[PDF]a methodological study

i Applicability of
FACTOR ANALYSIS
IN THE BEHAVIORAL SCIENCES


А methodological study


by


STEN HENRYSSON


тед
Calcutta D


tB "A


ZA


AWE


ALMQVIST & WIKSELL
STOCKHOLM


—-- чоиза 9
«nid


UPPSALA 1957
ALMQVIST & WIKSELLS BOKTRYCKERI AB


PREFACE


'The purpose of this monograph is to determine the kind of
problems occurring in the behavioral sciences, which can be
solved either completely or partly by factor analysis, and how
this can best be accomplished.

As a background to our study, we shall first discuss the charac-
teristics of the factor analytical models and the assumptions on
which they are based. That factor analysis has a wide field of
application as a descriptive condensing aid would appear to be
indisputable. There are differencies of opinion, however, about
the possibility of finding factors which can be included in explan-
atory models. This is partly due to the fact that the meaning of
explanatory factor analysis is not clear. Consequently an attempt
will be made to find a meaning for this kind of analysis. We
shall discuss, from this point of view, the question of how to
conduct an explanatory factor analysis, particularly when the
principles of simple structure are applied.

'The theory of factor analysis embraces a number of compre-
hensive and complex problems, e.g., of a statistical nature. Some
of these problems still require to be solved. This monograph will,
however, be concerned with the questions which are essential to
the main problem, and will disregard a number of important
but secondary problems. The monograph is not intended to
cover the discussion of the whole field of factor analysis. Nor
will the comments and criticisms, which have already appeared
and been refuted in psychological and statistical literature, be:
considered here.

The monograph mainly deals with questions of current interest
in order to clarify them, to discuss the problems yet unsolved,
to describe the attempts that have been made to solve them,
and to indicate along which lines the solutions are likely to be
found.


The problems dealt with are related to various sciences, such
as mathematics, statistics, psychology, and philosophy. From
the viewpoint of each of these branches of science, this account is
perhaps incomplete and imperfect and cannot claim to be “water-
tight". An attempt is made to integrate the results obtained in
different special sciences by investigating how factor analytical
methods can be used to solve problems in the behavioral sciences.
In any case a useful purpose may be served if someone, who is
fairly well acquainted with the sciences referred to above, can
formulate problems and suggest solutions in such a way, that
scientists, who specialize in a more limited field, will obtain
a better position from which they can attack some of these
problems.

It is obviously impossible, within the scope of this monograph,
to give a complete account of factor analytical theory. Only
the background which is essential for the discussion can be given.
For a fuller account, the reader is referred to classical handbooks
such as Thurstone’s Multiple Factor Analysis or Thomson’s
Factorial Analysis of Human Ability or Holzinger & Harman,
Factor Analysis.

It has been found expedient to devote the greater part of this
monograph to a purely mathematical treatment of factor analyt- ‘
ical theory, and not to make any attempt to go fully into questions |
connected with sampling fluctuations. This aspect of the problem |
is discussed in the latter part of this study. This does not at all
imply that the statistical viewpoint is of less importance, but is
due to heuristical reasons. The monograph will be clearer and
more readily understandable if the mathematical theory and its
significance are presented at the outset without the complica-
tions which are inevitable, if sampling fluctuations are included, It
should be pointed out however, that since the main aim of this
monograph is to consider how factor analysis can solve problems
in the behavioral sciences, the treatment of the purely statistical
discussion is limited to what is necessary in order to fulfil this
purpose.

During the time—since 1948 —that I have devoted myself
to the study of factor analytical problems, personal contacts |


7


have been established with a number of scientists working in
this field. I have worked under Thomson in Edinburgh, Thurstone
in Chicago, and Wold in Uppsala. When I was abroad, and also
on other occasions, I have had the opportunity of personal
discussions with Ahmavaara, Bargmann, M. S. Bartlett, Burt,
Cronbach, Eysenck, J. W. French, Holzinger, Lawley, Lubin,
Peel, Saunder, Tucker, Vernon, Whittle and other investigators.
During our discussions some of these scientists have expressed
ideas and points of view, which are not always to be found in
their published works.

Professors Husén, Malmquist and Wold have read different
sections of the manuscript and made suggestions and comments.
It should be pointed out, however, that I am personally respons-
ible for the statements made and the results given.


From Statens Psykologisk-Pedagogiska Institut, Statens Sam-
hälls- och Ráttsvetenskapliga forskningsråd, and Swerige- Amerika
Stiftelsen I have received grants for travelling, and for assistance
with calculation, typing and translation.

Many persons and institutions—besides those mentioned
above—have helped me in various ways in the writing of this
monograph. They are so numerous that it is impossible to name
them all here. The gratitude I feel towards all of them—from

. my daughters Mari and Kersti to my teacher and friend Louis
Leon 'Thurstone—is none the less great.


Stockholm, March 1957
Sten Henrysson


CONTENTS


PREFACE .
CONTENTS . š
SYMBOLS AND "TERMINOLOGY


I. Tue Basic THEORY or Factor ANALYSIS


The Rise of Factor Analysis

‘The Basic Equations. "En
Component Analysis and Factor Analysis
'The Geometrical T'heory and Rotation .
Assumptions about the Nature of the Data .


П. THe Most Important FACTOR MODELS


The Need for Restrictive Assumptions .
The Forerunners of Factor Analysis .
The Spearman School . i
Burt's Factor Analytical Methods .
'Thomson's Contribution .

The Thurstone School .


III. DIFFERENT APPLICATIONS OF FACTOR ANALYSIS


The Problem of Finding the Number of Dimensions
'The Problem of Finding Descriptive Factors
Descriptive Factor Analysis and Its Applications.
Prediction and Factor Analysis &
Differential Prediction and Factor Analysis è
Conclusions about Factor Tests as Predictors .

The Problem of Finding Explanatory Factors .


IV. FACTOR ANALYSIS AND THEORY OF SCIENCE


Description and Explanation in the Theory of Science .
Intervening Variables and Hypothetical Constructs.


The Distinction between Descriptive and Explanatory Factor


Analysis .


ou


10


V. ExPLaNATORY Facron ANALYSIS


The Testing and the Generating of Hypotheses


br s ow x og 89

Hypothesis-Generating Factor Analysis and the Principle of
ТРГУ н, 91
VI. Factor ANALYSIS ACCORDING TO THE PRINCIPLES oF SIMPLE
STRUCTURE
The Meaning of Simple Structure... буа к, уу
Оби ецца, a a . 100
"he Selection of "Pest We онен ае 101
bier Pa ОТР 104
Analytical Methods for Rotation to Simple Structure... | | 108
VII. THE PROBLEM or INVARIANCE

The Concept of "ics EE E E 111
Methods for Minding Invariante ыз; з, „лу, о, 115
Simultaneous Rotation to Invariant Solutions. | | | | | 5 7 € 118
Conclusions about the Methods of Finding Invariance , |. | | 121
Nonfactorial EUM Исе Б ree ае 122


VIII. Facron ANALYSIS FROM А Statistica, Pont or View


The Need of Statistically Acceptable Methods... . 2... 124


Maximum Likelihood Solutions and Least Squares Solutions . . 127
Tests of Significance f


or Factor Loadings ©... 1... 137
ССОРЕ 139
ОИ ЧОРТОН 143
REFERENCES $


11


SYMBOLS AND TERMINOLOGY


Although certain sections require a special set of symbols, generally
the following notations and terminology are used in this monograph.


raw score in test j for individual 7
subscript for individual
two subscripts designating tests


total number of persons in a group
mean

standard deviation

standard score in test j for individual 7
correlation between test j апа?

common factor score in the kth common factor for person i
specific factor score in test j for person 7
error factor score in test j for person i
unique factor score in test j for person i
loading of factor / in test 7
communality of the jth test


matrix containing standard scores

complete correlation matrix with unities in the diagonal
matrix containing all loadings

matrix containing all factor scores

reduced correlation matrix with communalities in the diagonal
matrix containing common factor loadings

matrix containing common factor scores

diagonal matrix containing specific parts of test variances
diagonal matrix containing specific loadings

matrix containing specific factor scores

diagonal matrix containing error parts of test variances
diagonal matrix containing error factor loadings

matrix containing error factor scores

diagonal matrix containing unique parts of test variances
diagonal matrix containing unique factor loadings
matrix containing unique factor scores

matrix containing correlations between all factors
matrix containing correlations between common factors
diagonal matrix containing communalities

matrix containing common factor loadings before rotation
transformation matrix

length of test vectors j апа?

angle between test vectors j and t


|


|


^


CHAPTER I


THE BASIC THEORY OF FACTOR ANALYSIS


The Rise of Factor Analysis


Factor analysis has its roots in the psychological and biolog-
ical discussions around Spencer, Galton, McDougall et al.,
during the later part of the nineteenth century. It was Spear-
man, however, who started the growth of factor analysis through
a paper in 1904 on the nature of intelligence. In this paper he
described a factor analytical investigation of various cognitive
tests. He started out from a psychological theory which he tried
to verify with the help of a factor analytical method. The basic
idea in this method was to trace back the relations between suit-
ably chosen cognitive variables to a general ability factor—the
so-called G-factor. Each observed variable was also assumed to
measure a specific factor not to be found in any of the other
observed variables. For some decades discussion revolved prin-
cipally around the validity of this simple model.

Through Thurstone particularly, development during the
1930's went beyond Spearman's limited model. More general
models were developed and, at present, research efforts are
mostly directed upon these models.

From the very beginning of factor analysis, an intensive debate
has been going on about its basis and usefulness. This debate is
still continuing and is characterized by the fact that the various
factor analysts have different views as to the value of the methods.
Even the same investigator's standpoint may vary. The present
methodological development is characterized by working with
different special problems connected with the various methods
or with the general theories, and by seeking new fields of
application. Since Thurstone’s book, Multiple Factor Analysis
(1947), and the third edition of Thomson’s book, Factorial Ana-


14


lysis of Human Ability (1948), were published, no general texts,
more important from a methodical viewpoint, have come out.
Thus, Vernon's The Structure of Human Abilities (1950 а) is
mainly of interest as to the application of the factor analysis.
Spearman-Jones’s book Human Ability (1950) added scarcely
anything new to the discussions in Spearman’s earlier work.
Cattell’s Factor Analysis (1952) and Fruchter’s Introduction to
Factor Analysis (1954) include accounts of the factor analytical
methods and the technical procedures.

A growing interest in factor analysis has lately been noticed
among statisticians. However, they have concentrated themselves
mainly on statistically efficient but unrotated solutions, which
need to be adapted and further developed in order to be really
useful in the behavioral sciences.


The Basic Equations


Factor analysis supplies methods for reducing a large number
of observed variables to a lesser number, in some way more
fundamental variables or, as they are usually called, factors. This
is usually done through an analysis of intercorrelations between
the observed ‘Variables.

In this section of the monograph an account is given of the
mathematical theory on which factor analysis is based. This
account is not intended to be complete but will only bring out
such characteristics of the factor analysis as are of importance
in the following discussion. A more complete account is given,
for example, by Holzinger-Harman (1941) or by Thurstone
(1947). Elmgren (1955), who introduced factor analysis in Swe-
den, has published a textbook in Swedish on the subject.

The present monograph is concentrated on analyses of correla-
tions between observed variables applied on large samples of
individuals at a certain time, that is, on the so-called R-technique,
which, at the present time, is the most developed and used
technique. It would carry too far to go into various other prob-
lems, such as analysis of correlations between persons. Many of
the principal questions regarding R-technique are, to a greater


“ide


—


—


15


or lesser degree, of importance also for the other techniques.
Further details about different techniques have been given by
Cattell (1952, pp. 88-126). The discussion here is also limited
to continuous variables having linear relations with each other.
The treatment of discontinuous and non-linear cases is discussed,
for example, by Coombs (1952), Lazarsfeld (1950), and Burt
(1953).

This chapter contains the mathematical theory. The purely
statistical discussion concerning factor analysis as a problem
of estimation and significance has, as mentioned in the preface,
been reserved to chapter VIII.

The basis for the theory are the different observations X;; for
N individuals in n different observed variables. For convenience
the word “соге” is used in the following instead of observation
and “test” instead of observed variable. This, of course, does not
imply that the methods can only be used on test scores.

Each one of the scores is expressed as the standard score =,
in each test according to the following formula


som раі аа-а O


Bj
5;


M, and s, are the mean and standard deviation for the jth test.
That means that


1 N T 1 N 3
M, 2,89 0 апа per р. (2)


The test scores expressed in standard scores for the N different
individuals in the z tests can be shown in a test score matrix like


the following


"Su Yu 213 8i
LO # з Sox
23 732 75s Эзу
Z= (3)


Zni Sng Фаз +++ Зам


16


Since the test scores are expressed in standard scores, the cor-
relation between the tests j and ¢ for the N individuals ac-
cording to the formula for the product moment correlation will be


t=- C (4)


All the intercorrelations between the different variables сап be
included in the following correlation matrix:


Tu Tio Tis soe Tin
Wan Жы Жы wave б
Ta Ts Tas +++ Tsa
К, = (5)
Tui Tre Trg +++ Tan


The matrix algebraic expression for the product moment
correlations will then be


R,= i ZZ. (6)
'The aim of factor analysis is, expressed in mathematical
language, to describe the results in the tests as functions of a
few variables, which from certain standpoints are more “Ғипда-
mental” or convenient than the tests. These variables searched
for, are called factors and can be indicated by Fy, Fy, Fy, ... Г,.
Each one of the different test scores х; is thus regarded as a
function of the factors according to the general formula z; =
Е.Е, ЕЁ»... Fy).
Factor analysis has hitherto practically only dealt with the
case, when х; is a linear combination of factors as


Zj =а Ё. +аз Fu +... + а Fei +... Faq Fy (7)


Fy; is the factor score of individual 7 in the Ath factor, and а
is the Ath factor loading in test j. Formula (7) can be regarded


17


as a regression equation where х; is the dependent variable,
whereas the various Fp; and ау refer to q independent variables
and their respective regression weights.

The fully developed system of factor equations for n tests
and N individuals in expressed by


Z - AF. (8)


The matrix A includes the loadings of the q factors in the n
tests.


dg Hess eoe |
uy (dass a ee Uy
A= (9)
Qj у... а... Ag
Way eg mas gx 63 Gag


The matrix F includes the N individuals’ scores in the q
factors.


Be hy va ee on
Wal xx а Re dit
>>>