e s e n t ( 1 2 5 ) c h a p t e r , we d e v e l o p e d a f o r m u l a , e q u a t i o n ( 2 ) , f o r t h e a b o v e c o r r e l a t i o n c o e f f i c i e n t i n t e r m s o f t h e f o l l o w i n g p r o d u c t . r u - Vlvcc-se^ . S i n c e , i n t h e a b o v e e q u a t i o n , t h e two t e r m s h t a n d h 2 a r e a l w a y s p o s i t i v e a n d n e v e r z e r o , t h e f o l l o w i n g r e l a t i o n s h i p m u st e x i s t b e t w e e n r l z a n d c o s e u . I f r l z < 0, t h e n c o s e t t < 0 , r i a = 0, t h e n c o s Qtz=0 and r 1 J t =» 0, t h e n c o s © l z = » 0 . B u t t h e f o l l o w i n g r e l a t i o n s h i p e x i s t s b e t w e e n t h e c o s i n e o f an a n g l e a n d t h e a n g l e i t s e l f f o r o u r f o r m o f d i a g r a m i n w h i c h © 1 Z i s a l w a y s l e s s t h a n 180° . I f cos e t t<;0, t h e n 0 U :>• 9 0 ° , c o s © i 2 = 0, t h e n e M = 90° a n d c o s © l i t > 0, t h e n © l t < 90° . Thus we c a n e s t a b l i s h t h e f o l l o w i n g r e l a t i o n s h i p b e t w e e n t h e s i g n o f t h e c o r r e l a t i o n c o e f f i c i e n t a n d t h e m a g n i t u d e o f t h e a n g u l a r s e p a r a t i o n o f t h e two t e s t v e c t o r s when p l o t t e d o n a d i a g r a m . I f * r l t < : 0, t h e n e l t > 9 0 ° , r i a = 0 , t h e n e l t = 90" a n d r « = » °> t h e n e 1 5,<^90°. ( 1 2 6 ) L e t u s a p p l y t h e f o r e g o i n g r e s u l t t o t h e p r o b l e m o f e s -t i m a t i n g t h e a n g u l a r d i s p e r s i o n o f t h e t e s t v e c t o r s i n F i g u r e 9. E x a m i n a t i o n o f t h a t d i a g r a m shows t h a t t h e t e s t v e c t o r s f o r t e s t s two a n d e i g h t a p p e a r t o s u b t e n d t h e l a r g e s t a n g l e a t t h e o r i g i n . The c o r r e l a t i o n b e t w e e n t h e s e t e s t s i s r » = a Z . l a 8 1 + a Z 2 a 8 Z = ( . 827) (.315 ) + (~ . 355 ) (. 736) = .260505 - . 2 6 1 2 8 0 = - .000775 . When we r e c a l l t h a t t h e o r i g i n a l f a c t o r l o a d i n g s w e r e r o u n d e d o f f t o t h e t h i r d p l a c e o f d e c i m a l s , i t a p p e a r s t h a t t h e v a l u e o f r i g i s d i f f e r e n t f r o m z e r o b y a n e g l i g i b l e a m o u n t . T h e r e f o r e , i t seems v a l i d t o c o n c l u d e t h a t r z g = 0 a n d t h a t eZ9 - 9 0 ° . The t e s t v e c t o r s a r e d i s p e r s e d o v e r a 90° a n g l e . I f we r o t a t e t h e a x e s i n a c l o c k w i s e d i r e c t i o n , r e -t a i n i n g t h e o r t h o g o n a l p r o p e r t y o f t h o s e a x e s , i t i s p o s s i b l e t o i n c l u d e a l l t h e v e c t o r s i n t h e p o s i t i v e q u a d r a n t a n d t o h a v e t h e r o t a t e d a x e s p a s s i n g t h r o u g h t e s t s two a n d e i g h t . F i g u r e 10 ( 1 9 , p . 108) shows t h e new p o s i t i o n o f t h e a x e s a f t e r t h i s r o t a t i o n h a s b e e n p e r f o r m e d . ( 1 2 7 ) F i g u r e 10 R o t a t e d F a c t o r s I T ft*****' / / ® o \ / h \ \ . X •\ \ / / (P / °® % \ I 1 i i \ M J t • . 1 0 © 1 ! 1 1 ^ % \ \ !/ \ \ r / X *»— ** -"~^ L e t u s r a e n t i o n a n i m p o r t a n t f e a t u r e o f t h e d i a g r a m o f F i g u r e 1 0 . The c i r c l e i s d r a w n u s i n g t h e same u n i t m e a s u r e a s t h a t e m p l o y e d f o r t h e o r i g i n a l a x e s . I t i s t h e n s i m p l y n e c e s s a r y t o a l l o w t h e new a x e s t o i n t e r s e c t t h a t c i r c l e i n o r d e r t o be c e r t a i n t h a t we r e t a i n t h i s same u n i t m e a s u r e . ( 1 2 8 ) Our p u r p o s e i s t o e x p r e s s t h e c o o r d i n a t e s o f t h e t e s t p o i n t s i n t e r m s o f t h i s new a x i s s y s t e m . T h e s e c o o r d i n a t e s w i l l r e p r e s e n t t h e f a c t o r l o a d i n g s o f t h e t e s t s u p o n t h o s e a x e s . A c u r s o r y e x a m i n a t i o n o f t h e f i g u r e shows t h a t t h e two t e s t s , two .and f i v e , h a v e z e r o l o a d i n g s on t h e new f a c t o r a x i s B, w h i l e t e s t s s i x a n d e i g h t h a v e z e r o l o a d i n g s on t h e new a x i s A. The r e m a i n i n g t e s t s a l l h a v e p o s i t i v e l o a d i n g s on t h e new r e f e r e n c e f r a m e . I n o r d e r t o e s t a b l i s h t h e new c o o r d i n a t e v a l u e s i t i s n e c e s s a r y t o c a r r y o u t t h e f o l l o w i n g m a t r i x m u l t i p l i c a t i o n . x l c o s a> - s i n w K x * y z s i n w c o s a? x k X 3 y 3 x ; Where t h e ( x , y ) ' s a r e t h e o r i g i n a l c o o r d i n a t e s , LP i s t h e a n g l e o f r o t a t i o n a n d t h e ( x 1 , y ' ) ' s a r e t h e new c o o r d i n a t e s . The ( x , y ) ' s i n t h i s c a s e a r e — t h e known f a c t o r l o a d i n g s a n d t h e ( x ' , y ' ) ' s a r e t h e f a c t o r l o a d i n g s we w i s h t o f i n d . I n o r d e r t o o b t a i n t h e s e v a l u e s we m u s t f i r s t e v a l u a t e t h e s i n e a n d c o s i n e o f t h e a n g l e o f r o t a t i o n . I f we remember t h e d e -f i n i t i o n o f t h e s e v a l u e s a s r a t i o s o f p a i r s o f s i d e s o f a r i g h t t r i a n g l e a n d i f we f u r t h e r r e c a l l t h a t t h e l o n g e s t s i d e o f t h e t r i a n g l e OMA i n q u e s t i o n i s o f u n i t l e n g t h , we c a n f i n d t h e two n e e d e d v a l u e s f r o m F i g u r e 10 by s i m p l y m e a s u r i n g t h e o t h e r two s i d e s . By m e a s u r e m e n t o f t r i a n g l e OMA we o b -t a i n t h e f o l l o w i n g v a l u e s . a n d Hence a n d ( 1 2 9 ) . 0M= .917 , MA= -.400 0A = 1. s i n ( p . M . ^ 4 0 0 = - . 4 0 0 c o s | | = - s ^ i 7 - = .917 . T h e s e v a l u e s c a n be v e r i f i e d by m e a s u r i n g t h e a n g l e W w i t h a p r o t r a c t o r a n d l o c a t i n g t h e v a l u e o f t h e s i n e a n d c o s i n e f r o m t h e a p p r o p r i a t e t r i g o n o m e t r i c t a b l e s . The m a g n i t u d e o f t h e a n g l e i s 23° 35' a n d , s i n c e t h e r o t a t i o n i s c l o c k w i s e , , t h e a n g l e i s n e g a t i v e . T h e r e f o r e , f r o m t h e t a b l e s , s i n u>= -.4001 a n d c o s ip = .9165 . E m p l o y i n g t h e s e v a l u e s r o u n d e d o f f t o t h r e e p l a c e s o f d e c i -m a l s we t h e n c a r r y o u t t h e m u l t i p l i c a t i o n r e c o r d e d i n T a b l e I X . T a b l e I X M a t r i x P r o d u c t t o R o t a t e A x e s I I I 1 .762 .000 2 .827 -.355 3 .604 .394 4 .670 .039 5 .460 -.197 6 .236 .552 7 .591 .617 8 .315 .736 A B I I I .917 .400 -.400 .917 A B 1 .6988 .3048 2 .9004 .0053 3 .3963 .6029 4 .5988 .3038 5 .5006 .0034 6 -.0044 .6006 7 .2951 .8022 8 -.0055 .8009 ( 1 5 0 ) The m a t r i x o n t h e r i g h t c o n t a i n s t h e f a c t o r l o a d i n g s o f o u r t e s t s u p o n t h e new r o t a t e d a x e s . S i n c e t h e o r i g i n a l c o r r e l a -t i o n m a t r i x , w h i c h T h u r s t o n e f a c t o r e d , was e x p r e s s e d o n l y t o t h e s e c o n d p l a c e o f d e c i m a l s , l e t u s r o u n d o f f t h e f a c t o r l o a d i n g s t o t h a t number o f d e c i m a l p l a c e s . The m a t r i x ( 1 9 , p. 107) t h e n becomes T a b l e X F a c t o r L o a d i n g s on t h e R o t a t e d A x e s A B 1 .70 .30 2 .90 .00 3 .40 .60 4 .60 .30 5 .50 .00 6 .00 .60 7 .30 .80 8 .00 .80 Thus we h a v e c o m p l e t e d t h e p r o b l e m o f r o t a t i o n o f a x e s i n two d i m e n s i o n s . . T h i s p h a s e o f t h e p r o b l e m h a s b e e n g i v e n m i n u t e c o n -s i d e r a t i o n s i n c e a s T h u r s t o n e ( 1 9 , p . 1 1 0 ) s a y s , "The e x t e n -s i o n o f t h e r e a s o n i n g i l l u s t r a t e d h e r e t o t h r e e o r more d i -m e n s i o n s i n v o l v e s e s s e n t i a l l y n o new i d e a s , a l t h o u g h t h e h i g h e r - d i m e n s i o n a l p r o b l e m s a r e more c o m p l e x . " The h i g h e r d i m e n s i o n p r o b l e m s , w h i c h u s u a l l y a r i s e i n a p r a c t i c a l ( 1 5 1 ) s i t u a t i o n , c a n be h a n d l e d a s f o l l o w s . L e t u s s u p p o s e t h a t , o u r o r i g i n a l f a c t o r i z a t i o n o f t h e c o r r e l a t i o n m a t r i x h a s d e r i v e d a f a c t o r m a t r i x w i t h l o a d i n g s on t h r e e common f a c t o r s w h i c h we w i l l d e s i g n a t e I , I I a n d I I I . Our p r o b l e m i s t o r o t a t e t h e s e a x e s , o b t a i n e d by t h e i n i t i a l f a c t o r i z a t i o n , i n t o a new s e t o f a x e s w h i c h w i l l f u l f i l l t h e r e q u i r e m e n t s i m p o s e d b y T h u r s t o n e . T h i s p r o b l e m c a n be d e a l t w i t h b y means o f two d i m e n s i o n a l m e t h o d s . I t i s s i m p l y n e c e s s a r y t o p l o t two d i m e n s i o n a l s e c t i o n s , one s e c t i o n f o r e a c h p a i r o f c o o r d i n a t e a x e s . T h u s , we w o u l d p l o t t h r e e d i a g r a m s s i m i l a r t o t h e one w i t h w h i c h we h a v e a l r e a d y d e a l t . The f i r s t o f t h e s e w o u l d be t h e p l o t o f t h e t e s t l o a d i n g s on f a c t o r s I a n d I I , t h e s e c o n d t h o s e u p o n I I a n d I I I , a n d t h e t h i r d t h e l o a d i n g s u p o n t h e f a c t o r s I a n d I I I . ¥e t h e n p e r f o r m a r o t a -t i o n o f a x e s i n e a c h o f t h e s e two d i m e n s i o n a l d i a g r a m s . S i n c e t h e r o t a t i o n s a r e c a r r i e d o u t s u c c e s s i v e l y i t i s n e c e s -s a r y t o r e p l o t t h e new f a c t o r l o a d i n g s i n a s i m i l a r manner t o a s c e r t a i n w h e t h e r f u r t h e r r o t a t i o n i s n e e d e d . T h u s , t h e r o -t a t i o n o f a x e s c a n be g e n e r a l i z e d t o a n r d i m e n s i o n a l f a c t o r m a t r i x w i t h t h e r o t a t i o n s t i l l b e i n g p e r f o r m e d i n a two d i -m e n s i o n a l p l o t . The m e t h o d h a s t h e a d v a n t a g e o f f u r n i s h i n g a g r a p h i c r e c o r d o f t h e e x i s t i n g r e l a t i o n s h i p s . H o w e v e r , i t ha s t h e d i s a d v a n t a g e t h a t many d i a g r a m s m u s t be p l o t t e d i f t h e number o f d i m e n s i o n s i s l a r g e . F o r e x a m p l e , i f t h e r e w e r e f i v e common f a c t o r s i n o u r f a c t o r m a t r i x , we w o u l d h a v e t o p l o t t e n i n i t i a l d i a g r a m s . C o n c e r n i n g t h e p u r p o s e o f t h i s w h o l e p r o c e d u r e Thomson ( 1 7 , p . £47) h a s t h e f o l l o w i n g t o s a y , ( 1 3 2 ) I t becomes i n c r e a s i n g l y c l e a r t h a t t h e w h o l e p r o c e s s i s one by w h i c h a d e f i n i t i o n o f t h e p r i m a r y f a c t o r s i s a r r i v e d a t by s a t -i s f y i n g s i m u l t a n e o u s l y c e r t a i n m a t h e m a t i c a l p r i n c i p l e s a n d c e r t a i n p s y c h o l o g i c a l i n t u i -t i o n s . When t h e s e two s i d e s o f t h e p r o c e s s c l i c k i n t o a g r e e m e n t , t h e w o r k e r h a s a s e n s e o f h a v i n g made a d e f i n i t e s t e p f o r w a r d . The two s u p p o r t one a n o t h e r . S i n c e t h e m a t r i x t h a t we e m p l o y e d f o r i l l u s t r a t i v e p u r -p o s e s was n o t t h e r e s u l t o f a n a c t u a l e x p e r i m e n t , t h e r e i s n o p o s s i b i l i t y o f p e r f o r m i n g t h e f i n a l s t e p i n t h e f a c t o r i n g p r o c e d u r e w h i c h i s t h a t o f i n t e r p r e t i n g t h e f a c t o r s o b t a i n e d . F o r t h i s r e a s o n we w i l l l e a v e t h e p r o b l e m o f f a c t o r i d e n t i -f i c a t i o n u n t i l t h e f o l l o w i n g c h a p t e r , w h e r e we w i l l d i s c u s s e x a m p l e s o f a c t u a l i n t e r p r e t a t i o n s t h a t h a v e b e e n made o f f a c t o r s d e r i v e d f r o m e x p e r i m e n t a l d a t a ( 1 3 S ) C h a n t e r V I  I n t e r p r e t a t i o n o f F a c t o r s I n t h e p r e c e d i n g c h a p t e r s , we h a v e s t u d i e d a f e w o f t h e t e c h n i q u e s w h i c h a r e a v a i l a b l e f o r - d e a l i n g w i t h t h e f a c t o r i -z a t i o n p r o b l e m . I n t h e p r e s e n t c h a p t e r i t i s o u r p u r p o s e t o d i s c u s s t h e f i n a l s t a g e o f a n y i n d i v i d u a l f a c t o r s t u d y . T h i s i s t h e p r o b l e m o f a t t a c h i n g p s y c h o l o g i c a l m e a n i n g t o t h e d e -r i v e d f a c t o r p a t t e r n . I n t h i s r e g a r d we w i l l c o n s i d e r o n l y t h e i n t e r p r e t a t i o n o f t h e r e s u l t s o f s p e c i f i c f a c t o r a n a l -y s e s . H o w e v e r , i t m u s t be b o r n e c l e a r l y i n m i n d t h a t t h e r e i s a f u r t h e r i m p o r t a n t s t e p i n any s c i e n t i f i c i n v e s t i g a t i o n . T h i s i s t h e p r o c e s s o f v e r i f y i n g t h e o b t a i n e d r e s u l t s a n d s o i n c r e a s i n g t h e c o n f i d e n c e w i t h w h i c h t h e y c a n be a c c e p t e d . S i n c e t h i s i s a n i n t e g r a l p a r t o f a n y s c i e n t i f i c p r o c e d u r e i t w i l l n o t be d i s c u s s e d f u r t h e r i n t h i s w o r k . I t w i l l be • a c c e p t e d a s a p h a s e o f t h e m e t h o d o l o g y w h i c h i s n e c e s s a r y i f we a r e t o o b t a i n t h e maximum p l a u s i b i l i t y f o r t h e f a c t o r s d e -r i v e d i n a n y s p e c i f i c s t u d y . L e t u s , t h e r e f o r e , t u r n o u r a t t e n t i o n t o t h e s u b j e c t m a t t e r o f t h i s c h a p t e r , t h a t i s , t h e i n t e r p r e t a t i o n o f t h e f a c t o r p a t t e r n . C o n c e r n i n g t h e n a t u r e o f f a c t o r a n a l y s i s a n d t h e r e l a t i o n s h i p e x i s t i n g b e t w e e n t h e a n a l y s i s a n d t h e i n t e r p r e t a t i o n o f f a c t o r s , C h a n t ( 3 , p . 2 6 8 ) w r i t e s , I t s y s t e m a t i z e s a b o d y o f d a t a s o t h a t c e r t a i n i n t e r p r e t a t i o n s a r e made p o s s i b l e ; b u t i t d o e s n o t s u p p l y t h e s e i n t e r p r e t a t i o n s . T h u r s t o n e i n d i c a t e s t h i s when he s t a t e s j "By i n s p e c t i o n i t i s p o s s i b l e t o name t h e f i r s t f a c t o r a l -t h o u g h t h e s t a t i s t i c a l p r o c e d u r e s do n o t o f ( 1 3 4 ) c o u r s e c o n c e r n t h e s e m a t t e r s o f d e s c r i b i n g o r n a m i n g t h e f a c t o r s . " S i m i l a r t o o t h e r t y p e s o f s t a t i s t i c a l a n a l y s i s , T h u r s t o n e ' s m e t h o d i s a s p e c i a l t y p e o f c l a s s i f i c a t o r y t e c h n i q u e . I t s v a l u e l i e s i n t h e o r g a n i z a t i o n o f d a t a , r a t h e r t h a n i n t h e e x p l a n a t i o n o f t h e s e d a t a . R e d u c e d t o a s i m p l e s t a t e m e n t t h e m e t h o d c l a s s i f i e s a s e t o f i n t e r c o r r e l a t i o n s a n d on t h i s b a s i s makes p o s s i b l e t h e d e s c r i p t i o n o f c e r t a i n g e n e r a l c a t e g o r i e s o r t y p e s o f b e -h a v i o u r . The f i n a l v a l i d a t i o n o f t h e c l a s s i -f i c a t i o n l i e s , a s T h u r s t o n e s t a t e s , i n t h e e x t e n t t o w h i c h t h e c l a s s i f i c a t i o n i s p s y -c h o l g o c i a l l y [sic] i n t e l l i g i b l e . Thus s o m e t h i n g o v e r a n d a b o v e t h e t e c h n i q u e s o f f a c t o r a n a l y -s i s m u s t be e m p l o y e d a n d t h i s e l e m e n t i s f u r n i s h e d by t h e k n o w l e d g e and acumen o f t h e i n v e s t i g a t o r . T h i s e l e m e n t o f s u b j e c t i v i t y h a s b e e n w i d e l y c r i t i c i z e d b u t T h u r s t o n e ( 1 9 , Pi 144) h a s a n a n s w e r f o r t h e s e c r i t i c s . The a t t e m p t s t o i n t e r p r e t t h e p r i m a r y f a c t o r s h a v e o f t e n b e e n c r i t i c i z e d b e c a u s e t h e y a r e n o t s u f f i c i e n t l y o b j e c t i v e . L e t u s be c l e a r a b o u t two d i s t i n c t p r o b l e m s t h a t a r e i n v o l v e d here.. I t i s one p r o b l e m t o i s o l a t e a p r i m a r y f a c t o r a n d t o d e t e r m i n e by r e p e a t e d e x p e r i m e n t s t h a t i t h a s some f u n c t i o n a l u n i q u e n e s s . T h a t i s f a c t o r a n a l y s i s p r o p e r . I t i s a n o t h e r p r o b l e m t o f i n d t h e p s y c h o l o g i -c a l o r p h y s i o l o g i c a l m e a n i n g o f a f u n c t i o n a l u n i q u e n e s s when i t h a s b e e n d e t e r m i n e d . T h a t i s a m a t t e r o f i n t e r p r e t a t i o n , a n d c o n s e q u e n t -l y i t i s n e c e s s a r i l y s u b j e c t t o d e b a t e , w i t h c o n f l i c t i n g i n t e r p r e t a t i o n s . We m u s t remember t h a t t h e i n t e r p r e t a t i o n o f e v e r y s c i e n t i f i c e x p e r i m e n t i s s u b j e c t i v e . T h e r e i s no k i n d o f s c i e n t i f i c e x p e r i m e n t i n w h i c h t h e i n t e r -p r e t a t i o n r o l l s o u t " o b j e c t i v e l y . " B e c a u s e o f t h i s s u b j e c t i v i t y i t i s n o t p o s s i b l e t o d e v e -l o p r u l e s , w h i c h , i f l e a r n e d by r o t e , w i l l s e r v e t o e n a b l e one t o b l i n d l y i n t e r p r e t t h e r e s u l t s o f a f a c t o r a n a l y s i s . The o n l y way i n w h i c h f a c i l i t y a t i n t e r p r e t a t i o n c a n be o b -t a i n e d i s by p r a c t i c e w i t h a c t u a l e x p e r i m e n t a l d a t a . A t t h e ( 1 3 5 ) same t i m e t h e f a c t o r i s t m u s t w o r k t o d e v e l o p a b r o a d a n d sound k n o w l e d g e o f t h e t h e o r e t i c a l b a s e s o f t h e s c i e n c e w i t h i n whose b o u n d a r i e s he w o r k s . T h i s w i l l s e r v e t o f u r n i s h a f r a m e o f r e f e r e n c e a g a i n s t w h i c h t o e v a l u a t e h i s r e s u l t s a n d i n t e r p r e t a t i o n s . F o r t h i s r e a s o n t h e p r e s e n t c h a p t e r w i l l be d e v o t e d t o p r e s e n t i n g t h e r e s u l t s o f t h r e e a c t u a l f a c t o r i a l s t u d i e s w h i c h h a v e b e e n s e l e c t e d f r o m t h e j o u r n a l s . T h e s e w i l l e n a b l e t h e r e a d e r t o e x a m i n e t h e r e a s o n i n g p r o c e s s e s em-p l o y e d i n i n t e r p r e t i n g s u c h m a t r i c e s . T h e r e w e r e two b a s i c c r i t e r i a e m p l o y e d i n s e l e c t i n g t h e s e i l l u s t r a t i v e s t u d i e s . The f i r s t was t h e c l a r i t y w i t h w h i c h t h e y d e m o n s t r a t e d w h a t m i g h t be t e r m e d e i t h e r a d e q u a t e o r i n a d e q u a t e i n t e r p r e t a t i o n s . I n t h i s r e s p e c t t h e f i r s t two s t u d i e s w e r e s e l e c t e d a s r e p r e -s e n t a t i v e o f r e l a t i v e l y a d e q u a t e f a c t o r i n t e r p r e t a t i o n , w h e r e a s t h e t h i r d was f e l t t o d e m o n s t r a t e f a c t o r n a m i n g w h i c h was i n a d e q u a t e . O v e r a n d abo v e t h i s c r i t e r i o n t h e r e was t h e s e c o n d a r y c o n s i d e r a t i o n t h a t t h e f a c t o r p a t t e r n s i n v o l v e d be r e l a t i v e l y s i m p l e a n d c l e a r c u t i n o r d e r t h a t t h e i n t e r p r e -t a t i v e m e t h o d s c o u l d be more r e a d i l y f o l l o w e d a n d u n d e r s t o o d . B e f o r e p r o c e e d i n g t o a d i s c u s s i o n o f t h e a c t u a l e x a m p l e s l e t u s c o n s i d e r t h e f o l l o w i n g a d v i c e f r o m T h u r s t o n e ( 1 9 , p. 145) c o n c e r n i n g t h e n a m i n g o f f a c t o r s , I n n a m i n g t h e p r i m a r y f a c t o r s i t i s a b e t t e r p o l i c y t o name t h e f a c t o r s i n t e r m s o f w e l l - k n o w n c o n c e p t s , s u c h a s Number, S p a c e , V e r b a l , a n d Memory f a c t o r s t h a n t o name them i n some n o n c o m m i t t a l way, s u c h a s x l f xz a n d s o o n . I f we name a f a c t o r "Number," i t w i l l p r o v o k e e x p e r i m e n t a t i o n w i t h number t e s t s a n d w i t h n o n n u m e r i c a l t e s t s , a n d t h e e x p e r i m e n t s a r e l i k e l y t o be made i n t e r m s o f p s y c h o l o g i c a l ( 1 3 6 ) h y p o t h e s e s t h a t c a n be s u s t a i n e d o r d i s -p r o v e d e x p e r i m e n t a l l y . I n t h i s way we s h a l l a d v a n c e f a s t e r t h a n i f t h e p r i m a r y f a c t o r s a r e l e f t a s i n t e r e s t i n g s t a t i s t i c a l c u r i o s i -t i e s . They s h o u l d be r e c o g n i z e d a s p s y c h o -l o g i c a l l y c h a l l e n g i n g e x p e r i m e n t a l e f f e c t s . A s u r v e y o f t h e l i t e r a t u r e r e v e a l s t h a t , a s w i t h t h e s e l e c t e d e x a m p l e s , t h i s a d v i c e h a s b e e n g e n e r a l l y o b s e r v e d . L e t u s now e x a m i n e t h e i l l u s t r a t i o n s g a r n e r e d f r o m t h e l i t e r a t u r e i n o r d e r t o g a i n a c l e a r e r p i c t u r e o f t h e p r o c e s s e s i n v o l v e d . The f i r s t o f o u r t h r e e e x a m p l e s c o n s i s t s o f t h e r e s u l t s o f a f a c t o r i a l s t u d y c a r r i e d o u t on t h e t e s t " s c o r e s o f f o u r g r o u p s o f C h i n e s e s t u d e n t s . F o r o u r p u r p o s e we w i l l e m p l o y t h e d a t a g a t h e r e d f r o n \ two o f t h e s e g r o u p s . T h e s e two sam-p l e s c o n s i s t r e s p e c t i v e l y o f 140 j u n i o r h i g h s c h o o l s t u d e n t s , a v e r a g e age 14§ y e a r s , a n d 372 h i g h s c h o o l g r a d u a t e s who t h r o u g h o u t t h e s t u d y w i l l be d e s c r i b e d a s c o l l e g e f r e s h m e n . A b a t t e r y o f t e s t s w e r e u s e d w h i c h a r e d e s c r i b e d b y t h e e x -p e r i m e n t e r s ( 4 , p p . 1 8 8 - 1 8 9 ) i n t h e f o l l o w i n g t e r m s . Our b a t t e r y c o n s i s t s o f n i n e - t e s t s . They w e r e w o r d a n a l o g i e s , w o r d c l a s s i f i c a t i o n s , w o r d c o m p l e t i o n s , f l a g t e s t , f o r m c l a s s i f i c a -t i o n s , X-0 s e r i e s , a r i t h m e t i c problems., number s e r i e s , a n d number c o m p l e t i o n s . M o s t o f t h e s e t e s t s a r e w e l l - k n o w n a n d o n l y a s h o r t d e s c r i p -t i o n o f s e v e r a l o f t h e l e s s f a m i l i a r o n e s w i l l h e r e be i n s e r t e d . The w o r d c o m p l e t i o n t e s t i s n o v e l i n C h i n a . C h i n e s e w o r d s c o n s i s t o f r a d i -c a l s i n s t e a d o f l e t t e r s . So a r a d i c a l was g i v e n a n d t h e t e s t e e s w e r e r e q u e s t e d t o com-p l e t e t h e w o r d i n o r d e r t o make c e r t a i n s e n s e . T h i s t e s t i s a C h i n e s e a d a p t a t i o n o f T h u r -s t o n e ' s w o r d c o m p l e t i o n t e s t . The f l a g t e s t was d i r e c t l y c o p i e d f r o m T h u r s t o n e . The f o r m o r f i g u r e c l a s s i f i c a t i o n t e s t i s w e l l - k n o w n i n Spearman's b a t t e r y . The number c o m p l e t i o n t e s t c o n s i s t s o f p r o b l e m s i n t h e f o u r r u l e s o f a r i t h m e t i c . I n e a c h p r o b l e m c e r t a i n f i g -u r e s w e r e t o be f i l l e d a n d t h e p r o b l e m was s o ( 1 3 7 ) c o n s t r u c t e d t h a t o n l y one a n s w e r was p o s s i b l e f o r e a c h o m i s s i o n . One w i l l n o t i c e t h a t t h e n i n e t e s t s f a l l i n t o t h r e e c a t e g o r i e s : t h e f i r s t t h r e e a r e v e r b a l , t h e n e x t t h r e e a r e n o n - v e r b a l , a n d t h e l a s t t h r e e d e a l w i t h n u m b e r s . So we h a v e t h r e e d i f f e r e n t t y p e s o f m a t e r i a l f o r o u r t e s t s . B u t t h e t e s t s c a n a l s o be a r r a n g e d i n -t o p a i r s o f d i f f e r e n t f u n c t i o n s , two c l a s s i -f i c a t i o n t e s t s , two s e r i e s t e s t s - , a n d two com-p l e t i o n t e s t s . The i n t e r c o r r e l a t i o n s b a s e d on t h e t e s t s c o r e s o f t h e g r o u p s a r e r e c o r d e d i n T a b l e I ( 4 , p . 1 9 0 ) a n d T a b l e I I ( 4 , p . 1 9 1 ) . T a b l e I N e t C o r r e l a t i o n s Among T e s t s * ( J u n i o r H.S. G r o u p ) T e s t 1 2 3 4 5 6 7 8 2 37 3 45 38 4 12 25 23 5 30 44 43 43 6 29 44 35 42 5 5 7 30 30 40 26 34 40 8 3 1 30 39 35 36 60 63 9 3 1 39 39 39 38 47 5 4 49 * A 1 1 d e c i m a l s b e i n g o m i t t e d h e r e a s w e l l a s i n a l l t h e t a b l e s f o l l o w i n g . (158) Table II Net Correlations Among Tests (College Freshman Group)* Test 1 2 5 4 5 6 7 8 2 399 3 297 400 4 290 289 300 5 393 479 303 374 6 216 273 144 350 329 7 365 330 253 324 347 233 8 304 276 318 264 372 145 418 9 397 401 344 389 459 265 438 388 *Because the population i n the college group was more numerous, a l l figures had been calculated to three decimal places. The investigators analysed these two matrices by Thur-stone fs Centroid Method and then carried out an orthogonal rotation of axes. The resultant centroid and rotated factor matrices are l i s t e d i n Table III (4, p. 191) and Table IV (4, p. 192). ( 1 3 9 ) T a b l e I I I The F a c t o r M a t r i c e s o f t h e J u n i o r H.S. G r o u p a . C e n t r o i d F a c t o r M a t r i x b . R o t a t e d F a c t o r M a t r i x T e s t I I I I I I h 2 A B C h 2 1 5 1 -36 23 44 37 54 12 44 2 5 7 - 2 1 - 0 9 38 55 28 - 0 5 38 3 61 - 2 7 16 4 7 4 9 46 13 47 4 4 9 14 -25 32 56 -12 00 33 5 66 -12 -30 54 71 12 - 1 4 5 4 6 73 09 - 2 4 - 60 77 00 04 5 9 7 68 28 37 68 54 12 6 1 68 8 72 33 15 65 65 - 0 1 48 65 9 67 12 02 46 63 06 24 46 c . D i r e c t i o n c o s i n e s o f d . C o s i n e s o f A n g l e s R e f e r e n c e V e c t o r s s e p a r a t i n g R e f e r e n c e V e c t o r s A -•: B C A B C I 94 25 24 A 1 0 1 I I 09 - 8 4 55 B - 0 1 1 0 1 I I I -34 49 80 C 00 - 0 1 100 ( 1 4 0 ) T a b l e I V F a c t o r M a t r i c e s o f t h e C o l l e g e F r e s h m a n G r o u p a . C e n t r o i d F a c t o r M a t r i x b . R o t a t e d F a c t o r M a t r i x T e s t I I I h* A B h z 1 5 7 2 1 4 1 347 5 8 7 - 0 0 5 345 2 636 157 429 653 - 0 0 6 426 3 499 118 263 5 1 1 0 0 1 2 6 1 4 547 058 303 5 1 8 182 3 0 1 5 668 034 448 656 1 2 1 445 6 5 5 2 •809 959 349 912 954 7 587 099 354 592 039 352 8 535 200 326 566 - 0 7 7 326 9 675 098 465 678 060 463 c . D i r e c t i o n c o s i n e s o f d . A n g l e s s e p a r a t i n g R e f e r e n c e V e c t o r s R e f e r e n c e V e c t o r s A B A B I 97 23 A 99 I I 23 - 9 7 B 00 99 B e f o r e p r o c e e d i n g t o r e p o r t t h e a u t h o r s 1 i n t e r p r e t a t i o n o f t h e a b o v e two r o t a t e d m a t r i c e s , t h e r e a r e s e v e r a l p o i n t s c o n c e r n i n g t h e s e t a b l e s t h a t s h o u l d b e ^ d i s c u s s e d . We n o t e t h a t e a c h o f t h e f a c t o r t a b l e s c o n t a i n s a n e x t r a c o l u m n h e a d e d h * . T h i s c o l u m n c o n t a i n s t h e v a l u e f o r e a c h t e s t ' s c o m m u n a l i t y o r common f a c t o r v a r i a n c e w h i c h i s o b t a i n e d by summing t h e s q u a r e s o f a l l t e r m s i n t h e g i v e n r o w . T h e s e ( 1 4 1 ) v a l u e s s e r v e a d o u b l e p u r p o s e . F i r s t l y , t h e y show t h e p r o -p o r t i o n o f t h e t o t a l t e s t v a r i a n c e t h a t i s a c c o u n t e d f o r by t h e g i v e n common f a c t o r s y s t e m . T h e n , s e c o n d l y , t h e y e n a b l e u s t o c h e c k t h a t o u r r o t a t e d r e f e r e n c e s y s t e m d e s c r i b e s a p -p r o x i m a t e l y a s l a r g e a p r o p o r t i o n o f t h e t o t a l t e s t v a r i a n c e a s d i d t h e i n i t i a l f a c t o r s . B e l o w e a c h o f t h e f o r e g o i n g f a c -t o r m a t r i c e s a r e r e c o r d e d two s u b - m a t r i c e s . The m a t r i x m a r k e d c , i n e a c h c a s e , i s t h e r o t a t i o n m a t r i x by w h i c h we m u s t p o s t - m u l t i p l y t h e c e n t r o i d l o a d i n g s t o o b t a i n t h e r o -t a t e d f a c t o r s a t u r a t i o n s . The s u b - m a t r i x m a r k e d d , i n e a c h o f t h e f o r e g o i n g t a b l e s , c o n t a i n s t h e n u m e r i c a l v a l u e o f t h e c o s i n e s o f t h e a n g l e s s e p a r a t i n g t h e v a r i o u s p a i r s o f r e f e r -e n c e a x e s . The a x e s s o d e s i g n a t e d a r e l i s t e d a t t h e t o p o f t h e c o l u m n a n d t o t h e l e f t o f t h e r o w i n w h i c h t h e number o c c u r s . T h e s e e n t r i e s a r e o b t a i n e d by p r e - m u l t i p l y i n g m a t r i x c b y i t s t r a n s p o s e . T h i s m a t r i x f u r n i s h e s u s w i t h i n f o r m a -t i o n c o n c e r n i n g t h e o r t h o g o n a l i t y o r l a c k o f o r t h o g o n a l i t y i n t h e new r o t a t e d a x i s s y s t e m . I n s p e c t i o n o f t h e two t a b l e s shows t h a t a l l t h e v a l u e s a r e e i t h e r a p p r o x i m a t e l y z e r o o r u n i t y . S i n c e , i f c o s t f = 0 , t h e n (0=90° a n d i f c o s < P = l , t h e n cp=O a, t h e s e e n t r i e s t e l l u s t h a t t h e r e f e r e n c e s y s t e m s a r e o r t h o g o n a l . The two e x p e r i m e n t e r s ( 4 , p. 1 9 3 ) p r o c e e d e d t o i n t e r p r e t t h e r o t a t e d f a c t o r l o a d i n g s f o r t h e j u n i o r h i g h s c h o o l g r o u p a s f o l l o w s . The n a t u r e o f t h e r o t a t e d f a c t o r s o f t h e j u n i o r h i g h s c h o o l p u p i l s i s o b v i o u s e n o u g h . They a r e ( a ) t h e g e n e r a l f a c t o r , i n t h e n a t u r e ( 1 4 2 ) o f S p earman's g ( b ) t h e v e r b a l f a c t o r V, a n d ( c ) t h e number f a c t o r N. They a r e r a t h e r c l e a r a n d c l e a n c u t . T h e s e t h r e e f a c t o r s made u s b e l i e v e t h a t i t i s t h e m a t e r i a l a n d n o t t h e f u n c t i o n t h a t d e t e r m i n e s t h e n a t u r e o f a f a c t o r . Thus t h e f i r s t t h r e e t e s t s w h i c h a r e c h a r a c t e r i z e d by t h e i r common v e r -b a l c o n t e n t h a v e a common f a c t o r V a n d t h e l a s t t h r e e t e s t s w h i c h a l l d e a l w i t h n u m bers h a v e a number f a c t o r N i n common. B u t t h e r e i s no c l a s s i f i c a t i o n f a c t o r , n o c o m p l e t i o n f a c t o r , a n d n o i n d u c t i o n f a c t o r ( f o r t h e s e r i e s t e s t s ) t h o u g h t h e y a r e s u c h common f u n c t i o n s . T h a t we d i d n o t f i n d a p e r c e p t u a l f a c -t o r P f o r t h e s e c o n d g r o u p o f t e s t s i s n o t s u r p r i s i n g . I t i s q u i t e i n harmony w i t h S p e a r m a n ' s v i e w , who w o u l d r e g a r d s u c h t e s t s a s p u r e l y n e o g e n e t i c . I f one s c r u t i n i z e s t h e n a t u r e o f t h e t e s t s i n t h i s g r o u p i n d i -v i d u a l l y , one a t o n c e s e e s t h a t t h e s e t h r e e t e s t s a r e n o t e x a c t l y t h e same. The f i r s t i s s p a t i a l , t h e s e c o n d b e i n g o f f o r m c l a s s i -f i c a t i o n , a n d t h e t h i r d b e i n g o f X-0 s e r i e s a r e e a c h u n i q u e . S o we c a n n o t e x p e c t a g r o u p f a c t o r r u n n i n g t h r o u g h them. So a p a p e r f o r m - b o a r d t e s t was i n t r o d u c e d t o t a k e t h e p l a c e o f t h e X-0 s e r i e s . We s h a l l s e e l a t e r t h a t we w e r e f u l l y j u s t i f i e d i n d o i n g s o . The i n t e r p r e t a t i o n s e m p l o y e d i n t h i s p o r t i o n o f t h e s t u d y a r e f u r t h e r v a l i d a t e d b y t h e f a c t t h a t s u c h f a c t o r s h a v e b e e n i s o l a t e d f r o m b a t t e r i e s o f t h i s n a t u r e i n many e x p e r i m e n t s . The f a c t t h a t a g e n e r a l f a c t o r a p p e a r s d o e s n o t c o m p l e t e l y a g r e e w i t h T h u r s t o n e ' s r e s u l t s . A l t h o u g h he d i d g e t a g e n e r -a l f a c t o r i n a g r o u p c o m p a r a b l e w i t h t h i s i n a g e , t h e f a c t o r he o b t a i n e d was a s e c o n d o r d e r one w h i c h o c c u r r e d when t h e c o r r e l a t i o n s b e t w e e n p r i m a r y f a c t o r s w e r e a n a l y s e d . The r e -s u l t s a r e n o t i n c o m p a t i b l e , h o w e v e r , s i n c e t h e b a t t e r y h e r e e m p l o y e d i s v e r y l i m i t e d i n s c o p e a n d w h a t may a p p e a r a s a g e n e r a l f a c t o r c o u l d w e l l be a common f a c t o r i n a l a r g e r ( 1 4 3 ) b a t t e r y . The p a p e r f o r m - b o a r d was s u b s t i t u t e d f o r t h e X-0 t e s t when t e s t i n g t h e c o l l e g e g r o u p . The r o t a t e d f a c t o r s r e c o r d e d i n T a b l e I V w e r e i n t e r p r e t e d ( 4 , p . 1 9 4 ) a s f o l l o w s . The f a c t o r p a t t e r n o f t h e c o l l e g e g r o u p i s e v e n more s i m p l e . I f n o t f o r t h e s i x t h t e s t , t h a t o f t h e p a p e r f o r m - b o a r d , one g e n -e r a l f a c t o r seems e n o u g h t o a c c o u n t f o r t h e w h o l e c o r r e l a t i o n t a b l e . As t h e c e n t r o i d f a c t o r s r e v e a l , t h e s e c o n d f a c t o r w o u l d be o f n o s i g n i f i c a n c e , i f n o t f o r t h e p r o m i n e n t n e g a t i v e l o a d i n g s o f t h e s i x t h t e s t . The r o t a t e d f a c t o r s t e l l much t h e same t h i n g . A s p a t i a l f a c t o r S w i t h t h e o n l y l a r g e l o a d i n g , f r o m T e s t 6 i s t h e o n l y f a c t o r r u n -n i n g b e s i d e t h e d o m i n a n t g. We f e e l r a t h e r s a t i s f i e d w i t h o u r ' i n t e r p r e t a t i o n , b e c a u s e T e s t 6 h a s a p r o m i n e n t l o a d i n g - i n f a c t , t h e h e a v i e s t - i n t h i s f a c t o r w i t h a l l o u r g r o u p s . I t h a s a l o a d i n g o f 0.92 w i t h t h e s e n i o r h i g h s c h o o l p u p i l s a n d t i e s f o r f i r s t p l a c e w i t h T e s t 4 i n t h e p r i m a r y s c h o o l g r o u p . The e x c e p t i o n o f t h e j u n i o r s c h o o l g r o u p s e r v e s w e l l t o p r o v e t h e r u l e . A s was p o i n t e d o u t e a r l i e r , t h e s i x t h t e s t i n t h i s g r o u p i s o f a d i f f e r e n t n a t u r e , t h a t o f X-0 s e r i e s . Now a s t h e f o r m - b o a r d t e s t i s t h e m a i n d e t e r m i n e r o f t h e s p a t i a l f a c t o r , i t s a b s e n c e may t h u s be v e r y w e l l e x p l a i n e d . H ence we deem t h a t o u r p i c t u r e f i t s n i c e l y a l l t h e f a c t s we g a t h e r e d . The f o r e g o i n g i n t e r p r e t a t i o n s seem q u i t e a d e q u a t e a n d s e r v e t o i l l u s t r a t e t h e g e n e r a l m e t h o d e m p l o y e d i n n a m i n g f a c t o r s . T h i s c o n s i s t s i n r e t u r n i n g t o t h e t e s t s t h a t e x -h i b i t m a r k e d p o s i t i v e o r n e g a t i v e l o a d i n g s on t h e f a c t o r s a n d s t u d y i n g t h e n a t u r e o f t h e s e t e s t s . T h e r e d o e s seem t o be one i n a d e q u a c y i n t h e i n t e r p r e t a t i o n o f t h e s p a t i a l f a c t o r s i n c e i t i s named on r a t h e r s l i g h t e v i d e n c e . R e g a r d i n g t h i s p o i n t T h u r s t o n e ( 1 8 , p. 75) s a y s , ( 1 4 4 ) No m e a n i n g f u l component c a n be i d e n t i f i e d u n -l e s s e a c h f a c t o r i s o v e r d e t e r m i n e d w i t h t h r e e o r f o u r o r more t e s t s . I n o r d e r t o make a p s y c h o l o g i c a l i d e n t i f i c a t i o n o f a common f a c -t o r i t i s e s s e n t i a l t o h a v e a number o f d i f -f e r e n t t e s t s t h a t i n v o l v e t h e f a c t o r . O n l y i n t h i s manner c a n we h a v e c o n f i d e n c e i n i d e n t i f y i n g t h e n a t u r e o f e a c h common f a c t o r . M e r e l y t o know t h a t c e r t a i n t e s t s h a v e one o r more f a c t o r s i n common i s n o t s a t i s f a c t o r y f o r p s y c h o l o g i c a l i n t e r p r e t a t i o n . We w a n t t o know t h e p s y c h o l o g i c a l n a t u r e o f e a c h f a c t o r . The p u r p o s e a n d t h e t e n t a t i v e h y p o t h e s i s t h a t o c c a s i o n e d t h e s t u d y w h i c h c o n s t i t u t e s o u r s e c o n d e x a m p l e c a n b e s t be e x p r e s s e d i n t h e a u t h o r ' s own w o r d s ( 2 0 , p p . 3 9 5 - 3 9 6 ) . The p r e s e n t p a p e r i s c o n c e r n e d c h i e f l y w i t h " m o t o r " a b i l i t i e s , p a r t i c u l a r l y t h o s e w h i c h may be c a l l e d " m a n u a l . " I t i s b a s e d p r i m a r i l y on d a t a f r o m t h e E x p e r i m e n t P r o p e r o f t h e M i n n e s o t a M e c h a n i c a l A b i l i t y p r o g r a m o f r e s e a r c h . . . . The 16 v a r i a b l e s . . . w e r e s e l e c t e d w i t h c e r t a i n e x p e c t a t i o n s . I t was b e l i e v e d , f o r e x a m p l e , t h a t t h e p a t t e r n o f t h e i r i n t e r -c o r r e l a t i o n s m i g h t c o n f i r m t h e t e n d e n c y f o r m e a s u r e s i n v o l v i n g a h i g h d e g r e e o f m a n u a l d e x t e r i t y t o f o r m a n i n d e p e n d e n t f u n c t i o n a l c l a s s i f i c a t i o n . I t was e x p e c t e d , m o r e o v e r , t h a t a n a n a l y s i s o f t h e s e l e c t e d i n t e r c o r -r e l a t i o n s w o u l d c o n t r i b u t e t o o u r u n d e r -s t a n d i n g o f t h e g e n e r a l n a t u r e o f t h e d e x -t e r i t y f a c t o r . I t seemed t o be p a r t i c u l a r l y d e s i r a b l e t o know t o wh a t d e g r e e m e a s u r e s o f s t r e n g t h a n d p h y s i c a l d e v e l o p m e n t c o n t r i b u t e t o a n a b i l i t y s u c h a s m a n u a l d e x t e r i t y . The t a b l e o f c o r r e l a t i o n s ( 2 0 , p. 3 9 7 ) i s l i s t e d i n T a b l e V. (145) T a b l e V* I n t e r c o r r e l a t i o n s o f 16 S e l e c t e d V a r i a b l e s ( N = 1 0 0 ) i . w -p </> * - <° f I t i . 5s ^ p. a. 3 4 5 6 7 8 9 11 12 14 16 18 19 20 25 37 3 4 52 5 34 23 6 14 14 63 7 18 10 42 37 8 21 24 39 30 54 9 30 13 56 49 46 40 11 09 13 11 -01 25 11 15 IS -03 -04 -05 -10 08 -12 04 66 14 -06 -06 -09 -09 15 -10 06 70 84 16 -07 03 -01 03 16 -02 04 54 50 60 18 -02 -09 01 -08 16 -11 -01 48 58 60 72 19 -09 -11 -01 -05 18 -04 04 59 67 68 74 78 20 26 19 53 52 24 31 55 04 05 02 03 -09 02 25 00 -12 22 24 24 19 40 11 15 09 04 09 10 30 37 12 09 46 " 39 32 28 42 08 09 04 03 - o i 03 64 30 The d e c i m a l p o i n t s i n t h i s t a b l e w e r e n o t o m i t t e d i n t h e o r i g i n a l t e x t . ( 1 4 6 ) T h i s m a t r i x was f a c t o r e d by t h e c e n t r o i d m e t h o d . The r e s u l t i n g m a t r i x ( 2 0 , p. 5 9 8 ) i s r e c o r d e d i n T a b l e V I . T a b l e V I C e n t r o i d M a t r i x 4 I I I I I I I V h* 3 P a c k i n g B l o c k s .31 ..33 .39 .36 .49 4 C a r d S o r t i n g .24 .26 -.45 .42 .50' 5 M i n n . S p a t i a l .56 .52 -.12 .12 .61 R e l a t i o n s 6 P a p e r Form .46 .48 -.25 .09 .51 B o a r d 7 S t e n q u i s t .56 .23 .12 -.21 .43 P i c t u r e I 8 S t e n q u i s t .40 .43 .23 -.15 .42 P i c t u r e I I 9 M i n n . A s s e m b l y .59 .45 -.10 -.16 .58 1 1 B a c k dynamometer .61 -.47 .27 -.09 .67 12 R i g h t - B a n d .53 -.64 .10 -.17 .73 dynamometer .80 14 L e f t - H a n d .54 -.68 .13 -.18 dynamometer 16 S p i r o m e t e r .53 -.58 -.16 .23 .70 18 H e i g h t .49 -.64 -.17 .19 .71 19 W e i g h t .55 -.67 -.17 .08 .79 20 ' Shop o p e r a t i o n s .55 .47 -.25 .01 .59 q u a l i t y c r i t e r i o n -.35 .32 25 S o n ' s mech. .35 .14 -.23 o p e r a t i o n s -.14 .46 37 I n t e r e s t A n a l y s i s .50 .37 -.22 B l a n k (new) By means o f t h e t r a n s f o r m a t i o n m a t r i x o f T a b l e V I I ( 2 0 , p. 3 9 8 ) , W i t t e n b o r n r o t a t e d t h e a x e s a n d o b t a i n e d t h e m a t r i x r e c o r d e d i n T a b l e V I I I ( 2 0 , p . 3 9 8 ) . ( 1 4 7 ) T a b l e V I I T r a n s f o r m a t i o n M a t r i x I I I I I I I V I .58 -.72 -.08 .39 I I .39 -.18 .34 -.83 I I I .67 .59 -.44 .00 I V .25 .32 .82 .39 T a b l e V I I I R o t a t e d F a c t o r M a t r i x I I I I I I I V h * 3 P a c k i n g B l o c k s .05 -.11 .23 .65 .49 4 C a r d S o r t i n g .07 -.16 .11 .67 .49 5 M i n n . S p a t i a l .02 -.01 .74 .26 .62 R e l a t i o n s 6 P a p e r Form -.02 -.07 .70 .10 .51 .Board 7 S t e n q u i s t .06 .39 .47 .23 .43 P i c t u r e I 8 S t e n q u i s t -.17 .28 .42 .37 .42 P i c t u r e I I 9 M i n n . A s s e m b l y -.03 .25 .71 .15 .58 1 1 B a c k dynamometer .61 .48 .00 .18 .63 12 R i g h t - H a n d .69 .49 -.06 -.07 .73 dynamometer 14 L e f t - H a n d .72 .52 -.10 -.05 .80 dynamometer 16 S p i r o m e t e r .83 .07 .05 -.10 .71 18 H e i g h t .82 .05 .03 -.16 .70 19 W e i g h t .84 .20 .05 -.19 .79 20 Shop o p e r a t i o n s .00 .03 .75 -.19 .60 q u a l i t y c r i t e r i o n 25 S o n ' s mech. .03 .32 .41 -.15 .29 o p e r a t i o n s 37 I n t e r e s t A n a l y s i s - .02 .17 .66 .01 .47 B l a n k (new) ( 1 4 8 ) • A t t h i s p o i n t W i t t e n b o r n ( 2 0 , p p . 3 9 7 - 5 9 9 ) r e m a r k s t h a t , A l t h o u g h a n o r t h o g o n a l s o l u t i o n i s g i v e n t o t h e p r e s e n t p r o b l e m , i t i s a p p a r e n t t h a t F a c t o r s I a n d I I a r e n o t t r u l y i n d e p e n d e n t . The v a r i a b l e s w h i c h c l u s t e r t o g e t h e r t o f o r m F a c t o r I I h a v e h i g h e r l o a d i n g s on F a c t o r I t h a n on F a c t o r I I . I t i s a p p a r e n t , t h e r e f o r e , t h a t p r e s e n t a t i o n o f F a c t o r I I a s a f a c t o r i n -d e p e n d e n t o f F a c t o r I i s n o t i n s t r i c t c o n -f o r m a n c e w i t h t h e n u m e r i c a l s o l u t i o n i n t h e p r e s e n t s t u d y . As t h e f a c t o r s a r e d i s c u s s e d v a r i a b l e by v a r i a b l e , t h e d a t a w i l l be p r e -s e n t e d i n t h e f o r m o f f a c t o r i a l e q u a t i o n s . The a u t h o r ' s u s e o f w h a t he t e r m s f a c t o r i a l e q u a t i o n s i s w o r t h y o f c l o s e r e x a m i n a t i o n b e f o r e we p r o c e e d w i t h h i s r e -p o r t e d i n t e r p r e t a t i o n s . A f a c t o r i a l e q u a t i o n i s s e t up by l i s t i n g t h o s e v a r i a b l e s t h a t h a v e m a r k e d l o a d i n g s on a g i v e n f a c t o r . H o w e v e r , t h e e n t r i e s e m p l o y e d i n t h e f a c t o r c o l u m n s a r e t h e s q u a r e s o f t h e i n d i v i d u a l f a c t o r l o a d i n g s . Thus t h e c e l l e n t r i e s i m m e d i a t e l y c o n v e y i n f o r m a t i o n c o n c e r n i n g t h e p r o p o r t i o n o f a g i v e n t e s t ' s v a r i a n c e t h a t i s due t o t h e f a c t o r whose number i s a t . t h e h e a d o f t h a t c o l u m n . A n o t h e r c o l u m n i s a d d e d t o t h e t a b l e a n d d e n o t e d a s U*. T h i s c o n -t a i n s t h e v a l u e f o r e a c h t e s t ' s u n i q u e n e s s a n d s o i n d i c a t e s t h e p r o p o r t i o n o f t h e t e s t ' s v a r i a n c e t h a t i s n o t e x p l a i n e d by o u r common f a c t o r ' r e f e r e n c e s y s t e m . The v a l u e f o r U* i s o b t a i n e d by s u b t r a c t i n g t h e t e s t ' s c o m m u n a l i t y f r o m u n i t y . L e t u s now p r o c e e d w i t h W i t t e n b o r n ' s i n t e r p r e t a t i o n ( 2 0 , p p . 3 9 9 - 4 0 1 ) o f t h e r o t a t e d f a c t o r s y s t e m . F a c t o r I a p p e a r s t o be a s i z e o r m a t u r a -t i o n a l f a c t o r . I t i s d e t e r m i n e d p r i m a r i l y by v a r i a b l e s 1 6 , 1 8 , a n d 1 9 . (149) F a c t o r I - S i z e I I I I I I I V U* 11 B a c k dynamometer .37... .23 .00 .03 .37 12 R i g h t - H a n d .48 .24 .00, .00 .28 dynamometer 14 L e f t - H a n d .52 .27 (-).OL. .00 .20 dynamometer (-).oi 16 S p i r o m e t e r .69 .00 .00 .30 18 H e i g h t .67 .00 .00 (").03 .30 19 W e i g h t .71 .04 .00 (").04 .21 A p p r o x i m a t e l y 70 o f e a c h o f t h e s e p e r c e n t v a r i a b l e s o f t h e t o t a l i s f o u n d i n v a r i a n c e F a c t o r I a n d no s i g n i f i c a n t amount o f v a r i a n c e i s c o n -t r i b u t e d b y t h e s e v a r i a b l e s t o a n y o t h e r f a c t o r . T e s t s 11, 12, a n d 14, w h i c h s u g g e s t a s t r e n g t h f a c t o r , F a c t o r I I , a c t u a l l y h a v e m o s t o f t h e i r common f a c t o r v a r i a n c e a n d a p p r o x i m a t e l y 50 p e r c e n t o f t h e i r t o t a l v a r i a n c e i n F a c t o r I . T h i s f i n d i n g i s o f i n t e r e s t b e c a u s e i n a n a n a l y s i s o f d a t a f o r 328 y o u t h s who w e r e o l d e r t h a n t h e p r e s e n t g r o u p i t h a s b e e n f o u n d t h a t s t r e n g t h a n d s i z e a r e i n d e p e n d e n t o f e a c h o t h e r . B e c a u s e o f t h i s , F a c t o r I a n d F a c t o r I I a r e t r e a t e d i n t h e p r e s e n t s t u d y a s i n d e p e n d -e n t o f e a c h o t h e r . The w r i t e r o f f e r s a s a d d i -t i o n a l j u s t i f i c a t i o n f o r t h i s t r e a t m e n t t h e c o n s i d e r a t i o n t h a t no. a d d i t i o n a l u n d e r s t a n d i n g o f t h e o r g a n i z a t i o n o f t h e v a r i a b l e s w o u l d r e -s u l t f r o m r i g o r o u s l y d e f i n i n g F a c t o r I I a s h i g h l y c o r r e l a t e d w i t h F a c t o r I . ( S i n c e t h e d a t a o f t h e p r e s e n t study- d o - n o t c a l l f o r a s t r e n g t h f a c t o r i n d e p e n d e n t o f t h e s i z e f a c t o r , t h i s i n d e p e n d e n c e c a n o n l y be c o n s i d e r e d as" -h y p o t h e t i c a l . I t i s r e a s o n a b l e t o f i n d s i z e -a n d s t r e n g t h h i g h l y c o r r e l a t e d among y o u n g b o y s a n d t o e x p e c t t h e s e v a r i a b l e s t o become i n c r e a s i n g l y i n d e p e n d e n t a s m a t u r a t i o n i s a t t a i n e d . I t i s h o p e d t h a t t h e r e s u l t s o f t h i s s t u d y w i l l h a v e i m p l i c a t i o n s f o r t h e . u s e o f c e r t a i n t y p e s o f t e s t s i n t h e s e l e c t i o n a n d g u i d a n c e o f y o u n g a d u l t s . I t i s h y p o t h e s i z e d , t h e r e f o r e , t h a t f o r s u c h y o u n g a d u l t g r o u p s b o d y s i z e a n d s t r e n g t h o f t h e u p p e r p a r t s o f t h e b o d y a r e r e l a t i v e l y i n d e p e n d e n t o f e a c h o t h e r . (150) F a c t o r I I , t h e p o s t u l a t e d s t r e n g t h f a c t o r , i s o f c o n s i d e r a b l e i n t e r e s t i n t h e p r e s e n t s t u d y b e c a u s e i t s v a r i a b l e s do n o t c o n t r i b u t e t o t h e m a n u a l d e x t e r i t y f a c t o r , F a c t o r I V . F a c t o r I I -r S t r e n g t h I I I I I I I V U* 11 B a c k ;37 .23 .00 .03 .37 dynamometer 12 R i g h t - H a n d . .48 .24 .00 .00 .28 dynamometer ( - ) . o i 14 L e f t - H a n d .52 .27 .00 .20 dynamometer The c o n t r i b u t i o n w h i c h F a c t o r I I makes t o c e r -t a i n o t h e r v a r i a b l e s s u c h a s Son's M e c h a n i c a l O p e r a t i o n s v a r i a b l e a n d t h e S t e n q u i s t A s s e m b l y t e s t s i s m e a n i n g f u l i n s o f a r a s s t r e n g t h o f ha n d s among b o y s w o u l d be e x p e c t e d t o be a s s o -c i a t e d w i t h t h e u s e o f t h e h a n d s e i t h e r a s i n -d i c a t e d d i r e c t l y by t h e Son's o p e r a t i o n s q u e s t i o n n a i r e o r i n d i r e c t l y by t h e S t e n q u i s t A s s e m b l y t e s t s w h i c h s a m p l e m e c h a n i c a l know-l e d g e . The. f a c t t h a t v a r i a b l e s 3 a n d 4, t h e m a n u a l - d e x t e r i t y v a r i a b l e s , do n o t c o n t r i b u t e t o t h i s f a c t o r i n a n y way i s t a k e n a s a d d i -t i o n a l e v i d e n c e t h a t m a n u a l d e x t e r i t y i s a c l a s s i f i c a t i o n o f a b i l i t y q u i t e i n d e p e n d e n t o f o t h e r t y p e s o f m a n u a l a b i l i t y . F a c t o r I I I , t h e s p a t i a l r e l a t i o n s f a c t o r i s d e f i n e d b y t h e M i n n e s o t a M e c h a n i c a l A s s e m b l y T e s t , t h e M i n n e s o t a P a p e r Form B o a r d T e s t a n d t h e M i n n e s o t a S p a t i a l R e l a t i o n s T e s t . F a c t o r I I I - S p a t i a l V i s u a l i z a t i o n I I I I I I I V 5 M i n n . S p a t i a l .00 .00 .55 .07 .38 R e l a t i o n s 6 P a p e r Form B o a r d .00 .00 .49 .01 .50 7 . S t e n q u i s t .00 .15 .22 .05 .58 P i c t u r e I (-).03 8 S t e n q u i s t .08 .18 .14 .57 P i c t u r e I I 9 M i n n . A s s e m b l y .00 .06 .37 .02 .55 20 S h o p o p e r a t i o n s .00 .00 .56 H . 0 4 .40 q u a l i t y c r i t e r i o n 2.5 .'Son's mech. .00 .10 .17 (-).02 .71 o p e r a t i o n s .53 37 I n t e r e s t A n a l y s i s .00 .03 .44 .00 B l a n k (new) (151) I t i s m o s t i n t e r e s t i n g t o o b s e r v e t h a t v a r i -a b l e 20, t h e Shop O p e r a t i o n s C r i t e r i o n , h a s • a l l o f i t s common f a c t o r v a r i a n c e and. o v e r 50 p e r c e n t o f i t s t o t a l v a r i a n c e i n t h i s p a r -t i c u l a r f a c t o r . I n a d d i t i o n , m e c h a n i c a l i n -t e r e s t s , 37, a n d S o n ' s O p e r a t i o n s i n m e c h a n i -c a l a c t i v i t i e s , 25, a r e a l s o h i g h l y c o r r e l a t e d w i t h t h i s f a c t o r . The i m p o r t a n c e o f m e a s u r e s o f s p a t i a l a b i l i t y a s i n d i c e s o f m e c h a n i c a l p r o m i s e i s s t r i k i n g l y i n d i c a t e d by t h e n a t u r e o f t h i s f a c -t o r . N o t o n l y t h e c r i t e r i o n b u t i n t e r e s t i n m e c h a n i c a l a c t i v i t i e s a p p e a r s t o be q u i t e i n -d e p e n d e n t o f t h e t h r e e a d d i t i o n a l f a c t o r s w h i c h a p p e a r i n t h i s s t u d y a n d w h i c h m i g h t on a n a p r i o r i b a s i s be e x p e c t e d t o c o n t r i b u t e t o a s h o p o p e r a t i o n s c r i t e r i o n o f m e c h a n i c a l a b i l i t y . F a c t o r I V i s p e r h a p s t h e m o s t i n t e r e s t i n g f a c t o r i n t h e p r e s e n t s t u d y . F a c t o r I V - M a n u a l D e x t e r i t y I I I I I I I V U a 3 P a c k i n g B l o c k s .00 ( - ) .o i .05 .42' .52 4 C a r d S o r t i n g .00 (-).03 .01 • 4 5 .51 I t i s d e f i n e d by two t e s t s w h i c h a p p e a r t o c a l l f o r a t y p e o f m a n u a l d e x t e r i t y . H o w e v e r , t h e s e t e s t s do n o t c o n t r i b u t e t o t h e s p a t i a l - v i s u a -l i z i n g f a c t o r w h i c h i n t h e l i g h t o f t h i s s t u d y i s t h e m e c h a n i c a l - a b i l i t y f a c t o r . P e r h a p s more s u r p r i s i n g i s t h e f a c t t h a t n e i t h e r t h e s t r e n g t h n o r t h e s i z e f a c t o r s c o n t r i b u t e i n any way t o f a c i l i t y i n m a n u a l d e x t e r i t y a s i d e n t i -f i e d b y t h i s f a c t o r . A l t h o u g h m a n u a l d e x -t e r i t y i s a n a b i l i t y w h i c h h a s l o n g b e e n c o n -s i d e r e d a s a d e f i n a b l e a t t r i b u t e , p r i o r t o t h e i n v e s t i g a t i o n s o f t h i s s e r i e s i t s e x i s t e n c e a s a f u n c t i o n a l c l a s s i f i c a t i o n o f a b i l i t y h a d n o t b e e n s a t i s f a c t o r i l y d e m o n s t r a t e d . As a m a t t e r o f f a c t t h e d a t a p r e s e n t e d i n t h e p r e s e n t a n d i n t h e p r e c e d i n g s t u d y may n o t be r e g a r d e d a s a d e q u a t e t o d e f i n e s a t i s f a c t o r i l y a n a b i l i t y s u c h a s m a n u a l d e x t e r i t y . T h i s r e s e r v a t i o n i s r e a s o n a b l e s i n c e b l o c k p a c k i n g a n d c a r d s o r t i n g w e r e p r i n c i p l e v a r i a b l e s i n d e f i n i n g t h i s f a c -t o r i n b o t h o f t h e s e s t u d i e s . A l t h o u g h t h e s t u d i e s w e r e done on two s a m p l e s a n d t h e f a c t o r o c c u r s i n two d i f f e r e n t t e s t b a t t e r i e s , i t s e x i s t e n c e r e q u i r e s f u r t h e r d e m o n s t r a t i o n . ( 1 5 2 ) The p r e c e d i n g e x a m p l e was c h o s e n a s b e i n g r e p r e s e n t a t i v e o f a d e q u a t e f a c t o r i n t e r p r e t a t i o n . T h e n , t o o , i t was f e l t t h a t t h i s s t u d y was e x c e p t i o n a l l y v a l u a b l e f o r i l l u s t r a t i v e p u r p o s e s s i n c e t h e a u t h o r s e t down i n d e t a i l t h e r e a s o n i n g u n d e r l y i n g t h e f a c t o r names t h a t w e r e e m p l o y e d . Our l a s t e x a m p l e was c h o s e n a s one w h i c h i l l u s t r a t e d w h a t m i g h t be t e r m e d i n a d e q u a t e i n t e r p r e t a t i o n . The t o t a l s t u d y , o f w h i c h t h e s e f a c t o r i n t e r p r e t a t i o n s f o r m a p a r t , a p p e a r s t o be a n a d e q u a t e p i e c e o f r e s e a r c h . H o w e v e r , t h e manner i n w h i c h t h e f a c t o r s w e r e named i n t h i s p h a s e o f t h e e x p e r i m e n t d o e s n o t seem t o o s a t i s f a c t o r y . The a u t h o r s ( 2 1 , p . 18) e x p r e s s -the p u r p o s e o f t h a t p o r t i o n o f t h e s t u d y i n w h i c h we a r e i n t e r e s t e d a s f o l l o w s , "To d e s c r i b e t h e i n t e r -n a l d y n a m i c s o f R e v i s e d E x a m i n a t i o n M ( F o rm A) i n t e r m s o f p s y c h o l o g i c a l l y m e a n i n g f u l o r t h o g o n a l f a c t o r s . " C o n c e r n i n g t h e s a m p l e e m p l o y e d , t h e r e p o r t ( 2 1 , p . 19) s a y s , "A s a m p l e o f b% o f a l l Army p e r s o n n e l t e s t e d i n C a nada d u r i n g M a r c h a n d A p r i l 1942 was d r a w n f r o m a v a i l a b l e r e c o r d s a t r a n d o m . A p o r t i o n o f t h i s s a m p l e was u s e d i n c a l c u l a t i n g c o r r e l a t i o n s a n d f a c t o r l o a d i n g s . (N= 6 2 5 ) . " The o b t a i n e d c o r r e l a t i o n c o e f f i c i e n t s ( 2 1 , p. 19) a r e l i s t e d i n T a b l e I X . ( 1 5 3 ) T a b l e I X I n t e r c o r r e l a t i o n s o f S u b - T e s t s — R e v i s e d . E x a m i n a t i o n M S u b - T e s t 1 2 3 4 5 6 7 8 1 .557 .462 .434 .500 .474 .509 .523 2 .532 .502 .587 .541 .584 .608 3 .479 .550 .556 .475 .598 4 .752 .486 .534 .581 5 .587 .661 .665 6 .694 .696 7 .684 8 T h i s m a t r i x was f a c t o r e d by t h e c e n t r o i d m e t h o d a n d t h e f a c t o r m a t r i x (21,- p . 2 0 ) r e c o r d e d i n T a b l e X was o b t a i n e d . T a b l e X ' T e s t F a c t o r s I I I I I I 1 .649 -.116 -.211 2 .744 - .114 -.211 3 .689 - . 0 6 4 -.139 4 .739 .391 -.042 5 .843 .345 .045 6 .782 -,,215 .219 7 .801 - . 0 8 6 .250 8 .836 - . 1 1 4 .074 T o t a l 6.081 1.445 1.191 V a r i a n c e 4.654 .368 .228 % V a r i a n c e 58.2 4.6 2.9 ( 1 5 4 ) The r e p o r t ( 2 1 , p . 20) f u r n i s h e s t h e f o l l o w i n g i n t e r -p r e t a t i o n o f t h i s f a c t o r p a t t e r n . C l e a r l y t h e f a c t o r s p r e s e n t e d h e r e c a n n o t be r e g a r d e d a s c o r r e s p o n d i n g t o p s y c h o l o g i c a l f u n c t i o n s commonly d e f i n e d by a s i n g l e o r s i m p l e d e s c r i p t i v e t e r m . The number o f t e s t s i n t h e b a t t e r y i s i n s u f f i c i e n t t o d e f i n e t h e f a c t o r s c l e a r l y i n s u c h t e r m s . None the. l e s s a n i n t e r p r e t a t i o n c a n be made. (a) F a c t o r I i s p o s i t i v e t h r o u g h o u t a n d i s r e g a r d e d a s g e n e r a l i n t e l l e c t u a l a b i l i t y . ( b) F a c t o r I I h a s b o t h p o s i t i v e a n d n e g a t i v e l o a d i n g s , w h i c h r e p r e s e n t d e v i a t i o n s f r o m a n a v e r a g e p r a c t i c a l - m e c h a n i c a l c o m p o n e n t . ( c ) F a c t o r I I I a l s o h a s p o s i t i v e a n d n e g a -t i v e l o a d i n g s , r e p r e s e n t i n g d e v i a t i o n s f r o m a n a v e r a g e c u l t u r a l o r e d u c a t i o n a l c o m p o n e n t . S i n c e t h e o p i n i o n h a s b e e n e x p r e s s e d t h a t t h e f o r e g o i n g i n t e r p r e t a t i o n c a n n o t be c o n s i d e r e d t o o a d e q u a t e , i t i s n e c e s s a r y t h a t we now e x a m i n e t h e r e a s o n s f o r t h i s s t a t e m e n t . F i r s t l y , t h e a u t h o r s n e i t h e r d e s c r i b e n o r name_ t h e s u b - t e s t s e m p l o y e d i n n a m i n g t h e f a c t o r s a n d i n f a c t g i v e n o r e a l r a t i o n a l e f o r t h e t e r m s u s e d . Thus t h e r e i s no i n f o r m a t i o n f u r n i s h e d t h e r e a d e r t o u s e i n c h e c k i n g t h e a d e q u a c y o f t h e s e i n t e r p r e t a t i o n s . S e c o n d l y , t h e c e n t r o i d f a c t o r s w e re u s e d w i t h o u t r o t a t i o n o f a x e s . A d m i t t e d l y t h e r o t a t i o n p r o b l e m i s one a b o u t w h i c h t h e r e i s no c o m p l e t e a g r e e m e n t , b u t i n t h i s c a s e t h e m a j o r e r r o r c o n s i s t s i n e m p l o y i n g t h e c e n t r o i d p a t t e r n o f s i g n s i n d e s c r i b i n g t h e f a c t o r s . H o w e v e r , a s was m e n t i o n e d i n C h a p t e r V, f o r any b a t t e r y w h a t s o e v e r t h e m o s t l i k e l y s i g n p a t t e r n t o a r i s e f r o m t h i s m e t h o d o f f a c t o r i z a -t i o n i s one i n w h i c h t h e f i r s t i s a p o s i t i v e g e n e r a l f a c t o r a n d a l l t h e r e m a i n i n g a r e b i - p o l a r f a c t o r s . Thus t h e a b o v e ( 1 5 5 ) f a c t o r d e s i g n a t i o n s h a v e b e e n b a s e d u p o n a p a t t e r n t h a t i s , i n a c t u a l f a c t , t h e one m o s t g e n e r a l l y p r o d u c e d by t h e method. F o r t h i s r e a s o n i t d o e s n o t seem f e a s i b l e t o p l a c e much c o n -f i d e n c e i n any p s y c h o l o g i c a l s i g n i f i c a n c e f o u n d i n t h i s a r -r a n g e m e n t o f s i g n s . T h i s b e l i e f i s i n k e e p i n g w i t h T h u r -s t o n e ' s c o n t e n t i o n ( 1 8 , p p . 74-75) t h a t , Wo m a t t e r now t h e c o r r e l a t i o n m a t r i x i s f a c -t o r e d , t h e a x e s m u s t be r o t a t e d i n t o a s i m p l e c o n f i g u r a t i o n b e f o r e any p s y c h o l o g i c a l i n t e r -p r e t a t i o n c a n be made. . . T h i s i s t h e m o s t s e r i o u s a n d t h e m o s t f r e q u e n t e r r o r i n c u r r e n t f a c t o r i a l s t u d i e s . Time a f t e r t i m e t h e a u t h o r o f a n a r t i c l e r e -p o r t s t h a t h i s p s y c h o l o g i c a l t e s t s show a s i n g l e common f a c t o r w i t h ^ s a t u r a t i o n s t h a t a r e a l l p o s i t i v e . The s u b s e q u e n t f a c t o r s a l l h a v e some p o s i t i v e a n d some n e g a t i v e s a t u r a t i o n s , a n d t h e i n t e r p r e t a t i o n o f t h e s e s u b s e q u e n t f a c t o r s i s l e f t i n d e t e r m i n a t e . S o m e t i m e s a n a t t e m p t i s made t o p u t p s y c h o l o g i c a l m e a n i n g i n t o t h e s e c o n d a n d s u b s e q u e n t f a c t o r s b u t a l -ways w i t h some h a z y r e s e r v a t i o n s a b o u t t h e f a c t t h a t t h e s a t u r a t i o n s a r e b o t h p o s i t i v e a n d n e g a t i v e . T h i s o p i n i o n i s u p h e l d by G u i l f o r d ( 6 , p. 4 8 8 ) who s t a t e s , " I t w o u l d be g r a t u i t o u s t o a t t e m p t t o g i v e p s y c h o l o g i c a l m e a n i n g o r s i g n i f i c a n c e t o a c e n t r o i d a x i s . I t p r o b a b l y d o e s n o t c o r r e s p o n d t o a n y r e a l v a r i a b l e i n human n a t u r e . " As was i n d i c a t e d a t t h e b e g i n n i n g o f t h i s c h a p t e r , i t i s n o t p o s s i b l e t o s t a t e c a t e g o r i c a l r u l e s by means o f w h i c h t h i s s u b j e c t i v e s t a g e o f f a c t o r a n a l y s i s c a n be p e r f o r m e d . F o r t h i s r e a s o n t h e f o r e g o i n g e x a m p l e s w e r e q u o t e d i n d e t a i l i n o r d e r t o f u r n i s h t h e r e a d e r w i t h a c l e a r e r p i c t u r e o f t h e g e n e r a l r e a s o n i n g i n v o l v e d . S i n c e s k i l l i n t h e i n t e r p r e t a -t i o n o f f a c t o r s c a n o n l y be a t t a i n e d by p r a c t i c e , t h e s t u d e n t ( 1 5 6 ) o f f a c t o r i a l m e t h o d s m i g h t u s e t h e f o l l o w i n g means o f o b t a i n -i n g s u c h p r o f i c i e n c y . The j o u r n a l l i t e r a t u r e c o n t a i n s many e x c e l l e n t e x a m p l e s o f t h e a p p l i c a t i o n o f f a c t o r a n a l y s i s . The f a c t o r m a t r i c e s r e p o r t e d i n s u c h s t u d i e s c o u l d s e r v e a s a s o u r c e o f p r a c t i c e m a t e r i a l . The f a c t o r s c o u l d be i n t e r -p r e t e d a n d t h e r e s u l t s c h e c k e d a g a i n s t t h e r e p o r t e d i n t e r -p r e t a t i o n s . H o w e v e r , i f s u c h m a t e r i a l i s e m p l o y e d , two f a c t s m u s t be b o r n e i n m i n d . F i r s t l y , i t i s v e r y u n l i k e l y t h a t t h e p e r s o n r e i n t e r p r e t i n g t h e s t u d y w o u l d h a v e a s much b a c k g r o u n d k n o w l e d g e a s t h e o r i g i n a l i n v e s t i g a t o r . S e c o n d l y , i t m u s t be remembered t h a t t h e a c t u a l d e s c r i p t i v e names may d i f f e r w h i l e t h e u n d e r l y i n g m e a n i n g r e m a i n s t h e same. F o r t h i s r e a s o n t h e two i n t e r p r e t a t i o n s s h o u l d be c o m p a r e d a s t o b a s i c m e a n i n g . We h a v e now c o m p l e t e d a s u r v e y o f a s e r i e s o f t e c h n i q u e s w h i c h make p o s s i b l e t h e f a c t o r i z a t i o n o f a n y g i v e n c o r r e l a -t i o n m a t r i x . H o w e v e r , i n c l o s i n g , t h e p o i n t m u s t be s t r e s s e d t h a t we h a v e o n l y d i s c u s s e d v e r y f e w o f s u c h t e c h n i q u e s a n d t h e n o n l y i n c o n n e c t i o n w i t h a s i n g l e f o r m o f f a c t o r a n a l y s i s . Two o f t h e o t h e r f a c t o r m e t h o d s w i l l be m e n t i o n e d i n t h e f o l -l o w i n g c h a p t e r . ( 1 5 7 ) C h a p t e r V I I C o n c l u s i o n I n t h e c o u r s e o f t h e p r e s e n t s t u d y we h a v e i n v e s t i g a t e d a f e w o f t h e f u n d a m e n t a l f o r m u l a e a n d t e c h n i q u e s o f T h u r -s t o n e ' s M u l t i p l e - F a c t o r A n a l y s i s . The p u r p o s e o f t h i s w o r k h a s b e e n t o e x p r e s s t h e s e m e t h o d s i n t e r m s t h a t may be u n d e r -s t o o d b y p s y c h o l o g y s t u d e n t s p o s s e s s i n g o n l y a k n o w l e d g e o f e l e m e n t a r y c o l l e g e m a t h e m a t i c s . I t i s e s s e n t i a l t h a t we r e c o g n i z e t h e l i m i t a t i o n s o f t h i s s t u d y . F i r s t l y , t h e r e i s t h e s h o r t c o m i n g e n t a i l e d b y t h e f a c t t h a t , a l t h o u g h we h a v e d e v e l o p e d s u f f i c i e n t f o r m u l a e t o e n a b l e one t o s o l v e a f a c t o r i a l p r o b l e m , we h a v e , i n r e a l i t y , o n l y d i s c u s s e d a s m a l l p o r t i o n o f t h e m e t h o d o l o g y d e v e l o p e d b y T h u r s t o n e . S e c o n d l y , t h e r e i s a n a d d i t i o n a l l i m i t a t i o n i n t h a t we h a v e c o n s i d e r e d n e i t h e r t h e b i - f a c t o r m e t h o d d e v i s e d b y H o l z i n g e r n o r t h e p r i n c i p a l component t e c h -n i q u e e v o l v e d b y H o t e l l i n g . B o t h t h e s e l i m i t a t i o n s w e r e due t o t h e n e c e s s i t y o f r e s t r i c t i n g t h e s c o p e o f t h e p r e s e n t i n -v e s t i g a t i o n . I n r e g a r d t o t h e i m p o r t a n c e o f u n d e r s t a n d i n g a v a r i e t y o f m e t h o d s , t h i s w r i t e r h o l d s t h e same o p i n i o n a s t h a t e x -p r e s s e d b y H o l z i n g e r a n d Barman (1 0 ) who b e l i e v e t h a t t o be p r o f i c i e n t i n t h e f a c t o r i a l f i e l d i t i s n e c e s s a r y t o be c o n -v e r s a n t w i t h a l l a v a i l a b l e m e t h o d s . O n l y t h e n c a n one s e l e c t t h e b e s t p o s s i b l e t o o l w i t h w h i c h t o s o l v e a s p e c i f i c p r o b l e m . T h e n , t o o , q u i t e f r e q u e n t l y one o r o t h e r o f t h e m e t h o d s i s ( 1 5 8 ) s e v e r e l y c r i t i c i s e d a n d o f t e n by someone w i t h a n i n c o m p l e t e k n o w l e d g e o f t h e t e c h n i q u e w i t h w h i c h f a u l t i s f o u n d . I t seems o n l y t o o o b v i o u s t h a t no c r i t i c i s m i s v a l i d u n l e s s t h e s u b j e c t m a t t e r w i t h w h i c h one d i s a g r e e s h a s b e e n c a r e f u l l y s t u d i e d . F o r i n f o r m a t i o n c o n c e r n i n g t h e s e o t h e r two m e t h o d s t h e r e a d e r i s r e f e r r e d t o c u r r e n t w o r k s by t h e a u t h o r s i n q u e s -t i o n . H o t e l l i n g ( 1 1 ) h a s w r i t t e n a j o u r n a l a r t i c l e d e s -c r i b i n g h i s t e c h n i q u e . Much v a l u a b l e i n f o r m a t i o n , c o n c e r n i n g t h i s m e t h o d i s a l s o f u r n i s h e d by H o l z i n g e r a n d Herman ( 1 0 ) . The b i - f a c t o r t e c h n i q u e h a s b e e n v e r y c l e a r l y p r e s e n t e d b y H o l z i n g e r a n d Harman (10) a n d i n a n e a r l i e r w o r k by H o l -z i n g e r a n d h i s a s s o c i a t e s ( 9 ) . F a c t o r a n a l y s i s h a s s o m e t i m e s b e e n c r i t i c i s e d b e c a u s e s e v e r a l s u c h m a t h e m a t i c a l t h e o r i e s a r e p r e s e n t e d a s a b a s i s f o r v a r y i n g m e t h o d s o f f a c t o r i z a t i o n . T h i s i s n o t a t r u l y l e g i t i m a t e c r i t i c i s m , h o w e v e r , s i n c e i t i s v e r y p o s s i b l e t h a t a n y one o f t h e s e t h e o r i e s may w e l l s u i t t h e a s s u m p t i o n s n e c e s s a r y i n a s p e c i f i c a p p l i c a t i o n . The p r i m e n e c e s s i t y i s t h a t t h e i n v e s t i g a t o r be c e r t a i n t h a t t h e p s y c h o l o g i c a l a s s u m p t i o n s n e c e s s a r y i n a n y s t u d y a r e c o n s i s t e n t w i t h t h e m a t h e m a t i c a l a s s u m p t i o n s made when d e v e l o p i n g t h e t h e o r y o f t h e p a r t i c u l a r m e t h o d he i n t e n d s t o e m p l o y . H o l z i n g e r a n d Harman ( 1 0 , p . 7) d i s c u s s t h i s p o i n t i n t h e f o l l o w i n g t e r m s . E v e n t h e s u b j e c t o f g e o m e t r y , w h i c h m i g h t seem t o d e p e n d on a u n i q u e m a t h e m a t i c a l t h e o r y , c a n be d e s c r i b e d by means o f many d i f f e r e n t t h e o r i e s . T h u s , t h e p h y s i c a l c o n f i g u r a t i o n s i n a p l a n e c a n be i n t e r p r e t e d i n t h e l i g h t o f ( 1 5 9 ) E u c l i d e a n g e o m e t r y , R i e m a n n i a n g e o m e t r y , o r v a r i o u s o t h e r t y p e s o f n o n - E u c l i d e a n geome-t r y . T h e r e f o r e , t h e a p p l i e d s c i e n c e o f g e o m e t r y c a n h a v e s e v e r a l a l t e r n a t i v e t h e o r i e s a s i t s b a s i s . A s i n t h e f o r e g o i n g i l l u s t r a t i o n s t h e r e a r e d i f f e r e n t t h e o r i e s , o r f o r m s o f s o l u t i o n , w h i c h may a r i s e i n t h e f a c t o r i a l a n a l y s i s o f a p a r t i c u l a r s e t o f d a t a . The u s e f u l n e s s o f f a c -t o r a n a l y s i s a s a s c i e n t i f i c t o o l h a s b e e n q u e s t i o n e d b y some w o r k e r s b e c a u s e o f t h i s i n d e t e r m i n a c y . I t s h o u l d be e v i d e n t , h o w e v e r , t h a t t h i s i s t a n t a m o u n t t o i n d i c t i n g a l l a p -p l i e d s c i e n c e s b e c a u s e t h e y do n o t d e p e n d u p o n u n i q u e t h e o r i e s . The i m p o r t a n c e o f m a k i n g c e r t a i n t h a t t h e a n a l y t i c m e t h o d a p p l i e d i s s u i t e d t o t h e p r o b l e m t o be s o l v e d c a n n o t be o v e r e m p h a s i z e d . The i n v e s t i g a t o r m u s t be r e a s o n a b l y s u r e t h a t t h e u n d e r l y i n g m a t h e m a t i c a l a s s u m p t i o n s a r e s a t i s f i e d . Of c o u r s e , t h i s r e q u i r e m e n t a p p l i e s e q u a l l y w e l l t o t h e u s e o f a n y o f t h e s t a t i s t i c a l t e c h n i q u e s a n d i s a p o i n t t h a t i s t o o f r e q u e n t l y f o r g o t t e n . R e g a r d i n g t h e i m p o r t a n c e o f c h e c k -i n g t h a t t h e d a t a i s s u i t e d t o t h e a s s u m p t i o n s e m p l o y e d , G u i l f o r d ( 7 , p . 5 7 2 ) s a y s , C o m p u t a t i o n a l p r o c e s s e s , i n t h e a b s e n c e o f com-p u t a t i o n a l e r r o r s , c a n n o t i n a n d o f t h e m s e l v e s p r o d u c e s o - c a l l e d s t a t i s t i c a l a r t i f a c t s . S u c h o u t c o m e s a r e t h e r e s u l t o f e r r o n e o u s a p p l i -c a t i o n o f t h e m a t h e m a t i c a l v o c a b u l a r y a t t h e b e g i n n i n g . H e r e s c i e n t i s t s a l w a y s t a k e a c h a n c e . T h i s c h a n c e d e p e n d s u p o n w h e t h e r o r n o t t h e s c i e n t i f i c p r o b l e m h a s b e e n p r o p e r l y r a t i o n a l i z e d , i n c l u d i n g t h e p r o p e r a s s u m p t i o n s . Thus e v e n a t t h i s i n i t i a l s t a g e t h e s c i e n t i s t ' s k n o w l e d g e o f b o t h p s y c h o l o g i c a l a n d f a c t o r i a l t h e o r y i s a c r u c i a l e l e m e n t i n t h e p r o c e d u r e . I n c o n c l u s i o n l e t u s c o n s i d e r v e r y b r i e f l y t h e p o s s i b l e t r e n d s t h a t may o c c u r i n f a c t o r a n a l y s i s i n t h e i m m e d i a t e ( 1 6 0 ) f u t u r e . I t i s p o s s i b l e t o d e s c r i b e t h e w o r k done up t o d a t e i n t e r m s o f two' s e p a r a t e t h o u g h o v e r l a p p i n g c a t e g o r i e s . The f i r s t o f t h e s e c o n s i s t e d i n t h e d e v e l o p m e n t o f m a t h e m a t i c a l f o r m u l a e "and t e c h n i q u e s a n d h a s a d v a n c e d v e r y r a p i d l y d u r i n g t h e p a s t f e w y e a r s . The s e c o n d g r o u p i n g c o n s i s t e d i n t h e a p p l i c a t i o n o f t h e s e t e c h n i q u e s t o s p e c i f i c p r o b l e m s . A t t h e moment i t w o u l d a p p e a r t h a t more a n d more a t t e n t i o n w i l l be t u r n e d t o t h e a p p l i e d p r o b l e m s i n o r d e r t o c o n s o l i d a t e a n d a p p l y t h e v a s t t h e o r e t i c a l s t e p s t h a t h a v e b e e n d e v e l o p e d . A t t h e same t i m e one o f t h e m a j o r t h e o r e t i c a l r e q u i r e m e n t s i s a more r i g o r o u s c r i t e r i o n b y w h i c h t o e s t a b l i s h t h e u n i q u e n e s s o f a f a c t o r s t r u c t u r e . I n t h i s r e s p e c t t h e s i m p l e s t r u c t u r e r e q u i r e m e n t c o n c e i v e d by T h u r s t o n e a p p e a r s t o be s o u n d l y d e -v e l o p e d a n d may w e l l s e r v e a s t h e b a s i s u p o n w h i c h s u c h f u r -t h e r t h e o r y i s c o n s t r u c t e d . H a v i n g c o m p l e t e d t h i s p r e l i m i n a r y s t u d y o f m u l t i p l e - -f a c t o r a n a l y s i s , i t m i g h t . b e w o r t h w h i l e t o c l o s e w i t h a q u o -t a t i o n f r o m K e l l e y w h i c h s e r v e s t o e x p r e s s t h e t r u e p u r p o s e o f f a c t o r a n a l y s i s a n d t h e r e a l m e a n i n g o f f a c t o r i a l r e s u l t s . T h i s a u t h o r ( 1 2 , p. 120) w r i t e s , T h e r e i s n o s e a r c h f o r t i m e l e s s , s p a c e l e s s , p o p u l a t i o n l e s s t r u t h i n f a c t o r a n a l y s i s ; r a t h e r , i t r e p r e s e n t s a s i m p l e , s t r a i g h t f o r -w a r d p r o b l e m o f d e s c r i p t i o n i n s e v e r a l d i m e n -s i o n s o f a d e f i n i t e g r o u p f u n c t i o n i n g i n d e f i n i t e m a n n e r s , a n d he who a s s u m e s t o r e a d more r e m o t e v e r i t i e s i n t o t h e f a c t o r i a l o u t -come i s c e r t a i n l y doomed t o d i s a p p o i n t m e n t . ( 1 6 1 ) B I B L I O G R A P H Y 1. A n a s t a s i , A n n e , a n d J o h n P. F o l e y , J r . D i f f e r e n t i a l P s y c h o l o g y . R e v i s e d e d . New Y o r k : . . . M a c m i l l a n , 1 9 4 9 . 894 p p . ' 2. B u r t , C y r i l . The F a c t o r s o f t h e M i n d . L o n d o n : U n i v . o f L o n d o n P r e s s , 1 9 4 0 . . 509 p p . 3. C h a n t , S.N.F. " M u l t i p l e F a c t o r A n a l y s i s a n d P s y c h o l o g i c a l C o n c e p t s , ! ! J . e d u c . P s y c h o l . . X X V I ( A p r i l 1 9 3 5 ) , 2 6 3 - 7 2 . I " 4. C h e n , T s o - Y u L . , a n d H u a i - H s u i Chow. "A F a c t o r S t u d y o f a T e s t B a t t e r y a t D i f f e r e n t E d u c a t i o n a l L e v e l s , " J . g e n e t . P s y c h o l . . 73 ( D e c . 1 9 4 8 ) , 1 8 7 - 9 9 . 5. C o n r a d , H e r b e r t i S . I n v e s t i g a t i n g a n d A p p r a i s i n g I n t e l l i -g e n c e a n d O t h e r . A p t i t u d e s . I n T.G. A n d r e w s ( E d . ) f  M e t h o d s o f P s y c h o l o g y . .New Y o r k : W i l e y , 1 9 4 8 . 4 9 8 - 5 3 8 . 6. G u i l f o r d , J . P . P s y c h o m e t r i c M e t h o d s . New Y o r k : McGraw-H i l l , 1 9 3 6 . 5 6 6 p p . 7 # 9 "Human A b i l i t i e s , " P s y c h o l . R e v . . 47 ( S e p t . 1 9 4 0 ) , 3 6 7 - 9 4 . 8. • =— . F u n d a m e n t a l S t a t i s t i c s i n P s y c h o l o g y a n d E d u c a t i o n . New Y o r k : M c G r a w - H i l l , 1 9 4 2 . 3 3 3 p p . 9. H o l z i n g e r , K a r l J . , a s s i s t e d b y F r a n c e s S w i n e f o r d a n d H a r r y Harman. S t u d e n t M a n u a l o f F a c t o r A n a l y s i s . C h i c a g o : D e p t . o f E d u c . , U n i v . o f C h i c a g o , 1 9 3 7 . 102 p p . 1 0 . H o l z i n g e r , K a r l J . , a n d H a r r y H. Harman. F a c t o r A n a l y s i s . A S y n t h e s i s o f F a c t o r i a l M e t h o d s . C h i c a g o : U n i v . o f C h i c a g o P r e s s , 1 9 4 1 . 417 p p . 1 1 . H o t e l l i n g , H a r o l d . " A n a l y s i s o f a C o m p l e x o f S t a t i s t i c a l . ._- V a r i a b l e s i n t o P r i n c i p a l C o m p o n e n t s , " J . educ_^ P s y c h o l . , X X I V ( S e p t . a n d O c t . 1 9 3 3 ) , 4 1 7 - 4 1 , 4 9 8 -3 2 0 " 1 2 . K e l l e y , Truman L. "Comment on W i l s o n a n d W o r c e s t e r ' s " N ote on F a c t o r A n a l y s i s " , " P s y c h o m e t r i k a . 5 ( J u n e 1 9 4 0 ) , 1 1 7 - 2 0 . 1 3 . M u r s e l l , James L. P s y c h o l o g i c a l T e s t i n g . New Y o r k : Longmans, G r e e n , 1 9 4 7 . 4 4 9 p p . ( 1 6 2 ) 1 4 . P e t e r s , C h a r l e s C., a n d W.R. V a n V o o r h i s . S t a t i s t i c a l P r o c e d u r e s a n d T h e i r M a t h e m a t i c a l Base's. - New Y o r k : M c G r a w - H i l l , 1 9 4 0 . 516 pp.. 1 5 . .Sisam, C h a r l e s H. C o l l e g e M a t h e m a t i c s . New Y o r k : Henry-H o l t , 1946.. 546 p p . : 1 6 . S p e a r m a n , C h a r l e s . The A b i l i t i e s o f Man. L o n d o n : Mac-m i l l a n , 1 9 2 7 . 415-pp.. 1 7 . Thomson, G o d f r e y H. The F a c t o r i a l A n a l y s i s o f Human A b i l i t y . B o s t o n : H o u g h t o n , M i f f l i n , 1939.. 326 p p . 1 8 . T h u r s t o n e , L . L . " C u r r e n t M i s u s e o f t h e F a c t o r i a l M e t h o d s , " P s y c h o m e t r i k a . 2 ( J u n e 1 9 3 7 ) , 7 3 - 7 6 . 1 9 . '• . M u l t i p l e - F a c t o r A n a l y s i s . C h i c a g o : U n i v . "of C h i c a g o P r e s s , 1 9 4 7 . 535 pp.. 2 0 . W i t t e n b o r n , J.R. " M e c h a n i c a l A b i l i t y , I t s N a t u r e a n d M e a s u r e m e n t . I I . M a n u a l D e x t e r i t y , " E d u c . p s y c h o l . Measmt., 5 ( W i n t e r 1 9 4 5 ) , 3 9 5 - 4 0 9 . -2 1 . "A F a c t o r i a l A n a l y s i s o f R e v i s e d E x a m i n a t i o n M " D i r e c t o r -a t e o f P e r s o n n e l S e l e c t i o n ( A r r a y ) , B u l l . C a n . P s y c h o l . A s s o c . . I V ( A p r i l 1 9 4 4 ) , 1 8 - 2 2 . 

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