EFFECTS UPON THE FACT ORIAL SOLUTION OF
ROTATING VARYING NUMBERS OF FACTORS
WITH DIFFERING INITIAL COMMUNALITY ESTIMATES

Thesis for the Degree of M. A.
MICHIGAN STATE UNIVERSITY
Donald F. KieI
I966

.1.

 

C I 93775
W ,. .,'.;,12_I.’]

ABSTRACT
EFFECTS UPON THE FACTORIAL SOLUTION
OF ROTATING VARYING NUMBERS OF FACTORS

WITH DIFFERING INITIAL COMMUNALITY ESTIMATES

by Donald F. Kiel

The most crucial problem in the entire field of factor
analysis has been its subjectivity. ‘While a number of issues
must be resolved in order to arrive at a truly objective method
of factor analysis this study concerned itself with two key
issues -- (1) the number of factors which should be considered
for a particular set of data and, (2) the effect of different
initial communality estimates on the final solution.

Four correlation matrices, well known in factor analytic
literature, were selected for study -- the eight physical variables,
eight political variables and twenty-four psychological variable
matrices from Harman (1) and the eleven.Air-Force classification
tests from.Fruchter (2).

Three principal axes factor analyses were carried out on
each matrix, one using unities (1.0) in the diagonals of the
correlation matrix, another using "Guttman communalities" and
a third using squared multiple correlations. The unrotated factors
from each of the twelve analyses were ordered from largest to
smallest on their corresponding latent roots and an extensive
series of rotations were calculated using the normalized Varimax
method. The two largest factors were rotated, then the three

largest, and so on until all real factors were included in the

I'll! {I'll A I ‘ w
1"

Donald F. Kiel

set of solutions.

The findings indicated that there is very little difference
in the numeric values of the largest factors regardless of the
initial communality estimates used. A point is reached, however,
at which unique factors begin to appear when unities are used
which are not present when squared multiple correlations are used
as initial communality estimates.

The factors have a tendency to split as additional factors
are rotated, emerging in a definite hierarchical pattern. For
example, a large verbal-deductive factor present in one solution
may Split into verbal and deductive factors when an additional
unrotated factor is included in the solution.

As a result of this study, a criterion has been proposed
(frequently called the KielJWrigley Criterion) for the number of
factors which should be included in a factor analytic report,
i.e. "when to stop factoring," It suggests that unities are
satisfactory as diagonal entries in the correlation matrix to
be factored. A series of rotations should be carried out, starting
with the two largest factors, then the three largest, etc. until
a solution is reached which includes a rotated factor on which
fewer than three of the variables have their highest loadings.
This is based on the theory that three or more points are

necessarily to define a hyperplane in n-dimensional space.

EFFECTS UPON THE FACTORIAL SOLUTION
OF ROTATING VARYING NUMBERS OF FACTORS

'WITH DIFFERING INITIAL COMMUNALITY ESTIMATES

By 5.,

'.

,e
Donald F. Kiel

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
‘MASTER OF ARTS

Department of Communication

1966

ACKNOWLEDGMENTS

While the writing of theses is never regarded by graduate
students as a simple and enjoyable task, the extraction of the
present document from the author has been a particularly arduous,
frustrating and frequently painful chore. It has been produced
through years of alternate patience, threats, kindness, cooperation
and cajoling by many wonderful individuals -- too many, in fact,
to mention all of their names.

My deepest appreciation goes to Professor Charles F. Wrigley,
director of the Computer Institute for Social Science Research at
Michigan State University, who has been most deeply involved in the
progress of this research and has been most responsible for guiding
and influencing my thoughts for a number of years, both profession—
ally and personally.

Professors David K. Berlo and Hideya Kumata of the Department
of Communication at Michigan State University and Professor Malcolm
MacLean of the Department of Journalism, University of Iowa have
been wise counselors, stimulating teachers, good friends, and long-
suffering advisors. The late Professor Paul Deutschmann gave me
many hours of excellent advice and encouragement. I sincerely regret
that his untimely death in 1963 prevented him from.seeing the culmin-
ation of this effort.

.A Special word of heart-felt appreciation to my wife, Mary,
without whose love, patience and frequent persuasion this thesis

would never have been completed.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS -------------------------------------------

LIST OF TABLES ---------------------------------------------

LIST OF FIGURES --------------------------------------------

Chapter
I.

II.

INTRODUCTION """""""'""""“"""* .....
DATA.AND.ANALYTICAL METHODS ----------------------

Introduction

Correlation Matrices Used in the Study
Computing Procedures Used in this Investigation
Factor Rotation

III. THE EFFECT ON THE FACTORIAL STRUCTURE OF INCREASING
THE NUMBER.OF FACTORS IN THE ROTATED FACTOR
SOLUTION -----------------------------------------
IV. THE EFFECT OF DIFFERENT COMMUNALITY ESTIMATES
0N VARYING ROTATIONAL SOLUTIONS ------------------
V. CONCLUSIONS --------------------------------------
LITERATURE CITED -------------------------------------------

iii

Page
ii
iv

vi

10

26

51
71

82

LIST OF TABLES

Page
Intercorrelations Among Eight Physical Variables
for 305 Girls --------------------------------------- 9
Intercorrelations of Eight Political Variables
for 147 Election Areas (1932 Presidental Election) -- ll

Intercorrelations of Eleven Air Force Aptitude Tests- 13

Intercorrelations of Twenty-four Psychological
Tests for 145 Children ------------------------------ l6

unrotated Principal Axes Factors for Eight Physical
Variables Unities Used in Leading Diagonals of
Correlation Matrix ---------------------------------- 27

Two, Three, Four, Five and Six Factor Varimax
Rotation Solutions for Eight Physical Variables ----- 28

Eight Political Variables -- Unrotated Principal
Axes Factors Unities Used in Leading Diagonals of
Correlation Matrix ---------------------------------- 34

Eight Political Variables -- Varimax Rotation Solutions
for Two, Three, Four, Five and Six Factors ---------- 35

Eleven Air Force Classification Tests -- Unrotated
Principal Axes Factors Unities Used in Leading

Diagonals of Correlation Matrix --------------------- 40
Eleven Air Force Classification Tests -- Varimax
Rotational Solutions for Two, Three, Four, Five

and Six Factors ------------------------------------- 42
TWenty-four Psychological Tests -- Unrotated Principle
Axes Factors Unities in Leading Diagonals of

Correlation Matrix ---------------------------------- 46

TWO, Three and Four Factor Varimax Rotation Solutions
for Twenty-four Psychological Tests ----------------- 47

Communality Estimates for Eight Physical Variables -- 55

Effects of Differing Communality Estimates on the
First Four Unrotated Factors ------------------------ 56

Remaining Unrotated Principal Axes Factors with
Guttman Communalities ------------------------------- 57

Communality Estimates for Eight Political Variables - 59

iv

LIST OF TABLES (cont.)

Table Page

4.5 Effects of Differing Communality Estimates on the
First Three Unrotated Factors for Eight Political

Variables ------------------------------------------- 59
4.6 Communality Estimates for Eleven Air Force

Classification Tests -------------------------------- 61
4.7 Eleven Air Force Classification Tests Effects of

Different Communality Estimates on the First Five

Unrotated P.A. Factors ------------------------------ 62

4.8 Twenty-four Psychological Variables Initial

Communality Estimates ------------------------------- 66
4.9 Variance Contribution (Eigenvalues) of First Eight

Unrotated Factors as a Percent of Total Original

Communality for Twenty-four Psychological Variables - 67

Figure

3.

1

LIST OF FIGURES

Hierarchical Structure of Eight Physical Variables
Two Through Eight Factor Solutions Varimax

Rotation -- Unities --------------------------------

Hierarchical Structure of Eight Political Variables
Two Through Eight Factor Solutions Varimax

Rotation -- Unities --------------------------------

Hierarchical Structure of Eleven Air Force
Classification Tests --'With Unities as Initial

Communality Estimates ..............................

Hierarchical Structure of Twenty-four Psychological
Tests -- Starting with Unities in Correlation

Matrix ---------------------------------------------

Hierarchical Structure of Eight Physical Variables

Starting with Guttman Communalities ................

Hierarchical Structure of Eight Physical Variables

Starting with Squared Multiple Correlations --------

Hierarchical Structure of Eight Political Variables

Starting with Guttman Communalities ----------------

Hierarchical Structure of Eight Political Variables

Starting with Squared Multiple Correlations --------

Hierarchical Structure of Eleven Air Force
Classification Tests -- Starting with Guttman

Communalities --------------------------------------

Hierarchical Structure of Eleven.Air Force
Classification Tests -- Starting with Squared

Multiple Correlations ------------------------------

Hierarchical Structure of Twenty-four Psychological

Tests -- Starting with Guttman Communalities -------

Hierarchical Structure of Twenty-four Psychological
Tests -- Starting with Squared Multiple

Correlations .......................................
Eight Physical Variables ...........................
Eight Political Variables ..........................

Eleven Air Force Classification Tests --------------

Twenty-four Psychological Variables ................

vi

Page

32

36

41

48

58

58

6O

60

63

64

69

7O
76
77
78

79

CHAPTER I
INTRODUCTION

Factor analysis is a mathematical method by which a large
number of variables can be resolved, i.e. described, in terms of a
small number of underlying categories or "factors". The technique
is often mistakenly labeled as psychological theory primarily because
it had its foundations among psychologists who were attempting to
develop mathematical models for the description of human intellec-
tual ability and its deve10pment for many years continued to be a
psychological problem. Only in fairly recent years has there been
a concerted effort to bring factor analysis into the realm of a
recognized statistical theory.

The father of factor analysis is generally conceded to
be Charles Spearman, who in 1904 published a paper (41) on the
nature of intelligence in which he described a factor analytical
investigation of various cognitive tests. He Spent the rest of his
life developing his Two-Factor Theory which stated that intellect
consists of a general ability factor and a number of Specific
factors -- one for each test used. In other words, his theory was
that each test contains two factors, a factor common to all tests
and a factor Specific to the individual test.

By the 1930's it became apparent that a single factor
was not adequate to describe many batteries of psychological tests
and group factor theory develoPed. While many investigators con-

tributed to the develOpment of multiple-factor theory, the name of

 

L. L. Thurstone is most widely recognized for popularization of
the theory. His pioneering work, "Vectors of the Mind", published
in 1935 (42) set forth a set of principles which have become pop-
ular among factor analysts and known as Simple Structure Theory.

Implicit in this brief history is the controversy which
has been generated over the years as to the prOper system.of factor
analysis. The controversy has been primarily between British and
American schools of factor analytic thought and -- of particular
salience to this thesis -- the key issue has been the number of
factors worth extracting from a given battery of variables. The
British have traditionally tended to stop with fewer factors than
have the Americans.

Excellent presentations of details of the various factor
theories are available elsewhere (17, 7, 11, 18), and will be dis-
cussed where applicable in this thesis. The overriding problem,
however, with the entire field of factor analysis has been its
subjectivity. .As'Wrigley wryly states it (52):

... In most Statistical work two persons who start with the
same data and calculate correctly will reach the same answer.
This is not necessarily the case for factor analysis. This
remains a method which depends upon arbitrary judgments by
the investigator, so that skill is acquired only after long
experience in estimating communalities, deciding upon the nums
ber of factors to be extracted, selecting pairs of factors for
rotation, and so on. ... In general, mathematical statisticians
have ignored factor analysis, treating it as a jungle of the
psychologists' making, into which any self-resPecting statisti-
cian would be most unwise to stray.

'Hhile there are a number of issues which must be resolved
in order to arrive at an objective method of factor analysis -- that

is, a method which does not depend primarily on human judgment --

this study will concern itself with only two of these issues, namely,

 

the determination of the number of factors which should be conSidered
for a particular set of data and the effect of different initial
communality estimates on the solution.

Various criteria have been proposed in the past for either
determining the number of factors which should be extracted in fac-
toring a correlation matrix or detenmining the number of extracted
factors which should be rotated. Philip Vernon, in an unpublished
manuscript (44) in 1949, listed twenty-four indices of significance
for centroid factors. Other indices have since been advanced. The
question of how many factors to extract was more important with non-
computerized, short-cut methods of factor analysis when factors were
determined serially, one at a time. Since desk calculator computing
methods were extremely laborious for even small selections of
variables, investigators usually preferred to Stop with as few factors
as possible. Present day analytic methods using digital computers
usually extract as many factors as variables, making more important
the question of determining which of these factors should be con-
sidered meaningful for further analysis, i.e. how many Should be
rotated?

Two general classes of criteria have been suggested:

(1) Judgmental methods which are based on arbitrary judgments of

such things as proportion of variance accounted for by each factor
and accepting only those which account for more than, say, five or
ten or twenty percent of the total variance; clarity of the factor
solution which, unfortunately, depends upon some vague definition

of the concept of clarity; or size of the sums of squares of the

loadings. Kaiser, for example, proposes that only those principal-

axes factors, obtained from a correlation matrix with unities in
the diagonals, which have latent roots greater than 1.0 should be
accepted on the theory that it is a doubtful gain to accept into
the system.any factors which contribute less information than a
single test. (25)
(2) Statistical criteria which involve tests with an associated
probability level to determine if factor loadings or residuals are
Significantly different from zero. Such Statistical criteria have
the disadvantage of being dependent upon the size of the sample from
which the correlation matrix was calculated and usually result in a
number of statistically significant factors which have no practical
value (28, 19). A.number of factor analysts have found that
empirical tests of significance frequently lead to about the same
results as the more proper statistical tests (17, p. 363).

Factor analysts, following the Thurstonian concept of
simple structure, have tended to regard each factor as having
equal status with every other factor, and that there is therefore
only one "correct" solution. It has also been the tradition for
published factor analyses to provide only one of a series of inter-
pretable solutions -- the particular one depending upon the inves-
tigator's criterion for, or judgment about, completeness of factor
extraction. However, little seems to be known about the effects
upon the final solution of varying the number of factors (52).
There have been a few suggestions in the literature which suggest
that a hierarchical organization would be more effective (40, 15,
3, 48) and_a few prior studies have noticed that when the number

of factors is increased the larger factors may Split into smaller

ones (45, 51). No Systematic study of this phenomenon has been
made.

Some newer methods of factor analysis introduced in the
past twenty-five years, such as Lawley's Maximum.Like1ihood method
(30) and Rao's Canonical Analysis (38), attempt to bypass the sep-
arate extraction of factors followed by rotation to a meaningful
and interpretable solution by going directly to a "final" solution.
Such methods have the disadvantage that they also depend upon a
statistical criterion making use of sample size and in actual
practice tend to produce far more factors than are considered mean-
ingful by most practicing psychologists or others using factor
analytic methods.

The problem.of the number of factors which Should be
accepted in a factor analysis is difficult to isolate from.the
problem of communality, that is, the initial entries in the diagonal
cells of the correlation matrix. In the Thurstonian simple
structure model the two problems are interrelated. This problem is
discussed in greater detail in Chapter IV.

The controversy between psychologists about the factors
of mental ability and the foundations of factor analysis among
psychologists should not be construed as limiting factor analysis
to usefulness only by psychologists. Indeed, quite the contrary
is the case. .Among the examples used in this study is an early
application in the field of physiology and one from political
science. Some stimulating applications have been and are being
made in the fields of medicine for classification of symptoms and

clinical tests as an aid to diagnosis, (4, 8, 35) in urban and

regional planning for comparison of American cities (37, 2), in
advertising and marketing research (6), political science (53) and
in many other disciplines.

The use of factor analysis in communications research
is not new, probably because many of the theoreticians in commun-
ications are also psychologists and have introduced the techniques
into other communications oriented areas. Extensive use of factor
analysis has been made by C. E. Osgood, et. a1. (34) in the study
of the measurement of meaning and the develOpment of the widely
used semantic differential technique, by Berlo, Lemert and Mertz
in the study of source credibility (1) and Kumata (29) in the cross-
cultural analysis of meaning. In the field of journalism.some of
the studies using factor analysis are those of Nafziger, MacLean
and Engstrom on tools for newsPaper readership data (32) and
Deutschmann and Kiel on attitudes toward the mass media (9).

The present study was undertaken for two reasons:

(1) "traditional" methods of factor analysis involving the
rotation of factors are most common and have a vast background

in empirical studies -- they will not soon be completely obsolete;
and (2) no systematic Study of the effect of rotating increasing
numbers of factors has been done. The results of such a study may
shed more light on efforts to reach the ephemeral goal of a
"definitive" direct and completely objective method of factor
analysis or at least encourage researchers to report factor
analytic results in a way which would make possible more and better

comparisons between similar studies.

CHAPTER II

DATA.AND.ANALYTICAL METHODS

Introduction

Since the purpose of this study was to investigate various
factorial solutions in the hope that an objective criterion for
"when to stop factoring" could be developed, particularly one which
was not dependent upon strictly Statistical criteria, methods of
analysis were selected which did not require subjective decisions
at any point in the analysis; and, Since it was also intended that
the findings of this investigation should have general applica-
bility to the wide range of disciplines now using factor analysis,
the data for this study were chosen not because of the relevance
of the variables to any particular discipline but, rather, because
the matrices had been intensively studied and reported on by other
investigators and were therefore well-known, even classics, in

factor analytic literature.

Correlation.Matrices Used in the Study

Initially ten matrices were considered for analysis and
some work was done on all of them. However, it became evident early
in the study that many of them would merely duplicate the results
of other matrices and unnecessarily confuse the presentation of
results, so the list was narrowed to four matrices which clearly
illustrated different types of factor solutions, and the major '

emphasis was placed upon the intensive study of these four.

In two of the four examples the matrices were made up of
subsets of variables chosen from larger sets of variables for
computational convenience by earlier investigators. This fact does
not reduce their value for the purposes of the present study, but it
would be of interest to apply the methods of this study to the
larger matrices at some future time to validate the results of this
study and to further insure that the criterion of "factorial in-

variance" is met.

Matrix I: Eight Physical Variables. -- This is a matrix of
intercorrelations of eight physical measurements made on 305 girls
between the ages of seven and seventeen. They were chosen by
Holzinger and Harman (21, p. 80) from a larger set of seventeen
variables reported by Mullen (31) to be representative of two
distinct factors which were bi-polar. It has been intensively
analyzed by Holzinger and Harman and by Harman (17, p. 82).as an
example of a rank two matrix. The numbering and description of
the variables and the complete correlation matrix is shown in
Table 2.1. Most of the measurements are self-explanatory except
for "bitrochanteric diameter" which is, in laymen's terms, hip
measurement.

The first four variables were chosen to be measures of
longitudinal growth or "lankiness" and the second four as measures
of horizontal growth or "stockiness". Observe that the first four
are more highly intercorrelated with each other than are the second

four and that, in comparison with most empirical matrices, part-

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10

icularly in the fields of psychology and communications, the

correlations are very high.

Matrix II: Eight Political Variables. -- This set of variables
has also been used by Holzinger and Harman as an example of the
applicability of factor analysis to a non-psychological field.
It is, again, a subset of seventeen political variables collected
by Gosnell and Schmidt (13) in 147 Chicago election areas following
the 1932 presidential election. It is well to remember, in inter-
preting the results, that this was at the height of the Depression,
with high unemployment and a great deal of "it's time for a change"
sentiment among the population.
The complete matrix of intercorrelations is given in Table
2.2 and a brief description of each of the variables follows:
1. ‘ngig -- percentage of the total Democratic and
Republican vote cast for Lewis, the Democratic candi-
date for mayor of Chicago in the 1932 election.
2. Roosevelt -- the corresponding percentage of the total
vote cast for Franklin Delano Roosevelt, the Democratic
candidate for President of the U. S. (and land-slide

victor) in 1932.

3. Party Voting -- percentage that the straight-party votes
were of the total vote in the election area.

 

4. Median Rental -- median rental price in the election
area (in dollars).

 

5. Home Ownership -- percentage of the total families in
* the election area owning their own homes.

6. Unemployment -- percentage unemployed in the election
area in 1931 of the gainful workers ten years of age
and over. '

7. Mobility -- percentage of total families in the area
at the time of the election who had lived less than
one year at the present address.

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12

8. Education -- percentage of population, eighteen years
of age and older, which had completed more than ten
grades of school.

‘Matrix III: Eleven Air Force Classification Tests. -- In Table 2.3
are the intercorrelations of eleven tests which were part of a
battery of tests used by the United States Army.Air Force during
‘Horld‘war II to classify aviation cadets into training assignments
for air-crew positions such as pilot, bombardier, and naviagtor.
The product moment correlations are based on a sample of 8,158
unclassified aviation students. The complete description of these
tests may be found in Guilford (14) although the matrix has been
intensively studied by Fruchter (11, pp. 69-72) and it is Fruchter's
analysis which has been used for comparison. A brief description
of the eleven tests follows:

1. ‘Qial and Table Reading -- the test consists of two
parts, the first of which measures how quickly and
accurately the examinee can read the dials on an in-
strument panel; the second involves locating Specific
values within the body of tables.

2. Spatial Orientation I -- a perceptual-Speed test in
which the subject is required to locate small sections
of an aerial photograph within a larger picture.

3. Reading Comprehension -- a test designed to measure
understanding of paragraph material and the ability
to make inferences based on the material read. An
attempt was made to minimize mechanical and numerical
content in the material presented.

4. Instrument Comprehension -- each of the 60 items consisted
of pictures of two instruments, and artificial horizon
and compass, followed by pictures of a plane in five
different attitudes. The problem was to determine which
of the five planes had a position and direction consistent
with the instrument readings.

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5. Mechanical Principles -- each item pictured a mechan-
ical stiuation, with questions designed to test the sub-
ject's ability to understand mechanical forces and
movements.

6. <§peed of Identification -- the subject was required
to match airplane silhouettes.

7. Numerical Operations I -- 100 simple numerical-computation
items involving addition and multiplication.

8. Numerical Operations II -- same as number 7 except
the problems involved subtraction and division.

9. Mechanical Information -- a verbally stated mechanical
knowledge test, relating particularly to Operation of
parts of automobiles. The items were quite brief,
calling for only a limited amount of reading and re-
quiring quite Specific mechanical knowledge.

10. Practical Judgment -- a test requiring the subject to
determine the most practical course of action to a
verbally presented problem situation.

11. Complex Coordination -- an apparatus test of the Speed
and accuracy of hand and foot adjustments to a complex
perceptual stimulus. The subject was faced with a panel
containing three rows of red lights and three rows of
corresponding green lights. 'Hhen a particular stimulus
pattern of red lights was presented the subject was
required to move controls similar to those used in an
airplane in flight so as to turn on the green lights
correSponding to each of the red lights. AS soon as
the match had been completed, a new set of red lights
was automatically presented.

 

Matrix IV: Twenty-four Psychological Tests. -- This example con-
sists of twenty-four psychological tests administered to 145 seventh
and eighth grade pupils of a suburban Chicago school in the late
1930's. The data was gathered by Holzinger and Swineford (20) and
Subsequent analyses by Holzinger and Harman (21), Harman (17),
Kaiser (27), Neuhaus and‘wrigley (33) and others have made it a

classic in factor analytic literature. The complete correlation

'3

15

matrix is presented in Table 2.4 and a brief description of the

tests follows:

1.

10.

11.

12.

13.

.Visual Perception Test -- a non-language multiple-
choice test composed of items from Spearman's
Visual Perception Test.

Cubes --.A simplification of Brigham's test of
Spatial relations.

Paper Form Board --.A revised multiple-choice test
of Spatial imagery, with dissected squares, tri-
angles, hexagons, and trapezoids.

Flags -- Adapted from a test by Thurstone. Requires
visual imagery in two or three dimensions.

Genergl Information --.A multiple-choice test of
a wide variety of simple scientific and social facts.

Paragraph Comprehension -- Comprehension of written
material measured by completion and multiple-choice
questions.

Sentence Completion --.A multiple-choice test in
which "correct" answers reflect good judgment on
the part of the subject.

‘Hord Classification -- Sets of five words one of
which is to be indicated as not belonging with the
other four.

‘word Meaning --.A multiple-choice vocabulary test.
Add -- Speed of adding pairs of one-digit numbers.

Code --.A simple code of three characters is presented
and exercise therein given to measure perceptual
Speed.

Counting Groups of Dots -- Four to seven dots,
arranged in random patterns, to be counted by the
subject. .A test of perceptual Speed.

Straight and Curved Capitals --.A series of capital
letters. The subject is required to distinguish
between those composed of straight lines only and
those containing curved lines. .A test of perceptual
Speed.

 

 

 

TABLE 2.11 INTERCORRELATIONS OF TWENTY-FOUR PSYCHOLOGICAL TESTS FOR 1115 CHILI‘REN

 

 

 

 

 

 

 

Test 1 2 3 8 5 6 7 8 9 10 11 12 13 18 15 16 17 18 19 20 21 22 23 28
I
1 1.000
2 .318 1.000 ‘
3 .803 .317 1.000 ‘
8 .868 .230 .305 1.000 .
5 .321 .285 .287 .227 1.000
6 .335 .238 .268 .327 .611 1.000
7 .308 .157 .223 .335 .656 .722 1.000 .
8 .332 .157 .382 .391 .578 .527 .619 1.000 -; ‘
9 .326 .195 .188 .325 .723 .718 .685 .532 1.000 .; '
10 .116 .057 —.075 .099 .311 .203 .286 .285 .170 1.000 . w
11 .308 .150 .091 .110 .388 .353- .232 .300 .280 .888 1.000 -' ’
12 .318 .185 .180 .160 .215 .095 .181 .271 .113 .585 .828 1.000 f
13 .889 .239 .321 .327 .388 .309 .385 .395 .280 .808 .535 .512 1.000 1
18 .125 .103 .177 .066 .280 .292 .236 .252 .260 .172 .350 .131 .195 1.000 ;
15 .238 .131 .065 .127 .229 .251 .172 .175 .288 .158 .280 .173 .139 .370 1.000
16 .818 .272 .263 .322 .187 .291 .180 .296 .282 .128 .318 .119 .281 .812 .325 1.000 2,, .
17 .176 .005 .177 .187 .208 .273 .228 .255 .278 .289 .362 .278 .198 .381 .385 .328 1.000*«: . ;,u,..:
18 .368 .255 .211 .251 .263 .167 .159 .250 .208 .317 .350 .389 .323 .201 .338 .388 .888 1:060 “ “7*1
19 .270 .112 .312 .137 .190 .251 .226 .278 .278 .190 .290 .110 .263 .206 .192 .258 .328 3358 1.0001,
20 .365 .292 .297 .339 .398 .835 .851 .827 .886 .173 .202 .286 .281 .302 .272 .388 .262 2.301 .167 1.COO
21 .369 .306 .165 .389 .318 .263 .318 .362 .266 .805 .399 .355 .825 .183 .232 .388 .173 .357 .331V .113 1.000
22 .813 .232 .250 .380 .881 .386 .396 .357 .883 .160 .308 .193 .279 .283 .286 .283 .273 '.317 ;3821 .863 .378 1.000
23 .878 .388 .383 .335 .835 .831 .805 .501 .50‘4 .262 .251 .350 .392 .282 .256 .360 .287 .272. g303; .599 .851 .503 1.000
28 .282 .211 .203 .288 .820 .833 .837 .388 .828 .531 .812 .818 .358 .308 .165 .262 .326 .805e .378 1.366 .888 .375 .838 1.000
Harman, Harry H. Modern Factor Analysis. hicago, 111.: U. of Chicago Press. 1960. p. 138,. :’

 

—_—-

 

91

 

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

17

‘Hord Recognition -- Twenty-five four-letter words
are studied for three minutes. These words are then
to be checked from.memory on a hundreddword list.

Number Recognition -- Similar to test 14. Fifteen
three-digit numbers.

Figure Recognition -- Similar to test 14. Fifteen
geometric designs.

Object-Number -- Twenty pairs of names of familiar
objects and two-digit numbers are studied for three
minutes. The words only are then presented to the
subject, who is required to supply the pr0per numbers.

Number-Figure -- Similar to test 17. Ten pairs of
numbers and geometric figures.

FigureAHOrd -- Similar to test 17. Ten pairs of
geometric figures and words studied for one minute.

Deduction -- Logical deduction test using the symbols
) and ( and the letters A, B, C, and D.

Numerical Puzzles --.A numerical deduction test,
the object being to supply four numbers which will
produce four given answers employing the operations
of addition, multiplication, or division.

Problem Reasoning --.A reasoning test in completion
form. Each problem lists the steps in obtaining a

required amount of water using two or three vessels
of given capacity.

Series Completion -- From a series of five numbers
the subject is supposed to deduce the rule of
procedure from.one number to the next, and thus
supply the sixth number in the series.

Woody-McCall Mixed Fundamentals: Form I -- A series
of thirty-five arithmetic problems, graduated for
difficulty.

ngputing Procedures Used In This Investigation

The decision as to the procedures to be used in this invest-

igation was based on four criteria: (1) the solution produced

should be as mathematically precise as possible, (2) no subjective

18

decisions Should be required in arriving at a solution, such as
estimate of rank, significance of residual matrix, etc., (3) the
solution should be psychologically acceptable, and (4) the methods
should be programmable for computation on an electronic computer.
.All of the original calculation for this study were made
on MISTIC, Michigan State University's original electronic digital
computer. Several new computer programs were written for MISTIC
in conjunction with this study and, Since MISTIC is now obsolete,
most of them.have been reprogrammed for or incorporated into
existing programs for the Control Data Corporation 3600 Computer
currently in use at Michigan State University. A 3600 FORTRAN
program (called FACTORA), incorporating all of the methods used in
this study as well as provision for calculating from.the raw data
matrix, is available from.the Michigan State University Computer
Laboratory. .All of the data for this study have been recalculated
on the CDC 3600 and the complete set of results and a listing of
the program are on file in the Michigan State University Library.
It is interesting to compare calculation time as an
example of the growth of the computer "art". On MISTIC, which
required laborious punching of paper tape for input and output and
some desk calculator work between runs, the complete eigenvalue-
factor loading solutions for each of three different communality
estimates (discussed in Chapter 5) and all sets of rotations for
all three sets of factors required approximately fifteen hours
of computer time over a period of several months. .All of the

same results were obtained on the 3600 in one computer run in

a...

19

less than ten minutes.

For each of the four examples used in this study, fact-
oring started from a previously calculated matrix of product-
moment correlations. For the methods used in this study no
assumptions are made about the statistical distribution of the
variables, hence the matrices could have been any other of the
frequently used non-parametric indices of relationship, i.e. the
tetrachoric correlation coefficient, phi coefficient, biserial or
point biserial correlation, etc. ‘Hhenever the assumptions of the
product-moment correlations can be met, however, they are the
preferred starting point for factor analysis, since they allow
the calculation of additional statistical tests for the signif-
icance of factor loadings, etc. Most modern factor analysis
programs for digital computers provide for Starting with "raw"
data; that is, the individual measurements of each of the
variables for each observation point (subject).

For the initial phase of this investigation unities
were used in the leading diagonals of the correlation matrices.
The controversy over the appropriate initial diagonal entries,
the so-called "communality question", is a burning issue in
factor analytic literature and will be discussed further in
Chapter 5. Unities, which are the self-correlations of each
variable and represent the total variance of each variable in the
linear factor model, were chosen because (1) they were computa-
tionally most convenient since they‘were already in the diagonals

in the output of the correlation program on MISTIC, (2) they

20

preserve the Gramian property of the matrix, producinglg_real,
positive roots, and (3) they are one of the few unique communality
estimates.

From.both a mathematical and statistical point of view
the preferred method of extracting the initial factors is the
method of principal axes, first proposed by Pearson in 1901 (36)
and developed by Hotelling in the 1930's (22). ‘While this method
was recognized as superior for many years, the computational method
was so laborious that it was not until high-Speed electronic computers
came into general use in the 1960's that the method became practical
for use with other than very small matrices.

The principal axes method has a number of properties which
make it ideal from a factor analytic standpoint: (1) it produces
a unique solution for any given correlation matrix; (2) the first
factor extracted in the sequential method (the factor correSponding
to the largest eigenvalue in the Jacobi method) accounts for the
maximum possible pr0portion of variance in the matrix, the second
factor for the maximum.pr0portion of the remaining variance, the
third factor for the maximum.remaining after the first two were
extracted, and so on (usually a small number of the total roots
will account for almost all of the total communality); (3) the
resulting columns of factor coefficients are "orthogonal"; that is,
the correlation between any two pairs of factors is zero meaning
that all factors are independent. This prOperty, expressed mathe-

matically, is:

21

where ap is a vector of factor loadings, AP is an eigenvalue of
the correlation matrix and the Kronecker 6P9 = 1 if p = q and
6pq = 0 if p i q.

The original method proposed by Hotelling (22), involving
the sequential extraction of factors, is extremely arduous and
inefficient even on high-Speed calculators (although it has been
programmed and used on MISTIC for factoring very large matrices
when only a relatively Small number of the total possible factors
was desired and when capacity would otherwise have been exceeded).
For computer applications the relationship between factor analytic
theory and the mathematical problem of determining the eigenvalues
and eigenvectors of a square, symmetric matrix has been taken into
consideration. The most frequently used method, and the method
programmed originally for MISTIC and more recently for the CDC
3600, makes use of work done by Jacobi (24) in the 1840's --
commonly known as the Jacobi method. It is an iterative method
which produces the complete matrix of eigenvalues (latent roots)
and eigenvectors without producing a residual matrix after each
eigenvalue. The eigenvectors are transformed into factors by
scaling each eigenvector by the square root of the correSponding
eigenvalue.

Even the Jacobi method has disadvantages. For large
matrices the number of iterations necessary frequently leads to
very great rounding errors. Recent work by Householder (33),
Givens (12),‘Hilkinson (46), and others is leading to more effi-

cient methods of solving the eigenvalue-eigenvector problem on

22

electronic computers. These methods are being rapidly added to the

repertory of computer methods of factor analysis.

Factor Rotation

While an occasional factor analyst will stop once the
principal axes factors have been extracted, most psychologists as
well as users of factor analysis in other disciplines do not find
the end product of the principal axes method useful or acceptable
from the standpoint of interpretability; Most prefer to rotate
some subset of the principal axes factors to some reference structure
which provides more psychological "meaningfulness".

For years the methods of rotation were crude, subjective
methods in which the final solution depended on the judgment of the
investigator. Any two investigators analyzing the same data were
not likely to reach the same solution (52). Just over ten years
ago the first significant developments were reported of analytical,
objective methods of rotation. Carroll (5), Neuhaus and'Hrigley
(33), Saunders (39), and Ferguson (10) almost simultaneously,
although independently, developed criteria which were very similar.
Neuhaus and'Hrigley, who were the first to program.and use the
method on electronic computers, coined the now well-known name,
Quartimax, for this method. It attempts to minimize the complexity
of the individual variables, that is, to approach a unifactorial
structure in which each variable has a high loading on only one
factor. This is extremely difficult to achieve with empirical data,
capecially for orthogonal structures, and usually leads to a fairly

large general factor.

23

Since the quartimax method did not completely meet one
of the important requirements for simple Structure in the Thurstonian
sense, namely that the number of zero loadings on a factor should
be maximized, Kaiser (25) deve10ped a modification of the quartimax
method which he called the varimax method. 'Uhere the quartimax
method placed the emphasis on the Simplification of each-Egg
(variable) of the factor matrix, the varimax method places more
emphasis on Simplifying the columns (26).

It was also Kaiser who noted a disadvantage in both his
original varimax method as well as in the quartimax method, namely,
that even after rotation there was more diSparity in the variance
contributions (sums of squares of the loadings on a factor) of the
different factors than is desirable. In other words, one objective
of rotation should be to give equal weight to each factor. Kaiser
attributed this diSparity to the fact that in either method of
rotation, each variable contributes to the function being maximized
as the square of its communality. In other words, a variable with
a communality twice as great as another variable will influence the
rotation four times as much. For this reason, Kaiser modified his
original method (referred to as the “raw" varimax method) by
weighting each variable so that it contributes equally to the
rotation.

This procedure, known as normalizing, involves dividing
each loading before rotation by the square root of the sum of
squares of all of the loadings for that variable (the observed

communality of the variable for that solution) which extends the

24

vector representation of that variable in common factor Space to
unit length, performing the rotation on the "normalized" loadings,
and then reducing the vectors back to their original length by
‘multiplying all of the rotated loadings for a variable by the
original Scaling factor. Notice that under orthogonal rotation
the communalities remain constant even though the individual
loadings change. The method using this weighting process is known
as "normal" varimax and requires that the final loadings be such

as to maximize the following function:

mu m n
V.1n 2 2(b /h,)‘*- 2(Eb2/h2)2

p=1j=1 19 J p=1 1:1 12 J

The mathematical details for achieving this maximization is available
elsewhere (25).

The same weighting or normalizing process is applicable
to the quartimax method as well, and results in a considerable
improvement in evenness of the factors over the "raw" quartimax.
The normalized quartimax and varimax are the methods presently
programmed for the CDC 3600.

The question of which rotational method is "correct" is
unresolvable. The normal varimax method was selected for this
study because it results in a solution which is probably closest
to the concept of simple structure preferred by most American
factor analysts. .After comparing varimax, quartimax and subjective
graphical solutions for four factors of the twenty-four psychological
tests, Harman concludes (17, p. 306) that "the varimax solution

seems to be the 'best' parsimonious analytical solution in the

25

sense that it correlates best with the intuitive concept of that

term as exemplified by the graphical solution." It has an additional
attribute which is considered by Kaiser as of primary importance;
namely, that of "factorial invariance." AS Thurstone describes this
principle, it is that "the factorial description of a test must
remain invariant when the test is moved from one battery to another
which involves the same common factors." (43, p. 361)

For the purposes of this study, the unrotated principal
axes factors were ranked in order by decreasing size of their corres-
ponding eigenvalues and then the first two, the first three, four,
five, and so on, factors were rotated. In the case of the factors
obtained using squared multiple correlations as initial communalities
(Chapter 5), only those factors corresponding to positive eigenvalues

were rotated.

CHAPTER III
THE EFFECT ON THE FACTORIAL STRUCTURE OF INCREASING

THE NUMBER.OF FACTORS IN THE ROTATED FACTOR SOLUTION

Introduction

The initial phase of this investigation was concerned with
observing the effects of rotating the first two, then the first
three, etc. factors where first refers to the unrotated factor
corresponding to the largest eigenvalue, second to the next
largest eigenvalue, and so on as described in the preceding chapter.
In this phase only those factors obtained using unities in the
principal diagonals of the various test correlation matrices were

studied.

Eight Physical Variables

The eight variables in this matrix were Specifically chosen
by Holzinger and Harman to represent four "longitudinal" and four
"horizontal" variables. In previously published analyses of this
matrix only two principal axes factors have ever been given, and
these using communalities calculated by estimating the rank. Table
3.1 shows the unrotated principal axes factors for this matrix.
The maximum.discrepancy between the unrotated factors calculated by
Harman (17, p. 173) and those obtained in this study is only .09.
As might be expected, since the variables were chosen to represent
only two factors, slightly more than 80% of the total variance is
accounted for by the first two factors. The first factor is a

large general factor with the first four variables (hypothesized

26

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29

as "lankiness“ variables) most highly loaded. The second unrotated
factor is of the bipolar type; that is, the two groups of variables
have opposite signs.

Table 3.2 contains the rotational solutions for two, three,
four, five and six factors.

'Uhen rotated the two factors originally hypothesized are
clearly evident, the longitudinal variables (1, 2, 3, and 4)
appearing highly loaded on the first factor and the four horizontal
variables (5, 6, 7, and 8) highly loaded on the second factor.
Notice that the observed communalities (columns headed hz, the
usual abbreviation) indicate that 85% or more of the total
original communality of the first five variables has been extracted
in the first two factors. (Since the initial communality was 1.0
for each variable, the observed communality is a prOportion.)

Notice that only 62% of the total variance of variable 8
appears in the first two factors and that the loading of this
variable on the third unrotated factor is .60, an additional
variance contribution of approximately 36%. The effect of in-
cluding this third factor in a rotation is to Split the eighth
variable away from the group of "stockiness" variables to form a
new factor on which only the eighth variable has a high loading
although there is still an appreciable loading of variable 8 on
the second factor.

On the unrotated factors, variables 6 and 7 have appreciable
loadings on the fourth factor, although of opposite signs, and each
of the other variables also have relatively higher loadings on at

least one of the remaining factors. These additional non-zero

 

 

3O

loadings are probably attributable to "unique and Specific"
variance, since communality estimates were not used in the diagonals
of the correlation matrix.

In the four factor solution, variables 6 and 7 do not form
a new factor but, instead, because of the Opposite signs, cause
the formation of two new factors, doublets of variables 5 and 6
and another of variables 5 and 7 (variable 5 has its highest
loading on the factor with variable 6). Additional examples of
bipolar factors Splitting into either Specific factors or doublets
will be seen in other test matrices.

The rotation of the five largest factors produces no new
factors. The fifth factor is of the type we Shall call a "null"
factor, signifying that none of the variables have their highest
loading on that factor and, in most cases, none of the variables
have any appreciable loading on such a factor.

The addition of the sixth factor to the rotational solution
results in a Specific factor on which variable five has its highest
loading although variable 5 still retains appreciable loadings on
the two doublet factors produced in the four factor solution.

The seven and eight factor solutions produce no new factors on
which any other variable has its highest loading, although two
factors appear which have relatively high loadings on variables

1 and 4. Variables 1 through 4, the "lankiness" or longitudinal
variables, which had the highest correlation coefficients and

the largest loadings on the first unrotated factor, remain grouped

into a single factor.

31 —

Figure 3.1 gives a hierarchical diagram of the results of
the seven sets of rotations. The variable numbers in solid boxes
indicate that those variables had their highest loadings on that
particular factor. 'Hhen a variable had a loading greater than .40
(16% of the variance of that variable) on a particular factor it is
indicated in dotted lines for that factor. The symbol N indicates
a "null" factor, one on which no variable had any appreciable
loading. Factors on which at least three variables were most highly
loaded are noted by rectangular boxes, those with less than three
highest loadings are enclosed within circles.

From a physiological point of view the hierarchy seems to
be eminently sensible. The "lankiness" variables are related to
bone structure which normally is proportional in an individual and
is independent of leanness or obesity. The arm Span and length of
forearm (variables 2 and 3) would naturally be most closely related.
The "stockiness" variables, on the other hand, are less closely
related. It would not be unusual for girls seven to seventeen to
have varying chest and hip girths, and the chest width and chest
girth are measures of somewhat different types of bodily deve10pment.

Additional Speculation is not germane to this discussion.

32

FIGURE 3.1
HIERARCHICAL STRUCTURE OF EIGHT PHYSICAL VARIABLES --
STARTING WITH UNITIES IN CORRELATION MATRIX

Number of
Factors

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

®®®

 

 

 

33

Eight Political Variables

The unrotated principal axes factors for this matrix are
shown in Table 3.3. In this example over 85% of the total variance
is contributed by the first two factors, although almost 30% of the
variance of the first variable appears in the third factor.

The results when the two largest, the three largest, etc.
factors are rotated is presented in Table 3.4. The order of the
variables has been rearranged in the table to group the variables
in the factors in which they appear. Figure 3.2 presents the same
information in the form of a hierarchical structure. Notice that
it is much easier to interpret in this form.

When only two factors are rotated, six of the eight variables
have their highest loadings on the first factor, and the other two
have their highest loadings on the second factor. Variables 4 and
8 also have appreciable loadings on the second factor. Observation
of the signs of the loadings is necessary for interpretation of
these factors, since both are of the bipolar type. These two
factors have previously been identified by Holzinger and Harman
(17, p. 178) as a large "Traditional Democratic Voting" factor
(relating high Democratic party as well as straight ticket vote
to high unemployment, high residential mobility, and IOW'median rental
and low education) and a smaller factor which has been called a
"Home Permanency“ factor (high home ownership negatively related to

home mobility, education, and median rental).

34

 

 

 

 

 

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2

436

FIGURE 3. 2

HIERARCHICAL STRUCTURE OF EIGHT POLITICAL
VARIABLES -- STARTING WITH UNITIES IN
CORRELATION MATRIX

 

 

 

 

 

 

 

 

 

 

 

37

‘When three factors are rotated the six variables which
previously clustered on the first factor subdivide into two groups.
The first variable (percent voting for Lewis) shifts completely to
the third factor and "percent Roosevelt vote" also has its highest
loading on this third factor, although still retaining a high
loading on the first factor. 'While two high loadings do not uniquely
determine a factor, we can see that there is a tendency for the
"Traditional Democratic Vote" to subdivide into what might be inter-
preted as "Local" and "National" components. It is possible that
had there been additional measures of local voting behavior in the
selection of variables, this third factor might have been better
established. In Chapter 5 it will be shown that when lower commun-
ality estimates are used, rotational stability is established with
the three factor solution.

'When the four largest factors obtained with unities in the
diagonals of the original correlation matrix are rotated the first
factor, on which variables 3, 4, 6, and 8 had their highest loadings
in the three factor solution, splits with variables 4 and 8 forming
a new factor on which variable 6 also has an appreciable negative
loading. In this solution four factors are produced, on each of
which two variables have their highest loadings. It should be noted
that each of the four factors so obtained seem.to "make sense" from
the standpoint of interpretability.

The addition of a fifth factor produces only a "null" factor.
The six factor solution, however, results in the splitting of

variables 3 and 6 into two unique factors. Rotational stability is

—-

 

38

achieved at this point, with the seven and eight factor solutions

adding only additional null factors.

Eleven.Air Force Classification Tests

The analysis of this matrix is particularly interesting
because it illustrates quite a different effect than previously
observed, namely, the complete disintegration of the factorial
structure. In each of the two preceding examples, at least one
group of variables remained clustered in a single factor; that is,
at least one factor remained on which more than one variable had
its highest loading even after all factors had been included in
the rotational solution. In this example, when unities are used
in the diagonals of the original correlation matrix, complete
fissioning takes place, eventually resulting in eleven unique
factors.

The difference is apparent even with the unrotated factors
(Table 3.5). Unlike the two preceding examples, in which the first
two factors accounted for from eighty to ninety percent of the
total variance, with this matrix the first two factors account for
less than 50% of the total variance; in fact, the first five factors
(this matrix has been used elsewhere in the literature (11, p.
149-151) as an example of a five factor test battery) account for
only about 75% of the total variance. The total variance is
apportioned much more evenly among the eleven unrotated factors,
with the later factors accounting for a higher percentage of the

variance than had been the case in the two preceding examples.

39

Figure 3.3 shows the hierarchical structure for the Eleven
Air Force Classification Tests matrix. The effect of rotating the
first two factors is that those variables (1, 2, 6, 7, 8) which had
negative or zero loadings on the second factor grouped on one factor
and those with positibe loadings grouped on the other factor; this
in spite of the fact that variables 5 and 9 had higher loadings on
the second unrotated factor than on the first.

‘Uhen three factors are rotated, the three variables which had
the largest negative loadings on the third unrotated factor (2, 6, 11)
have split away from the two preceding factors and formed a new factor.
Variable 4, which was also negatively loaded on the third unrotated
factor, and variable 1 which had a zero loading on the third factor
also have appreciable loadings on this new third factor.

The four factor solution results in splitting of both the
second and third factors of the three factor solution. It is at
this point that the solution begins to stabilize. In this example
less than fifty percent of the total variance of variables 2, 3, 4,
6, 10 and 11 were accounted for in the first three factors. (The
communalities shown for each solution may be interpreted as prOportions,
since the total original communality was 1.0 for each variable.)
Notice that there was a great deal of factorial instability between
the two, three and four factor solutions -- that, in general, the
variables which had the greatest increase in proportion of variance
with the addition of a new factor tended to have the greatest influence
in the formation of a new factor.

The five factor solution;is that presented in the literature

by Fruchter. It would be of little value to present a detailed

 

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41

FIGURE 3. 3

HIERARCHICAL STRUCTURE OF ELEVEN AIR FORCE CLASSIFICATION
TESTS -- WITH UNITIES AS INITIAL COMMUNALITY ESTIMATES

 

 

 

 

 

 

 

 

 

 

 

Numberof
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43

interpretation in this study, but for reference they have been
identified by Fruchter as: (numbered in decreasing order of factor

variance)

I -- Numerical. The three highest loadings are for numerical
operations (1, 7, 8)

II -- Perceptual Speed, consisting of the Spatial orientation
test (a misnomer?) and Speed of identification test
(variables 2 and 6), both of which involve perception
of small detail working against a time limit.

III -- Mechanical Experience, with highest loadings on the
mechanical principles and mechanical information tests
(variables 5 and 9)

IV -- Verbal Comprehension, with highest loadings on the
reading comprehension (3) and practical judgment (10)
tests

V -- Spatial Relations, consisting of the tests of instrument
comprehension (4) and complex coordination (ll).

Fruchter points out that "in a new area of investigation it
would ordinarily not be feasible to identify five factors derived
from only eleven tests." (11, p. 149) This is certainly true.

This was a matrix based on a very large sample and consisting of
test items which had been validated in many preceding analyses,
therefore it was useful to demonstrate the effects of increasing the
number of factors.

The solutions for six through all eleven factors each result
in the splitting of one of the preceding factors. The first factor,
that composed of variables 1, 7 and 8, persists longest, through the
nine factor solution.

Table 3.6 gives the rotational solutions for two, three,
four, five and six factors. The remaining solutions are on file

in the Michigan State University library.

44

Twenty—four Psychological Tests

Each of the preceding examples has been a very well structured
matrix, with variables chosen from larger batteries of tests to
illustrate particular types of solutions. The twenty-four psychological
test matrix was for a number of years considered a prime example of
a large matrix for principal axes factor analysis, and has been
extensively studied and factored in different ways by Holzinger and
Harman,‘Wrigley and Neuhaus, Kaiser and many others. 'Hith the
comparatively recent advent of large-capacity electronic computers,
it can no longer be considered exceedingly large, since it is now
possible to accurately factor matrices of many more variables which
psychologists and others have long desired. It does, however, serve
as an excellent example of a large matrix containing variables which
are not necessarily clearly representative of any single factor and
for which more than one solution has been proposed in the past.

Table 3.7 gives the unrotated principal axes solution with
unities as initial communality estimates. Five factors have sums of
squares (eigenvalues) greater than 1.0 and account for approximately
60% of the total variance.

In prior analyses of this matrix it has always been presented
as either a four or five factor set of variables. It is interesting
to notice that three of the variables have their highest unrotated
loadings on factors other than the first four. Variable l9 (figure-
word) is most highly loaded on the fifth factor, variable 2 (cubes)
on the seventh factor, and variable 15 (number recognition) on the

eleventh factor.

4.5

The two, three and four factor solutions are shown in
Table 3.8.

The complete set of 23 different rotational solutions is
on file in the Michigan State University Library. 'Uith the rotation
of only the two largest factors, tests which are essentially verbal
and Spatial are divided from those which are perceptual and numeric.
Only a small proportion of the total variance of many of the
variables is included in the two-factor solution, however. 'Hhen
three factors are rotated, verbal and Spatial factors appear
separately with deductive tests common to both factors. 'Hith the
addition of a fourth factor, a group of memory and recognition tests
is isolated. The addition of a fifth factor results in a new factor
on which only variable 19 (figure-word) has its highest loading,
although variables 3 and 17 (Paper Form Board and Object-Number)
also have quite high loadings on this factor. The six factor
solution illustrates an effect which is frequently observed with
large matrices, namely, that segments of two or more former factors
will sometimes combine to form an additional factor. In the Six
factor solution, the factor containing variables 14 through 17 of
the five factor solution Splits into two new factors, isolating
the recognition tests (14 through 16) from the memory tests (17
through 19). Notice that variable 19 has recombined with the other
memory tests and that variable 3 has Split away from the other
Spatial tests into an additional factor on which variables 1 and
13 also have high loadings. The seven, eight and nine factor

solutions result in the Splitting of variable 2 away from the Spatial

   

 

TABLE 3.7 TWENTY-FOUR PSYCHOLOGICAL TESTS ~— UNROTATED PRINCIPLE AXES FACTORS
UNITIES IN LEADING DIAGONALS OF CORRELATION MATRIX

 

 

 

 

 

 

 

 

 

 

Variable I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII XVIII IXX XX XXI XXII XXIII XXIV
1. Visual Perception 62 ~01 43 ~20 ~01 07 20 22 ~15 16 05 13 ~04 32 ~04 02 12 ~27 12 ~01 -13 -08 05 04
2. Cubes 40 ~08 40 ~20 35 09 ~51 ~02 ~26 ~26 07 02 ~12 ~22 04 ~09 ~12 ~07 07 ~07 ~02 ~07 02 04
3. Paper Form Board 45 ~19 48 ~11 ~38 33 ~08 ~36 ~06 02 ~05 =13 ~03 ~01 10 24 07 00 -16 09 02 08 ~04 09
4. Flags 51 ~18 34 ~22 ~01 ~19 46 14 14 ~27 04 ~21 ~28 ~16 ~00 ~20 ~04 03 10 14 06 03 ~07 ~06
5. General Information 70 ~32 ~34 ~05 08 08 ~12 03 ~23 ~01 00 ~12 13 ~01 ~20 03 08 ~15 ~13 19 08 ~08 ~05 ~22
6. Paragraph Comprehension 69 ~42 ~27 08 ~01 12 00 13 ~05 ~13 02 16 ~11 04 21 10 ~02 05 02 ~04 ~08 25 20 ~11
7. Sentence Completion 68 ~43 ~36 ~07 ~04 01 08 01 00 ~09 ~10 ~03 01 08 14 -14 ~05 ~10 ~09 ~24 ~08 05 ~25 06
8. Word Classification 69 ~24 ~14 ~12 ~14 12 16 ~17 12 ~07 ~26 ~08 22 ~15 ~23 10 ~11 01 22 ~16 ~04 ~12 12 03
9. Word Meaning 69 ~45 ~29 08 ~01 ~07 ~01 12 ~12 03 07 11 ~02 02 ~07 ~11 06 08 01 15 23 ~02 06 26
10. Addition 47 54 ~45 ~20 08 ~09 ~01 ~08 08 ~10 ~06 04 ~10 ~03 ~10 08 ~17 ~11 ~10 22 ~24 06 01 12
11. Code 58 43 ~21 03 00 30 ~04 32 ~00 06 21 02 14 ~15 13 24 03 06 21 02 05 00 ~15 02 ;§
12. Counting Dots. 48 55 ~13 ~34 10 04 16 ~30 ~13 16 02 02 ~06 06 03 ‘ ~07 ~12 ~12 04 ~11 30 14 03 ~04
13. Straight-Curved Capitals 62 28 04 ~37 ~08 36 13 18 ~04 10 ~02 ~05 03 ~01 09 -26 ~04 26 ~18 01 ~08 ~13 07 ~03
14. Word Recognition 45 09 ~06 56 16 38 ~08 ~13 26 06 11 ~31 ~14 06 ~09 ~21 09 ~08 11 02 ~05 07 03 02
15. Number Recognition 42 14 08 53 31' ~06 13 07 ~30 21 ~44 ~07 ~18 ~01 07 14 ~06 08 ~04 ~02 01 ~07 ~02 01
16. Figure Recognition 53 09 39 33 17 17 08 13 30 ~21 02 29 11 08 ~21 06 ~16 ~00 ~20 ~05 11 03 ~03 ~01
17. Object-Number 49 28 ~05 47 ~26 ~11 25 ~21 ~15 ~14 18 19 ~00 ~29 12 ~08 13 ~13 ~07 ~04 ~04 ~13 04 ~01
18. Number—Figure 54 39 20 15 ~10 ~25 ~02 ~00 ~35 ~29 01 ~17 27 18 ~13 ~07 07 16 05 ~02 ~06 17 ~02 01
19. Figure-Word 48 14 12 19 ~60 ~14 ~34 19 10 11 ~21 05 ~04 02 06 ~14 ~21 ~09 09 09 06 00 ~02 ~06
20. Deduction 64 ~19 13 07 29 ~19 03 ~29 18 06 08 00 28 15 34 . ~02 ~14 03 08 17 ~02 ~10 ~00 ~03
21. Numerical Puzzles 62 23 10 ~20 17 ~23 ~16 18 32 ~00 ~27 ~07 13 ~15 12 ~01 32 ~10 ~10 ~05 06 06 07 03
22. Problem Reasoning 64 ~15 11 06 ~02 ~33 ~05 13 02 37 36 ~23 01 ~11 ~09 10 ~15 ~00 ~15 ~15 ~06 03 06 02
23. Series Completion 71 ~11 15 ~10 06 ~11 ~08 ~25 07 30 ~02 31 ~10 ~12 ~20 . ~08 14 18 10 03 ~11 10 ~12 ~07
24. Arithmetic Problems 67 20 ~23 ~06 ~10 ~17 ~23 ~12 15 ~19 10 ~02 ~29 29 ~01 L 16 10 14 ~00 ~11 06 ~21 01 ~04
Contribution of Factor 8.14 2.10 1.69 1.50 1.03 .94 .90 .82 .79 .71 0.64 0.54 0.53 0.51 0.48 0.39 0.38 0.34 0.33 0.32 0.30 0.27 0.19 0.17

(Eignevalue)
Percent of Total Variance 33.9 8.7 7.1 6.3 4.3 3.9 3.8 3.4 3.3 3.0 2.7 2.3 2.2 2.1 2.0 1.6 1,6 1,4 1,4 1,3 1_2 1,1 ,8 _7
Cumulative Percentage 33.9 42.6 49.7 55.9 60.2 64.1 67.9 71.3 74.6 77.5 80.2 82.5 84.7 86.8 88.8 ‘90_4 I92,O 93.4 94.8 96.1 97.4 98.5 99.3 100.

 

 

 

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49

tests, variables 11 and 13 away from the numerical tests and re-
isolation of variables 19 and 20 into unique factors. The solutions
for ten through thirteen factors are interesting because they
illustrate the stabilizing, for several solutions, of two new
combinations of variables not previously grouped. These are the
combinations of variables 1 (visual perception test), 11 (code)

and 13 (straight and curved capitals) into a much clearer Perceptual-
Speed factor, and the stabilizing of tests 10 (addition), 12
(counting dots), 21 (numerical puzzles), and 24 (Woody-McCall
"arithmetic") into a possibly clearer numerical ability factor.

Test 21 splits away on the thirteenth factor, since it is a deductive
test requiring numerical Operation. Most important, however, is the
strong relationship of test 12, which was intended as a perceptual
speed test, with test 10 (addition). These two tests remain on the
same factor until the twenty-one factor solution -- one of the
strongest linkages among the twenty-four tests.

The strongest factor, in the sense of being most resistent
to Splitting, is the verbal factor (variables 5 through 9). Variable
8 starts to Split away on the nineteen factor solution, but it is
not until the last two solutions that this factor breaks up into
unique factors.

This matrix is an additional example of the complete disint-
egration of the factorial structure, with the complete splitting into
twenty-four unique factors. In the process, however, there is a
great deal of structural instability, with variables Splitting away

from one factor and becoming part of another factor, etc.

50

Figure 3.4 shows the hierarchical structure of the first
sixteen rotational solutions obtained with unities in the diagonals
of the original correlation matrix. The diagram shows in solid
boxes those variables which have their highest loading on a particular
factor and, in dotted lines attached to the solid boxes, those other

variables which have loadings greater than 0.40 on the factor.

CHAPTER IV
THE EFFECT OF DIFFERENT COMMUNALITY ESTIMATES

0N VARYING ROTATIONAL SOLUTIONS

One of the most perplexing problems in the field of factor
analysis, and one of the greatest obstacles in reaching agreement
on an objective procedure for factor analyzing a given correlation
matrix, has been the question of what starting values should be
inserted in the principal diagonal elements of the matrix prior to
extracting any factors -- the so-called "communality problem”.

The problem.stems from basic factor theory which assumes that any
variable can be described in terms of two basic types of factors:
(1) common factors which are present in more than one variable of
a set of variables, and (2) unique factors which are present only
in that particular variable. This is further complicated by the
fact that the uniqueness of a variable can be subdivided into two
components: (1) error variance, due to imperfections in measure-
ment, and (2) Specific variance, which is reliable but due to the
Specific selection of tests in the battery. The Specific variance
is sometimes added to the common variance, the sum being termed the
reliability of the variable.

Factor analysts in the behavioral sciences have usually
preferred to start factoring with values in the diagonals of the
correlation matrix which represented the proportion of the variance
of each particular variable which was common to all of the other

variables in the test battery -- the "communalities". Unfortunately

51

52

there is no.g priori knowledge of the exact values for the commun-
alities, hence the proliferation of many suggestions as to the
approximations which should be used.
Distinction should be made here between different uses of
the word "communality". As‘Wrigley points out (51):
The term "communality" has been defined by various writers in
different and sometimes conflicting ways. Through all their
definitions, however, there runs the notion that communalities
are reduced values to be inserted in the leading diagonal, and

that, after extraction 0f.E common factors (k.< p, the number of
tests), they become either zero or very small.

Various sets of communalities, based on varying numbers of
common factors, will satisfy the theory, i.e. provide residual
correlations of exactly zero. The solution with the fewest
common factors is generally assumed to be the best.

Another use of the word communality is in referring to the
sum of squares of loadings across factors for each variable when
presenting a particular factor solution. These should perhaps be
called the "observed communalities" for a particular solution. In
actual practice, the exact meaning is usually clear from the context.

Because of the controversy over the appropriate initial
values to place in the principal diagonals of the correlation matrix
prior to factoring, and because it had been suggested by some authors
(17) that when the number of variables was large (say 20 or more)
the values in the diagonal made little difference, an additional
phase was added to this study to compare different rotational solu-
tions, obtained in the same manner as in the preceding chapter,

when communality estimates other than unities were used in the

diagonals of the test matrices.

53

Since the intention was still to use only methods which
were objective, two methods of directly calculating communality
estimates were chosen:

1. Squared multiple correlations -- Wrigley, in a number
of excellent papers (47, 48, 51), has urged the use of squared mult-
iple correlation of a variable with each of the other n-l variables
as the best value to be inserted in the diagonal for that variable
prior to factoring.

Until the communality problem.is stated in such a way that
exact and unique values can be found, we shall do better to
insert the squared multiple correlations in the diagonal. The
SMCS mdght be called the observed communalities, in distinction
from the theoretical communalities . . ., since they and not the
minimal rank communalities measure the predictable common var-
iance in the observed correlations. They are objective, unique
and obtainable rapidly and without iteration with modern com-
putational equipment. Then all factors with positive roots
should be rotated. That is to say, to avoid the downward bias
of dimensionality when sample size is small, decisions upon the
number of factors and the diagonal values will be made algebra-
ically rather than statistically. Since the SMCS can be proved
to be lower bounds for the minimal rank communalities, and the
number of SMC factors with positive roots is a lower bound for
the minimal rank number of factors, this preposal follows the
conservative policy of operating exclusively with observed
common variance instead of trying to determine what would happen
to common variance in a domain of tests. (52, p. 472)

The calculation of squared multiple correlations, SMCS for
short, as‘wrigley said is very simple on a modern electronic computer,
and the SMCs for all of the variables in a matrix can be calculated

simultaneously. The procedure is simple to calculate the inverse,

R-l, of the original correlation matrix with unities in the diagonals.

The SMC for variable 21 is then:
11
SMLCi =-£-I%-l ‘where r11 is the diagonal element of the
r

inverse correSponding to variable zi.

54

2. Guttman communalities -- As implied in the preceding
quotation, the insertion of SMC's in the diagonals of the correlation
matrix produces a matrix which is not Gramian, i.e. has a number of
eigenvalues which will be negative (imaginary). Guttman proposed a
method (16), very similar to the SMC calculations, which results
in communality estimates which have two very desirable properties:
(a) the matrix in which they are inserted remains Gramian, and (b)
the rank of the matrix is reduced by exactly one, i.e. the resulting
principal axes solution results in (n-l) positive roots and one
zero root.

The calculation is somewhat more complicated than that for
SMCS but very similar, it is direct, and the solution is unique for
any matrix. The starting point is, again, the inverse, R-l, of the
original correlation matrix, R, with unities in the diagonals. A

diagonal matrix, D, is then formed as follows:
d.. =I/ ii for i = j
13 r

d,. = O for i # j

1J
*
Let A be the smallest eigenvalue (latent root) of the triple-
product matrix DRD. The Guttman communality for variable zj is then
rjj - 1*
GCj =--33-- which is similar to the formula for SMCs.
r
As in the preceding chapter, the procedure for this phase of

the investigation was to insert the new communality estimates into
the diagonals of the test matrices, obtain the principal axes factors,
order the factors in descending order by size of the correSponding

latent root, and then rotate the two largest factors, then the three

55

largest factors, and so on until all factors with positive roots

had been rotated.

Eight Physical Variables

A comparison of the two communality estimates is given
in Table 4.1:

TABLE 4.1

 

COMMUNALITY ESTIMATES FOR
EIGHT PHYSICAL VARIABLES

 

 

 

 

 

Variable Guttman SMC
Communality

1 .9088 .8162
2 .9252 .8493
3 .9011 .8006
4 .8950 .7884
5 .8754 .7488
6 .8036 .6042
7 .7828 .5622
8 .7409 .4778

Total 6.8328 5.6475

% Total

Variance 85.5 70.7

 

 

 

'7

Since the SMCS are a measure of the predictable common
variance in the observed correlations for a particular variable,
they may also be considered a proportion of the total variance
factored. Notice that in the above example each of the first five
variables has at least 75% or mmre of its variance in common with
the other tests. The prOportion falls considerably for the last
three variables; in fact, less than 50% of the total variance of
variable eight is in common with the rest of the variables.

One might expect to find quite different factor loadings in

all of the solutions using the three different estimates. This is

not the case, however, Table 4.2 shows a comparison of the first
three unrotated principal axes factors from each of the three solu-
tions. It is evident that the additional unique variance present
when unities were used in the diagonals makes little difference in
the first two factors, and becomes evident primarily in the later
factors. This is particularly true of variable eight. 'With unities
in the diagonal, the total variance is almost equally divided between
the first and third factors, which this third "unique" factor pract-
ically disappears with the use of SMCS. The maximum.difference between
highest and lowest loading for any variable on either of the first two
factors is only 0.06 or .36% of the total variance. Notice that the
swm of the two largest eigenvalues, 5.87, is greater than the sum of
the original communalities in the SMC solution. Table 4.3 gives the
remaining four factors with Guttman communalities, only four positive
eignevalues resulted using SMCs.
TABLE 4.2
EFFECTS OF DIFFERING COMMUNALITY ESTIMATES

ON THE FIRST FOUR UNROTATED FACTORS
(EIGHT PHYSICAL VARIABLES)

 

 

Variable I II III

 

U GC SMC Diff U GC SMC Diff U GC SMC Diff

 

86 86 86 00 -37 -34 -30 05 -O7 -08 -06 04
84 84 84 01 -44 -42 -38 04 08 ll 14 O4
81 81 83 00 -46 -43 -42 06 01 05 03 08
84 84 86 01 -40 -37 -35 O6 -10 -14 -14 O4
76 74 72 03 52 53 55 02 -15 -12 -06 10
67 65 62 04 53 52 50 04 -05 03 04 O8
62 59 56 05 58 55 53 06 -29 -21 ~06 25
67 64 6O 06 42 39 37 06 6O 39 13 48

coucnuawaI-a

 

Eigenvalue 4.67 4.54 4.45 .26 1.77 1.62 1.49 .31 .48 .25 .07 .41

 

Z of Total 58.4 66.4 77.1 3.2 22.1 23.7 25.8 3.9 6.0 3.7 1.2 5.1
Original
Communality

 

 

 

 

57

TABLE 4.3

REMAINING UNROTATED PRINCIBAL AXES FACTORS
‘UITH GUTTMAN CGMMUNALITIES -- (EIGHT PHYSICAL VARIABLES)

 

 

 

 

 

Variable IV V VI VII

1 -09 16 05 -10

2 06 -10 10 -03

3 -03 -21 ~09 01

4 07 15 ~03 12

5 ~07 Ol -12 -06

6 -32 -04 08 07

7 27 -08 07 00

8 14 10 ~03 00

Eigenvalue .22 .12 .05 .04
Z of Total

Variance 3.2 1.8 0.7 0.6

 

Figures 4.1 and 4.2 show the hierarchical structures which
appear when the factors obtained with Guttman communalities and SMCs
are rotated. These may be compared with Figure 3.1, the hierarchical
structure of the solution starting with unities. In all three cases
variables one through four remain grouped into a single factor through
the entire series of rotations. The two factor solution is the same
with all three communality estimates. The basic differences between
the use of different communality estimates is that with unities var-
iables five through eight eventually Split into four unique factors.
‘Hith Guttman communalities, variable eight splits away from.variables
five through seven and with SMCs only two factors ever appear, the
rotation of the two additional factors with positive eigenvalues

producing only null factors.

Eight Political Variables
Tables 4.4 and 4.5 show similar analyses for the eight political

variable matrix. In this example there is even less discrepancy between

58

FIGURE 4. 1

HIERARCHICAL STRUCTURE OF EIGHT PHYSICAL VARIABLES --
STARTING WITH GUTTMAN COMMUNALITIES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Numberof
Factors
2 1...4 5...8
f 7
3 1 .4 5
4 1 .4 5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 1 . 4] 5
7 1 ,. 4 5
A FIGURE 4.2

HIERARCHICAL STRUCTURE OF EIGHT PHYSICAL VARIABLES --
STARTING WITH SQUARED MULTIPLE CORRELATIONS

Number of
Factors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

unities and SMCs over all eight of the variables, with the maximum
uniqueness of 33% for variable 5. Table 4.5 shows the comparison
of the first four factors with different initial communalities.

In this example the first factor is almost identical with all three
initial estimates. There is, however, slightly greater discrepancy
in at least one variable on each of the other three factors.

TABLE 4.4

COMMUNALITY ESTIMATES FOR
EIGHT POLITICAL VARIABLES

 

 

 

 

 

 

 

 

Guttman Squared Multiple
Variable Communality Correlation
l .9074 .7709
2 .9591 .8989
3 .9330 .8343
4 .9212 .8051
5 .8684 .6745
6 .9150 .7896
7 .9016 .7565
8 .9516 .8801
Total 7.3574 6.4099
1 of Total
Variance 91.97 80.13
TABLE 4.5
EFFECTS OF DIFFERING COMMUNALITY ESTIMATES
ON THE FIRST THREE UNROTATED FACTORS
FOR EIGHT POLITICAL VARIABLES
Variable I II III —

 

U GC SMC Diff U GC SMC Diff U GC SMC Diff

 

 

1 74 73 72 02 -36 -35 ~33 03 54 49 39 15
2 86 86 86 00 -43 -43 -43 00 16 15 14 02
3 89 89 88 Ol -21 -20 -19 02 -l6 -16 ~14 02
4 -89 ~88 -87 02 -04 -05 -O6 02 28 26 22 06
5 32 31 30 02 89 84 77 12 23 21 18 05
6 91 9O 88 03 --02 -02 -01 01 -19 -18 -14 05
7 -70 -70 ~68 02 -62 -61 -58 04 -06 -O8 -08 02
8 -94 -94 -94 00 -09 -10 -ll 02 09 O9 08 01
Eigenvalue 5.19 5.11 5.01 .18 1.54 1.43 1.28 .26 .53 .44 .31 .22

 

 

% of Total 64.8 69.5 78.2 19.2 19.5 19.9 6.6 5.9 4.8
orig. comm, -

 

 

 

60

Figures 4.3 and 4.4 diagram the hierarchical structures
obtained with the two communality estimates (they may be compared
with Figure 3.2, the structure with unities). Three factors stab-
ilize through the entire series of rotational solutions for both
Guttman communalities and SMCs, the addition of factors resulting

only in null factors.

F I GU R E 4.3
HIERARCHICAL STRUCTURE OF EIGHT POLITICAL VARIABLES
STARTING WITH GUTTMAN COMMUNALITIES '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of
Factors
. ——--.
2 1...4,6,8 5,7_4_,_8_;
3 [3.4.6,813‘21—31 0
4 [3, 2., 81 217:1 a: as
5 [3, 4, 6: BIKE—.1: 0 ® ®
6 {3, 4,46, 8:21:73 @ ® ® ®
7 l3, 45,3121}, @ ® ® ® @

 

FIGURE 4. 4

HIERARCHICAL STRUCTURE OF EIGHT POLITICAL VARIABLES --
STARTING WITH SQUARED MULTIPLE CORRE LATIONS

 

 

 

 

 

 

 

 

 

 

 

bhunbercfl
Factors
2 1...4,6, 5,7 4,5,:
3 3:5» hear}; an
4 1,2 {52 3,446,835.11 0 ®

 

 

 

61

Eleven Air Force Classification Tests
In the two preceding examples, both the Guttman communalities

and the SMCs were unusually large -- for most variables the initial
values in the diagonals of the correlation matrix were greater than
0.75. ‘Hith the eleVen variable classification test matrix there is
a much greater discrepancy between unities and the other two commun-
ality estimates, as illustrated in Table 4.6.

TABLE 4.6

COMMUNALITY ESTIMATES FOR ELEVEN
AIR FORCE CLASSIFICATION TESTS

 

 

 

 

Guttman Squared Multiple

Variable Communality Correlation

l .693 .500

2 .570 .301

3 .584 .323

4 .568 .296

5 .609 .364

6 .543 .257

7 .695 .503

8 .706 .522

9 .547 .262

10 .531 .237

11 .544 .258
Total 6.590 3.824
Z of Total 59.91 34.75

Variance

 

In Spite of this discrepancy in initial diagonal entries,
there is less difference in unrotated loadings on the first three
factors than might be expected. The difference is much more marked
in factors four and five. Table 4.7 give the comparison figures
for the first five factors for all three starting communalities.
Notice that the percentage of total original communality accounted
for by the first three factors actually increaSes with lower commun-

alities.

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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63

FIGURE 4. 5

HIERARCHICAL STRUCTURE OF ELEVEN AIR FORCE CLASSIFICATION
TESTS -- STARTING WITH GUTTMAN COMMUNALITIES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of
Factors
2 1,2,6,7,8 3: 3,4,5,9,1o,11|
3 1,7,8_3__;|1_2,6,11 4; 3,4,5,9,10
- j \
a "‘1
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64

FIGURE 4. 6

HIERARCHICAL STRUCTURE OF ELEVEN AIR FORCE CLASSIFICATION
TESTS -- STARTING WITH SQUARED MULTIPLE CORRE LATIONS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of
Factors
2 1,2, I, 8J—---6———43, 4, 5, 9, 10, 11
3 1, 7, 8 2, 6, 11 3, 4, 5, 9, 10
1 f 1
4 1, 7, 8 2, 6, 11 3, 4, 5, 9, 10 ®

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 1, 7, 8 2, 6, 11 3, 4, 5, 9, 10 ® ®

 

 

 

 

 

65

The hierarchical structure for different rotational solutions
for both initial communality estimates is shown in Figures 4.5 and
4.6. The structure starting with Guttman communalities is the same
as that with unities (Figure 3.3) through the six factor solution.

The addition of a seventh factor produces only a null factor. Variables
3 and 10 split on the eight factor solution rather than on the seven
factor solution as they did with unities in the diagonals. The
addition of other factors results in no further Splitting, merely the
addition of null factors.

Uith SMCS as the starting communality estimates only five
factors have positive latent roots. Rotation of these factors results
in only three factors, corresponding to the three factor solutions
with both unities and Guttman communalities. Of particular importance
is the fact that the five factors which have been presented in the

past for this matrix never appear.

TWenty-four Psychological Variables

Guttman communalities and the squared multiple correlations
for the twenty-four variables are given in Table 4.8. As in the
previous examples, the different initial communality estimates make
relatively little difference in the first few unrotated factors;
the maximum.difference between loadings on the fourth factor being
only 0.16 in this example.

Of greater significance is the proportion of the total original
communality represented by the eigenvalues (variance contributions)
of the unrotated factors with different communality estimates. Table

4.9 shows this comparison. Notice that for the first four factors,

66

TABLE h.7 TWENTY-FOUR PSYCHOLOGICAL VARIABLES

INITIAL COMMUNALITY ESTIMATES

 

 

 

 

 

Variable Guttman Squared Multiple
Communalities Correlations
1 .7835 .5107
2 .6900 .2996
3 .7528 .8805
8 .7385 .8092
5 .8551 .6726
6 .8569 .6766
7 .8602 .6880
8 .8069 .5638
9 .8731 .7133
10 .8137 .5790
11 .7969 .5810
12 .7950 .5368
13 .7961 .5393
11.1 o 7161 o 35 811
15 .6871 .2929
16 .7871 .8286
17 .7399 .8128
18 .7533 .8825
19 .7199 .3671
20 .7626 .8637
21 .7669 .8733
22 .7563 .8898
23 .9058 .5611
28 .7905 .5265
Total 18.6636 11.9828
Percent of
Total Variance 77.8 89.8

 

 

 

67

the percent of variance contribution increases as the size of commun-
ality decreases. From this point on the percentage decreases.

TABLE 4.9

VARIANCE CONTRIBUTION (EIGENVALUES) OF FIRST EIGHT UNROTATED FACTORS
AS.A PERCENT OF TOTAL ORIGINAL COMMUNALITY FOR TWENTY-FOUR PSYCHOLOGICAL

 

 

 

 

 

VARIABLES
Unities Guttman Squared Multiple
Communalities Correlations
, Z Total A Z Total A Z Total
Factor ' Comm. Comm. Comm.
1 8.14 33.9 7.93 42.5 7.66 64.2
2 2.10 8.7 1.91 10.2 1.67 14.0
3 1.69 7.1 1.48 7.9 1.21 10.1
4 1.50 6.3 1.24 6.7 .92 7.7
5 1.03 4.3 .76 4.1 .45 3.7
6 .94 3.9 .71 3.8 .41 3.4
7 .90 3.8 .64 3.4 .32 2.7
8 .82 3.4 .59 3.2 .31 2.6
* *
Total Orig. 24.00 100.0 18.66 77.8 12.94 53.9
Communality

 

 

 

 

*
Percent total original communality is of the total variance (24.0).

Figures 4.7 and 4.8 give the hierarchical structure of the
solutions with Guttman communalities and SMCs through the fifteen
and fourteen factor solutions, respectively. Through the four factor
solutions they are almost identical to the hierarchy obtained with
unities in the diagonals. (The appearance of variable 15 in the same
factor with variables 1 through 4 in the three factor solution with
SMCs is somewhat misleading -- actually it was almost equally loaded
on both the first and third factors.) The addition of a fifth factor

with all three communality estimates results in either a null factor

68

(with SMCs) or a factor with only one or two highest loadings (with
Guttman communalities or unities). This is primarily due to the
great discrepancy in the coefficients for variable 19 on the fifth
unrotated factor with the three different initial diagonals.
A.comparison of the complete hierarchical structure for all
three sets of solutions reveals that with unities the factors
continue fissioning until the final twenty-four factor rotation in
which each variable is loaded most highly on a unique factor. ‘Hith
Guttman communalities, there is still considerable factorial in-
stability, with some variables Splitting into unique factors on one
solution then combining again with other variables on a later
solution, etc. Eventually, however, when all twenty-three factors
with positive eigenvalues are rotated, variables 5 through 9 (the
verbal tests) remain in a single factor, tests 10 and 12 (addition
and counting dots) cluster on a single factor, the remaining seven-
teen variables each have their highest loading on a single unique
factor, and four null factors make up the total of twenty-three.
The complete set of rotations starting with SMCs in the diagonals
(only fourteen factors had positive eigenvalues and were included
in the solutions) shows much more stability. Some Splitting continues
through the ten factor solution. From this point on (factors 10
through 14) no further Splitting takes place, merely the addition
of null factors. The verbal-deductive, numerical and memory factors

remain undivided.

69

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CHAPTER V

CONCLUSIONS

One of the major results of this Study is the con-
firmation of previously published reports of factor splitting
when different numbers of factors are rotated and the hierarch-
ical structure which is therefore obtained.

In this study we have defined four types of factors:
(1) _group factors -- factors on which three or more of the
variables in the analysis have their highest loadings; (2)
binary factors -- factors on which only two of the variables
have their highest loadings and which are therefore insuffi-
ciently determined in three or higher dimensional hyperSpace;

(3) unique or Specific factors -- on which only one variable
has its highest loading; and (4) null factors -- those factors
on which none of the variables have their highest loadings.

The tendency for factors to emerge hierarchically as
additional factors are rotated follows several different patterns.
‘With few variables of high reliability such as those in the
physical and political variable matrices, the Splitting is
"clean", with little or no tendency for variables to Split away
from one factor and recombine with others. The point at which
no further Splitting takes place is usually reached before all
factors are rotated and usually the solution includes some null
factors. In the two examples mentioned, however, the matrices

consisted of variables eSpecially selected to illustrate two-

71

 

72

factor sets and from 80 to 84 percent of the total variance was
accounted for by the first two factors.

'With larger matrices consisting of sets of variables
with lower reliabilities (which is usually the case in exploratory
factor analysis) there are usually more than two factors inherent
in the data. In these cases variables which have small loadings
on all of the first few factors will be quite unstable until un-
rotated factors on which the variables have relatively high load-
ings are included in the rotational solution. Another phenomenon
observed is that segments of two or more earlier factors will
sometimes combine to form new factors. This effect is much more
noticeable when unities, which include a large amount of unique
variance, are used in the diagonals of the matrix than when
smaller initial communality estimates such as squared multiple

correlations are used.
The Effect of Different Communality Estimates on Factor Fissioning

With large diagonal entries such as unities in the
correlation matrix, the factors will usually continue to Split
into many unique factors. This effect seems to vary inversely
with the reliability of the variables, a rough measure of which
is the discrepancy between unities and squared multiple correl-
ations (l - R§(n-l)). ‘When there is relatively little difference
between unities and squared multiple correlations, as illustrated
by most of the variables in the physical and political variable
matrices, most of the variance for a particular variable tends

to be concentrated in a single, early factor and the addition of

73

later factors into the rotational solution adds little additional
variance (in the form of high loadings) which seems to be the
primary cause of factor fissioning.

'Hhen there is a great discrepancy between unities and
squared multiple correlations for any of the variables, there will
usually be at least two unrotated factors on which that variable
has a high loading when unities are used in the leading diagonals,
an early group factor and a later unique factor. The introduction
of the later factors containing unique loadings into the rotational
solutions results in unique rotated factors.

‘Hhen squared multiple correlations are used in the
leading diagonals as initial communality estimates, the rotation
of additional factors results in few unique factors. Instead
most of the later unrotated factors contain very small loadings
and result only in the formation of null factors.

'With Guttman communalities, which are numerically
less than unities but greater than squared multiple correlations,
the effect of increasing the number of factors in the rotational
solution is a combination of the two above types of results. There
are more unique factors formed than with squared multiple corre-
1ations but there are also a number of group factors which remain
in the final solution (with all factors rotated) as well as some
null factors.

The findings of this study indicate that-Ehg£g_i§
relative1y_little difference in the numerical values of the first

few factors extracted by thegprincipal axes method regardless of

the initial communality estimates. For example, a comparison of

the first four unrotated factors obtained with unities and squared
multiple correlations for the twenty-four psychological variables
indicated that of the ninety-six pairs of loadings only three had
a difference greater than 0.1, one of these was on the third
factor and the other two on the fourth factor. The maximum
discrepancy was 0.161 and the average difference over the ninety-
six pairs was 0.035.

One of the best indications of the point at which unique
variance begins to appear when unities are used, hence the point
at which the factorial structure will begin to Show differences
with different initial communality estimates, is to compare the
percentage of the total original communality accounted for by the
eigenvalues of each of the factors with different communalities.
The smaller the initial communality estimates the larger will be
the percentage of the total accounted for by the earlier factors.
This is illustrated graphically in Figures 5.1, 5.2, 5.3 and 5.4,
the data for which is compiled from Tables 4.2, 4.5, 4.7, and 4.9.
As long as the percentages for squared multiple correlations are
higher than those for unities the factorial solutions are almost
identical. The point at which the percentage for unities is
greater than that for the lower communality estimates is the point
at which the factors begin to include unique loadings with unities
which are not present in the factors derived from the matrix with
squared multiple correlations. This cross-over point is also
the point at which the factorial solution begins to differ.

Probably the most significant result of this study is

the confirmation that the factorial structure -- that is, the

75

grouping of variables on the different factors -- is identical
regardless of the initial communality estimates up to the cross-
over point where binary or unique factors begin Splitting off
with unities or where null factors are formed when starting with
squared multiple correlations. The controversy over the prOper.
communality estimates which should be used and the insistence on
the use of values other than unities now seems somewhat unnecessary.
The "common" factors are those corresponding to the largest latent
roots with the principal axes method. Any Specific variance
which is included due to the use of unities in the correlations
matrix appears only in the "later" factors correSponding to the
smaller latent roots and may be excluded by simply not including
such factors in the rotational solution.

The comparison of percentages such as those in Figures
5.1 through 5.4 might well be used as a criterion for "when to
stop rotating". This would involve doing two principal axes
analyses, one with unities as initial communality estimates and
another using squared multiple correlations. Those factors on
which the eigenvalues obtained with squared multiple correlations
accounted for a greater percentage of the total initial commu-
nalities than those with unities would be rotated. 'While this
method would be feasible for very large, fast electronic computers
on which it would be possible to save two different sets of
factors, it would be inordinately expensive and time consuming
for the average researcher. However, the results of this study

do suggest an alternative criterion which is both useful and

Percent of Total Initial Communality

40%

 

76

FIGURE 5.1
EIGHT PHYSICAL VARIABLES --

Percent of Total Initial Communality Accounted
for by Latent Roots of Four Largest Factors.

U nities
—— — — — Guttman Communalities
—-— Squared Multiple Correlations

 

30%- ‘
\
\
\
\
\-
\\
20%- \
\ \
\
10%. }\
\
\
\
\ \
\‘I ------- -x
O 4 I 1 - ~— - fl
1 2 3 1

Factor Number

69. 52 77 FIGURE 5. 2
50% ..
64. 8356 ‘ EIGHT POLITICAL VARIABLES '-
\ Percent of Total Initial Communality
Accounted for by Latent Roots of Four
\ Largest Factors.
\
\ \
\

40% 2 \

(«D

O

§
T

20% -

Percent of Total Initial Communality

10% N

 

 

\\ \ —-—— Unities
\ _ "' — — Guttman Communalities
\ - Squared Multiple Correls.
\ -
\
\\
\

l l

2 3
Factor Number

p

4;

50%

40%

Y.-
o
9Q

20%

Percent of Total Initial Communalit

10%

I

 

l

78

FIGURE 5. 3
ELEVEN AIR-FORCE CLASSIFICATION TESTS --

Percent of Total Initial Communality Accounted for
by Latent Roots of the Five Largest Factors.

Unities

 

— — — — Guttman Communalities

- Squared Multiple Correlations

 

l l

 

 

2 3 4
Factor Number

50%

40%

10%

 

 

 

80

practical. The results also suggest a guide for reporting the
results of factor analyses.

In every case so far studied, the solution following the
above cut-off point had at least one factor on which less than
three of the variables had their largest loadings. The following
method of factoring any set of variables is therefore recommended.
It is completely objective, with no need for subjective decisions
at any point during the calculations, and is practical and fast
on a modern electronic digital computer.

1. Since there is little difference between unities and
other communality estimates as far as the factorial
structure is concerned, unities are recommended for the
entries in the leading diagonals of the correlation
matrix to be factored.

2. Carry out a complete principal axes analysis by the best
eigenvalue-eigenvector method available.

3. Rank the principal axes factors from largest to smallest
on the basis of their correSponding eigenvalues.

4. Rotate the factors by an acceptable analytical procedure
(the varimax method is recommended) starting with the two
largest factors, then the three largest factors, and so on,
continuing to rotate with additional factors as long as
each factor in the solution includes the highest loadings
of at least three variables. In other words, terminate
rotations when a solution contains a factor on which less

than three variables have their largest loadings.

‘

5. In reporting the results of an analysis, the complete
hierarchical structure should be given as well as the

final factorial solution.

For descriptive or exploratory factor analyses, particularly
when little is known from other work about the factorial structure
of the variables, it might be useful to rotate several additional
factors beyond the cut-off point at which binary or unique factors
begin to Split off. The additional information obtained could
suggest additional factors which might be significant if additional
variables were to be included in the test battery.

The requirement that each interpretable factor should in-
clude the highest loadings for at least three variables is not
simply an arbitrary choice. Other investigators have also suggested
that each factor should be highly loaded on at least three variables
(18, 17). It is based on the mathematical axiom that at least
three points are rquired to define a plane in three dimensional
Space. (In N-dimensional Space we would theoretically require
at least N non-coplanar points to determine a hyperplane. An
investigation is currently underway to determine the feasibility
of using a variable test for cut-off rather than the constant
three. In each of the four examples used in this study the

cut-off point would be the same.)

10

11

12

13

14

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