The Scientific Use of Factor Analysis in Behavioral and Life Sciences

634 Pages·1978·13.041 MB·English

by  Raymond B. Cattell (auth.)

THE SCIENTIFIC USE
OF FACTOR ANALYSIS
in Behavioral and Life Sciences
THE SCIENTIFIC USE
OF FACTOR ANALYSIS
in Behavioral and Life Sciences
RAYMOND B. CATTELL
University of Hawaii
PLENUM PRESS • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Cattell, Raymond Bernard, 1905-
The scientific use of factor analysis in behavioral and life sciences.
Bibliography: p.
1. Factor analysis. 2. Psychometrics. I. Title. [DNLM: 1. Factor analysis, Statisti
cal. BF39 C368s]
BF39.C33 519.5'3 77-10695
ISBN-13: 978-1-4684-2264-1 e-ISBN-13: 978-1-4684-2262-7
DOl: 10.1007/978-1-4684-2262-7
© 1978 Plenum Press, New York
Softcover reprint of the hardcover lst edition 1978
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N.Y. 1O0ll
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
To Karen
whose mathematical interests called me
to make concepts and experiences
in this domain explicit
Preface
It needs no great scientific insight to see that such multitudinously multi
variate subjects as psychology, physiology, sociology, and history need
multivariate methods. As this book may show, those methods-multivari
ate analysis of variance, regression analysis, typal and discriminant func
tion analysis, multidimensional scaling, and factor analysis-belong to a
single structural arch which bears up conceptual and causal understanding
in all these subjects. But factor analysis is the keystone of that arch. Since
factor analysis has itself developed fantastically in thirty years [my first
small book (Cattell, 1952b) could almost cover the field in a few chap
ters!] , this book confines itself to that subject, with only brief connecting
asides on the related areas.
The purpose of a preface is to permit more personal comments and
to explain why the design is what it is. Among the former the author often
apologizes for writing in an already crowded library shelf, and, having con
fessed the crime, thanks his friends for their connivance. The area already
enjoys a truly excellent array of books, from extremely good elementary
introductions by Child, Henrysson, Lawlis, and Chatfield, through the
workbook emphasis of Fruchter, and the intermediates of Guertin and
Bailey and Comrey, to the comprehensive technical works of Anderson,
Ahmavaara, Gorsuch, Harman, Horst, Lawley and Maxwell, Mulaik, Rao,
Rummel, and Van de Geer, not to mention the undating books by Burt,
Thomson, and Thurstone. Seldom has there been such a contrast between
a feast of first-rate texts and the starvation level at which many depart
ments keep their students in formal instruction. It seems by no means
unusual for undergraduate majors to try to handle their psychology with
no instruction in factor analysis but, for some reason, an excessive drilling
in analysis of variance (ANOV A). Yet factor analysis is a gift of psychology
vii
viii Preface
to the life sciences, created by the needs of the subject, whereas ANOV A
began with the needs of farmers. 1
Factor analysis was fathered in psychology by Burt, Kelley, Spear
man, Pearson, and Thurstone, and mothered in mathematics by Hotelling
and the developments of the eigenvalue concepts going back to Jacobi.
The offspring shows the more mathematical inheritance in Anderson,
Ahmavaara, Harman, Horst, Lawley and Maxwell, Mulaik, and Rao, and
the more psychological or social science inheritance in Burt, Gorsuch,
Rummel, and Thurstone. This is not to say that the latter, in the least,
lack any mathematical precision and, indeed, the excellent balance in
Gorsuch and in Rummel came near to persuading me that a further book
in the field is uncalled for.
However, nearly fifty years of discussing the difficulties both of
students and of leading researchers in relation to statistics, and factor
analysis in particular, have convinced me that a radical new approach
is necessary. One must respect first the scientific interests of the beginner
and only slowly and with patience reorient to mathematical concepts.
The aphorism is as old as Herbart that good education moves from the
known to the unknown and from the simple to the complex-and no
psychodynamicist needs to be told that it must link to powerful interests
the whole way. My title The Scientific Use of Factor Analysis means just
this: that for the student the interest begins in his science as such; and for
the researcher the demands of viable scientific models are given prece
dence over mathematical neatness, which can easily become pedantry.
This radical change of emphasis from most existing texts connotes also
an art of gradation and a policy of keeping the final center of gravity in
the right place.
The gradation is shown in the fact that the levels of mathematical
sophistication and precision demanded are quite different at the beginning
and the end. I am assuming that in the one- or two-semester course (ac
cording to student ability) in which the book is completed the student
will grow-in general attitude to mathematical models. One or two of my
braver fellow authors have confessed that they are afraid to make simpli
fied, two-thirds true or insufficiently qualified statements at the beginning
and that the same high rigor of statement and formulation must apply
from the first page. They are afraid of their peers, which means they are
writing for their peers. The natural growth of an enquiring mind is quite
different from that: it is willing to believe at first that water is H 0, and
2
1 My emphasis is only to redress a balance; for here as elsewhere I have always argued
for a two-handed use of factor analysis and analysis of variance. I trace this to stu
dent years in which I shuttled across a little plot of grass between the laboratory
where Spearman was developing factor analysis and the Galton Laboratory where
Fisher was shaping with equal brilliance the analysis of variance.
Preface ix
talk of deuterium is at that stage distracting and offensive pedantry. If
the reader is not too aware of these transitions I am happy to have had
some success in the art that conceals art; but the procedure of "sketching
in" and then returning to more precise details may be mistaken for
repetitiousness (though good education requires planned repetition with
growth). It has also required more cross reference than usual, back and
forth, so that the reader explicitly recognizes amendments. I have, where
possible, followed a further principle, one that, other things being equal,
the historical order of development of a science is the best teaching
order. I followed this order quite closely in my first book on factor
analysis, but I have adhered to it only loosely in this book for reasons
connected with some peculiarities in the later growth of the subject,
which I have no space to explain here.
The book is divided into two parts and the teacher will have no dif
ficulty in recognizing that the second part is significantly more difficult
and involves a step from almost concrete exercises to abstract concep
tions on the advancing fringe of mathematical psychology in factor anal
ysis. Although the emphasis here is on the scientist's interests, so that
mathematical derivations present in some other factor analytic texts are
omitted, there has been no omission of the exact ultimate formulas by
which a given mode of analysis or mathematical model is properly repre
sented and conceived. If the reader is mathematically endowed, he will
not lack the comprehensive statement he desires. If he is not so endowed
he will get a verbal equivalent. Although reviewers of my 1952 book saw
each, according to his specialty, various shortcomings, I was gratified to
find that it was received as one of the clearly readable books they had
encountered on factor analysis. I have attempted to live up to that here,
though one result has been of some cost to the publisher, namely, that the
book's higher ratio of explanations in reading to condensed formulas has
made it consume more paper.
A further instance of adjusting to learning interests is the way in
which I have brought in matrix algebra concepts in a smooth sequence,
introducing them as they are needed, rather than starting with a solid
block chapter devoted entirely to that topic. One consequence of my
plan, which teacher and student must recognize, is that the chapters must
necessarily be read in the order given. However, I have explained to the
reader at one or two chapter beginnings that he may choose to skip all
or parts of the particular chapter if he is reading for a first general perspec
tive. Nevertheless, at no stage can he jump ahead very far-except in the
last four chapters which might be picked up in a freer order.
The above must not be taken as any implicit criticism of the more
purely mathematical presentations. Indeed, without the existence of these
handsome and painstaking mathematical volumes (not to mention Psycho-
x Preface
metrika, the British Journal of Mathematical Psychology, the Journal of
Mathematical Psychology, and Multivariate Behavioral Research) what I
have attempted here would be impossible. In part, I have written this
book as a signpost, directing the reader to them at appropriate points.
In particular, I have referred the reader to more detailed developments
in Gorsuch and Rummel, the scientific emphases in which dovetail more
readily with these chapters, but also to the mathematical-statistical over
views of Anderson, Harman, Horst, Lawley and Maxwell, Mulaik Van de
Geer, and other well-known texts. On the other hand such short introduc
tions as those of Child, Fruchter, Lawlis and Chatfield, and others men
tioned above may function as introductions to the present work.
As to the final "center of gravity" at which the book aims to leave
the student or researcher finally well balanced, it is best described as that
of a scientist in his own field. I have watched over decades several recur
rent well-meaning attempts to place the teaching of mathematical statistics
where it seems logically to belong-in the mathematics department. With
the exception of a few advanced graduate students who ended their ca
reers as mathematicians rather than as psychologists, these transfers of
courses-equally in psychology, economics, and physiology-from the
substantive department to the department of mathematics, were a fail
ure.2 In the first place, psychology is a big enough subject in itself over
2 Since some may wish more documentation of my meaning here I would say that
historically when mathematicians joined the development of factor analysis they
wanted to direct it according to the development of their own abstract systems
rather than according to what I have frequently contrasted here as a set of more
complex and flexible scientific models. The latter are equally precise, and largely
mathematical (though they demand other properties too, in an analogical basis).
However, these models respond to what the sensitively inquiring psychologist sees
as "a best fit," not to the logically possible permutations and combinations of
principles which the mathematician sees as the next logical development. For ex
ample, it is admittedly a simpler and cleaner job for the mathematician if he keeps
unities instead of "guessed" communalities in the diagonal of the correlation matrix
and if he keeps axes mutually orthogonal to rotation, whereas the scientific model
has every indication of requiring something a bit more complex and even a bit
messy. For some years, the mathematicians thus dragged the less perceptive psy
chologists from factors to the authority of components. They also called psychol
ogists to the apparent mathematical purity of orthogonal factors and it has taken
much argument, by the present writer and others, on behalf of psychological models
to draw the pack of researchers from an unreasoning pursuit of "mathematical
prestige" to what psychological research indicated as a more apt model.
On the one hand the pure mathematician's demands are a harsh intrusion, and
on the other they sometimes invite a wild goose chase or irrelevant developments
we do not need. For example, he may suggest that factor space be handled in Rieman
nian rather than Euclidean geometry. We should thank him for providing this pos
sibility, but many psychologists seem to need to be reminded that we do not have
to adopt it, just to be "up-to-date" mathematically, if it proves irrelevant. As a neces-
Preface
xi
which to stretch a student's interests, without his having to master a new
discipline from the ground up. Except for a few utterly rare instances,
even with the greatest attention in teaching to the relevance of math-stat
to substantive issues, the student's interest will move only slowly, over
months and years, to a real understanding of the need for mathematical
models in science, and a real enjoyment of their intrinsic beauty. If
psychological substantive research can recruit a few hundred such men,
it will be fortunate; but every psychologist, because he is in a field of
multiply determined behaviors, should grasp the general principles of
factor analysis and the behavior specification equation.
This book attempts a highly comprehensive survey in Part II of
sary illustration, though I trust not invidious, I would mention the developments
around "alpha factor analysis," made by mathematicians among psychologists, and
valid as a rejuggling of the possible ways of doing a factor analysis, but scarcely sug
gested by the needs of any psychological relations ever perceived. The juggler of
models has a right to juggle, and perhaps our concern should be rather with the
weakness of the average psychologist which leads him to jump on the latest band
wagon with no perception of the direction in which it is going. As one contemplates
a certain tour· de-force offered by a certain kind of mathematical statistician to his
brother psychologist one is reminded of the question of Macaulay regarding the
powerful intellect of Dr. Johnson, "How it chanced that a man who reasonned on
his premises so ably, should assume his premises so foolishly, is one of the great
mysteries of human nature."
What we should like the gifted mathematical statistician to give us, in terms of
helping us meet the most likely scientific model, can be illustrated by many urgent
needs. After the initial breakthrough of MacDonald we need a thoroughly pro
grammed nonlinear factor analysis, and we need a way of handling the reticular
model instead of having to restrict to the strata model (p. 201, below). For twenty
years we have called to mathematicians for a solution for oblique con factor rota
tion; and for more theoretically insightful significance tests for oblique rotated
factor loadings as well as the difference between an obtained and a hypothetically
stated factor structure (see, however, Joreskog, p. 482). With all due gratitude for
those things in which pure mathematical statisticians have turned aside to help us,
the situation familiar to scientists remains: that generally we ask for a solution to
one problem and get from the mathematician an answer to a different question, in
which he has shifted the assumptions from those we must make. For example, we
asked in the forties and fifties for a test of significance for the size of an oblique
rotated factor but got an answer only for an unrotated factor, indeed, not even for
a factor but for a principal component.
We have no right to demand of mathematicians that they be supermen, but
better cooperation from both sides in defining goals would have been fortunate.
Failing to get responses to their problems, psychologists have in fact resorted to
solutions for their methodological problems which may be approximate and often
look relatively clumsy to the mathematician. Sometimes (as in those publications
by the present writer) they have formulated the real problem but have had to answer
it by brute Monte Carlo methods, in order to get on with their work with some
assurance of knowing roughly where they are statistically.
xii Preface
advanced mathematical statistical techniques, so that the psychologist
knows what is going on. But it invites him to look over the fence rather
than to get over it, knowing that psychology is a vast enough field to re
quire him on this side of the fence. However, its contrast to some other
leading texts is not just in this restraint. It lies rather in the far greater
space given to scientific models, to the judicious choice of procedures
in relation to experiment, to factor interpretation, to relations that
broaden psychometric concepts, such as scaling, validity, and reliability,
to higher strata models, and to the whole strategy and tactics of the use
of factor analysis in research.
Finally, the book differs in venturing to convey, along with the
explicit mathematical and statistical formulations, some values of a more
intuitive kind arising from long and diverse experience in the field. In
justification of this liberty it must be stated, at the possible risk of im
modesty, that the Laboratory of Personality and Group Analysis at the
University of Illinois is to have done factor analyses, in more diverse sub
stantive areas (psychology, animal behavior, learning, socioeconomic
trends, group dynamics, educational achievement, motivational states, and
cultural history) and with more varied designs, than almost any laboratory
in the world. Inevitably that brings to the recommendations and decisions
certain perceptions and values not encompassed in the most thorough
mathematical-statistical texts.
With this statement of the book's design and intentions the teacher
and student can best judge for themselves its role in courses. The writer
believes that it will function as an attractive avenue to the more statistic
ally detailed treatments in the books indicated. Part I can stand on its own
feet as a craftsman's introduction to the basic concepts and processes of
factor analysis. Part II leads out on a new level to the more complex
statistical treatments which can be pursued as far as desired in the given
directions and with particular textbook aids.3 Since the horizon of new
3 If undergraduate senior courses are concerned I would advocate making a one-semes-
ter course from Part I (with associated illustrative examples and quizzes). If gradu
ates, or if a class of highly selected undergraduates, then I believe it is practicable to
cover the whole book in a semester. An alternative is to make Part I half of a first
semester graduate course in general statistics, in which ANOV A takes the other
half, and to keep Part II for a second-semester course specifically in factor analysis.
But there is also the lone reader to consider, in research or applied work, whose
statistical and general background is more mature, and this important individual
has often been in my mind, especially in Part II. Indeed, my dialogue after the
first five chapters is often with the seasoned researcher and philosopher of science.
In order not to talk down to him, I have attempted in the first five chapters (to
some degree in the rest of Part I) a palimpsest, in which the writing on the surface
will be easily followed by the student, but which hopefully will not offend the
specialist inasmuch as he sees writing between the lines which has implications of
a more subtle kind. Such a reader can be left to his own devices, but almost cer
tainly in Part II most readers will need the guidance and exegesis of a good teacher.