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Factorial Stability As A Function Of Analytical Rotational Method, Type Of Simple Structure, And Size Of Sample
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Factorial Stability As A Function Of Analytical Rotational Method, Type Of Simple Structure, And Size Of Sample
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This dissertation has been
microfilmed exactly as received 6 5 -3 1 0 6
HAMBURGER, Charles David, 1932-
FACTORIAL STABILITY AS A FUNCTION OF
ANALYTICAL ROTATIONAL METHOD, TYPE
OF SIMPLE STRUCTURE, AND SIZE OF SAMPLE.
U niversity of Southern C alifornia, Ph.D., 1965
Psychology, general
University Microfilms, Inc., A nn Arbor, M ichigan
FACTORIAL STABILITY AS A FUNCTION OF
ANALYTIC ROTATIONAL METHOD, TYPE OF SIMPLE
STRUCTURE*, AND SIZE OF SAMPLE
by
Charles David Hamburger
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(Psychology)
January 1965
UNIVERSITY O F S O U T H E R N CA LIFO RN IA
THE GRADUATE SCH O O L
UNIVERSITY PARK
LOS ANGELES. CA LIFO R N IA 9 0 0 0 7
This dissertation, written by
......
under the direction of h.l8—Dissertation Com
mittee, and approved by all its members, has
been presented to and accepted by the Graduate
School, in partial fulfillment of requirements
for the degree of
D O C T O R O F P H I L O S O P H Y
Dean
D a te...
ERTATION COMMITTEE
M z ,
ACKNOWLEDGEMENTS
The author wishes to express his indebtedness to
the members of his committee,-Professors J. P. Guilford .
(Chairman), Norman Cliff, and A. L. Whiteman, for the
valuable guidance and assistance received both throughout
the preparation of this dissertation and during his
graduate studies at the University of Southern California.
Special thanks are due Dr. Philip R. Merrifield, who
initiated interest in the general problem of factorial
invariance. Dr. Wayne S. Zimmerman, principal investigator
for NSP Grant G - 194-89, "Investigation and comparison of
analytical and graphical methods of rotation in factor
analysis" in which the author was Involved, and to the
staff of the Aptitudes Research Project, all of whom
provided help in many ways.
Finally, particular thanks are owed to my wife
for an inordinate amount of patience and encouragement.
ii
TABLE OP CONTENTS
ACKNOWLEDGEMENT.................................
Page
LIST OP TABLES .................................
Chapter
I.
DESCRIPTION OP THE PROBLEM ..............
The Method of Generating the
Sample Factor Patterns ..........
. 9
Methods of Rotation ...............
. 15
Type of Factor Pattern............
*
Sample Size .......... .......
*
H
H
INDICES OP FACTORIAL INVARIANCE ........ . 16
III. SIMPLE STRUCTURE .......................
IV. ANALYTIC AND SEMI-ANALYTIC ROTATIONAL
CRITERIA .............................
Oblique Solutions ................. . 56
Analytic Oblique Comparisons with
Other Methods................... . 58
Semi-analytic Methods .............
V. METHODOLOGY AND RESULTS ...............
Description of the Experimental
Variables .......................
Generation of the Sample Correlation
Matrices .......................
Method of Factoring ...............
. 59
Rotational Procedures .............
Description of the Stability
Criteria ....................... . 62
Procedures for Analysis of the
Stability Criteria ............. . 64
VI. DESCRIPTION OP RESULTS .................
. 85
Complete Summary Matrix with the
RMS as the Criterion ............
. 85
Complete Summary Matrix with the
Range as the Criterion .......... . 86
Summary Matrix Totals by Population
Interval ..................... . . 88
Page
Chapter
VI* (Continued)
Summary Matrix Totals by Population
Communality......................... 90
Cross-tabulation of Sample Cell
Means and Rotated Population
Loadings........................... 91
Cross-tabulation of Population
Pattern Positions and Rotated
Population Pattern Positions^ .... 95
VII. CONCLUSIONS.................................. 97
REFERENCES........................................100
APPENDIX: Summary Matrixes for all Treatment
Conditions; Sample Cell Means,
Ranges, and RMS Values................106
iv
LIST OP TABLES
Table
1.
2 .
3.
4.
5.
6.
7-
8.
9.
10.
11.
12.
Page
Contrived Population Factor Patterns with 12
Tests, 4 Common Factors, and 12 Unique Factors. 50
Population Correlation Matrices Reproduced
from the Population Factor Patterns with the
Exact Communalities (h^) in the Diagonals. 52
Equamax "Adjusted" Values Following Rotation
of the Population Common Factor Pattern. 54
Varimax "Adjusted" Values Following Rotation
of the Population common Factor Pattern. 55
Biquartimin "Adjusted" Values Following Rotation
of the Population Common Factor Patterns. 56
Procrustes "Adjusted" values Following Rotation
of the Population Common Factor Pattern.
Mean Summary Matrix Values with the Root Mean
Square as a Criterion of Stability for the
Sample Factor Loadings about their Population
Position.
Mean Summary Matrix Values with the Range as
an Estimate of Variance for the Sample Factor
Loadings about their Population Position.
Mean RMS Value by Population Factor Loading
Intervals of .10 for Each Summary Matrix.
Mean Range Value by Population Factor Loading
Intervals of .10 for Each Summary Matrix.
Mean RMS Summary Matrix Values by Population
Communality.
Mean Range Summary Matrix Values by Population
Communality. 74
57
66
67
68
70
72
v
Table Page
13. Three-Category (H, M, L) Cross-Tabulation of
Mean Sample Loadings Against the Population
Loadings Adjusted by Rotation. 77
Ik. ltoree-Category (H, M, L) Cross-Tabulation of
Rotated Population Pattern Position against
the Population Pattern Position. 79
15. Tabulation of Summary Cell Frequency
Distributions Into Symmetric (S) and Asymmetric
(A) categories.
vi
CHAPTER I
DESCRIPTION OP THE PROBLEM
One of the major problems In the analysis and
interpretation of results of factor analytical studies is
related to the invariance of the factorial solution. Ob
tained factors should not be entirely dependent upon
particular groups of variables and individuals that are
used in one study, but should be potentially replicable
in analyses based upon different experimental conditions.
In his excellent comprehensive analysis of the in
variance problem Henrysson (i960, p.Ill) identifies the
issue by stating
"Though the design and the test selection
are carefully made, some of the factors may
still be artificial products of the very con
stellation of tests contained in the battery.
A single factor analysis yields more or less
unverified hypotheses as to the factors which
must be proved Invariant through other factor
studies employing other tests and in respect
to other populations, before it can be said
that the factors found have the generality
required of factors with explanatory proper
ties."
Other investigators (e.g. Guilford and Zimmerman,
1963), also believe that the basis of the invariance con
siderations with rotations is essentially a test sampling
problem. They maintain that for successful factor analytic
1
verification of hypotheses in an empirical domain the
investigator would have to have at least a subset of well
substantiated variables as representatives of their con
structs. This specification would tend to negate the use
of both "blind" graphical rotations and the analytical
rotational methods without marker variables.
Thurstone (19^7) describes four basic types of
factorial invariance, all of which are mutually related.
The four cases arise from combinations of same and dif
ferent populations and identical and changed batteries.
When the same population is investigated, then numerical
invariance would result, with different populations, con
figurational invariance would be expected. The terms
numerical and configurational invariance will be defined
below.
Henrysson (i960) adds two additional cases based
upon splitting the changed battery case into partly and
entirely changed batteries. Henrysson1s descriptions of
the six cases are given below with some suggestions by
the present writer as to how they might be tested.
(1) Numerical invariance, with the same population,
same battery.
"If the same test battery is used for
the same population, there should be numeri
cal invariance in all samples of the popula
tion. That means that the factor structure
found in the different samples should be
Identical within the limits of the sampling
errors." (op. cit., p.112)
To test this type of invariance would either re
quire obtaining random samples from a population, or ran
dom division of an original sample into two or more parts,
or perhaps formation of samples by some type of matching
procedure. The indices of invariance in these cases
would be analogous to various types of reliability coef
ficients .
(2) Configurational invariance,with different
populations, same battery.
"If the same test battery is used for
different populations, then configurational
invariance may be expected to take place.
Thi3 signifies that the change of populations
affects the size of the factor loadings, but
in proportion to the changes in the variance
of the different tests administered to the
population. Thi3 implies that if a test
has zero loading in a certain factor in one
population, it will have zero-loading in
the same factor also in another population."
(op. cit., p.112)
It would be hypothesized here that the factors
would be the same in all analyses, but the numerical
size of the loadings would change in relation to the
variance of the population in the factors in each battery.
Thus the configuration of loadings would remain the same
but there would be a proportional change in the size of
the loadings. According to Henrysson, for factorial
composition of a battery to be comparable from population
to population, It must be assumed that Individuals In both
populations solve the items by the same procedures. An
example of this case could Involve testing adult and adol
escent groups (Oershon, et al, 1963), or two nationality
groups (Vandenberg, 1959)* with the same batteries.
(3) Numerical invariance, with the same population,
partly different batteries.
"If the test batteries, used on samples
from the same population, Include tests of
which only some are the same, the numerical
invariance requires that each of the tests
that is common to two or more batteries re
tains its factor pattern.”(op. cit., p.114)
This would mean that addition or deletion of test
vectors does not change the position of hyperplanes. The
hypothesis of no change in the factor loadings of a test
from battery to battery could be tested by varying the
number of tests representing a factor or by including the
same test in different batteries but with different com
panion tests representing the same factor. The size of
loadings on the test should remain unchanged within the
limits of sampling errors.
(4-) Configurational invariance, with different
populations, partly different batteries representing the
same hypothesized factors.
"The configurational invariance signi
fies, in this case that the same hyperplanes
are to be localized, but partly with the
help of other teat vectors; at the same
time the angles between the hyperplanes
can be changed," (op. cit., 114-115)
(5) Configurational Invariance, with the same
population, completely different tests In each battery
representing the hypothesized factors.
An example of a means of Investigating this type of
invariance is given by Tucker (195^)* The original purpose
of his inter-battery method of factor analysis was to pro
vide information relevant to the stability of factors over
different selections of tests.
(6) Configurational invariance using both dif
ferent populations and different tests.
Henrysson feels that the latter two cases represent
the most convincing forms of invariance in that the
chances of a particular grouping of tests produoing arti
ficial factors would be minimized through independent
replication. It is obvious, however, in these cases that
comparisons must be based upon subjective impressions that
the same factor exists in both studies or by some form of
indirect comparison. This might be obtained through the
use of experimentally independent criterion tests, not
inoluded in the battery, that are known to have high cor
relation with tests in each battery. An example of this
might be the inclusion of different "Structure of Intel
lect" tests (Guilford, 1959)# In each battery that
correlate highly with an Independent measure of verbal
comprehension, numerical ability, memory, etc. Every
battery would have the same number of tests representing
each hypothesized factor, each test having a high correla
tion with a criterion measure that is not In the battery.
Another approach might be through construction of artifi
cial factor matrices with a hypothetical factor pattern
based upon knowledge of the factor profile of each of its
variables from prior research. This factor pattern could
then be compared to each of the experimental batteries.
In dealing with empirical data it had never been
feasible to employ large numbers of replications of bat
teries. The major drawbacks have been with matching the
experimental variables underlying the hypothesized factor
pattern, equating experimental samples from populations,
and finally the formidability of computations involved.
The advent of large digital electronic computors has al
leviated these difficulties and made this type of study
possible.
It is readily seen that Invariance of a factor
pattern could be both defined and investigated in a number
of ways. Most of the present methods involve a matching
of Individual factors from one factor matrix against those
of another. Decisions regarding congruence of factors
are based upon obtained values of statistical indices.
7
But nowhere In the literature Is there any information
regarding the empirical sampling characteristics of in
dividual loadings under the combined conditions of rota
tional method, type of simple structure of underlying
factor pattern, and size of sample. The current trend
amongst factor analysts in using the analytic methods
would suggest the need for a study of this type. It was
hoped that this investigation might clarify the issue of
relative merit of the various methods in terms of an ob
jective standard.
The basic strategy of this study was to observe and
comment upon the sampling behavior of the factor loadings
of a pre-determined factor pattern. Most of the recent
research relating to factorial invariance investigates the
configurational relationship of a factor In one battery to
that in another. The present study is more concerned with
the more basic numerical type of invariance that could be
Involved in sampling from a given population of individuals.
What was actually studied was the relative amount of
fluctuation from population values that occurred through
direct and combined effects of the experimental conditions.
Perhaps a more apt term for the sampling character
istics of individual loadings about a population value
would be factorial stability. The latter term will be
used in this paper, as It Is felt by the writer that the
8
stability and numerical invariance of a sample factor
loading about its population value are similar. The
difference lies in the fact that invariance measures are
based upon the correlation of two factors, whereas
stability is more concerned with variability and standard
error. It would seem premature to speak of invariance
or matching of factors under various factor analytic ap
proaches until more generic knowledge of empirical sampl
ing characteristics of individual loadings is obtained.
A potential source of instability can be seen from
an examination of the general linear model U3ed in multiple
factor analysis. Each variable Zj is considered to be a
linear function of all common factors determined to be
necessary from reduction of the order of a correlation
matrix and a unique factor that supposedly does not enter
into the correlations with other variables. This expres
sion is given by Zj = aJ1P1+ aJ2F2 + * * * aJmFm+ a^uJ
where F-^, Fg, . . . . + Fm represents the common factors,
UJ is the unique component and the coefficients, ajk* are
the factor loadings or weights- . The general assumptions
made are that the factor scores have means of zero and
unit variance. The set of linear equations representing
all variables is called a factor pattern. It is assumed
that the common factors in a pattern may or may not be
correlated but that the unique components are uncorrelated
both with other unique factors and with the common factors.
(Thurstone, 19^7; Harman, i960)
It is quite possible that sampling from a popula
tion, though, should result in some correlation of the
unique components amongst themselves and with common
factors as they are represented in the sample. Sampling
might also cause random deviations from unity in the
variance of each of the factors. In this study these two
considerations were used to inject error into an obtained
"sample” correlation matrix, thus providing for subsequent
distortion in the final resolution of the factor pattern.
The Method of Generating the Sample Factor Patterns
By the Monte Carlo techniques it was possible to
generate random samples of a given factor pattern as fol
lows. Starting with the assumptions of unit variance and
zero intercorrelation amongst all factors in a factor
pattern, both common and unique, four different representa
tions of simple structure matrices were constructed. The
first of these was based upon Tucker13 definition of linear
independent constellations in the extreme case, i.e., un
ivocal tests. The second and third introduced varying
degrees of test complexity into simple structure patterns.
Finally, the fourth factor pattern was constructed as a
non-simple structure matrix. Each represented a complete
10
factor pattern as unique loadings were included.
Random error was Introduced by the generation of
sample factor covariance matrices. The elements of these
matrices were randomly selected by an IBM 709M- computer
program using a table of random normal deviates. These
values were used to simulate the covariance components.
The constructed factor covariance matrices were then used
to post-multiply the given factor patterns. These factors
structures were then post-multiplied by the transpose of
the original factor pattern. This procedure created
sample variable covariance matrices. The sample correla
tion matrices resulted from pre and post-multiplication
of the covariance matrix by the inverse of the square root
of its main diagonal. The desired end products, sample
factor matrices, were obtained by principal component ex
tractions with the squared multiple correlation as an
estimate of communality. A mathematical description of
these procedures is given in Chapter V.
It should be noted that the elements of the factor
covariance matrix were not always logically independent,
although the method of sampling used in this study could
by chance force them to be so. For example, it would not
be expected that two factors which were highly correlated
in a particular sample with a third factor, would not also
be correlated with each other. The sampling procedure
IX
used did not guarantee this could not happen. The more
appropriate procedure would be to consider the factor cor
relations as partial correlations coefficients and to
"reconvert" them into their respective correlations. On
practical grounds this was not considered necessary because
of the small values that emerged after division by the
square root of the degrees of freedom.
Two alternative procedures by Mosier (1939) and
Kaiser and Dickman (1962) for obtaining the sample correla
tion matrices were considered, both of which had the same
drawbacks for use in this study.
Mosierfs method generated random correlations
matrices in the following manner. He constructed a given
simple structure factor pattern, V . Prom this he repro
duced the population correlation matrix RQ. An error term
erjk/°rjk ^or each Jxk-th element of R0 was randomly
selected. These values were normally distributed and fell
between ±3.00. The standard error of a correlation co
efficient, was based upon an N of 100. A table of
these o ' j . j i c was prepared in order to obtain the error com
ponents of the correlation matrix R^. Thus we have the
total error for each rj^ given by arjic/°x»jic x ^rjk* T* 113
provided the error matrix E. Then we have RQ= R^-E and
finally R^ - RQ+E.
12
This method provides only for obtaining the off-
diagonal elements of R^. The diagonal elements are
considered unknown and were estimated by various methods.
Mosier1s procedures were not tenable for the present
study in that there was no way of allowing for the inter
action of unique components of the linear factor model in
perturbing a factor pattern.
Kai3er and Dlckman present a method for obtaining
sample correlations matrices from a given population cor
relation matrix. The technique makes use of the funda
mental postulate of component analysis,
Z=FX
where F represents any factoring of a population correla
tion matrix R of order nxn and X is the score matrix of
order nxN. An arbitrary sample score matrix, X, is
obtained through use of random numbers from a population
with mean zero and unit variance. Then we obtain
A A
Z=FX
A
where Z is the desired matrix representing observations
from a multivariate population with zero means, unit
standard deviations and with a population correlation
A
matrix, R. From Z the desired R can be easily obtained.
The problem with this approach is that R must be
given in advance. This is the same objection that is
raised for Mosier's method. Also the derivation of R
from a population pattern does not include the hnique
13
components of the general linear factor model. As the
present study incorporates the effects of the unique
components from population factor patterns the use of
either of the above techniques was precluded.
Methods of Rotation
In the past decade various analytic rotational
methods that lay claim to providing factorial invariance
if it exists in a population of individuals have been
developed under simple structure considerations. Effects
of three analytic methods that have been reported to give
the most favorable results were examined. They are the
normal varlmax, equamax, and blquartimln methods. (Kaiser,
1958; Saunders, 1962) The latter provides a basis of
comparison of an oblique method with the first two ortho
gonal techniques. A semi-analytic standard of comparison,
the Procrustes method (Hurley and Cattell, 1962, also was
utilized. This program transforms an unrotated matrix to
a least squares fit of a hypothesized factor pattern.
These four methods were used to rotate the obtained sample
factor matrices. Chapter IV of this study reviews some
of the literature concerning these and other rotational
methods.
Type of Factor Pattern
The nature of the configuration of loadings in an
idealized population factor pattern might also have an
effect upon the stability of the rotational solution. In
empirical studies there is no prior knowledge as to the
true pattern of loadings. Prior to 195°» rotations were
generally made using an investigator's knowledge of the
psychological domain as the main criteria for placement of
reference axes. The main objective criteria for a good
solution were Thurstone's simple structure rules. These
rules demanded a certain pattern of high and low loadings
as constraints for fitting the reference axes. In the
present study the effects of three different variations
of the simple structure criteria were examined. For
comparison a population pattern that did not adhere to
simple structure was included. The simple structure con
cept is discussed in Chapter III.
Sample Size
The size of the sample would be expected to have an
effect upon stability. The extent of this effect, combined
with the other variables under study was examined, using a
large (n=i+00) and a small (n=100) sample for typical
factor analyses.
Finally the effects of sizes of population loadings
and communalities were investigated under the conditions
mentioned.
Statistical indices of stability considered were the
range and root mean square value (RMS) of the cell values
of the matrix formed by combining rotated values of random
samples of factor patterns. The shape of the frequency
distributions of values in each of these cells were also
Inspected. A review of some of the issues involved in
invariance measures is given in Chapter II.
In summary, the purpose of this study was primarily
twofold. One aim was to observe the sampling character
istics of the elements of various types of simple structure
factor matrices. The other was an attempt to evaluate
various presently popular analytic rotational methods in
relation to a criterion of stability.
CHAPTER XI
INDICES OF FACTORIAL STABILITY
This chapter represents a review of some present
Indices and some suggestions for other potentially useful
measures. The main difficulty with existing indices of
factorial stability is that there is no exact information
regarding the sampling distributions Involved. Effective
parametric tests of significance have not been devised for
the matching of factor profiles. Burt (194-8), Tucker
(1951), and Wrigley and Neuhaus (1955), among others, have
developed indices for this purpose each of which can be
described as an unadjusted correlation coefficient; i.e.
one in which the mean is not subtracted out. This type of
index suffers In that there is not presently available
statistical inferential tests for measuring its signifi
cance. A measure of similarity of factor profiles and of
the total factor patterns of two batteries with the same
hypothesized factors might be based upon rank order
statistics. Kendall's tau or the coefficient of concord
ance could possibly be used for measures of similarity of
factor profiles and total factor patterns, respectively.
16
17
In any case. In lieu of an extant parametric
statistical test, the above types of non-parametric
ordinal method of measuring association, might be imple
mented. What they would lack in power and sensitivity
would be compensated for by the relaxation of assumptions
regarding distributions and their parameters.
Cattell and Baggaley (i960) have recently advocated
a nominal type of statistic for comparing profiles of
factors. They refer to this as the "salient variable
similarity index" s. Although the concept of a nominal
type of measure might be a fruitful one to investigate,
the statistic s would not be generally applicable to a
general study of invariance. This limitation arises from
its requirements of marker variables which are not always
available in exploratory studies, and their equal repre
sentation in factors to be compared. This index also does
not make full use of all of the entries in the factor
matrix; in particular it ignores the hyperplanar or low
loadings in measuring the relationship between two factors.
In conjunction with measuring the similarity pro
files of factors across tests for similarity, it might also
be profitable to consider an Index that measures the
similarity of test profiles across factors. This would
more closely adhere to Thurstone*s (19^7) basic concept of
invariance, I.e. that "the factorial description of a test
18
should remain Invariant when the test Is moved from one
battery to another which involves the same common factors."
An index derived to measure the relationship between rows
of different factor matrices would also be generally ap
plicable to all of the six types of invariance mentioned.
Consideration of transformation or comparison
matrices suggests a means for an indirect test of invari
ance. Attention could be focused on the characteristics
of the off-diagonal values of square transformation
matrices that convert the factor matrix representing one
battery into the factor matrix of the other battery. In
variance would be approached as convergence to a diagonal
matrix occurred. This would require the investigation of
statistical distributions and parameters of the off-
diagonal values. If identification of these theoretical
distributions should prove to be infeasible, an ordinal
or nominal approach might be implemented so as to provide
a means for conducting tests of significance regarding the
presence or absence of "transformation" invariance.
Ahmavaara (195*0 presents a criterion of numerical
invariance between factor solutions in two different popu
lations, a criterion which utilizes a measure of linear
relationship. The method entails the correction of
variances and covariances in one of the matrices so that
the tests in common to the two batteries are measured by
the same units of measurement. Then a comparison matrix
relating one battery to another is formed. If this com
parison matrix is a unit matrix, then factorial invariance
is said to exist. Although Ahmavaara1s formulations of
his transformation analysis has been severely criticized
on the basis of mathematical considerations, (Bargmarm,
i960) the general concept of invariance under transforma
tion would seem to be a fruitful area for investigation.
Tucker (1956), Raach (1953), Cattell (195*0, Kaiser (1961),
and Werdelin (1961), among others, have developed trans
formation matrices providing for the simultaneous rotation
of two batteries to one simple structure solution. The
major problem with the indices of factorial similarity
that emerge from these techniques is that, as mentioned
before, they fail to provide a statistical test of signi
ficance .
Because of these limitations, it was decided in
this investigation to use as a simple descriptive index
of instability, the root mean square deviation, RMS. This
index i3 a type of standard error; in this case the
standard error of sample (within cell) factor loadings
about a population factor loading. As the purpose of this
study Involved analyzing the fluctuation behavior of
individual loadings rather than measuring pairwise factor
20
congruence or matching, RMS was considered to be both an
appropriate and easily obtained index. It should be noted
that the root mean square Is also frequently used In the
latter context, i.e. where the summation is along variables
rather them across samples and the deviations are between
the paired factors rather than between population and
sample values. If factor matching were desired, RMS
should also probably be used Instead of Tucker's phi.
This is because RMS weights all pairwise deviations
equally whereas phi gives more emphasis to covariance of
the higher values. Phi has also been known to give mis
leading high values in matching factors (Bechtoldt, 1961).
The range is also Included as an auxiliary measure
of stability. This gives the extent of fluctuation in an
absolute sense as a function of different treatment and
parametric conditions. It provides an estimate of possible
limits that could occur through sampling.
CHAPTER III
SIMPLE STRUCTURE
The objective criterion underlying the various
types of invariance involves a limited segment of
Thurstone's (194-7) simple structure rules.
Thurstone's five criteria for the determination of
the reference vectors for a simple structure are as
follows: (op. cit., p.335)
1. Each row of the factor matrix should
have at least one zero.
2. If there are m common factors, each column
of the factor matrix should have at least m
zeros.
3. For every pair of columns of the factor
matrix there should be several variables
whose entries vanish in one column, but not
in the other.
4 - . For every pair of columns of the factor
matrix, a large proportion of the variables
should have vanishing entries in both columns
when there are four or more factors.
21
5. For every pair of columns of the factor
matrix there should be only a small number
of variables with non-vanishing entries in
both columns.
In general, only the first two rules are closely
adhered to in factor-analytic solutions. This Is because
of the relative ease of objectifying and Including them In
the formation of analytic rotational criteria.
Thurstone thought that placing restrictions of
simple structure upon reference axes provided a firm base
for numerical or metric invariance. This type of invari
ance would be found when a test is transferred to another
simple structure battery which has the same common factors.
He stated that other restrictions on the reference frame
providing for numerical invariance of the factorial
structure of a test are potentially identifiable. However,
at the time., that Multiple Factor Analysis was written
(19*4-7)» he felt that simple structure was the only re
striction on the reference frame that gave metric invari
ance .
Thurstone developed this concept so as to offer a
productive vehicle for generating hypotheses. A simple
structure factor analysis would permit more rigorously
23
formulated hypotheses relating to invariance, parsimony,
and uniqueness of factor structures* These factors could
be subsequently modified and empirically tested by both
factor analytic replication and by experimental procedures
other than factor analysis.
However, in trying to apply the simple structure
rules, investigators have found that they are really not a
definition of the concept (Crowder, 1955; Butler, 1961).
Rather they represent a description of what an Idealized
factor pattern would look like if it were possible for it
to be obtained. This need for more explicit and objective
definition has been the main reason for the controversies
that have revolved about the topic.
Butler (1961) discusses simple structure with the
view of offering a possible solution to the problem of
factorial invariance. He gives a method for specifying
and obtaining an optimal or best simple structure out of
a total spectrum of simple structures for any given
factorial solution. This simplest structure exists when
the test vectors represent a partition of the common
factor space into sub-sets of vectors such that the inter
sections of the sub-sets are empty. The disJuncture of
the sub-sets of test vectors then represents the highest
degree of over-determination of the factors. Furthermore,
the simplest structure is then unique; no alternative
24
simple structure is possible which represents as simple a
structure as that in which all te3ts are pure on a given
set of factors.
Several attempts have been made to present the total
concept in an explicit mathematical form. Carroll (1953)
originally intended to include all five rules, but he sub
sequently ruled out the possibility of their inclusion in
one functional statement. Others have formed analytic
criteria based upon the first, or second rule, or both
combined. The stability of these types of criteria forms
a portion of this investigation.
Tucker (1955) sets forth some rigid requirements
for an objective definition of simple structure. He makes
the distinction between exploratory and confirmatory
factor-analytic studies and concludes that an objective
definition of simple structure depends upon both study
design and objective criteria. He gives a definition of
simple structure in terms of linear constellations. The
procedure he gives for obtaining solutions based upon
these definitions would only be practical on high speed
digital computors. Tucker feels that the technique would
be successful with well designed studies although there is
no certainty that all linear constellations would be found.
Merrifield (1962) extends Tucker's simple structure
exposition to include the hypothesized relations of factor-
25
analytic measures to constructs in an empirical area. He
gives three criteria for naming factors in terms of the
number of constructs represented in the hyperplane of a
factor. Using these criteria as a base he offers some
stringent rules for defining the concepts of general,
proper, partial, singlet, and residual factors. The in
variance of factors and tests are defined in terms of the
stability of their hyperplanar description.
A statistical significance test of the overdetermin-
atlon of a simple structure was developed by Bargmann
(1953). The criteria of overdetermination is a function
of the number of null loadings found in a factor structure.
A null loading is defined as one in which a/h < ±.10, where
a is the value of the factor loading and h represents the
square root of the communality of the particular variable.
The values falling into this band constitute the values
defining the dimensional hyperplanes of the k dimensional
common factor space. For overdetermination of this space
each factor must have at least k null loadings in its
hyperplane. The probability that a variable (vector) will
fall within a defined null band by chance for a given
number of factors k is obtained by relating the ratio of a
sector of a hypersphere to the total hypersphere. This
probability, P^, is then used with the binomial distri
bution function to determine the least number of null
26
loadings necessary for overdetermination of hyperplanes at
a particular statistical significance level. This amount
would be determined for a particular combination of quan
tities of factors and variables.
Using empirical results as estimates for unavailable
sampling formulations, Bechtoldt (1961) observed the
effects of variation in the use of communality, type of
extraction procedure, and in samples from a given popula
tion. The extracted factors obtained from the various
treatment combinations show little agreement using both a
graphic mode of presentation and factor matching indices,
i.e., Tucker's phi and the second moments of the dif
ferences between pairs of corresponding columns of factor
matrices. But when an objective oblique simple structure
rotational method, oblimax, was utilized and compared with
graphic solutions that were guided by information from
the test variables, a high degree of factorial invariance
was found. In most cases the degree of simple structure
is also found to be statistically significant using
Bargmann’s test. Bechtoldt offers this as support for
Thurstone's belief in configuration and perhaps metric
Invariance of simple structure solutions.
An early empirical study by Mosier (1939) on the
effects of chance error on simple structure provided some
of the Impetus for the present study. He reconstructed
sample correlation matrices from a hypothetical structure.
The correlation coefficients were saturated with chance
error components. Rotations to least squares fits of
hypothesized simple structures were made following centroid
extractions using different methods of estimating communal-
ities. Mosier found that random correlational error and
mode of communality estimate made little difference in
establishing the primary factor pattern provided the
centroid matrix had a rank at least as great as that of
the hypothesized pattern matrix.
In a somewhat similar recent study, Cattell and
Gorsuch (1963) perturbed a given simple-structure pattern
with error components. Then both the original and error
pattern were randomly orthogonally ’ ’spun'1 on their re
ference axes to arbitrary positions. Subsequent results
of rotations to a criterion of simple structure demon
strated that data involving well established hypotheses
achieved this goal far better than data obtained from a
random source.
CHAPTER IV
ANALYTIC AND SEMI-ANALYTIC ROTATIONAL CRITERIA
There are two major reasons for the rapid develop
ment and implementation of analytic techniques. The first
is to eliminate subjective influences in rotation and to
provide for more agreement among investigators as to the
structure of a particular battery. The second is the ease
by which these techniques can be handled by electronic
computers, thus avoiding the tedious burden of graphical
rotations. The desirability of these two goals is not
unequivocally agreed upon by workers in the field. Much
evidence has been offered indicating the inadequacies of
a completely analytical technique. Compromise is often
sought in the form of semi-analytic methods. These issues
will be discussed later in this section.
All of the analytic methods now in use stem from
some algebraic or analytic formulation of some of the
parameters of a matrix that conforms to Thurstone*s simple
structure criteria. They are efforts to give a mathe
matical functional form to the descriptive statements of
the five rules. A transformation matrix is obtained re
lating to the analytic statements and is used for trans
forming the arbitrary factor matrix so that the rotated
28
29
matrix contains the specified property of the analytic
criterion.
Over the past 25 years a number of these criteria
have been developed. These have met with varying success
in the rotational area. A survey of some of the more im
portant analytic and semi-analytic methods will follow,
including a review of inter-comparisons of these tech
niques among themselves and with the graphical subjective
method. Additional recent material pertaining to these
methods and other related factor analytic topics can be
found in reviews by Thompson (1962) and Merrifield and
cliff (1963).
The first workable analytic criteria published in
the literature was that of Horst (19^1). This was based
upon Thurstone's principle that each column of the factor
matrix should have a minimum number of negative values
and a maximum number of nearly vanishing values. More
explicitly, it is stated that for a given factor the sum
of the squares of the significant (non-statistically)
factor loadings divided by the sum of squares of all the
loadings shall be a minimum.
Tucker (191 +i +) proposed a compromise between the U3e
of subjective Judgment methods of graphical techniques and
routine use of analytical methods. He locates the axes
for subgroups of tests by an analytical method. The
30
judgments used in the selection of the test subgroups are
based on graphic data that depict the interrelation of the
factors.
A single plane analytical method is given by
Thurstone (195*0. One starts with a hypothesis that a
given sub-group of test vectors lies in a hyperplane of
dimensionality (v-1) where v is the rank of the correspond
ing correlation matrix. Then successive guesses are made
as to which test vectors lie in the same hyperplane. The
desired reference vector would then be the normal which
defines the hyperplane. The method is described for the
three-dimensional case, but it is applicable to a factor
problem of any rank.
Ferguson (195*+)» Carroll (1953), Neuhaus and
Wrigley (195^-)» and Saunders (1953), all independently
derived, at approximately the same time, an analytic
criterion which in the orthogonal case is identical. It is
primarily based upon maximization of the sum of the fourth
powers of test loadings across factors or (identically)
the minimization of the sum of cross-products of squares
of factor loadings across factors. This criterion is com
pletely analytical and yields a unique solution. The
digital computer made practical the iterative solution
of the equations involved. Carroll’s method permitted
either orthogonal or oblique solution. This method was
31
later expanded to Include the oblimln class of solutions.
Among problems encountered was that when using empirical
data, the presence of factorially complex tests produced
a type of hyperplanar fit which needed to be adjusted by
graphical rotations. The fewer tests of this type the
closer the criterion came to approximating simple structure.
In a study (Wrigley, et al, 1958) that compared a
quartimax rotation that had been made from a centroid ex
traction with others that had been either graphically ro
tated from centroids or obtained from another factor-
analytic model, it was found that the analytic solution
agreed quite closely with the others. The correlation
matrix was obtained using Thurstone’s PMA test battery.
The major difference was the general factor provided by
the quartimax. It was felt that the overall factorial
structure was sufficiently close to that hypothesized to
warrant its use at least in the initial rotational stages.
The emergence of a general factor when one is not
indicated by the study design has also been found by other
factor analysts. It is generally concluded that quartimax
has a tendency to maximize the number of variables that
load on factors and to collapse underdetermined and ap
parently correlated factors. This gives the appearance
of a general or highly complex composite factor.
32
Kaiser (1958) developed his varimax criterion as a
means of maximizing the variance of the squared factor
loadings in each column of a factor matrix. This was in
keeping with adherence to the principles of simple
structure, In that emphasis is placed on a simplified des
cription of each column of the factor matrix. It is ap
parent that as the variance is maximized, the factor load
ings will tend toward a one-zero configuration. He defines
maximum simplicity for a complete factor matrix as the
maximization of the sum of the variances of each factor.
To avoid bias attributable to differential weighting of
variables by their communality size, Kaiser adjusted his
original "raw" varimax criterion to one in which the rows
are normalized. This provides for a more equitable dis
tribution of variance among the factors. Following rota
tion, the unit length row vectors are transformed back to
their original length. Normal varimax is shown to have
theoretical factorial Invariance for two factors by delet
ing variables representing the factors and maintaining the
same rotational angle and loading pattern. An example is
also given of invariance for more than two factors.
Several studies have been made testing the efficacy
of the varimax method with respect to the various claims
for its close adherence to simple structure, factorial
Invariance, and prior satisfactory graphic resolutions.
33
These would generally include some comparison to other
analytic techniques. — -
Kaiser (i960) rotated Thurstone's PMA battery using
varimax. Comparisons between factors of the original
subjectively rotated factor pattern, Zimmerman's revised
graphic solution, a quartimax solution, and those of
varimax were made, using correlations of the factors which
defined the solutions as a criterion. Varimax rotations
tended to identify basically the same factors as the sub
jective solutions, perhaps better than did the quartimax
method. However, Kaiser states that the value of the
solution by his criteria should be based on its psycho
logical meaningfulness and not on relationships to results
from other studies.
In a methodological study (Marks, et al., i960)
varimax and a graphic solution were employed in rotating
extractions from a 21-test intercorrelation matrix. Both
solutions yielded underdeterrained structures, i.e., too
few zero level loadings in the hyperplanes. The graphical
and analytical solutions yielded 11 and 5 identifiable
factors, respectively, with little similarity in psycho
logical meaningfulness.
In a three-year project being conducted in the
Psychometric Laboratory of the University of Southern
California on the investigation and comparison of analyti
cal and graphical methods of rotation in factor analysis
34
(Zimmerman, et al., 1963), several empirical findings have
been obtained on varimax effects. Two related matrices
with intercorrelations of 15 and 15 psychological tests
were selected for analysis. It was felt that previous
analyses of the test variables had shown that their factor
structure had been well established. The data were select
ed so that underdetermined, moderately well determined, and
overdetermined factors were included. Hie graphic rota
tions made to conform to previous findings were considered
satisfactory and were used as the criteria with which
analytic results were to be compared. The methods included
quartimin, biquartimin, and covarimin as oblique solutions
and quartimax as an orthogonal solution. Of the analyti
cal methods used, normal varimax results most closely
resembled those of the graphic solutions. Discrepancies
arose through the varimax method's tendency to separate and
collapse the graphically derived factors. These findings
seemed to be dependent upon the balance among them in
terms of over and underdetermination. In another study
using a 70-variable, 18 (graphically determined) factor
matrix, hypotheses relating to varimax analysis of the
data were substantiated. Again it was found that there
was a tendency for overdetermined factors to be built up
at the expense of those that were not as well determined,
35
for factors whose leading tests were moderately or signifi
cantly correlated to collapse, and for singlet, doublet,
and sometimes bipolar factors to emerge inappropriately.
Hamburger and Merrifield (1963) investigated the
effects of normal varimax and biquartimln methods upon
factor matrices contrived to have a particular structure.
Three matrices were constructed under consideration of
specific simple structure parameters. Criteria for evalua
tion of deviations in factor structure were based upon
changes in complexity of test, number of tests representing
a factor, hyperplane configurations, comparison of distri
bution functions, and Tucker*s coefficient of congruence.
It was found that when strong orthogonal simple structure
exists, varimax and biquartimin solutions tend to reveal
it. With the varimax method, the introduction of com
plexity into the factor profile of a test markedly reduces
the definability of factors, causing a decrease in the
frequency of both high loadings (i.e., beyond ±.30) and
hyperplane loadings (within the range of ±.10).
Saunders (1962) demonstrates that by changing one
parameter K of a quartic equation whose coefficients are
formed from fourth-degree moments of the data, all the
orthogonal analytic formulations can be obtained. When K
is equal to zero, quartimax is given, when equal to one,
then varimax. Empirical manipulation of this constant
with data from a number of sources revealed that with K
36
set equal to one-half the number of factors rotated gave
the most satisfactory solutions. Saunders named this
criterion equamax to denote the equating of Importance to
all factors rotated. He noted that equamax Is Identical
to varimax when only two factors are rotated but given in
creased weight to the second-degree moments In the equation
as the number of factors is Increased. Evidence was given
that this procedure equalizes the contribution of the
second-degree terms and distributes the total factor vari
ance more evenly amongst all of the factors rotated.
Examples were given of the effects of equamax on a 59-
variable, 12-factor problem. It was shown that equamax
converts a general factor and Its accompanying bl-polar
factors Into a simple structure with positive manifold.
For this example, It was also demonstrated that the
equamax method distributes the variance more equally among
all of the factors than does varimax for the same battery.
The success of this procedure has been found, in
dealing with empirical material, to be quite dependent
upon an accurate estimate of the number of factors to be
rotated (Saunders, 1962a).
Obliaue Solutions
Descriptive factor analysis from the viewpoint of
comprehension and ease of utilization originally made use
of a model in which the factors were defined to be un
correlated. This began with Spearman's assumptions that
specific factors were uncorrelated with each other and with
the general factor. Within the last 25 years, particularly
since the widespread implementation of Thurstone*s common
factor model, there have been Investigators who prefer the
premise that the primary factors can be correlated. They
would prefer a general methodology that would take this
possibility into consideration and accept orthogonality
as a special case. It has been maintained (Anderson and
Rubin, 1956) that if factorial Invariance is to hold from
population to population in an investigation of a particu
lar trait configuration that it is necessary to allow for
changes in the correlations between the factors.
Perhaps the most widely used oblique techniques at
the present time are the oblimin methods of Carroll (1958),
which all involve some function to be minimized. The
biquartimin method used in this study is a member of the
oblimin class. It was developed to balance out the in
adequacies of Carroll's original quartlmin method and
Kaiser's covarimln method. The quartlmin criterion, N, is
the same as that of the quartimax method, but without the
constraint of orthogonality. Again, this method involves
the minimization of the sums of cross products of squared
factor loadings. Kaiser (1958) developed the covarimin
method by simply relaxing the orthogonality restriction of
his varimax criterion. The covarimin criterion, C, is then
given as a function which involves the minimizing of the
covariances of squared elements of a factor structure.
Empirical studies by both investigators found that the
quartlmin method is biased toward highly correlated factor
axes while the covarimin method tends to move toward axes
which are too orthogonal. As a compromise Carroll
developed the biquartimin criterion, B = N + C/n, where n
is the number of variables, and N and C are defined above
as the expression to be minimized. This involves minimiza
tion of both C and N, each of which has a logical rationale
for inclusion and provides distinct advantages (Harman,
I960).
Analytic Oblique Comparisons with Other Methods
Several recent studies which have investigated the
congruence of orthogonal and oblique analytic solutions,
and analytic oblique with graphic solutions, are reviewed
below.
Using eight physical variables, Harman (i960) com
pared the three oblique methods described above. The
biquartimin method was interpreted as providing the most
adequate solution. Of the three methods, it gave the
best compromise between simple structure (well defined
zero loadings) and low correlation between primary factors.
This substantiated the basis for development of the bi
quartimin method.
39
In another study involving 24 psychological tests,
Harman (i960) compared biquartimin factors with those that
were graphically obtained. He found that the analytic
method tended to give more unequal distribution of
varlanoe contribution of factors than the subjective
graphic teohnlque. This was also found to be true by
Zimmerman and Merrlfield (1962) in a study comparing
various orthogonal and oblique methods to a graphic
solution, using a battery of 15 psychological tests.
In terms of adequate simple structure, however,
Harman (i960) provides evidence, using eight political
variables, that any one of four different oblique analytic
methods performs better than a graphic solution. He
suggests that the results of an analytic method could be
used to improve upon the original intuitive ^judgments,
stating "• • .this kind of rationalization is to be ex
pected if the objeotive definition of simple structure is
indeed a good explication of the intuitive concept."(p*333)
Coan (i960) made comparisons of independent oblique
(obllmax) and orthogonal (quartimax) factor rotations on
the same set of factors from a problem in which the
"subjects" were chicken eggs. The correlation matrix
variables consisted of six direct measurements and 15
ratios. It was found that the oblique solution gave two
second-order factors which were similar to two factors of
40
the orthogonal solution. Coan determined that the oblique
solution gave better simple struoture, provided a more
unambiguous or unique position for the primary axes, and
yielded factors that were easier to Interpret.
In a comparison of the biquartimin and varimax
methods with graphic rotations it was found that the
biquartimin method seemed to be more generally appropriate
If orthogonality was not assumed (fyman, et al, 1962).
This was supported by the orthogonal structure achieved
with biquartimin and what were interpreted to be superior
solutions when orthogonality was relaxed in order to
obtain better simple structure.
Resnick (1961) comparing varimax and oblimin
methods, found preoisely the same factors by both methods
except for a reversal of order of factors. The latter, of
course, represents only an artifact of the procedure used
by the computer programs involved. Resnlok also found a
near zero level of intercorrelation among the primary
factors. The conclusion was drawn that there is no real
difference between the solutions.
Finally, Cattell and Dlckman (1962) conducted a
study in which measurements were made on various types of
balls. They found that an oblique rotation method gave
simple structure solutions in which the number of zero
loadings were statistically significant with Bargmann* s
41
test. A varimax solution provided factors that did not
have a statistically significant number of zero loadings.
The reason for this was that many of the low loadings
present In the oblique solution increased in size with
the varimax method. However, the factors resulting from
both oblique and orthogonal methods could be identically
interpreted.
Semi-Analytic Methods
Several methods have arisen In which a transforma
tion matrix is found that maximally relates a factor
solution to a given hypothesized pattern or target matrix.
This generally involves a least squares best fit to this
hypothesis and thus combines initial subjective judgment
and subsequent analytical convergence.
The earliest semi-analytic method was proposed by
Thurstone (1936). This required an iterative least
squares solution for the projections on reference axes of
selected groups of variables. The variables were chosen
subjectively so as to maximize the location of hyperplanes.
The computing load required for handling large matrices
was too large to make this method feasible at that time.
Horst (1941) developed a modification of Thurstone1s
method which invokes the principle that each column of the
factor matrix should have a minimum number of negative
values and a maximum number of nearly vanishing values.
42
A similar technique for transforming a factor matrix
to a hypothesized simple structure matrix was proposed by
Mosler (1939) and Horst (1956)* There are no restrictions
on the number of simple structure factors which might
contribute significantly to the variance of a test. The
method is an approximation to a least squares solution if
a centroid matrix is used and is exactly a least square
solution if principal components are used.
Tucker (19^4, 1955) gives a method which locates
analytically the axes for subgroups of tests. The selec
tion of the subgroups is based on data obtained from
inspection of graphic plots of the factors.
Saunders (i960) developed a modification of his
analytic solutions that utilizes a hypothesized factor
pattern to be input to a computer program together with
factor data. The computer thus selects the sub-set of
variables as a basis for rotation in a given plane. This
is done so as to provide an orthogonal solution which
closely adheres to the hypotheses. A measure of goodness
of fit to the pattern is provided. Pattern matrices that
specify partial or incomplete hypotheses relative to the
factors, or those that would provide overdetermined simple
structure, can be used. Saunders (1962) also Included the
option of a total or partial pattern to supplement his
trans-varimax criteria.
^3
As a type of factor-analytic hypotheses testing
procedure Hurley and Cattell (1961) developed the
Procrustes computer program. This method fits a given
set of data to a previously stated hypothesis about the
factor pattern. This presumes that the number of factors
to be rotated has previously been determined. The method
Is essentially identical to independently developed pro
cedures of Ahraavaara (1957)» Mosier (1939)» Horst (1956),
Cliff (1962).
This procedure starts with an orthogonal, unrotated
factor matrix V . Then a hypothesized factor pattern Vrs
is stated. It is desired to find the transformation matrix
x . ' such that VQ is transformed to VpS, where vrs is the
best least squares fit to Vrs, i.e., VQ\ ' « V^3. To
forestall the clandestine use of the program as an easy
method of making data fit a hypothesis with subsequent
claims of theory substantiation, the authors strongly
recommend the use of a statistical significance test of
goodness of fit. Those recommended (e.g., Tucker’s phi
and the salient variable matching index) unfortunately,
as discussed earlier in this paper, are not completely
satisfactory.
Applications for Procrustes other than direct
hypothesis testing are given. One would be clarifying the
hyperplanes in the final stages of graphic rotation.
Another is the shifting of an unrotated factor matrix to a
im
position where the main structure has been previously
determined by a series of different graphic rotations.
Then the final position could be obtained graphically.
Finally, Hurley and Cattell suggest its possible use in
exploratory studies when an apparently maximized simple
structure is achieved. By using Procrustes with other
even better simple structure alternatives, it might be
possible to determine if the solution has actually been
maximized.
Recently, Butler (1963) tested his criteria of
simplest data factors against several other analytic
oblique and orthogonal criteria. As previously discussed,
this is based on the concept of simple structure requiring
the maximal separation of factors. Butler explicitly
defines in algebraic form the simplest data factors as unit
length vectors collinear with the most nearly orthogonal
data basis. In practice, this criteria will almost always
provide oblique primary factors.
Using data from Thurstone's box problem, Butler
concludes that simplest data bases were similar to other
accepted solutions in all types of variable combination
of the correlation matrix, i.e., regardless of whether the
correlation matrix had entries that were both positive and
negative in value. Equamax and varlmax identified the
primary factors only when the correlation matrix was
45
non-negative. Butler offers this as evidence that varimax
and equamax are not factorlally invariant. Quartlmax did
not even identify the primary factors in the non-negative
subset of the correlation matrix.
In comparison the box problem and other empirical
matrices against the various obllmln criteria, Butler
found that the simplest data bases gave quite similar
solutions. In conclusion he states (1963* P*5) ”...
the use of simplest data factors would result in factors
identical with primary factors or in factors in a highly
favorable position for graphical rotation to a final simple
structure position with a minimum of complications."
Because of the high probability that some graphical adjust
ment will usually be necessary, Butler describes his
criteria as semi-analytic.
In one of a group of studies (Oocka, 1959) emerging
from a project on analytic methods of rotation under the
direction of Horst, comparisons are made among several
solutions by a technique that could be classified as
semi-analytic and by approximations to varimax and
quartlmax solutions. The semi-analytic technique is a
slightly modified version of Thurstone's analytic method
(MTAP) which requires that the experimenter select the
trial transformation (row) vectors required for the
iterative solution. Horst*s analytic Iterative rotation to
a modified varimax criterion (AIR-V) includes an analytic
method of selecting trial attribute vectors. When
comparisons with a subjective-graphical standard solution
were made, it was concluded that the MTAP solution was
most congruent to the graphic solution and quartlmax least
If simple structure patterning is permitted, all three of
the procedures satisfactorily matched the graphic standard
CHAPTER V
METHODOLOGY AND RESUMS
The basic objective of this study was to observe
fluctuation characteristics of rotated sample factor load
ings about their population values. The particular mode of
attack to this problem was ohosen because It provided for
the Interaction of the unique portion of the linear factor
model as an additional source of variation due to sampling.
In theory the unique and oommon components of the factor
equation are defined to be orthogonal. But In practice,
we are never able to Isolate them. Both common and unique
components, through random intercorrelations resulting from
sampling variability, could quite possibly influence fac
tor solutions by affecting the obtained experimental cor
relation matrices and entering into the estimate of
communality•
As Indicated In the first chapter, the effects of
at least three relevant Independent considerations are of
praotioal concern to those who use factor analysis for ex
planatory purposes. These considerations Involve questions
relating to complexity of factor pattern, mode of trans
formation of principal axes, and sample size. The results
of this study hopefully provided at least some partial
answers to these questions.
47
Description of the Experimental Variables
The four types of population factor pattern used in
this study were constructed so as to observe a continuum
of complexity. This was done in order to examine the
stability in behavior of results, with respect to the
simple structure concept throughout its domain of defini
tion. The factor patterns were based upon 12 variables
k * •
and four common factors and can be described as follows:
(1) Univocal tests, strong simple structure (U).
The designation "strong" is defined to mean that each
column of the pattern has at least one more zero loading
than is necessary for overdetermination of the hyperplane,
i.e., with k factors, k + 1 zero level loadings would be
needed. A univocal test is one which is loaded on only one
factor.
(2) Complex tests, strong simple structure (G).
(3) Complex tests, weaker simple structure (CC).
A weaker simple structure is defined in the context of this
study as one in which some of the hyperplanes are defined
by only k zero level loadings.
(4) No simple structure (D). This represents a
pattern in which some of the hyperplanes are underdeter-
mlned, i.e., less than k zero level loadings in at least
one column.
49
Table (l) gives the four patterns with the unique
loadings Included. Table (2) presents the population
correlation matrices reproduced from the patterns.
The four rotational methods examined for stability
characteristics were described in Chapter IV. They are
listed below with an identification letter for ease of
description in tables and later discussions.
1. Equam&x (E)
2. Varimax (V)
3. Biquartimln (B)
4. Procrustes (P)
Tables 3-6 give the modified population pattern positions
of the four common factors when rotating by each of the
methods following reproduction of the population correla
tion matrix and extraction of principle factors. These
modified positions were not identical to those of the
population patterns, but were very similar.
The two sample sizes are given as and Ng. They
represent values of 100 and 400, respectively.
Generation of the Sample Correlation Matrices
To simplify the mathematical description, matrix
notation will be used. Values of subscripts will apply to
the actual dimensions of parameters in this study. As pre
viously mentioned, factors in the population are assumed to
have unit variance, zero means, and zero intercorrelations.
TABLE 1
Contrived Population Factor Patterns with 12 Tests,
4 Common Factors, and 12 Unique Factorsa
(U) Unique Tests, Strong Simple Structure (C) Complex Tests, Strong Simple Structure
Factor I II in IV
Uj
Factor I n in rv
Uj
Test 1 . 80 .60 Test 1 . 80 . 50 . 30 . 14
2 . 92 .40 2 . 92 . 20 . 33
3 . 50 #
00
-J
3 . 50 . 87
4 . 71 . 71 4 . 71 . 71
5 . 55 . 84 5 . 20 . 55 . 35 . 73
6 . 40 . 92 6 .40 . 30 . 86
7 . 84 . 55 7 . 35 . 84 . 30 . 28
8 . 45 . 89 8 . 35 . 45 . 82
9 . 25 . 97 9 . 25 . 97
10
. 35 . 94
10 . 30 . 35 . 89
11 .40 . 92 11 . 45 . 20 . 40 . 77
12 . 20 . 98 12 . 60 . 80
a The unique factors U., are presented in columnar rather than diagonal form. Since j=l, . . . , 12,
each matrix is of order 12X161 ? ui
o
Zero values are represented as blanks in these matrices.
TABLE 1--C ontinued
(CC) Complex Tests, Weaker Simple Structure (D) No Simple Structure
Factor I II in IV
Uj
Factor I n in IV
Uj
Test 1 . 80 50 .30
. 14 Test 1 .80 . 50 . 30 . 14
2 . 92 . 20 . 33 2 . 92 .20 . 33
3 . 50 . 87 3 . 50 . 87
4 • 71 . 20 . 68 4 . 71 .20
.19 . 65
5 . 20 55 . 35 . 73 5 . 20 . 55 . 25 . 35 .69
6 •
40 . 16 . 30 . 85 6 . 20 .40 .16 . 32 . 82
7 • 35 . 84 . 30 . 28 7 . 35 . 84 . 30 . 28
8 . 35 30 . 45 . 76 8 . 35 . 30 . 45 . 22 . 73
9 •24 . 25 . 94 9 . 24 . 25 . 23 . 91
10 . 30 . 25 . 35 .85
10 . 30 .16 . 25 . 34 . 84
11 . 45 . 20 . 40 . 77
11
. 45 .15 . 20 . 41 .75
12 . 21 . 60 . 77 12 . 21 . 60 . 77
TABLE 2
POPULATION CORRELATIONS MATRICES REPRODUCED FROM THE POPULATION
FACTOR PATTERNS WITH THE EXACT COMMUNALITIES (h2) IN THE DIAGONALS
(U) Unique Tests, Strong Simple Structure
Test 1 . 64 . 74 . 40 .00 .00 . 00 .00 .00 . 00 .00 .00 .00
2 . 74 . 84 . 46 .00 .00 . 00 . 00 .00 . 00 .00 .00 .00
3 .40 .4 6 . 25 . 00 . 00 . 00 . 00 .00 .00 .00 . 00 . 00
4 . 00 .00 . 00 . 50 . 39 . 28 . 00 .00 .00 .00 . 00 .00
5 . 00 . 00 . 00 . 39 . 30 . 22 .00 .00 .00 .00 . 00 .00
6 .00 .00 . 00 . 28 . 22 .16 . 00 .00 .00 . 00 .00 .00
7 . 00 . 00 . 00 .00 . 00 . 00 . 70 . 38 . 21 .00 . 00 .00
8 . 00 . 00 . 00 .00 .00 .00 . 38 . 20 .11 . 00 . 00 .00
9
.00 . 00 . 00 .00 . 00 . 00 . 21 . 11 .06 . 00 .00 .00
10 . 00 .00 . 00 . 00 . 00 . 00 .00 .00 . 00 . 12 . 14 . 07
11 . 00 .00 . 00 . 00 .00 . 00 . 00 .00 .00 . 14 .16 .08
12 .00 .00 . 00 .00 . 00 .00 . 00 .00 .00
. 07 . 08 .04
(C) Complex Tests, Strong Simple Structure
Test 1 . 98 . 74 . 40 . 36 . 54 . 29. . 27 . 28
.00 . 35 . 48 . 18
2 . 74 . 89 . 46 . 00 .18 .00 . 17 . 41 . 05 . 28 . 45 .00
3 . 40 . 46 . 25 . 00 . 10 . 00 .00 .18 .00 .15 . 23 .00
4 . 36 . 00 . 00 . 50 . 39 . 28 . 25 .00 .00 .00 . 00 . 00
5 . 54 . 18 . 10 . 39 . 47 . 33 . 30 . 07
.00
. 18 . 23 . 21
6 . 29 . 00 . 00 . 28 . 33 . 25 . 23 . 00 .00
. 11 . 12 . 18
7 . 27 . 17 . 00 . 25 . 30 . 23 . 92 . 38 . 21 . 11 . 29 .18
8 . 28 . 41 . 18
.00 . 07 . 00 . 38 . 33 .11 .11 . 25 .00
9 . 00 . 05 .00 .00 . 00 . 00 . 21 .11 .06 . 00 . 05 . 00
10
. 35 . 28 . 15
.00 . 18 . 11 .11 . 11
.00 . 21 . 28 . 21
11 . 48 .45 . 23 .00 . 23 . 12 . 29 . 25 .05 . 28 .40 . 24
12 .18 .00 . 00 .00 . 21 . 18 . 18 .00 .00 . 21 . 24 . 36
W
TABLE 2--C ontinued
(CC) Complex Tests, Weaker Simple Structure
Test 1 . 98 . 74 . 40 . 42 . 54 . 29 . 27 . 43 .12 . 35 . 48 . 35
2 . 74 . 89 .48 . 00
. 18 .03 . 17 .41 .05 . 33 . 45 .19
3 .40 . 48 . 25 .00 . 10 .00 .00 .18 .00 .15 . 23 .11
4 . 42 . 00 . 00 . 54 .46 . 34 . 31 . 21 . 17 . 07 . 08 .12
5 . 54 . 18 . 10 . 46 .47 . 33 . 30 . 24 .13 . 18 . 23 . 25
6 . 29 . 03 .00 . 34 . 33 . 28 . 36 . 19 .1 4 . 15 .15 .18
7 . 27 . 17 . 00 . 31 . 30 .36 . 92 .48 .29 . 32 . 29 . 18
8 . 43 . 41 . 18 . 21 . 24 .19 . 48 . 42 .18 . 22 . 25 . 07
9 . 12 . 05 .00 . 17 .13 . 14 . 29 . 18 .12 . 06 .05 .00
10 . 35 . 33 . 15 . 07 . 18 .15 . 32 . 22 .06 . 28 . 33 .27
11 . 48 . 45 . 23 . 08 . 23 .15 . 29 .25 . 05 . 33 .40 . 33
12 . 35 .19 .11 . 12 . 25 . 18 . 18 . 07 .00 . 27 . 33 .40
(D) No Simple Structure
Test 1 . 98 . 74 . 40 .41 . 54 . 46 . 27 . 50 .19 .42 . 56 . 35
2 . 74 . 89 . 46 . 04 . 23 . 22 . 17 .41 . 05 . 33 . 45 .19
3 . 40 . 46 . 25 .00 .10 . 10 . 00 .18 .00 .15 . 23 .11
4 . 41 .04 . 00 . 58 . 51 . 38 . 47 . 34 . 26 . 23 . 22 .11
5 . 54 . 23 . 10 . 51 . 53 . 41 . 51 .42 . 28 . 33 . 37 . 25
6 . 46 . 22 . 10 . 38 . 41 . 33 . 37 . 33 . 21 . 27 . 31 . 23
7 . 27 . 17 . 00 .47 . 51 . 37 . 92 . 55 . 36 . 37 . 34 . 18
8 . 50 .41 . 18 . 34 . 42 . 33 . 55 .46 . 24 . 34 . 38 . 21
9 .19 . 05 . 00 . 26 . 28 . 21 . 36 . 24 . 17 .18 . 18 . 14
10 . 42 . 33 . 15 . 23 . 33 . 27 . 37 . 34 . 18 . 29 . 35 . 27
11 . 56 . 45 . 23 . 22 . 37 . 31 . 34 . 38 . 18 . 35 . 43 . 34
12 . 35 . 19 . 11 . 11 . 25 . 23 . 18 . 21 . 14 . 27 . 34 .40
V J l
V > l
T A B L E 3
5*
EQUAMAX "ADJUSTED" VALUES FOLLOWING ROTATION
OF THE POPULATION COMMON FACTOR PATTERN®
Strong Simple Structure
(U) Univocal Teats
Strong Simple Structure
(C) Complex T ests
Factor I II i n IV Factor I n i n IV
T est 1 . 802 . 000 . 000 . 000 Test 1 . 763 . 561 . 044 . 290
2 . 918 . 000 . 000 . 000 2 . 918 . 005 . 195 . 089
3 . 500 . 000 . 000 . 000 3 . 498 . 009 . 000 . 054
4 . 000 . 707 . 000 . 000 4 . 000
. 695 . 048 - . 122
5 . 000 . 550 . 000 . 000 5 . 157 . 608 . 049 . 269
6 . 000 . 398 . 000 . 000 6 - . 032 . 442 . 037 . 229
7 . 000 . 000 . 838 . 000 7 - . 024 . 337 . 873 . 210
8 . 000 . 000 . 451 . 000 8 . 352 - . 026 . 450 . 026
9 . 000 . 000 . 249
. 000 9 . 001 -
. 017 . 247 - . 007
10 . 000 . 000 . 000 . 348
10 . 264 .0 6 6 . 018 . 374
11 . 000 . 000 . 000 . 401 11 . 405 . 062 . 216 . 433
12 . 000 . 000 . 000 . 200 12 . 063 . 101 . 024 . 587
(CC) Weaker Simple Structure (D) No Simple Structure
Complex T ests
F a c to r I n i n IV F a c to r I n i n IV
T e s t 1 . 735 . 564 . 106 . 333 T e st 1 . 729 . 549 .068 . 377
2 . 902 037 .150 . 236 2 . 916 . 006 . 167 . 151
3 . 495 . 006 020 . 117 3 . 488 . 012 016 . 105
4 003 . 705 . 207 -. 001 4 022 . 704 . 286 . 056
5 . 143 . 613 . 145 . 224 5 . 141 . 567 . 341 . 271
6 044 . 411 . 256 . 197 6 . 150 . 428 . 231 . 264
7 037 . 175 . 888 . 312 7 005 . 289 . 904 . 141
8 . 364 . 1 56 . 504 . 065 8 . 327 . 279 . 490 . 192
9 . 014 . 150 . 306 -. 024 9 028 . 237 . 299 . 163
10
. 218 . 039 . 211 . 432 10
. 244 . 179 . 301 . 335
11 . 350 . 073 . 153 . 498 11 . 373 . 191 . 248 . 441
12 .06 9 . 184 044 . 601 12 . 093 . 080 . 078 . 617
The "adjusted" positions are obtained by equamax rotation
subsequent to principal component extraction from the population
correlation m atrices with the exact hr in the diagonals. In Tables 4,
5, and 6 the "adjusted" positions are obtained by the varim ax,
biquartimin, and P rocru stes methods, respectively.
55
T A B L E 4
VARIMAX "ADJUSTED" VALUES FOLLOWING ROTATION
OF THE POPULATION COMMON FACTOR PATTERN
(U) Univocal T ests (C) Com plex T ests
Strong Simple Structure
F actor I II III IV F actor I II III IV
T est 1 . 802 . 000 . 000 . 000 T est 1 . 784 . 560 . 023 . 232
2 . 918 . 000 . 000 . 000 2 . 927 . 000 . 170 . 033
3 . 500 . 000 . 000 . 000 3 . 500 . 006 -. 013 . 023
4 . 000 . 707 . 000 . 000 4 001 .6 9 3 .0 4 6 135
5 . 000 . 550 . 000 . 000 5 . 178 . 611 . 043 . 248
6 . 000 . 398 . 000 . 000 6 014 . 446 . 037 . 222
7 . 000 . 000 . 838 . 000 7 . 014 . 342 . 872 . 204
8 . 000 . 000 . 451 . 000 8 . 364 027 . 441 . 004
9 . 000 . 000 . 249 . 000 9 . 007 017 . 247 007
10 . 000 . 000 . 000 . 348 10 . 287 . 071 . 001 . 356
11 . 000 . 000 . 000 . 401 11 . 437 . 067 . 205 . 406
12 . 000 . 000 . 000 . 200 12 026 .111 . 025 . 588
(CC) Com plex T ests (D) No Sim ple Structure
Weaker Simple Structure
F actor I H IH IV F actor I II IH IV
Teat 1 . 748 . 566 . 095 . 303 T est 1 . 754 . 558 . 045 . 314
2 . 911 - . 036 . 144 . 205 2 . 924 . 009 . 164 . 100
3 . 498 . 006 - . 024 . 101 3 . 493 . 009 - . 017 . 080
4 . 000 . 707 . 199 - . 008 4 . 010 . 719 . 253 . 019
5 . 153 . 616 . 1 38 . 214 5 . 163 . 592 . 318 . 230
6 034 . 416 . 254 . 194 6 . 169 . 448 . 21 5 . 231
7 . 018 . 187 . 888 . 306 7 . 011 . 338 . 892 . 104
8 . 371 . 162 . 499 . 048 8 . 343 . 306 . 477 . 150
9 . 016 . 154 . 304 - . 028 9 - . 01 5 . 258 . 291 . 146
10 . 235 . 044 . 211 . 423 1 0 . 264 . 205 . 299 . 306
11 . 368 . 077 . 152 . 484 11 . 399 . 218 . 246 . 406
12 . 089 . 187 - . 042 . 597 12 . 125 . 109 . 089 . 605
56
TA B L E 5
BIQUARTIMIN "ADJUSTED" VALUES FOLLOWING ROTATION
OF THE POPULATION COMMON FACTOR PATTERN
(U) Univocal Test
(C) Complex Tests
Strong Simpl e Structure Strong Simple Structure
F actor I II HI IV Factor I II HI IV
T est 1 . 802 . 000 . 000 . 000 T est 1 . 750 . 511 . 045 . 108
2 .918 . 000 . 000 . 000 2 . 909 . 035 . 109 . 051
3 . 500 . 000 . 000 . 000 3 . 497 . 011 . 046 . 019
4 . 000 . 707 . 000 . 000 4 . 013 . 702 .037 . 205
5 . 000 . 550 . 000 . 000 5 . 143 . 578 .01 3 . 168
6 . 000 . 398 . 000 . 000 6 . 042 . 421 . 022 . 175
7 . 000 . 000 . 8 38 . 000 7 . 064 . 301 . 857 . 136
8
. 000 . 000 . 451 . 000 8 . 335 . 048 .4 1 6 . 039
9 . 000 . 000 . 249 . 000 9 . 008 . 022 . 246 . 015
10 . 000 . 000 . 000 . 348 10 . 260 . 027 . 019 . 321
11 . 000 . 000 . 000 . 401 11 . 393 . 010 .1 6 2 . 352
1 2
. 000 . 000 . 000 . 200 12 . 069 . 054 . 006 . 572
(CC) Complex Tests (D) No Simple Structure
Weaker Simple Structure
F actor I H HI IV F actor I H HI IV
Test 1 675 . 498 .046 . 152 T est 1 . 675 . 498 . 046 . 152
2 868 . 096 . 088 . 074 2 .868 .096 . 088 . 074
3 482 . 013 . 056 . 036 3 .4 8 2 .013 . 056 . 036
4 . 039 . 671 .089 . 076 4 .03 9 .671 . 089 . 076
5 . 092 . 565 . 026 . 134 5 . 092 .565 . 026 . 134
6 . 090 . 359 . 181 . 144 6 .09 0 .359 . 181 . 144
7 122 . 037 . 836 . 216 7 .1 2 2 .037 . 836 . 216
8 322 . 079 . 445 . 054 8 . 322 .079 . 445 . 054
9 . 007 .113 . 277 . 066 9 .007 .11 3 . 277 . 066
10 . 165 . 030 . 172 . 366 10 .165 .030 . 172 . 366
11 293 . 000
.0 9 9 . 411 11 . 293 .00 0 . 099 . 411
12 . 012 . 1 31 . 098 . 567 1 2 .012 .131 . 098 . 567
57
TA BL E 6
PROCRUSTES "ADJUSTED" VALUES FOLLOWING ROTATION
OF THE POPULATION COMMON FACTOR PATTERN
ate— ^ ■—
(U) Univocal Teste (CC) Complex T ests
Strong Simple Structure Weaker Simple Structure
F actor I II i n IV Factor I U HI IV
T est 1 802 . 000 . 000 . 000 T est 1 800 500 001 305
2 918 . 000 . 000 . 000 2 924 000 197 -002
3 500 . 000 . 000 . 000 3 509 -004 002 001
4 . 000 707 . 000 . 000 4 004 707 001 196
5 . 000 550 . 000 . 000 5 199 553 001 349
6 . 000 398 . 000 . 000 6 001 400 157 300
7 . 000 . 000 838 . 000 7 001 349
840 301
8 . 000 . 000 451 . 000 8 350 301 449 -003
9 . ooo . 000 249 . 000 9 -002 239 246 002
10 . 000 . 000 . 000 348 10 303 -001 254 352
11 . 000 . 000 . 000 401 11 450 002 201 396
12
. 000 . 000 . 000 200 12 212 -001 -005 598
. 000 . 000 . 000 . 000
(C) Complex T ests (D) No Simple Structure
Strong Simple Structure
Factor I H m IV F actor I n in IV
T est 1 801 502 001
299 T e st 1 802 498 005 302
2 921 000 197 -001 2 921 005 199 -007
3 501 -002 001 002 3 500 -003 -001 007
4 002 707 002 -001 4 000 713 197 187
5 197 553 -001 352 5 197 554 250 353
6 001 397 -002 303 6 203 407 158 312
7 001 351 840 299 7 001 352 842 298
8 350 -002 452 002 8 350 296 448 225
9 001 000 248 000
9 001
239 248 234
10 303 001 004 350 10
299 159 252 340
11 449 -001 200 401 11 449 152 193 411
12 001 -001 -001
599 1 2 211 000 002 597
58
First, we define the following:
1. The population pattern matrix A^j, i-1, . . . ,
12 (variables); J-l, . . ., 16 < * + common factors plus 12
unique factors).
2. A symmetric matrix,Z^j of order 16 containing
random normal deviates. Hie off-diagonal elements z±j,
i * J, of this matrix are randomly selected, and unities
are placed in the diagonal.
3. A sample factor intercorrelation matrix, R( )j*
(symmetric) of order l6, where R (f) j * k - 1, 2.
4- . A diagonal matrix of normal random deviates,
Sj, of order l6, where the diagonal elements Sj are
randomly selected.
5. A symmetric sample factor standard deviation
matrix vf(j)# where Vf (j j * Ij + where I is an
identity matrix of order 16. Hie value Sj/\J2N^ is taken
to represent the standard error of a (factor) standard
deviation. This formulation of v(f)j allows the desired
effect of a small amount of variability about an assumed
population standard deviation of unity.
6. The sample factor variance-covariance matrix,
Cf(j) of order 16, where C(f)j- V(f)J R(f)j v'r)J
It should be noted that the sampling procedure from
a random normal deviate table was used to produce 20 Zj
matrices. Hiis set of matrices was then differentially
59
weighted as a function of sample sizes of 100 and 400.
Thus two sets of 20 c(f)j matrices were formed.
7. The sample variable covariance matrix of order
12 is then given: m Aijc(f)jAij
8. Finally, the sample variable correlation matrix,
R^, is given by: R^ = where D^ 1/2 is the
inverse of the square root of the diagonal elements of C^.
These operations provided 160 sample correlation
matrices, of order 12 for each combination of pattern and
sample size.
A computer program was written for the IBM 7094 to
obtain the samples. The sampling procedure required the
input of a table of normal random deviates by means of the
program (Churchman, et al., 1957)*
Method of Factoring
The first step in a multiple factor analysis is the
linear transformation of a correlation matrix, R, to an
orthogonal basis in which the order of R is generally
reduoed, i.e. the extraction of factors* The columns of
the new matrix F, represent the common factors inherent in
the variables of the correlation matrix. Each diagonal
2
cell value of R is the communality, h^, and is defined as
the proportion of common factor variance in the ith
variable•
60
2
In empirical research, h is not known in advance,
and some estimate of its value must be provided by the
factor analyst. The means of estimating this value has
caused much controversy in psychometrics. It can be
seen readily that the number of factors obtained is quite
dependent upon this estimate. This issue was not under
Investigation here. As it was known that the rank of the
population correlation matrices was always four, only four
factors in each sample were rotated even though there were
a number of cases in which sizable variance remained in a
fifth factor, e.g., sometimes 10% or more of the total
2
trace, sh.^ .
The principal-factor method is the extraction pro
cedure used in this study. It is theoretically most
palatable to mathematicians, and with electronic computers
is no more difficult to obtain than a centroid solution.
The latter was in vogue until recently and has served as an
approximation for the principal-factors method. The basis
for the principal-factors technique Involves selection of
a set of principal axes of ellipsoids formed by the point
representation of a set of variables in the factor space.
These axes are the representations of the common factors.
They are selected so that their contribution to the total
61
communallty will be Inversely related to the order of
extraction. This procedure continues until no portion of
the total communallty remains. Essentially, the solution
Involves finding the eigenvalues and eigenvectors of a
characteristic equation.
2
The squared multiple correlation, R , of each
variable with the other variables in the population factor
2
pattern was used as a communallty estimate. R has many
important theoretical properties including that of being
the lower bound for the actual communallty. It is com
pletely objective and is easily obtained on any electronic
computer capable of obtaining the inverse of a fairly
large matrix of intercorrelations. This study required
the modification of an extant principal components program
O
(BIMD computer program manual, 1961) so that R could be
computed from the sample correlation matrices that emerged,
with unities in the diagonal.
Rotational Procedures
Existing programs for the varimax, blquartlmln, and
Procrustes methods (BIMD computer program manual, 19&1,
Carroll, 1958; Miller, 1962) were utilized with necessary
modifications added when needed for this study. An equamax
program was obtained by modifying the algebraic statements
for the varimax criterion.
62
A problem arose with the output of these programs.
The final order of the factors was not always consistent
with their original position in population sequence. A
similar problem existed with sign changes as reference
axes were frequently rotated l80° from their actual maxi
mized positions in the convergence process. In many cases
this required either a reordering to the population
sequence, a reflection of signs for a factor or both, after
inspection of the rotated matrices. Factor matching to the
population pattern usually did not present a problem.
However, in a relatively few instances two factors in a
sample matrix would become difficult to differentiate,
and in a few cases factors became bipolar. The subjective
pairing might account for some inordinately large ceil
range and RMS values, although it was felt that a "best"
match was always obtained.
Description of the Stability Criteria
Subsequent to rotations by each of the analytic and
semi-analytic rotational methods, the matrices were
collapsed into sets of 20. There were 32 of these sets,
each constituting a rotational solution within a type of
simple structure within a sample size. These sets will be
referred to as summary matrices. Each set had 4-8 cells
and 20 sample values within each cell. The latter pro
vided the raw data for the summary sample statistics, mean.
63
range, and RMS for each cell, and for within cell frequency
distributions. A computer program was written to generate
these statistics and distributions. The criteria may be
described as follows:
Sample matrix, A^j^, representing the 1th rotation
al method, jth population pattern, and kth sample size.
The operation of Joining, J, or collapsing the 20 samples,
20
J [ An(Uk)|
The sample mean of cell m (where subscripts ijk
have been dropped for notational convenience),
20 a
gl j r"" * 1' * * **^8)
n - 1
Range of cell m, Rm, as determined by the differ
ence between the highest and lowest value for cell m for
the 20 sample values.
Root mean square of cell mi, RMSmi, from the
population position as adjusted by rotation of the popula
tion pattern by the 1th rotational method, where
aiJml " ami \
RMS
_ ^ ^ amnl^~ ami^
mi
n * 1
and is the sample value, and am^ the "adjusted”
population value.
The appendix contains the summary matrices for all
treatment conditions.
64
Procedures for Analysis of the
------Stability Criteria------
The data that resulted from the sampling procedure
are presented In tabular form. Because the method of
generating the summary matrices and the form of these
matrices were not independent from condition to condition,
hypothesis testing assumptions were not tenable for test
ing of differences between matrices. The only randomness
that existed was within the cells of a particular summary
matrix, but not between cells of different summary matrices
or the cells within Individual summary matrices. The
possibility of testing the deviation of cell means from
their population values for statistical significance and
tabulating the frequency of significant differences for
each summary matrix was considered. But even with this
type of relative frequency analysis no true inferential
statistical statements could be made. This fact becomes
apparent when one considers that there were only two sets
of random factor variance-covariance matrixes, Cf(j)>
generated, one set for each sample size. This would again
obviate a classical statistical analysis of the effects of
different factor patterns and rotational methods. There
fore, a more direct descriptive approach as given in (4),
(5), and (6) below was considered as more appropriate for
this study.
65
The following conditions were evaluated:
(1) Mean summary matrix values (Tables 7-8).
Each summary matrix contains information relating
to the general stability of the sample factor loadings.
Table 7 uses the root mean square (RMS) as the criterion
measure for total variability of the cell factor loadings;
Table 8 uses the range (R) of the cell factor loadings as
an estimate of variance. The mean value for a particular
matrix is the criterion index averaged over 48 cells (four
factors x 12 variables); i.e.,
Since there are four rotational methods, four population
factor patterns, and two sample sizes, each criterion
measure has 52 entries in its table.
(2) Mean matrix values by population interval
(Tables 9-10).
Each of the summary matrices is broken down into
intervals of population factor loadings interval size =.10.
The average (mean) of the criterion values that fall into
each population interval was computed. This was done
for large and small sample size and for both RMS (Table 9)
and range (Table 10) for all treatment conditions.
(5) Mean summary matrix values by population com-
munality (Tables 11-12)
66
T A B L E 7
MEAN SUMMARY MATRIX VALUESa , WITH THE ROOT
MEAN SQUARE AS A CRITERION OF STABILITY FOR
THE SAMPLE FACTOR LOADINGS ABOUT
THEIR POPULATION POSITION
Small Sample Size
Rotation Method
E V B P
c
In
I D
ts
n )
O h
u i
c i
. 126
. 138
. 126
. 148
. 130
. 136
.118
.115
O
, p 4
c t f
ft
0
&
CCj
D 1
. 133
. 144
. 138
. 142
. 140
. 152
.113
.113
Large Sample Size
Rotation Method
E V B P
d
4)
U2
. 068 . 068 . 066 . 061
nl
PU
C 2
. 092 . 095 . 094 . 079
c
o
•p4
ft
o
d i
c c 2
D 2
. 094
. 067
. 098
. 092
. 097
. 095
. 077
. 057
aA sum m ary m atrix contains the 48 values (12 tests x 4 factors)
for each of the experim ental conditions. These 48 values w ere then
averaged to give the entries in tables 7 and 8.
T A B L E 8
67
MEAN SUMMARY MATRIX VALUES, WITH THE RANGE
AS AN ESTIMATE OF VARIANCE FOR THE SAMPLE
FACTOR LOADINGS ABOUT THEIR POPULATION POSITION
Small Sample Size
Rotation Method
E V B P
G U. .470 .472 .480 .415
I * A
0 )
+ - >
£ Cx .435 .4 8 4 .442 .373
G
o
5 CC, .385 .413 .4 4 2 .332
■3 1
a
o
ft Dx .5 1 6 .5 2 4 .530 .35 4
Large Sample Size
Rotation Method
E V B P
G U , .221 .221 .2 1 2 .187
H C
CD
c
C2 .200 .208 .2 4 4 .192
G
O
rt C C , .1 85 .1 9 4 .1 9 6 .1 5 4
a
o
ft D 2 .212 .315 .310 .163
TABLE 9. MEAN RMS VALUE BY POPULATION FACTOR LOADING
INTERVALS OF . 10 FOR EACH SUMMARY MATRIX
Population
Loading UE1 UV1 UB1
Small Sample Size
UP1 CE1 CV1 CB1 CPI
Inter vai
. 91-1.00 .114 .114 .116 . 132 .074 . 074 . 076 . 102
.8 1 -. 90 . 349 . 344 . 349 . 314 . 394 .428 . 385 . 274
. 71-. 80 . 103 . 104 .111 . I l l .068 . 070 .065 .069
. 61-. 70
. 51-. 60 .117 . 118 . 121 . 122 . 100 .136 .103 . 104
.4 1 -. 50 . 146 . 144 . 147 .094 .134 . 147 . 145 .090
. 31-. 40 . 104 . 140 . 170 .126 .133 . 147 . 158 . 107
. 21-. 30 . 178 . 182 . 193 . 152 .153 .161 .151 .118
.1 1 -. 20 . 147
.150 . 143 . 139 .128 .127 .116 . 102
.0 0 -. 10 .126 .119 .119 . I l l .139 . 147 .129 .118
Population
Loading CCE1 CCV1 CCB1 CCP1 DEI DV1 DB1 DPI Mean
Interval
.9 1 -1 .0 0 .112 . 115
.150 .092 . 196 .124 . 176 .094 . 116
.8 1 -. 90 . 257 . 277 . 349 . 262 . 294 . 357 .371 . 174 . 323
. 71-. 80 . 074 . 088 . 106 . 073 . 170 .122 . 130 .062 .095
. 61-. 70
. 51-. 60 .094 .098 . 116 .099 . 173 .169 . 216 .110 .125
.4 1 -. 50 .125 . 131 . 146 . 082 .129 . 149 .162 . 188 . 128
. 31-. 40 . 127 . 135 . 132 . 107 .156 .156 .156 .131 .131
. 21-. 30 . 173 . 182 . 173
.130 .152 .153 .167 .127 .159
. 11-. 20 .150 . 148 . 137 .128 .127 .126 .124 .114 .132
. 00-. 10 . 117
.120 .116 .098 . 125 .115 .121 . 105
. 120
a The value in this colum n is the m ean for the six teen exp erim en tal conditions for a
given population loading in terval.
c r \
00
TABLE 9. — Continued
Large Sample Size
Population
Loading UE2 UY2 UB2 UP 2 CE2 CY2 CB2 CP2
Interval
. 91-1.00 . 101 . 101 . 103 . 113 . 056 .056 . 055 .098
.8 1 -.9 0 . 297 . 297 . 298 . 303 . 334 . 338 . 313 . 278
. 71-. 80 . 085 . 085 . 086 .094 . 047 . 051 . 044 . 076
. 61-. 70
. 51-. 60 . 045 .045 . 046 .052 .070 .072 . 087 .099
.4 1 -. 50 . 047 . 047 . 047 . 047
.060 .061 .079 .068
. 31-.40 .077 . 077 . 078 .080 .090 .092 .110 .065
. 21-. 30 . 076 . 076 . 075 . 072 . 084 .090 .090 .064
. 11-. 20
.101 . 102 . 101 .093 . 087 .083 .084 .060
.0 0 -. 10 .060 .059 . 056 .049 .094 . 101 .092 . 088
Population
Loading CCE2 CCV2 CCB2 CCP2 DE2 DV2 DB2 DP 2 Mean'
Interval
. 91-1.00 .093 . 095 . 142 . 089 . 102 . 100 . 133 .091 . 095
. 81-. 90 . 238 . 239 . 297 . 284 . 216 . 272 . 286 j . 240 . 283
.7 1 -.8 0
.049
. 054 . 101 . 069 . 054 .092 . 105 . 063 . 072
.6 1 -.7 0
. 51-. 60 .060 .063 . 088 . 068 . 088 . 128 . 142 .099 . 078
.4 1 -. 50 .076 . 081 .091 . 056 . 067 . 078 .084 . 047 .065
. 31-. 40 .081 . 087 . 109 . 072 . 071 . 105 . 114 . 055 .085
. 21-. 30 . 128 .133 .114 . I l l .063 . 094 .096 . 052 .089
. 11-. 20 . 114 .114 . 101 . 087 .059 .095 .093 . 050 .089
. 00-. 10 .081 . 084 . 072 .053 . 058 .063 .058 .048 .069
a The value in this colum n is the m ean for the six teen exp erim en tal conditions for a
given population loading in terval.
TABLE 10. MEAN RANGE VALUE BY POPULATION FACTOR LOADING
INTERVALS OF . 10 FOR EACH SUMMARY MATRIX
Small Sample Size
Population
Loading VE1 UV1 UB1 UP1
Interval
. 91-1.00 . 173 . 172 . 170 . 183
, 81-. 90 . 775 . 741 . 740 .466
. 71-. 80 . 241 . 238 . 269 . 258
. 61-. 70
. 51-. 60 . 528 . 537 . 522 . 499
.4 1 -. 50 . 375 . 562 . 582 . 308
. 31-. 40 . 432 . 446 . 615 .413
. 21-. 30 . 518 . 575 . 603 , 469
. U - . 20 . 549 . 528 . 505 .456
. 00-. 10 .475 . 479 . 470 . 432
Population
Loading CCE1 CCV1 CCB1 CCP1
Interval
.9 1 -1 .0 0 . 234 . 259 . 323 . 182
. 81-. 90 . 383 .432 . 538 . 286
. 71 -.80 . 209 . 205 . 405 . 238
. 61-. 70
. 51-. 60 . 336 . 385 .406 . 319
.4 1 -. 50 .427 .462 . 476 . 268
. 31-.40 . 376 . 413 . 458 . 341
. 21-. 30 .412 . 446 .490 . 383
. 11-. 20 . 385 . 386 . 362 . 335
.0 0 -. 10 . 399 . 434 . 454 . 338
CE1 CV1 CB1 CPI
. 212 . 223 . 281 . 210
. 910 .911 . 900 . 442
. 209 . 215 . 229 . 268
. 322 . 536 . 337 . 350
.452 .476 . 484 . 312
. 427 .486 .460 . 348
. 557 . 602 . 564 . 429
. 504 . 507 . 482 . 389
. 418 .466 . 552 .390
DEI DV1 DB1 DPI Meai
. 523 . 258 . 537 .161 . 256
.474 . 732 . 802 . 214 . 608
. 563 .468 . 446 . 216 . 292
. 543 . 612 . 755 . 346 . 458
. 455 . 534 . 601 . 309 . 443
. 565 . 571 . 547 .416 . 457
. 578 . 553 . 577 .414 . 511
.464 . 500 . 475 . 472 . 455
. 448 . 477 .440 . 324 .436
-4
O
TABLE 10, --C ontinued
Large Sample Size
Population
Loading
Interval
VE2 UV2 UB2 UP2 CE2 CV2 CB2 CP2
. 91-1.00 .089 .089 . 088 .114 .099 .099 .161 .120
.8 1 -. 90 . 189 . 187 .191 . 199 .198 . 198 . 208 . 173
. 71-. 80
. 61-. 70
.111 .111 .119 . 140 .110 .124 .139 . 141
. 51-. 60 . 141 . 142 . 140 . 148 .155 . 148 . 261 . 223
.4 1 -. 50 .199 . 198 .199 . 195 .165 .195 . 293 . 215
.3 1 -.4 0 . 241 . 242 . 245 . 236 . 204 . 230 . 287 . 175
. 21-. 30 . 224 . 225 . 229 . 225 . 238 . 253 . 264 . 205
. 11-. 20 . 383 . 381 . 378 . 371 . 220 . 231 . 238 . 184
.0 0 -. 10 . 228 . 231 . 218 . 183 . 205 . 209 . 235 .196
Population
Loading
Interval
CCE2 CCV2 CCB2 CCP2 DE2 DV2 DB2 DP2
. 91-1.00 . 123 .115 .162 . 108 .099 . 112 . 186 .110
. 81-. 90 . 135 . 150 . 217 .155 .125 . 562 . 523 . 145
.7 1 -.8 0
. 61-. 70
. 124 . 139 . 159 .116 .160 . 272 . 352 . 115
. 51-. 60 . 141 . 142 .161 . 149 . 176 . 364 . 324 . 188
.4 1 -. 50 . 180 . 175 . 183 . 148 . 205 . 258 . 258 . 141
. 31-. 40 . 180 . 205 . 219 . 146 . 228 . 370 .400 .211
. 21-. 30 . 203 . 217 . 223 . 175 . 236 . 348 . 340 . 174
.1 1 -.2 0 . 210 . 218 . 209 .169 . 225 . 338 . 360 .168
.0 0 -. 10 . 189 . 193 . 185 . 149 .198 . 234 .199 . 140
Mean
.117
. 222
. 154
. 187
. 200
. 238
. 236
. 267
.199
TABLE 11
MEAN RMS SUMMARY MATRIX VALUES BY POPULATION COMMUNALITYa
(U) Strong Simple Structure, (C) Strong Simple Structure,
Univocal Tests Complex Tests
Rotational Method Rotational Method
h2 E V B P
Method^
Mean h2 E V B P
Method
Mean
. 84 .062 .062 . 060 .046 . 057 . 98 .054 .062 .063 .064 . 061
. 70 . 115 .115 .113 . 105 . 112 . 92 .166 .169 . 193 . 118 . 162
. 64 .052 .051 .050
.038 .048 .89 .098 .097 .086 .061 . 080
. 50 .081 .081 . 078 .068 . 077 . 54 .089
.090 . 061 . 075 . 079
. 30 .053 .053 . 051 . 044 .050 . 47 .071 .074 . 074 .075 . 074
. 25 .060 .060 .056 .060 .059 . 42 . 096 .098 . 115 .094 . 101
. 20 .061 .061 .060 . 047 .057 . 40 .080 .082 .085 . 058 . 076
.16 . 077 . 077 .076 .073 .076 . 40 .098 .097 . 097 .093 .096
.16 .061 . 061 .059 .060 .060 . 28 .117 . 122 .119 . 087 .111
. 12 .047 .048 . 047 .049 . 048 . 28 .097 . 104 .111 . 089 . 100
.06 .064 .064 .062 .066 . 064 . 25 .061 .063 . 054 . 056 . 059
.0 4 .081 .082 .079 .075 . 079 .12 .082 .081 .075 . 082 .080
aEach value given in tables 11 and 12 is the mean of criterion m easures across factors for
each variable. The totals for each rotational method for a given communality are then averaged to
give a composite row mean. Only large sample size results are given in tables 11 and 12.
no
^T his colum n g iv es the m ean value of a ll rotational m ethods for a given com m unality.
T A B L E -11 - -Continued
(CC) Weaker
Comple:
Simple Structure
s Tests
(D) No Simple Structure
Rotational Method Rotational Method
h2 E V B P
Row
Mean h2 E V B P
Row
Mean
. 98 .081 . 093 . 089 . 058 .080 . 98 . 070
•095 .099 . 047 .078
. 92 . 131 . 132 .168 .133 . 141 . 92 . 103 • 149 .160 . 083 .124
. 89 . 100 . 106 .111 . 060 .094 . 89 . 078 •079 . 084 .060 . 075
. 50 . 107 . 110 .094 .061 . 093 . 58 . 064 «119 .115 . 044 . 085
.47 . 062 .062 . 056 . 055 . 059 . 53 . 053 *107 . 103 . 050 . 078
.40 . 072 . 078 . 092 . 066 . 077 . 46 . 056 *088 . 105 . 046 . 074
. 36 .115 . 109 .115 . 106 . I l l .43 .061 •077 . 078 .049 .066
. 33 . 107 . 112 . 085 . 068 .093 . 40 . 087 075 . 072 .069 . 075
. 25 . 104 . 107 . 102 . 075 . 097 . 33 .063 091 .089 .059 . 076
. 25 . 049 . 052 .040 . 053 . 048 . 29 .067 091 . 102 .064 .081
. 21 . 106 . 115 .127 . 106 .113 . 25 . 043 044 . 038 . 054 . 045
.0 6 .097 . 097 . 086 .086 . 091 . 17 .063 089 .094 . 060 . 077
TABLE 12
MEAN RANGE SUMMARY MATRIX VALUES BY POPULATION COMMUNALITY
(U) Strong Simple Structure
Univocal Tests
(C) Strong Simple Structure
Complex Tests
Rotational Method Rotational Method
h2 E V B P
Method
Mean h2 E V B P
Method
Mean
. 84 .176 .175 . 175 .163 . 172 . 98 .169 . 179 . 188 . 148 . 171
. 70 . 184 . 183 . 183 . 176 . 182 . 92 .196 . 201 . 298 . 178 . 218
. 64 . 208 . 207 . 207 . 143 .191 .89 . 218 . 225 . 225 . 123 .198
. 50 . 227 . 226 . 226 . 213 . 223 . 54 . 183 . 186 . 149 .153 .168
. 30 . 185 . 184 . 184 . 179 . 183 .47 . 181 .194 . 253 . 181 . 202
. 25 . 257 . 256 . 256 . 216 . 246 .42 . 223 . 208 . 456 . 326 . 303
. 20 . 245 . 244 . 244 . 241 . 244 .40 . 208 . 225 .197 .161 . 198
.16 . 210 . 208 . 208 . 204 . 208 .40 . 203 . 218 . 229 . 178 . 207
.16 . 284 . 285 . 285 . 284 . 285 . 28 . 234 . 251 . 362 . 285 . 283
.12 . 183 . 182 . 182 . 182 . 182 . 28 . 202 . 207 . 215 . 186 . 203
.06 . 224 . 224 . 224 . 217 . 222 . 25 . 185 .190 . 175 . 187 . 184
. 04 . 295 . 295 . 295 . 295 . 293 .12 . 192 . 190 . 185 . 198 . 191
■P
TABLE 12--C ontinued
(CC) Weaker Simple Structure (D) No Simple Structure
Complex Tests
Rotational Method Rotational Method
h2 E V B P
Method
Mean h2 E V B P
Method
Mean
.98 .158 . 177 .196 .113 . 161 . 98 .193 . 316 . 280 . 118 . 227
.92 .190 . 214 . 231 .121 . 189 . 92 . 171 . 433 . 442 . 128 . 294
. 89 . 221 . 210 . 210 . 095 . 184 . 89 . 201 . 204 . 220 . 103 . 182
. 50 .197 . 222 . 216 .135 . 218 . 58 . 212 . 412 .426 . 134 . 296
.47 .163 . 175 .168 . 143 .162 . 53 . 188 . 396 . 371 .167 . 281
.40 .163 .171 .195 . 170 . 175 . 46 . 195 . 361 .407 .132 . 274
. 36 . 209 . 212 . 186 . 143 . 188 .43 . 233 . 281 . 260 . 179 . 238
. 33 . 170 .166 .170 .134 .160 . 40 . 269 . 216 . 174 . 116 .194
. 25 .196 . 204 . 216 .163 .195 . 33 . 218 . 322 . 318 . 200 . 265
. 25 .169 . 170 .136 . 205 . 170 . 29 . 256 . 351 . 376 . 247 . 308
. 21 . 188 . 201 . 213 . 215 . 204 . 25 . 177 . 178 .152 . 198 . 176
. 06 .192 . 198 . 223 . 205 . 205 . 17 . 227 . 306 . 305 . 404 . 311
VJI
76
Each row of the summary matrices represents a
variable with a given population communality value* The
mean of the criterion measures averaged across factors for
each variable represents an index by which effects of
communality size were investigated. Only large sample
size effects were studied.
(4) Cross-tabulation of sample means against
rotated population loadings (Table 13)*
The absolute value of the possible range of factor
loadings is between zero and unity. This range is often
broken down by factor analysts into high, medium, and low
categories, roughly corresponding to significant, non
significant, end hyperplane loadings, respectively. The
mean sample loadings for each of the conditions
were cross-tabulated against the population loadings
^aijk^ rota^ed one °f methods under study. This
gave an additional view of possible variability and the
direction it might take. The high category was given as
a^j^ s * 3^5i the medium category as .105 - aj_jic < *2^5;
and low is given as .000 £ |a
of both the small sample size and the large sample size
(Table 13) are given.
(5) Cross-tabulation of rotated population pattern
position against the population pattern position (Table 14X
To obtain an estimate of the amount of distortion
the rotational methods inflict upon the original pattern,
ijkl < Results from use
TABLE 13
THREE-CATEGORY (H, M, L)a C ROSS - TAB U LA T IONb OF MEAN SAMPLE
LOADINGS AGAINST THE POPULATION LOADINGS ADJUSTED BY ROTATION
Small Sample Size
Sample Mean Position
Structure
Pop.
Position H
E
M L H
Y
M L H
B
M L H
P • •
M L
H 9 1 0
9 1 0
9 1 0
9 1 0
U M E 1 1 0 V 1 1 I 0 B 1 1 0 P 1 1 0
L 0 0 36 0 0 36
0 0 36 0 0 36
H 14 0 0 14 0 0 11 0 0 16 0 0
c M E 1 9 1 V 1 10 1 B 3 9 1 P 3 5 0
L 0 1 22 0 0 22 0 1 23 0 7 17
H 14 0 0 14 0 0 12 0 0 14 2 0
cc M E 2 17 1 V 1 16 1 B 2 8 3 P 1 13 0
L 0 0 14 0 0 16 0 3 20 0 1 17
H 13 0 0 13 0 0 9 1 0 15 0 0
D M E 5 18 1 Y 4 18 1 B 0 15 3 P 5 17 0
L 0 3 8 0 2 10 0 5 15 0 2 9
aEach of the experimental conditions are broken down into high (H), medium (M), and low (L)
categories to roughly denote the range corresponding to significant, non-significant and hyperplane
loadings, respectively. -> J
^Table entries are frequencies.
TABLE 1 3 --Continued
Small Sample Size
Sample Mean Position
Structure
Pop.
Position H
E
M L H
V
M L H
B
M L H
P
M L
H 8 2 8 2 8 2 8 2
U M E 2 V 2 B 2 P 2
L 36 36 36 36
H 13 1 13 1 12 1 15 1
C M E 11 1 V 11 1 B 16 P 7 1
L 1 21 22 19 3 21
H 13 1 13 1 12 15 3
CC M E 1 18 1 V 1 17 B 11 2 P 14
L 14 2 14 3 20 1 15
H 13 12 8 2 14 1
D M E 2 21 1 V 24 B 15 1 P 22
L 3 8 2 10 2 20 11
co
TABLE 14
THREE-CATEGORY (H, M, L) CROSS-TABULATION OF ROTATED POPULATION
PATTERN POSITION AGAINST THE POPULATION PATTERN POSITION
Population Pattern Position
Structure
Rot.
Pattern
Position
H
E
M L H
V
M L H
B
M L H
P
M L
H 10 10 10 10
U M E 2 V 2 B 2 P 2
L 36 36 36 36
H 14 2 14 2 12 4 16
C M E 8 V 8 B 8 P 8
L 1 23 2 22 1 23 24
H 14 2 14 2 12 3 1 16
CC M E 12 2 V 12 2 B 8 6 P 14
L 6 12 4 14 2 16 18
H 12 3 12 3 10 4 1 15
D M E 1 20 1 V 20 2 B 11 11 P 22
L 1 10 1 10 1 10 11
vo
80
a cross-tabulation, using the same categories as In (4)
above, was made. The rotated pattern positions were
obtained following reproduction of the population
correlation matrix and extraction by principal components.
It was not possible to apply Bargmann'a test of
stability of simple structure. This was because the
construction of the population factor patterns with com
plexity involved giving several factors only enough zero
loadings necessary for overdetermination, I.e., four. The
null positions of the population pattern were also con
founded by rotation to the various criteria. Thus in many
Instances the rotated population hyperplanes became under-
determined. Sampling from these new positions could not
then be logically expected to meet the requirements of
Bargmann*s tests where significance at the 5<# level for
four factors and 12 variables requires at least five
loadings in the null range. It should be noted, however,
that the univocal test simple structure sample mean
positions nearly always have at least 7 loadings in the
null range for their a/h value. This would give signifi
cance at the .05 probability level.
(6) Tabulation of summary cell frequency distri
butions (Table 15)•
A rough analysis of the shape of the cell
distributions was attempted. A sample size of 20 was too
small to make accurate estimates, but it was hoped that
8l
T A B L E 15
TABULATION OF SUMMARY C E L L FREQUENCY
DISTRIBUTIONS INTO SYMMETRIC (S)
AN D ASYMMETRIC (A) CATEGORIES
Sm all Sample Size Large Sample Size
(S) Sym m etric (A) A sym m etric (S) Sym m etric (A) A sy m m etric
EU 33 15 42 6
EC 36 12 38 10
ECC 38
0 40 8
ED 33 1 5 40 8
vu 36 12 35 13
VC 35 13 32 16
vcc 29 19 31 17
VD 28
20 33 15
BU 29 19 34 14
BC 28
20 32 16
BCC 32 16 28 20
BD 14 34 30 18
P U 24 14 37 11
PC 25 23 31 17
PCC 2t- 26 31 17
P D 25 23 31 17
82
some possible leads as to the underlying theoretical
sampling distribution of factor loadings could be obtained.!
The interval size of .02 was found to be most satisfactory
for producing mesokurtic distributions after some experi
mentation with other Intervals.
A two-way classification was made with the
categories given as symmetric, S and asymmetric, A.
Ratings were made by the author without knowledge of the
treatment category of the summary matrix. However, it was
possible that many unforeseen errors of rating could have
entered into the evaluation. It was hoped that potentially
useful information might be apparent enough to override
possible bias.
CHAPTER VI
DISCUSSION OP RESUMS
(l) Complete summary matrix with the RMS as the
orlterion (Table 7).
Rotational solutions: Judging from the analytic
solutions the equamax method (E) has a decided edge In
terms of stability as compared with both the varlmax (V)
and blquartimln (B) methods. Its total RMS Indices for
each cell are smaller than those of the varlmax and bl
quartimln methods V and B In three out of four cases
(across type of pattern) for both the small and large
samples. Within the large sample (Ng) the advantage of
the equamax method Increases as the pattern becomes more
stable. This would tend to substantiate Saunder's claims
for the equamax method when the Investigator has a good
estimate of the number of factors to be rotated. In an
exploratory type of study when there is Incomplete know
ledge of the factorial domain, it is possible that the
equamax method would not out-perform the varlmax method
quite as consistently. The varlmax method Is generally
superior to the blquartimln method In the small sample
case, but they are roughly on a par with large samples.
It was expected that the blquartimln method would show an
83
m
advantage over the orthogonal methods as the structure
complexity Increased. By allowing angular flexibility In
rotation It has been claimed by some factor analysts that
simple structure and approach to satisfaction of the
rotational criteria can be better achieved. However, It
was found here that the sample primary factors tended to
over-correlate as the amount of correlation between the
population primary factors Increased, nils tended to
create additional bias by decreasing the size of the
sample factor loadings from the blquartimln method, and
thereby Increasing RMS's for the large loadings.
Hie Procrustes (P) solution was Included as a
standard with which to compare the performance of the fully
analytic solutions. Because the population pattern was
used as the hypothetical target matrix, it was not surpris
ing that the Procrustes method results were most stable in
all cases. If a less exact hypothetical matrix were the
target, e.g., a one-zero configuration, it is quite likely
the Procrustes method would not fare as well.
Type of pattern: Within small samples (M^), the
only positive conclusion was that strong simple structure
with univocal tests (U) provides opportunity for the most
stable solution. The fact that the solutions for the
better-determined pattern C were less stable than those
for pattern CC In three out of four cases (across methods)
could not be accounted for on any theoretical basis. Only
the blquartimln method behaved as expected, with stability
decreasing as the pattern became more complex.
In the large sample category (N2), the most stable
patterns were found for the U pattern as expected and for
the case of lack of simple structure with D pattern, which
was not expected. This result would lead one to believe
that instability first increases and then proceeds to
decrease with continued increase in pattern complexity.
The continuing increase of complexity of pattern would
lead to additional rank reduction of the correlation
matrix. The logical extreme, of course, would be the
convergence to a general factor G. According to Spearman's
formulations there is every reason to believe that G would
be more stable regardless of the type of invariance under
investigation.
Sample size: As expected, the larger sample size
(N2 = 4-00) showed considerably higher stability than the
smaller (N1 = 100). The square root of sample sizes
entered into the error generation program. This led to a
belief that stability measures should fluctuate about a
difference of magnitude proportional to the difference of
the square root of the sample size. It was surprising
however, that the differences were not larger. The ratios
of RMS values in and N2 ranged from about 1/2 to about
86
2/3, with a preponderance of the latter, and not of the
former value as expected. These findings would indicate
a somewhat slower convergence rate to the population posi
tion than that given as a function of the square root of
the sample Blze.
(2) Complete summary matrix with the range as the
criterion
Table (8) was constructed parallel to Table (7) but
with the average range of sample values of each cell in
stead of the RMS as a criterion measure. This value was
Intended to serve as an index for the bounds of variation.
Rotational solutions: In every case but one
(i.e., under Ug) the equamax method yields the smallest
average range of the analytic solutions. The varlmax and
blquartimln methods perform about equally well, with a
slight advantage to the varlmax method in the small sample
case.
Type of pattern: The largest ranges in the pat
terns most often occurred in U and D. This was a complete
reversal of findings with the RMS index. An explanation
„ for the result might be as follows: Consider the ex
pected value of the deviations of sample factor loadings
a* from the population position of the rotated solution
a11,E(a* al1). This expression can be separated into
the following components,
E(a^-ai) = E
87
where a * . . Is the sample mean value of the factor loading.
^ J
The second term on the right hand side la the expected
value of deviations due to sampling and the first would be
analogous to true variability. Then the observed vari
ability of the factor loadings from a population position
might be a linear additive function of sampling error and
real variance. The total expression would be a function of
what Is measured by RMS. As previously mentioned, the
unlvocal test (U) simple structure and non-simple structure
(D) patterns had smaller RMS values but larger range values
than the two complex test simple structure patterns, C
and CC. This finding then would imply that the sampling
error component is more heavily weighted than the vari
ability component In the RMS Index as used in this study.
An ad hoc explanation might be based upon the
interaction of type of simple structure with analytic
rotations. The analytic rotational methods, particularly
the orthogonal ones described in this study all tend to
maximize extreme loadings. As the U and D patterns have
more total extreme loadings, respectively, in the high
and low range than the C and CC patterns, a smaller
separation from the population positions of the former
patterns would be expected of their sample patterns
88
following rotations. Although the hypothesis might
account for the smaller RMS values of U and D, no logical
rationale could be provided for the discrepancies in
range.
Sample size: When the range was used as the
criterion, the sample size was found to affect ratios of
two to one, as originally hypothesized.
(2) Summary matrix totals by population interval
In order to evaluate the effect of size of indi
vidual population loadings upon the stability of the
sample factor loadings this analysis was performed. It
consisted of an ordering of the population loadings in
intervals of .10, and the tabulation of the mean RMS and
range values in each Interval for every summary matrix.
RMS (Table 9); Inspection of the Information in
these tables would Indicate a strong tendency to greater
stability in the Intervals less than .10 and greater than
.40. This would be in keeping with the general tendency
for investigators to rely more upon the extreme loadings
in making their interpretations. This finding generally
held within each of the experimental conditions and
became readily apparent when the average values for a
population interval were summed across all categories
within a sample size. It was found to hold for both
sample sizes.
89
An unusual finding was the very large RMS value for
the loading interval, .81 to .90. This probably has no
theoretical import, other than the fact that it is an
artifact of a gross underestimation of communality. The
particular grouping of the variables Involved in the
determination of the squared multiple correlation for this
variable led to wide deviations between that value for the
variable and Its population communality when combined with
sampling effects. The particular loading affected was
that of the seventh variable on the third factor for all
patterns (see appendix). The effect might have been some
what compensated for if there were other loadings in that
population interval with which to be averaged. Unfortu
nately this was not the case. Perhaps the general picture
of the stability of loadings greater than .50 would have
been clearer with more values to be averaged in these
upper intervals.
Range (Table 10); The same conclusion held for the
range as for the RMS. The most noticeable difference was
the more stable appearance within the interval .81 to .90.
Perhaps measures of variance of loadings are less sus
ceptible to artifacts of communality estimate and
sampling than measures of total variability.
90
(4) Summary matrix totals by population
2
communality (h )•
2
No consistent effect of h could be obtained from
analysis of Tables 11 and 12. This was true with both
RMS (Table 11) and range (Table 12). Perhaps there was
2
some tendency for the highest h to be somewhat more
stable than the lowest, as would certainly be logically
assumed. But no other consistent pattern could be
discerned. Again, the second highest communality, that of
variable (row) 7» performed strangely, although less so
in the case of range indices.
A possible explanation for the ineffectiveness of
2
h as a relevant independent variable might lie in the
normalization process that occurs for all of the rotation
al methods. For the three analytic techniques the rows of
the orthogonal unrotated factor matrix are normalized so
as to weight each variable equally in the determination of
the final position of the reference frame. After rotation
the rows are denormalized, thus restoring the original
commonalities in the principal components matrix (taking
into consideration deletion of unrotated principal
components and rounding due to approximation techniques).
The Procrustes method achieves the same result by normalis
ing the columns of the transformation matrix which fits an
91
arbitrary solution to a hypothesized one. It was expected
that the denormalization process would preserve the mono
tonic increase in variability or decrease in stability
2
that would be hypothesized as h decreased. At least in
the context of this study this was not the case.
It should be noted that this analysis was attempted
only for the large sample size. As these results appeared
inconclusive, it was not expected that positive causative
relationships would emerge within N-^.
(3) Cross-tabulation of sample means and rotated
population loadings (Table 13).
Same sample size: Within N.^ varlmax sample load
ings are most commensurate with the population loadings
for all of the rotational methods. Except for structure
U, in which all methods show about the same deviations,
varlmax deviations are less than or equal to those of the
other methods. All solutions showed an increase in the
number of deviations with increased complexity of factor
structure (with the exception of a reversal of results of
PC and PCC).
The direction of the bias was generally upward,
i.e., samples from population loadings in the low and
medium categories would fall in medium and high sample
92
categories. Thus it might be said that there was a
tendency for rotational methods to produce more
"significant" loadings than appears in the population.
In a statistical hypothesis testing paradigm this would
represent a tendency for type 1 error to occur. It
should be noted that in no case was there a high fre
quency of downward deviation and in nine cases the bias
was upward.
Large sample sizes: In large samples the results
were not as unequivocal as in small. Again the varlmax
method shows equal or smaller deviations than the equamax,
but more than the blquartimln method in the case of C, and
more than in the case of the Procrustes method in D. On
an overall evaluation, however, it would seem that the
varlmax method performs best in this type of analysis.
A possible interpretation for this reversal in
performance when compared with previous findings with
the RMS and range as criteria would be that the varlmax
method has greater total variability but has at least no
more of the sampling error component, perhaps less, than
the equamax method. Although both the equamax and varlmax
methods are based on criteria that would maximize the
variance of the factors, the equamax characteristic of
distributing variance more equally could both constrict
the total variability of both sample factors and sample
93
loadings and Increase the sampling error. On the other
hand, the varlmax method, Intending toward extreme values,
would provide variance to contribute to total variability
from the population positions.
Deviations from monotonicity occur in reversals of
the following: varlmax strong complex test, strong simple
structure and varlmax non-simple structure; biquartlmln
univocal test and biquartlmln complex test, weaker simple
structure; and Procrustes univocal test simple structure
and non-simple structure solutions.
The direction of bias changed In the large sample
solutions. In 11 out of the 16 conditions, the bias was
downward, I.e., means of samples of population values
tended to decrease. There were five cases of bias upward.
This again could be construed as potential type I error,
but In this Instance ooourrlng at both tails of the
sampling distribution. As expected, the total frequency
of these off diagonal events decreased from to Ng,
i.e., 71 to 48 respectively.
(6) Cross-tabulation of population pattern posi
tion and rotated population pattern position (Table 14).
This table gives an estimate of the distortion of the
original pattern Invoked by the rotational methods.
94
All of the solutions maintain their high, medium,
and low categories within U. As the structures become
more complex E and V show deviations In both directions.
In CC a large proportion of loadings defining hyperplanes
were lost for both solutions. The biquartlmln solution,
because of Its obliqueness, caused a general reduction of
loadings with subsequent Increase In hyperplane definition.
The Procrustes method generated sample mean values that
maintained the same population category trichotomy across
s '
all structures.
Results of the orthogonal methods Improved In the
non-simple structure pattern, perhaps because of a regres
sion artifact. There were certainly more middle level
loadings In the non-slmple structure pattern than in the
other patterns. Because of the higher correlations of
primary factors with additional complexity, the biquartlmln
method showed an Increase of deviations with increasing
complexity of structures. This might serve as a warning
to those Investigators who would permit obliqueness In
analyzing complex batteries, hoping to use the "simpler"
structure of correlated primary factors as a guide In their
Interpretations.
These results tend to verify some of the previously
mentioned findings regarding the varlmax and biquartlmln
methods and their effects upon different structures
95
(Hamburger and Merrifield, 1963), i.e., as patterns
become more complex, solutions from these methods become
more ambiguouB.
(7) Tabulation of summary cell frequency distribu
tion (Table 15).
In most of the solutions, increasing the size of
the sample gave c clearer picture of the shape of the cell
distributions. It becomes apparent within small sample
size the distributions tend toward a symmetric type, most
likely the normal. With increased sample size, the range
diminishes and there is a relative switch from flatness to
steeply peaked appearances.
Among the analytic solutions, the equamax method
provided the most normal appearing distributions followed
by the varlmax method. The Procrustes method loadings were
generally less normally distributed than any of the latter.
This might be because of constraints on variability
provided by the rotation or transformation imposed by a
fixed pattern. It would be equally possible for loadings
to accumulate at the extremes or in the center of the
distribution, and perhaps even to become multimodal when
degrees of freedom are diminished. At the same time,
the primary factors lose their Independence when the
requirement of orthogonality of the reference axes is
relaxed and correlation permitted, which might more than
overbalance the effect of losing this constraint, *nie
relative asymmetry of the biquartlmln method would tend to
support this hypothesis. But only further study of this
phenomenon with much larger samples will completely clarify
the Issue.
The only relationship that could be readily
observed between type of structure and distribution of
loadings was that simple structure with unlvocal tests
tended to provide relatively more normality and symmetry.
This finding would imply that with well defined orthogonal
factors and univocal tests in a population, adding normal
error functions leads to normally distributed deviations
in sample factor loadings; otherwise, errors and loadings
are not, empirically, uncorrelated.
CHAPTER VII
CONCLUSIONS
The results of this study would seem to have
provided some interesting leads in exploration of the
problem of rotated factorial stability. The following
general conclusions summarize the most relevant findings:
(l) In comparing orthogonal analytic solutions it
appeared that the equamax method gave greater stability or
less total variability than did the varimax method about
population positions (rotated) of factor loadings. A
possible explanation was given by assigning categories of
sampling error and variance as linear components of total
variability. The equamax method's characteristic of equat
ing factor variances is offered as an explanation for a
smaller variance (range) component and somewhat greater
sampling component than for the varimax method. This
sampling component was measured by deviation of the mean
sample cell loading from the rotated population position.
Both orthogonal solutions performed better than the
biquartlmln method in terms of stability as the type of
factor structure became more complex. This is probably due
to the increased correlation of simple primary factors
which resulted in generally decreased factor loadings in
the biquartlmln method.
98
The semi-analytic Procrustes solution gave the most
stable results. This was as expected because the exact
population patterns were provided as target matrices.
(2) The increase of complexity of factor structure
did not show the expected linear relationship with
stability. The most stable solutions were for the univo
cal test, strong simple structure factor pattern and the
pattern that did not possess simple structure. This led
to the suggestion that a plateau of instability ocours as
complexity is increased.
(2) Increased sample size resulted in greater
stability. But the decrease in error, in the case of the
RMS criterion, was not proportional to the square root of
the sample size. The ratio of rms values for the two
sample sizes tended to 3 to 2, rather than the expected 2
to 1. This result might be indicative of a decreasing
convergence rate, for some not readily apparent reason,
with increasing sample size. The 2 to 1 ratio tended to
appear with the range as the criterion.
(4) Greater stability was found at the extreme
population factor loadings; i.e., factor loadings less
than or equal to .10 and greater than or equal to .25*
This would tend to support the general practice of sub
jectively weighting the extreme loadings in interpretation
of factors. This conclusion held for both RMS and range.
99
(5) No consistent or logical relationship could be
observed between oommunality size and the criterion
measures of stability. An explanation is offered in terms
of the balancing out of any possible effect of oommunality
by the normalization process prior to rotation.
(6) The comparison of the rotated pattern with
the original population pattern revealed instability
increasing with complexity for all of the analytic
criteria. An exception to this was the Improvement of the
orthogonal solutions in the non-simple structure pattern.
This was probably due to the increased number of inter
mediate loadings that existed in the population pattern.
(7) The frequency distributions of sample-factor
loadings within methods were usually symmetrical,
particularly with larger samples. The orthogonal solu
tions gave the most normal-appearing distributions. This
was especially true for the equamax method. Correlation
of primary : actors was posited as the source of
asymmetry in distributions for the obllmln solution.
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APPENDIX
Summary matrices for all treatment conditions;
sample cell means, ranges, and RMS values.
EQUAMAX: STRONG SIMPLE STRUCTURE UNIVOCAL TEST
Mean
1 2 3 4
1 0. 800 -0. 002 0. 030 0. Oil
2 0. 812 -0. 009 0. Oil -0. 015
3 0. 572 -0.026 0. 047 0. 008
4 -0. 065 0. 574 -0.013 0. 003
5 0.011 0. 533 0. 002 -0.022
6 0. 023 0. 418 0. 041 -0. 026
7 0. 031 0. 020 0. 527 0. 041
8 0. 043 0. 030 0. 460 0. 030
9 0. 014 -0. 018 0. 352 0.074
10 0. 005 -0.017 -0.023 0. 397
11 0. 015 -0. 019 -0. 043 0. 313
12 -0. 022 0. 003 -0. 018 0. 239
1 2 3 4
1 0. 791 0. 002 0. 006 -0. 027
2 0.819
-0. 003 0. 003 -0. 004
3 0. 500 0. 001 -0. 004 0. 008
4 -0.012 0. 566 0. 007 0. 016
5 -0.001 0. 518 0. 007 0. 014
6 0. 012 0. 414 -0. 013 0. 024
7 0. 002 -0.012 0. 545 0. 001
8 -0. 012 -0.000 0. 479 0. 015
9 0. 010 0. 010 0. 298 -0.020
10 -0. 010 0. 014 0. 004 0. 321
11 0. 003 -0. 000 -0. 008 0. 311
12 -0.010 0. 026 -0. 024 0. 250
Small Sample Size
Range
1 2 3 4
0. 176 0. 320 0. 396 0. 333
0. 173 0. 261 0. 370 0. 222
0. 384 0. 510 0. 620 0. 703
0. 544 0. 305 0. 332 0. 364
0. 244 0. 528 0. 442 0. 372
0. 220 0. 398 0. 302 0. 754
0. 502 0. 427 0. 775 0. 436
0. 389 0. 399 0. 766 0. 679
0. 410 0. 530 0. 518 0.878
0. 548 0. 531 0. 424 0. 428
0. 385 0. 692 0.742 0. 471
0. 490 0. 633 0. 726 0. 549
Large Sample Size
1 2 3 4
0. 075 0. 239 0. 145 0. 373
0. 089 0. 282 0. 143 0.188
0. 223 0. 213 0. 215 0. 275
0. 227 0. 148 0. 214 0. 318
0.150 0. 141 0. 258 0. 190
0. 155 0. 239 0. 173 0. 272
0. 126 0. 193 0. 189 0. 229
0.132 0. 273 0. 175 0. 399
0. 207 0. 209 0. 224 0. 254
0. 143 0. 261 0. 157 0. 171
0. 226 0. 177 0.418 0. 313
0. 183 0. 368 0. 247 0. 383
RMS
1 2 3 4
0. 048 0. 093 0.
109
0.
091
0. 114 0. 076 0. 091 0. 053
0. 101 0. 120 0. 137 0. 156
0. 125 0. 158 0. 095 0. 083
0. 076 0. 117 0. 125 0. 107
0. 065 0. 115 0. 097 0. 158
0. 108 0. 111 0. 349 0. 125
0. 109
0. 124 0. 190 0. 165
0. 093 0. 132 0. 178 0. 264
0. 126 0. 127 0. 110 0. 142
0. 112 0. 174 0. 167 0. 160
0. 119
0. 151 0. 168 0. 147
1
0. 024
2
0. 061
3
0. 039 0.
4
082
0.101 0. 062 0. 038 0. 047
0. 044 0.058 0.064 0. 073
0. 056 0. 145 0. 050 0. 072
0. 042 0.045 0.065 0. 060
0. 045 0. 066 0. 058 0. 076
0.039 0.063 0. 297 0. 062
0. 038 0.062 0. 050 0. 094
0. 052 0.057 0. 076 0. 071
0. 044 0. 053 0. 042 0. 049
0. 057 0. 053 0. 082 0. 115
0. 059 0. 098 0. 066 0. 101
o
-a
EQUAMAX: STRONG SIMPLE STRUCTURE, COMPLEX TESTS
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 714 0. 507 0. 102 0. 261 0. 160 0. 342 0. 509 0. 595 0. 051 0. 102 0. 130 0. 145
2 0. 847 0. 013 0. 184 0. 069 0. 212 0. 374 0. 548 0. 589 0. 074 0. 104 0. 133 0. 216
3 0. 569 -0.039
0. o n -0. 010 0. 361 0. 402 0. 359 0.764 0. 108 0.104 0. 097 0. 202
4 -0. 050 0. 651 0. 049 -0.068 0. 420 0. 258 0.445 0. 342 0. 107 0. 084 0. 203 0.116
5 0.159 0. 604 0. 083 0. 245 0. 317 0. 410 0. 405 0. 421 0. 088 0. 107 0. 129 0. 101
6 -0. 009 0.451 0. 071 0. 180 0. 232 0. 470 0. 337 0. 591 0. 067 0.119 0. 211 0.146
7 0. 057 0. 314 0. 536 0. 205 0. 397 0. 493 0. 910 0. 446 0. 141 0.180 0. 394 0. 153
8 0. 401 0. 012 0. 446 0. 003 0. 512 0. 368 0. 682 0. 349 0. 158 0.179 0. 180 0.113
9 -0. 012 -0. 054 0. 384 0. 017 0. 200 0. 330 0.712 0. 724 0.063 0. 228 0. 199 0.156
10 0. 256 0. 034 0. 076 0. 446 0. 442 0. 500 0. 482 0. 249 0. 121 0. 117 0. 172 0. 084
11 0. 453 0. 057 0. 199 0. 401 0. 420 0. 445 0. 649 0.414 0.146 0. 107 0. 164 0.164
12 -0. 054 0. 107 0. 023 0. 538 0. 281 0. 343 0. 432 0. 234 0.140 0. 115 0. 127 0.093
Large Sample Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 728 0. 510 0. 103 0. 288 0. 119 0. 193 0. 192 0. 171 0. 034 0. 074 0. 050 0. 062
2 0. 850 0. 010 0. 226 0. 097 0.099 0. 262 0. 298 0. 212 0. 056 0. 081 0. 103 0. 150
3 0.
509
0. 001 -0. 012 0. 040 0. 195 0. 173 0. 201 0. 169 0. 043 0. 049
0. 065 0. 086
4 -0. 007 0. 653 0. 066 -0. 064 0. 206 0. 101 0. 127 0. 297 0. 051 0. 060 0. 149
0. 094
5 0. 166 0. 595 0. 058 0. 293 0. 139 0. 148 0. 208 0. 230 0. 049 0. 040 0. 105 0. 088
6 -0. Oil 0. 448 0. 040 0. 264 0. 169 0. 234 0. 198 0. 333 0. 061 0. 078 0. 222 0. 107
7 0. 034 0. 296 0. 557 0. 215 0. 184 0. 165 0. 198 0. 235 0. 084 0. 132 0. 334 0. 115
8 0. 313 -0. 016 0. 495 0. 028 0. 225 0.
279
0. 160 0. 229 0. 080 0. 184 0. 044 0. 077
9 -0. 019 -0. 004 0. 304 -0. 018 0. 147 0. 192 0. 220 0. 210 0. 051 0. 161 0. 061 0. 053
10 0. 261 0. 068 0. 029 0. 383 0. 229 0. 223 0. 186 0. 168 0. 075 0. 054 0. 188 0. 069
11 0. 397 0. 061 0. 241 0. 433 0. 204 0. 177 0. 224 0. 207 0. 078 0. 049
0. 109 0. 083
12 -0. 043 0. 117 0. 015 0. 510 0. 219 0. 237 0. 214 0. 163 0. 124 0. 090 0. 077 0. 100
M
O
CD
EQUAMAX: WEAKER SIMPLE STRUCTURE, COMPLEX TESTS
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 694 0. 482 0. 126 0. 357 0. 185 0. 355 0. 442 0. 430 0. 083 0.123 0. 130 0. 119
2 0. 823 -0. 037 0. 129 0. 253 0. 234 0. 378 0.439 0. 395 0. 112 0. 113 0. 124 0. 198
3 0. 575 -0. 040 -0. 009 0.
069
0. 359 0. 466 0. 354 0. 581 0. 111 0. 112 0. 083 0. 127
4 -0. 046 0. 680 0. 169
0. 041 0. 364 0. 232 0. 334 0. 400 0. 108 0. 065 0. 156 0. 187
5 0. 161 0. 622 0. 121 0. 220 0. 256 0. 383 0. 342 0. 295 0. 073 0. 103 0. 120 0. 096
6 -0. 010 0. 415 0. 252 0. 148 0. 197 0. 432 0. 402 0. 603 0. 062 0. 123 0. 243 0. 167
7 0. 032 0. 267 0. 632 0. 262 0. 452 0. 388 0. 383 0. 451 0. 135 0. 118 0. 257 0. 125
8 0. 409
0. 185 0. 483 0. 075 0. 495 0. 412 0. 585 0. 564 0. 154 0. 237 0. 153 0. 139
9 0. 010 0. 106 0. 429 -0. 010 0. 285 0. 268 0. 561 0. 474 0. 078 0. 149 0. 237 0. 113
10 0. 190 0. 017 0. 250 0. 492 0. 324 0. 521 0. 415 0.
309
0. 116 0. 125 0. 260 0. 150
11 0. 390 0. 055 0. 150 0. 492 0. 409 0. 425 0. 428 0. 338 0. 112 0. 118 0. 123 0.
119
12 0. 067 0. 163 -0. 009 0. 550 0. 246 0. 231 0. 379 0. 290 0. 145 0. 089 0. 098 0. 085
Large Sample; Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 710 0. 506 0. 165 0. 318 0. 138 0. 223 0. 121 0. 149 0. 062 0. 080 0. 127 0. 054
2 0. 828 -0. 031 0. 179 0. 248 0. 123 0. 257 0. 284 0. 219 0. 093 0. 070 0. 066 0. 170
3 0. 502 0. 004 -0. 020 0. 097 0. 184 0. 163 0. 173 0. 157 0. 039 0. 046 0. 056 0. 055
4 -0. 006 0. 670 0. 234 0. 018 0. 175 0. 111 0. 229 0. 271 0. 049 0.036 0. 193 0. 151
5 0. 157 0. 611 0. 137 0. 227 0. 171 0. 129 0. 203 0. 148 0. 049 0. 033 0. 104 0. 060
6 -0. 036 0. 420 0. 254 0. 232 0. 154 0. 222 0. 164 0. 244 0. 046 0. 070 0. 224 0. 074
7 0. 000 0. 225 0. 638 0. 302 0. 142 0. 209 0. 135 0. 274 0. 048 0. 124 0. 238 0. 113
8 0. 323 0. 139 0. 548 0. 099
0. 153 0. 143 0. 140 0. 243 0. 057 0. 169 0. 105 0. 096
9
-0. 000 0. 141 0. 350 -0. 022 0. 195 0. 220 0. 175 0.
179
0. 050 0. 167 0. 113 0. 056
10 0. 202 0. 053 0. 221 0. 451 0. 185 0. 184 0. 210 0. 172 0. 078 0.046 0. 212 0. 088
11 0. 347 0. 076 0. 153 0. 502 0. 176 0. 139 0. 158 0. 178 0. 081 0. 044 0. 079 0. 085
12 0. 097 0. 213 -0. 031 0. 510 0. 224 0. 262 0. 198 0. 150 0. 169 0. 132 0. 072 0. 087
EQUAMAX: NO SIMPLE STRUCTURE
1
Mean
2 3 4
Small Sample Size
Range
1 2 3 4 1
RMS
2 3 4
1 0. 662 0. 422 0.168 0. 387 0. 502 0. 828 0. 405 0. 771 0. 142 0. 208 0. 146 0. 173
2 0. 774 0.057 0.127 0. 198 0. 523 1. 239 0.491 0. 831 0.196 0. 215 0. 136 0. 173
3 0. 574 -0.021 -0. 024 0. 084 0. 284 0.738 0. 437 0. 383 0. I l l 0. 162 0. I l l 0. 092
4 -0. 042 0. 614 0. 276 0. 130 0. 387 0. 623 0. 688 0. 517 0. 094 0. 199 0. 188 0. 150
5 0. 157 0. 509 0. 343 0. 297 0. 246 0.491 0. 616 0. 530 0. 073 0. 152 0.147 0. 143
6 0.157 0. 388 0. 229 0. 260 0. 373 0. 543 0.478 0. 568 0.086 0. 155 0. 122 0. 155
7 0. 025 0. 383 0. 631 0. 216 0. 503 0. 688 0. 474 0. 563 0. 109 0.181 0. 294 0. 151
8 0. 379 0. 317 0. 463 0. 169 0. 472 0. 766 0. 445 0. 535 0. 137 0. 180 0. 120 0. 128
9
-0. 015 0. 211 0. 360 0. 182 0. 194 0. 679 0. 622 0. 614 0.066 0.169 0. 206 0.181
10 0. 200 0. 175 0. 333 0. 382 0. 286 0. 384 0. 572 0. 590 0.089 0.118 0.164 0.164
11 0. 392 0.163 0. 270 0. 443 0.401 0. 626 0. 370 0. 318 0. 110 0. 155 0. 106 0.098
12 0. 093 0. 087 0. 125 0. 485 0. 331 0. 270 0. 536 0. 595 0. 086 0.064 0. 141 0.193
Large Sample Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 702 0. 467 0. 136 0. 392 0. 193 0. 229
0. 160 0. 189
0. 052 0. 103 0. 082 0. 044
2 0. 817 0. 014 0. 179 0. 200 0. 099 0. 255 0. 251 0. 198 0. 102 0. 069 0. 066 0. 075
3 0. 506 -0. 001 -0. 010 0. 088 0. 171 0. 147 0. 126 0. 262 0. 043 0. 038 0. 034 0. 057
4 -0. 018 0. 660 0. 303 0. 101 0. 215 0. 127 0. 217 0. 289
0.
059
0. 055 0. 063 0. 078
5 0.
139
0. 546 0. 334 0. 309 0, 177 0. 148 0. 204 0. 221 0.
049
0. 040 0. 058 0. 066
6 0. 154 0. 426 0. 219 0. 314 0. 165 0. 213 0. 188 0. 304 0.
049
0. 065 0. 049 0. 090
7 -0. 003 0. 350 0. 692 0. 193 0. 142 0. 246 0. 125 0. 170 0. 041 0. 085 0. 216 0. 069
8 0. 313 0. 269 0. 534 0. 182 0. 165 0. 247 0. 139 0. 230 0. 049 0. 054 0. 057 0. 063
9
-0. 032 0. 236 0. 315 0. 159 0. 223 0. 235 0. 224 0. 225 0. 059 0. 066 0. 066 0. 061
10 0. 230 0. 174 0. 340 0. 347 0. 222 0. 295 0. 226 0. 280 0. 057 0. 065 0. 070 0. 074
11 0. 351 0. 165 0. 262 0. 463 0. 297 0. 172 0. 275 0. 187 0. 074 0. 051 0. 058 0. 059
12 0. 127 0. 124 0. 101 0. 489 0. 420 0. 196 0. 257 0. 204 0. 089
0. 067 0. 054 0. 136
M
M
O
VARIMAX: STRONG SIMPLE STRUCTURE, UNIVOCAL TEST
Mean
1 2 3 4
1 0. 801 i 0.000 0. 026 0. 013
2 0. 812 -0. 006 0. 008 -0. 013
3 0. 574 -0. 022 0. 045 0.009
4 -0. 067 0. 572 -0. 012 0. 005
5 0. 008 0. 532 0. 001 -0. 024
6 0. 020 0. 418 0. 040 -0.026
7 0. 032 0.019 0. 528 0. 046
8 0. 044 0. 031 0. 465 0. 035
9 0. 015 -0. 020 0. 351 0. 073
10 0. 004 -0. 018 -0.021 0. 395
11 0. 015 -0.020 -0.042 0. 309
12 -0. 022 0. 001 -0.017 0. 235
1 2 3 4
1 0. 791 0. 001 0. 006 -0.027
2 0. 819 -0. 003 0. 003 -0. 003
3 0. 501 0.001 -0. 004 0. 008
4 -0. 012 0. 567 0. 007 0. 014
5 -0.001 0. 518 0. 006 0. 012
6 0. 012 0.415 -0. 013 0. 023
7 0. 002 -0. 012 0. 545 0. 000
8 -0. 012 0.000 0. 479 0. 014
9
0. 010 0. 010 0. 297 -0.021
10 -0. 010 0. 015 0. 005 0. 321
11 0. 003 0. 000 -0.007 0. 311
12 -0. 010 0. 027 -0.024 0. 249
Small Sample Size
Range
1 2 3 4
0. 175 0. 311 0. 389 0. 321
0. 172 0. 261 0. 412 0. 209
0. 382 0. 504 0. 565 0. 689
0. 571 0. 302 0. 336 0. 392
0. 258 0. 537 0. 436 0. 387
0. 221 0.403 0. 294 0. 758
0. 504 0. 427 0. 741 0. 451
0. 388 0. 397 0. 741 0. 640
0.409 0. 541 0. 575 0. 885
0. 547 0. 560 0. 404 0. 446
0. 391 0. 717 0. 692 0.489
0. 482 0. 680 0. 743 0. 528
Large Sample Size
1 2 3 4
0. 074 0. 240 0. 147 0. 365
0.089 0. 282 0. 143 0. 186
0. 223 0. 213 0. 215 0. 271
0. 227 0. 148 0. 215 0. 315
0.149 0. 142 0. 255 0. 187
0. 154 0. 239 0.174 0. 266
0. 127 0. 196 0. 187 0. 221
0. 132 0. 273 0. 174 0. 396
0. 207 0. 210 0. 225 0. 254
0. 142 0. 257 0.156 0.171
0. 227 0. 176 0. 420 0. 316
0. 184 0. 373 0. 250 0. 381
RMS
1 2 3 4
0. 048 0.
089
0.
109 0. 086
0. 114 0. 074 0. 096 0. 047
0. 103 0.
118 0. 130 0. 155
0. 129 0. 159
0. 096 0. 088
0. 079 0. 118 0. 123 0. 111
0. 064 0. 116 0. 094 0. 160
0. 109 0. 113 0. 344 0. 124
0. 108 0. 124 0. 185 0. 160
0. 093 0. 133 0. 182 0. 262
0. 129 0. 128 0. 109
0. 140
0. 113 0. 180 0. 164 0.
165
0. 118 0. 156 0.
169 0. 150
1 2 3 4
0. 024 0. 061 0.039 0. 081
0. 101 0. 062 0. 038 0.045
0. 044 0.058 0.064 0. 073
0. 056 0.145 0. 050 0. 071
0. 042 0. 045 0. 065 0. 060
0. 045 0.066 0. 059 0. 075
0.039 0.063 0. 297 0. 060
0. 038 0.062 0. 050 0.093
0. 052 0.057 0. 076 0. 072
0. 045 0. 053 0. 042 0. 050
0. 058 0. 053 0. 083 0. 115
0. 059 0.099 0. 066 0. 102
» - »
VARIMAX: STRONG SIMPLE STRUCTURE. COMPLEX TESTS
Mean
1 2 3 4
1 0. 730 0. 511 0.066 0. 220
2 0. 856 0. 015 0.145 0. 042
3 0. 570 -0. 039 -0. 012 -0.029
4 -0. 048 0. 650 0. 035 -0. 074
5 0. 174 0. 610 0.065 0. 221
6 0. 000 0. 455 0.064 0. 167
7 0.086 0. 326 0. 504 0. 216
8 0. 417 0. 020 0.400 0.019
9 0. 000 -0. 048 0. 363 0. 040
10 0. 275 0. 044 0. 085 0. 411
11 0. 476 0. 068 0.190 0. 364
12 -0. 032 0. 119 0. 045 0. 505
1 2 3 4
1 0. 745 0. 508 0.078 0. 247
2 0. 860 0. 007 0. 202 0. 058
3 0. 509 -0. 002 -0.027 0. 017
4 -0. 004 0. 651 0.062 -0.071
5 0. 183 0. 597 0. 048 0. 277
6 0. 005 0. 448 0.036 0. 256
7 0.061 0. 300 0. 552 0. 210
8 0. 328 -0. 015 0. 485 0. 014
9
-0. Oil -0.002 0. 304 -0. 017
10 0. 278 0. 071 0. 020 0. 367
11 0.423 0. 065 0. 227 0. 413
12 -0. 019 0. 124 0. 014 0. 509
Small Sample Size
Range
1 2 3 4
0. 172 0. 356 0. 475 0. 600
0. 223 0. 381 0. 525 0. 597
0. 366 0. 403 0. 369 0. 721
0, 425 0. 258 0. 483 0. 309
0. 364 0. 393 0. 383 0. 397
0. 243 0. 481 0. 337 0. 587
0. 436 0. 521 0. 911 0. 634
0. 505 0. 380 0. 741 0. 718
0. 208 0. 336 0. 741 0. 727
0. 447 0. 529 0. 700 0. 522
0.
439 0. 585 0. 631 0. 487
0. 234 0. 456 0. 752 0. 678
Large Sample: Size
1 2 3 4
0. 145 0. 185 0. 189 0. 195
0. 099 0. 251 0. 323 0. 226
0. 193 0. 175 0. 205 0. 187
0. 223 0. 104 0. 130 0. 285
0. 147 0. 135 0. 192 0. 300
0. 191 0. 255 0. 182 0. 377
0. 211 0. 170 0. 198 0. 225
0. 239 0. 270 0. 170 0. 253
0. 140 0. 193 0. 222 0. 206
0. 246 0. 201 0. 197 0. 182
0. 234 0. 179
0. 223 0. 235
0. 241 0. 278 0. 220 0. 161
RMS
1 2 3 4
0. 052 0. 102 0. 127 0. 151
0. 074 0. 104 0. 125 0. 214
0. 107 0. 104 0. 094 0. 198
0.107 0.087 0. 213 0. 117
0.093 0. 107 0.131 0. I l l
0. 067 0. 124 0. 217 0. 148
0. 153 0. 185 0. 428 0.169
0. 158 0.177 0. 224 0. 155
0.063 0. 226 0. 215 0. 181
0. 123 0. 116 0. 196 0. 118
0. 153 0. 127 0. 164 0.185
0. 137 0. 125 0. 183 0.165
1 2 3 4
0. Q38 0. 075 0. 057 0. 078
0.056 0. 076 0.095 0. 160
0. 042 0. 048 0. 067 0.096
0.052 0.064 0. 145 0.100
0. 054 0. 044 0. 106 0.093
0.068 0.083 0. 224 0.114
0.094 0.128 0. 338 0.117
0. 077 0.188 0. 048 0. 078
0. 047 0. 163 0. 060 0.053
0. 082 0.056 0. 198 0. 079
0. 087 0.050 0. 100 0.092
0. 121 0.093 0. 075 0.099
M
M
ro
VARIMAX: WEAKER SIMPLE STRUCTURE, COMPLEX TESTS
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0, 706 0. 483 0. 103 0. 329 0. 181
0. 375 0. 421 0. 536 0. 093 0. 126 0. 131 0. 149
2 0. 831 -0. 036 0. 117 0. 225 0.
259 0. 420 0. 379 0. 485 0. 115 0. 116 0. 112 0. 229
3 0. 574 -0. 040 -0. 012 0. 052 0. 398 0. 481 0. 385 0. 599
0. 113 0. 111 0. 092 0. 130
4 -0. 040 0. 683 0. 152 0.037 0. 352 0. 230 0. 414 0. 403 0. 105 0. 063 0. 156 0. 192
5 0. 172 0. 627 0. 100 0. 206 0. 293 0. 372 0. 351 0. 278 0. 076 0. 107 0.
109 0. 091
6 -0. 003 0. 421 0. 234 0. 142 0. 200 0.
449 0. 391 0. 653 0. 059 0. 132 0. 226 0. 177
7 0. 047 0. 282 0. 615 0. 256 0. 482 0. 492 0. 432 0. 454 0. 136 0. 134 0. 277 0. 144
8 0. 417 0. 195 0. 467 0. 057 0. 499 0. 414 0. 620 0. 678 0. 155 0. 249 0. 157 0.151
9
0. 015 0. 114 0. 426 -0.008 0. 265 0. 264 0. 591 0. 506 0. 076 0. 157 0. 238 0.109
10 0. 206 0. 027 0. 242 0. 481 0. 387 0. 527 0. 477 0. 380 0. 128 0. 122 0. 266 0.162
11 0. 403 0. 060 0. 139 0. 472 0. 455 0. 539 0. 465 0. 381 0. 127 0. 134 0. 134 0.133
12 0. 084 0. 168 -0. 016 0. 543 0. 234 0. 347 0. 357 0. 298 0. 127 0. 098 0. 102 0. 088
Large Samplei Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 720 0. 506 0. 159 0. 295 0. 158 0. 222 0. 148 0. 178 0. 073 0. 078 0. 140 0. 081
2 0. 836 -0.
029 0. 176 0. 224 0. 115 0. 228 0. 285 0. 213 0. 095 0. 062 0. 065 0. 201
3 0. 504 0. 002 -0. 023 0. 084 0. 183 0. 164 0. 171 0. 161 0. 039 0. 045 0. 052 0. 070
4 -0. 002 0. 671 0. 227 0.013 0. 187 0.
119 0. 289 0. 294 0. 050 0. 035 0. 193 0. 161
5 0. 166 0. 612 0. 132 0. 219 0. 178 0. 133 0. 202 0. 187 0. 051 0. 034 0. 106 0. 057
6 -0. 027 0. 421 0. 252 0. 229 0. 153 0. 238 0. 175 0. 250 0. 051 0. 075 0. 223 0. 080
7 0. 014 0. 231 0. 637 0. 295 0. 158 0. 265 0. 150 0. 284 0. 045 0. 129 0.
239 0. 115
8 0. 330 0. 143 0. 545 0. 085 0. 153 0. 153 0. 111 0. 246 0. 060 0. 175 0. 109 0. 104
9 0. 002 0. 144 0. 347 -0.025 0. 193 0. 230 0. 202 0. 168 0. 049 0. 172 0. 112 0. 054
10 0. 216 0. 055 0. 222 0. 442 0.
199 0. 171 0. 244 0. 191 0. 090 0. 048 0. 222 0.098
11 0. 362 0. 078 0. 153 0. 490 0. 182 0. 144 0. 157 0. 199
0. 098 0. 044 0. 070 0.100
12 0. 112 0. 214 -0. 029 0. 506 0. 213 0. 289 0. 195 0. 151 0. 148 0. 126 0. 071 0.092
H *
H
VARIMAX: NO SIM PLE STRUCTURE
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 717 0. 435 0. 102 0. 325 0. 294 0. 603 0. 491 0. 545 0. 083 0. 193 0. 140 0. 125
2 0. 820 0. 036 0. 094 0. 172 0. 258 0. 707 0. 463 0. 428 0. 124 0. 149 0. 137 0. 131
3 0. 559
-0.
019
0. 006 0. 073 0. 489 0. 702 0. 557 0. 723 0. 127 0. 143 0. 130 0. 146
4 -0. 047 0. 659
0. 225 0. 083 0. 372 0. 643 0. 743 0. 464 0. 095 0. 160 0. 174 0. 118
5 0. 174 0. 569 0. 276 0. 242 0. 244 0. 579 0. 517 0. 425 0. 073 0. 167 0. 137 0. 129
6 0. 180 0. 432 0. 184 0. 219 0. 332 0. 509 0. 606 0. 614 0. 085 0. 133 0. 141 0. 141
7 0. 018 0. 437 0. 578 0. 200 0. 425 0. 798 0. 732 0. 467 0. 107 0. 211 0. 357 0. 158
8 0. 357 0. 345 0. 412 0. 190 0. 443 0. 775 0. 603 0. 658 0. 124 0. 190 0. 186 0. 181
9
-0. 003 0. 194 0. 396 0. 151 0.
209 0. 599 0. 637 0. 638 0. 066 0. 153 0. 230 0. 166
10 0. 232 0. 167 0. 312 0. 375 0. 297 0. 426 0. 566 0. 635 0. 089 0. 122 0. 150 0. 199
11 0. 428 0. 197 0. 220 0.
409 0. 433 0. 517 0. 706 0. 544 0. 115 0. 134 0. 149 0. 124
12 0. 129
0. 106 0. 090 0.
491
0. 388 0, 330 0. 307 0. 645 0. 101 0. 081 0.
079
0. 170
Large Sample Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 722 0. 462 0. 131 0. 344 0. 222 0. 274 0. 513 0. 254 0. 062 0. 116 0. 139 0. 061
2 0. 828 0. 024 0. 167 0. 164 0. 112 0. 234 0. 233 0. 235 0. 100 0. 065 0. 063 0. 086
3 0.
509
-0. 004 -0. 016 0. 067 0. 172 0. 146 0. 124 0. 269 0. 043 0. 040 0. 034 0. 057
4 -0. 004 0. 634 0. 292 0. 069 0. 237 0. 321 0. 846 0. 243 0. 066 0. 121 0. 208 0. 080
5 0. 162 0. 529
0. 323 0. 273 0. 215 0. 402 0. 718 0. 247 0. 057 0. 121 0. 176 0. 073
6 0. 175 0. 422 0. 215 0. 284 0. 175 0. 337 0. 456 0. 318 0. 057 0. 091 0. 118 0. 097
7 0. 021 0. 341 0. 662 0. 160 0. 155 0. 793 0. 562 0. 221 0. 043 0. 201 0. 272 0. 076
8 0. 333 0. 267 0. 512 0. 146 0. 175 0. 598 0. 422 0. 249 0. 046 0. 140 0. 099 0. 068
9
-0. 017 0. 225 0. 302 0. 142 0. 205 0. 375 0. 426 0. 219 0. 055 0. 115 0. 128 0. 058
10 0. 251 0. 176 0. 327 0. 319 0. 243 0. 472 0. 340 0. 347 c. 061 0. 118 0. 099 0. 086
11 0. 389
0.
179
0. 254 0. 430 0. 199 0. 331 0. 371 0. 222 0. 062 0. 089 0. 088 0. 068
12 0. 137 0. 126 0. 100 0. 480 0. 183 0. 224 0. 231 0. 225 0. 054 0. 056 0. 055 0. 134
M
£
BIQUARTIMIN: STRONG SIMPLE STRUCTURE, UNIVOCAL TESTS
Mean
Small Sample Size
Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 798 0. 010 0. 013 0. 020 0.182 0. 350 0. 399 0. 309 0. 049 0. 085 0. 101 0. 071
2 0.809 -0. 003 0. 000 -0. 024 0. 170 0. 291 0. 407 0. 264 0. 116 0.065 0. 092 0. 060
3 0. 567 -0. 017 0. 034 0. 005 0.402 0.452 0. 558 0. 673 0. 100 0. 106 0. 123 0. 151
4 -0. 068 0. 563 -0. 005 0. 012 0. 544 0. 355 0.412 0. 381 0. 126 0.172 0. I l l 0. 088
5 0. 004 0. 523 0. 006 -0.016 0. 269 0. 522 0. 415 0. 371 0. 078 0. 121 0. 129 0.096
6 0. 022 0. 421 0. 038 -0.007 0. 185 0. 325 0. 273 0. 683 0. 060 0. 108 0.089 0. 135
7 0. 028 0. Oil 0. 522 0. 069 0. 474 0. 417 0. 740 0. 426 0.104 0. 104 0. 349 0. 133
8 0. 042 0. 035 0.453 0. 067 0. 404 0. 386 0. 762 0. 589 0. 105 0. 118 0.194 0. 175
9 0. 008 -0.038 0. 349 0. 075 0. 395 0. 365 0. 603 0. 881 0. 090 0.118 0. 193 0. 268
10 0. 005 -0. 035 -0. 010 0. 346 0. 549 0. 771 0. 380 0. 906 0. 125 0.174 0.106 0. 195
11 0. 013 -0.037 -0. 035 0. 262 0. 360 0. 745 0. 687 0. 614 0. 106 0. 201 0.166 0. 207
12 -0. 023 0. 009 -0.020 0. 237 0.510 0.639
Large SampL
0. 708
e Size
0. 505 0. 117 0.144 0. 161 0.143
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 790 0. 001 0. 006 -0. 026 0.079 0. 230 0.145 0. 323 0.025 0. 054 0. 040 0. 080
2 0. 818 -0. 003 0. 003 -0.002 0. 088 0. 258 0.145 0.160 0.103 0.055 0. 038 0. 042
3 0.499
0. 000 -0.004 0.009 0. 222 0. 199 0. 213 0. 230 0.044 0. 054 0.060 0.066
4 -0. 012 0. 564 0. 007 0.010 0. 205 0.148 0. 206 0. 292 0. 050 0. 147 0.045 0. 068
5 0. 000 0. 517 0. 008 0. 008 0.137 0. 140 0. 234 0. 203 0. 040 0.046 0.059 0. 059
6 0. Oil 0. 413 -0. 013 0. 019 0. 154 0. 243 0. 173 0. 247 0. 042 0.066 0. 058 0.068
7 0. 002 - 0 . Oil 0. 543 0. 002 0. 132 0. 186 0. 191 0. 194 0. 039 0.062 0. 298 0. 052
8 -0. 012 0. 001 0. 478 0. 016 0. 140 0. 269 0. 176 0. 379 0. 039 0. 060 0. 050 0.089
9
0. 010 0. Oil 0. 295 -0. 019 0. 194 0. 199 0. 229 0. 247 0. 049 0. 055 0. 075 0. 070
10 -0. 009 0. 013 0. 007 0. 320 0. 139 0. 262 0. 157 0. 170 0. 042 0. 053 0. 040 0. 051
11 0. 003 -0. 002 -0. 005 0. 309 0. 215 0. 186 0.413 0. 322 0. 056 0. 052 0. 080 0. 117
12 -0. 009 0. 026 -0. 021 0. 245 0. 175 0. 360 0. 232 0. 378 0. 057 0. 095 0. 063 0. 101
H
BIQUARTIMIN: STRONG SIMPLE STRUCTURE, COMPLEX TESTS
Means
1 2 3 4
1 0. 686 0. 466 -0.026 0. 117
2 0. 826 -0. 014 0. 072 -0. 041
3 0. 558 -0.050 -0.058 -0. 076
4 -0. 050 0. 650 0. 021 -0. 125
5 0. 135 0. 572 0. 020 0. 152
6 -0. 025 0. 430 0. 040 0. 126
7 0. 029 0. 277 0.493 0. 140
8 0. 380 - 0 . Oil 0. 388 -0. 065
9
-0. 024 -0. 061 0. 375 0. 015
10 0. 231 -0. 005 0. 022 0. 398
11 0. 423 0. 012 0. 120 0. 318
12 -0. 076 0. 068 0. 002 0. 516
1
1
0. 697
2
0. 450
3
-0. 002
4
0. 116
2 0. 825 -0. 032 0. 139
-0. 030
3 0. 501 -0. 015 -0. 062 -0.029
4 -0. 013 0. 648 0. 038 -0. 142
5 0. 132 0. 538 0. 002 0. 193
6 -0.032 0.406 0. 034 0. 195
7 -0. 001 0. 231 0. 526 0. 139
8 0. 276 -0. 024 0. 436 -0. 019
9 -0. 032 -0. 013 0. 302 -0. 026
10 0. 236 0. 018 -0. 013 0. 326
11 0. 359 -0.007 0. 182 0. 353
12 -0.071 0. 061 -0.004 0. 488
Small Sample Size
Range
1 2 3 4
0. 217 0. 362 0. 373 0. 514
0. 281 0. 360 0. 485 0. 565
0. 411 0. 309 0. 470 0. 669
0. 257 0. 242 0. 507 0. 318
0. 371 0. 435 0. 438 0. 366
0. 243 0. 512 0. 354 0. 588
0. 419 0. 525 0. 900 0. 503
0. 524 0. 359 0. 695 0. 423
0. 146 0. 332 0. 762 0. 747
0. 456 0. 468 0. 554 0. 373
0.469 0. 517 0. 591 0.460
0. 301 0.460 0. 351 0. 244
Large Sample Size
1 2 3 4
0. 161 0. 197 0. 194 0. 199
0. 161 0. 231 0. 273 0. 233
0. 202 0. 164 0. 185 0. 150
0. 146 0. 116 0. I l l 0. 221
0. 223 0. 339 0. 166 0. 285
0. 171 0. 325 0. 601 0. 350
0. 339 0. 379 0. 208 0. 264
0. 290 0. 620 0. 493 0. 421
0. 117 0. 179 0. 233 0. 212
0. 272 0. 211 0. 161 0. 214
0. 279 0. 190 0. 219 0. 231
0. 235 0. 220 0. 151 0. 182
RMS
1 2 3 4
0. 061 0. 091
0. 102 0. 113
0. 076 0. 113 0. 111 0. 173
0. 116 0. 089
0. 097 0. 177
0. 074 0. 068 0. 147 0. 098
0. 096 0. 116 0. 111 0. 096
0. 088 0. 144 0. 173 0. 145
0. 190 0. 270 0. 385 0. 147
0. 179
0. 142 0. 190 0. 100
0. 050 0. 201 0. 212 0. 169
0. 140 0. 111 0.
189 0. 101
0. 183 0. 120 0. 142 0. 159
0. 116 0. 122 0. 136 0. 090
1 2 3 4
0. 050 0. 070 0. 067 0. 064
0. 055 0. 085 0. 084 0. 121
0. 046 0. 040 0. 054 0. 076
0. 050 0. 038 0. 064 0. 091
0. 065 0. 080 0. 057 0. 095
0. 076 0. 104 0. 190 0. 105
0. 142 0. 211 0. 313 0. 107
0. 093 0. 166 0. 100 0. 101
0. 039 0. 132 0. 067 0. 062
0. 107 0. 071 0. 190 0. 074
0. 100 0. 050 0. 104 0. 085
0. 097 0. 093 0. 103 0. 093
H
H
o\
I
BIQUARTIMIN: WEAKER SIMPLE STRUCTURE, COMPLEX TESTS
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 638 0. 420 -0. 020 0. 172 0. 236 0. 388 0. 314 0. 421 0. 129 0. 129 0. 086 0. 123
2 0. 780 -0. 089
0. 054 0. 102 0. 323 0. 385 0. 337 0. 501 0. 150 0. 111 0. 096 0. 198
3 0. 548 -0. 055 -0. 042 -0. 013 0. 326 0. 390 0. 796 0. 426 0. 095 0. 097 0.
139 0. 088
4 -0. 063 0. 655 0. 062 -0. 027 0, 296 0. 268 0. 429
0. 375 0. 091
0. 082 0. 119 0. 198
5 0. 122 0. 584 -0. 001 0. 109 0. 274 0. 388 0. 440 0. 346 0. 075 0. 110 0. 103 0. 112
6 -0. 050 0. 372 0. 155 0. 092 0. 243 0. 507 0. 413 0. 652 0. 065 0. 152 0. 171 0. 176
7 -0. 037 0. 182 0. 533 0. 188 0. 549 0. 508 0. 538 0. 564 0. 150 0. 175 0. 349 0. 175
8 0. 365 0. 115 0. 389
-0. 046 0. 510 0. 466 0. 729 0. 712 0. 162 0. 200 0. 182 0.
159
9
-0. 015 0. 071 0. 396 -0. 032 0. 241 0. 309 0. 630 0. 520 0. 065 0. 128 0. 227 0. 101
10 0. 125 -0. 053 0. 189 0. 417 0. 3jB5 0. 545 0. 755 0. 425 0. 166 0. 134 0. 270 0. 152
11 0. 317 -0. 008 0. 073 0. 390 0. 459
0. 622 0. 413 0. 453 0. 154 0. 143 0. 145 0. 138
12 0. 012 0. 120 -0. 062 0. 493 0. 231 0. 397 0. 361 0. 423 0. 096 0. 111 0. 122 0. 122
Large Sample Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 641 0. 433 0. 024 0. 122 0. 154 0. 214 0. 205 0. 210 0. 115 0. 099 0. 085 0. 056
2 0. 773 -0. 087 0. 125 0, 092 0. 162 0. 195 0. 275 0. 206 0. 142 0. 075 0. 068 0. 157
3 0. 482 -0. 008 -0. 051 0. 013 0. 176 0. 125 0. 135 0. 109
0. 041 0. 037 0. 039 0. 043
4 -0. 033 0. 623 0. 114 -0. 073 0. 162 0. 164 0. 305 0. 233 0. 046 0. 087 0, 103 0. 140
5 0. 108 0. 553 0. 012 0. 117 0. 166 0. 148 0. 188 0. 171 0. 055 0. 050 0. 048 0. 070
6 -0. 090 0. 348 0. 165 0. 165 0. 131 0. 275 0. 208 0. 251 0. 062 0. 112 0. 155 0. 077
7 -0. 086 0. 098 0. 564 0. 218 0. 180 0. 239
0. 217 0. 288 0. 052 0. 214 0. 297 0. 109
8 0. 265 0. 049 0. 489
-0. 017 0. 168 0. 153 0. 111 0. 249
0. 088 0. 105 0. 081 0. 065
9 -0. 022 0. 094 0. 314 -0. 063 0. 180 0. 266 0. 267 0. 178 0. 044 0. 133 0. 095 0. 071
10 0. 125 -0. 033 0. 171 0. 379 0. 207 0. 184 0. 220 0. 239 0. 146 0. 077 0. 203 0. 083
11 0. 265 -0. 008 0. 090 0. 409 0. 229 0. 162 0. 164 0. 223 0. 144 0. 053 0. 087 0. 085
12 0. 030 0. 151 -0. 097 0. 457 0. 142 0. 274 0. 153 0. 174 0. 105 0. 117 0. 114 0. 125
H
H
BIGUARTIMIN: NO SIM PLE STRUCTURE
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 614 0. 368 -0. 024 0. 115 0. 345 0. 597 0. 348 0. 555 0. 100 0. 154 0. 164 0. 132
2 0. 756 -0. 039 0. 038
0. 052 0. 537 0. 645 0. 463 0. 436 0. 176 0. 143 0. 130 0. 138
3 0. 526 -0. 060 -0. Oil 0. 014 0. 576 0. 288 0. 573 0. 690 0. 153 0. 082 0. 140 0. 140
4 -0. 106 0. 598 0. 117 -0. 027 0. 430 0. 548 0. 512 0. 393 0. 104 0. 161 0. 148 0. 110
5 0. 084 0. 490 0. 150 0. 087 0. 312 0. 530 0. 539 0. 420 0. 080 0. 147 0. 142 0. 112
6 0. 104 0.
391 0. 079 0. 084 0. 283 0. 559
0. 487 0. 393 0. 090 0. 144 0. 135 0. 107
7 -0. 074 0. 295 0. 464 0. 111 0. 389
0. 806 0. 802 0. 598 0. 101 0. 251 0. 371 0. 191
8 0. 266 0. 193 0. 337 0. 075 0. 446 0. 551 0. 795 0. 772 0. 132 0. 176 0. 207 0. 195
9
-0. 047 0. 138 0. 300 0. 107 0. 197 0. 607 0. 689 0. 667 0. 067 0. 170 0. 248 0. 205
10 0. 146 0. 074 0. 230 0. 278 0. 437 0. 467 0. 564 0. 657 0. 106 0. 122 0. 171 0. 190
11 0. 329 0. 104 0. 126 0. 272 0. 439 0.
709
0. 648 0. 598 0. 129 0. 160 0. 144 0. 169
12 0.
049
0. 073 0. 019
0. 379 0. 408 0. 549 0. 297 0. 979 0. 102 0. 165 0. 082 0. 285
Large Sample Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 615 0. 343 -0. 052 0. 166 0. 249 0. 230 0. 429 0. 212 0. 066 0.
129
0. 137 0. 063
2 0. 766 -0. 055 0. 106 0. 026 0. 186 0. 215 0. 299 0. 179 0. 133 0. 057 0. 071 0. 075
3 0. 486 -0. 026 -0. 040 0. 001 0. 160 0. 116 0. 125 0. 208 0. 044 0. 029 0. 033 0. 046
4 -0. 071 0. 558 0. 120 -0. 050 0. 211 0. 455 0. 822 0. 214 0. 059 0. 147 0. 184 0. 070
5 0. 060 0. 419 0. 137 0. 134 0. 196 0. 412 0. 630 0. 246 0. 050 0. 134 0. 159 0. 069
6 0. 082 0. 324 0. 056 0. 173 0. 158 0. 355 0. 457 0. 300 0. 047 0. 111 0. 107 0. 092
7 -0. 079 0. 216 0. 527 0. 019 0. 161 0. 892 0. 523 0. 193 0. 050 0.
249
0. 286 0. 055
8 0. 243 0. 154 0. 387 -0. 005 0. 227 0. 741 0. 404 0. 257 0. 064 0. 175 0. 104 0. 075
9 -0. 076 0. 153 0. 217 0. 069 0. 231 0. 393 0. 359 0. 237 0. 062 0. 127 0. 125 0. 063
10 0. 155 0. 070 0. 217 0. 208 0.
249
0. 555 0. 322 0. 377 0. 074 0. 138 0. 098
0.
099
11 0. 277 0. 061 0. 127 0. 310 0. 245 0. 306 0. 236 0. 252 0. 078 0. 094 0. 072 0. 066
12 0. 042 0. 034 -0. 010 0. 417 0. 145 0. 148 0. 1 66 0. 235 0. 044 0. 051 0. 042 0. 150
( — 1
M
00
PROCRUSTES: STRONG SIMPLE STRUCTURE, UNIVOCAL TESTS
Mean
Small Sample Size
Range
RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 782 -0.004 0. 013 0. 002 0. 206 0. 289 0. 286 0. 359 0. 058 0. 075 0.079 0. 096
2 0. 797 -0. 012 -0. 002 -0.022 0. 183 0. 234 0. 216 0. 281 0. 132 0. 050 0. 060 0. 065
3 0. 548 -0.032 0. 043 0. 004 0. 281 0. 417 0. 377 0.479 0. 082 0. 102 0. 103 0. 137
4 -0.062 0. 568 0. 004 0. 010 0. 329 0. 310 0. 402 0.401 0. 097 0. 165 0.095 0. 102
5 0. 015 0. 517 0. Oil -0. 026 0. 206 0.499 0. 612 0. 360 0. 064 0.122 0. 154 0. 113
6 0. 016 0. 414 0. 042 -0.035 0. 340 0. 370 0. 331 0. 766 0. 075 0.104 0.100 0. 182
7 0.015 0. 014 0. 544 0. 010 0. 367 0. 335 0. 466 0. 355 0. 075 0.091 0. 314 0.091
8 0. 030 0. 025 0. 478 0. 017 0. 423 0. 396 0. 335 0. 655 0. 115 0. 114 0.105 0.176
9 0. 003 -0.019 0. 346 0. 028 0. 395 0. 445 0. 469 0. 830 0. 093 0. 121 0. 152 0. 209
10 0. 000 -0. 006 0. 030 0. 407 0. 410 0. 443 0. 748 0. 421 0. 102 0. 128 0. 147 0.131
11 0. 005 -0.016 0. 021 0. 336 0. 306 0. 515 0. 588 0. 450 0.091 0. 147 0. 132 0.143
12 -0. 034 0.017 0. 008 0. 248 0.482 0.531
Large Sample
0. 658
Size
0.456 0. 121 0.149 0.162 0. 139
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 781 0. 001 0. 003 -0. 018 0. 128 0. 123 0. 120 0. 162 0. 035 0. 031 0. 035 0. 052
2 0. 809 -0. 005 0. 001 0. 007 0. 114 0. 102 0.
099
0. 076 0. 113 0. 024 0. 024 0. 022
3 0. 490 -0.001 -0.005 0. 016 0. 220 0. 248 0. 221 0. 245 0. 045 0. 058 0. 060 0. 077
4 -0. 012 0. 558 0. 005 0. 001 0. 110 0. 151 0. 126 0. 183 0. 032 0. 154 0. 137 0. 047
5 0. 003 0. 510 0. 007 -0. 002 0. 110 0. 142 0. 169 0. 184 0. 031 0. 052 0. 043 0. 049
6 0. 008 0.404 -0. 012 0. 009 0. 137 0. 247 0. 201 0. 266 0. 039 0. 068 0.062 0. 070
7 0. 003 -0. 004 0. 539 -0. 003 0. 093 0. 135 0. 199 0. 248 0. 025 0.
039 0. 303 0. 051
8 -0. 009 0. 004 0. 473 0. 010 0. 084 0. 161 0. 170 0. 231 0. 028 0. 043 0. 049 0. 066
9 0. Oil 0. 018 0. 290 -0. 021 0. 212 0. 213 0. 225 0. 316 0. 052 0. 061 0. 072 0. 078
10 -0. 006 0. 007 0. 010 0. 314 0. 159 0. 274 0. 151 0. 163 0. 040 0.
059
0. 040 0. 057
11 0. 005 -0. 010 -0.002 0. 309
0. 154 0. 192 0. 368 0. 297 0. 050 0. 053 0. 073 0. 115
12 -0. 003 0. 020 -0.019 0. 242 0. 196 0. 316 0. 219 0. 371 0. 056 0. 087 0.065 0.
£119
o
PROCRUSTES: STRONG SIMPLE STRUCTURE, COMPLEX TESTS
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 777 0.459 0. 131 0. 384 0. 187 0. 256 0. 746 0. 263 0. 055 0. 076 0. 205 0.096
2 0. 833 0. 082 0. 183 0. 025 0. 210 0. 171 0. 449 0. 219 0. 102 0.098 0.096 0.059
3 0. 541 -0.023 0. 000 -0.056 0. 378 0. 351 0. 378 0. 285 0. 086 0.086 0. 098 0. 098
4 -0. 041 0. 615 0. 039 0.177 0. 248 0. 349 0. 483 0. 282 0. 074 0.123 0. 107 0. 071
5 0. 239 0. 524 0. 115 0. 436 0. 258 0.419 0. 734 0. 218 0. 072 0.108 0. 199 0. 105
6 0. 031 0. 394 0. 110 0. 333 0. 334 0. 343 0. 456 0.462 0. 078 0.103 0. 128 0. 126
7 0. 082 0. 454 0. 584 0. 346 0. 217 0. 391 0. 412 0. 394 9. 101 0. 131 0. 274 0.093
8 0. 364 0.188 0. 461 0. 017 0. 392 0. 325 0. 337 0.462 0.108 0. 147 0. 089 0. 103
9 -0. 017 0.095 0. 343 0. 030 0. 303 0. 373 0. 671 0. 527 0. 085 0. 182 0. 176 0. 132
10 0. 350 -0.016 0. 133 0. 407 0. 356 0. 375 0. 615 0. 407 0.100 0.096 0. 176 0. 116
11 0. 523 0.056 0. 270 0. 382 0. 277 0. 335 0. 459 0. 338 0. 110 0. 110 0. 138 0. 080
12 0. 080 0.016 0. 116 0. 536 0.326 0.268
Large Sample
0. 538
Size
0. 281 0. 155 0. 077 0.174 0. 101
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 757 0. 411 0. 055 0. 318 0. 144 0. 142 0. 131 0. 173 0. 056 0.094 0. 064 0. 040
2 0. 833 0. 063 0. 155 0. 003 0. 120 0.108 0. 187 0. 077 0. 098 0. 067 0.056 0. 021
3 0. 496 -0. 018 -0.053 -0.024 0. 165 0. 215 0. 185 0. 181 0. 043 0.052 0. 074 0. 056
4 -0. 039 0. 615 0. 033 0. I l l 0. 131 0. 138 0. 169 0.172 0. 054 0. 097 0. 057 0. 093
5 0. 209 0. 464 0. 062 0. 414 0. 131 0. 272 0. 137 0.182 0.039 0. 109 0. 071 0. 081
6 0. 027 0. 356 0. 090 0. 372 0. 123 0. 241 0. 560 0. 217 0. 044 0.078 0. 131 0.093
7 0. 057 0. 413 0. 564 0. 302 0. 086 0. 245 0. 173 0. 207 0. 060 0. 084 0. 278 0.049
8 0. 295 0.182 0.439 0. 022 0. 118 0. 374 0.410 0.403 0.067 0. 143 0. 085 0.082
9 -0. 035 0. 119 0. 288 0. 003 0. 192 0. 143 0. 229 0. 228 0. 063 0.127 0. 078 0.058
10 0. 340 -0. 023 0. 061 0. 349 0. 197 0. 276 0. 139 0.133 0. 062 0. 055 0, 197 0. 041
11 0. 478 0.036 0. 262 0. 392 0. 143 0. 208 0. 233 0. 128 0. 048 0. 059 0. 085 0. 040
12 0. 082 -0. 017 0. 090 0. 517 0. 162 0. 166 0. 143 0.174 0. 135 0.045 0. 102 0. 090
ro
o
PROCRUSTES: WEAKER SIMPLE STRUCTURE, COMPLEX TESTS
Small Sample Size
Mean Range RMS
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 742 0. 470 0. 060 0. 322 0. 205 0. 200 0.
249
0. 203 0. 074 0. 058 0. 089
0. 055
2 0. 843 -0. 041 0. 177 0. 092 0. 182 0. 220 0. 282 0. 195 0. 092 0. 068 0. 067 0. 110
3 0. 562 -0. 033 0. 013 -0. 042 0. 260 0. 362 0. 352 0. 357 0. 090 c. 087 0. 094 0. 108
4 -0. 032 0. 676 0. 033 0. 182 0. 165 0. 271 0. 282 0. 324 0, 056 0. 071 0. 075 0. 196
5 0. 204 0. 598 -0. 001 0. 318 0. 220 0. 355 0. 416 0. 248 0. 056 0. 100 0. 105 0. 075
6 -0. 005 0. 404 0. 179
0. 235 0. 337 0. 467 0. 366 0. 518 0. 070 0. 109
0. 203 0. 145
7 0. 046 0. 256 0. 587 0. 310 0. 242 0. 274 0. 286 0. 369 0. 079
0. 116 0. 262 0. 083
8 0. 410 0. 182 0. 487 0. 034 0. 362 0. 371 0. 388 0. 582 0. 117 0. 202 0. 102 0. 129
9
0, 008 0. 119 0. 361 0. 020 0. 299 0. 329 0. 563 0. 474 0. 090 0. 160 0. 179 0. 133
10 0. 234 -0. 010 0. 241 0. 455 0. 262 0.
439
0. 456 0. 403 0. 097 0. 102 0.
259 0. 142
11 0. 433 0. 931 0. 158 0. 421 0. 225 0. 313 0. 484 0. 297 0. 078 0. 099 0. 118 0. 084
12 0. 125 0. 138 -0.
039 0. 535 0. 375 0. 332 0.
459
0. 283 0. 151 0. 157 0. 106 0. 099
Large Sample Size
1 2 3 4 1 2 3 4 1 2 3 4
1 0. 720 0. 467 0. 002 0. 236 0. 104 0. 106 0. 128 0. 113 0. 085 0. 046 0. 031 0. 070
2 0. 837 -0. 048 0. 166 0. 047 0. 108 0. 079
0. 131 0. 063 0. 089 0. 053 0. 045 0. 051
3 0. 493 0. 000 -0. 032 -0. 004 0. 202 0. 231 0. 194 0. 191 0. 043 0. 056 0. 062 0. 052
4 0. 001 0. 664 0. 024 0. 115 0. 118 0. 127 0. 117 0. 178 0. 030 0. 053 0. 037 0. 123
5 0. 166 0. 572 -0. 041 0. 281 0. 146 0. 119 0. 161 0. 146 0. 047 0. 036 0. 057 0. 081
6 -0. 025 0. 388 0. 132 0. 276 0. 114 0. 186 0. 175 0. 177 0. 043 0. 057 0. 143 0. 058
7 0. 029 C. 200 0.
559 0. 257 0. 097 0. 056 0. 155 0. 175 0. 037 0. 152 0. 284 0. 057
8 0. 343 0. 148 0. 473 -0. 007 0. 121 0. 149
0. 127 0. 138 0. 036 0. 155 0. 041 0. 039
9 0. 010 0. 155 0. 284 -0. 035 0. 172 0. 220 0. 211 0. 216 0. 047 0. 163 0. 066 0. 068
10 0. 220 -0. 004 0. 216 0. 373 0. 199 0. 245 0. 202 0. 214 0. 096 0. 046 0. 218 0. 062
11 0. 363 0. 009 0. 145 0. 404 0. 156 0. 156 0. 215 0. 154 0. 095 0. 046 0. 077 0. 046
12 0. 108 0. 128 -0. 063 0. 508 0. 132 0. 141 0. 122 0. 178 0. 115 0. 138 0.
069 0. 100
M
l\»
H
PROCRUSTES: NO SIMPLE STRUCTURE
Mean
1 2 3 4
1 0. 772 0. 470 0. 133 0. 385
2 0. 839 0. 056 0. I l l 0. 116
3 0. 548 -0.031 -0. 003 -0. 021
4 -0. 034 0. 692 0. 318 0. 289
5 0. 214 0. 592 0. 326 0. 423
6 0. 221 0. 448 0. 198 0. 339
7 0. 028 0.479 0. 677 0. 414
8 0. 385 0. 372 0. 518 0. 278
9 -0. 006 0. 246 0. 358 0. 299
10 0. 298 0. 175 0. 307 0.461
11 0. 487 0. 198 0. 249 0. 449
12 0. 213 0.081 0. 073 0. 506
1 2 3 4
1 0. 758 0.436 0. 012 0. 280
2 0. 833 0. 022 0. 138 0. 037
3 0. 500 0. 007 -0. 024 -0.020
4 -0.010 0. 649 0. 200 0. 187
5 0. 185 0. 515 0. 214 0. 347
6 0. 205 0. 390 0. 113 0. 328
7 0. 014 0. 352 0. 606 0. 286
8 0. 329 0. 275 0. 460 0. 181
9 -0.016 0. 227 0. 254 0. 213
10 0. 282 0. 146 0. 263 0. 334
11 0. 438 0.129 0. 168 0. 408
12 0. 204 0. 052 0. 009 0.464
Small Sample Size
Range
1 2 3 4
0. 152 0.163 0.436 0. 282
0.161 0. 218 0. 263 0. 308
0. 345 0. 295 0. 382 0. 326
0. 209 0. 280 0. 686 0. 272
0. 202 0. 361 0. 555 0. 348
0. 339 0. 388 0. 387 0. 534
0. 268 0. 385 0. 214 0. 323
0. 304 0. 356 0. 363 0. 467
0. 241 0.450 0. 629 0. 549
0. 261 0. 397 0. 393 0. 540
0. 270 0. 350 0. 454 0. 407
0. 292 0. 406 0.473 0. 332
Large Sample Size
1 2 3 4
0. 093 0.118 0. 122 0. 139
0. 110 0. 118 0. 095 0.089
0. 168 0. 217 0. 202 0. 206
0. 090 0.137 0. 143 0.165
0. 112 0.164 0. 181 0. 211
0.136 0. 250 0.158 0. 257
0.096 0. 120 0. 145 0. 150
0.107 0. 139 0.124 0. 156
0.170 0. 212 0.199 0. 235
0. 196 0. 287 0.182 0. 321
0. 128 0. 205 0. 213 0. 169
0. 121 0. 122 0. 108 0. 212
RMS
1 2 3 4
0. 052 0. 052 0. 171 0. 109
0. 094 0. 084 0. 112 0. 149
0. 088 0. 090 0. 094 0. 106
0. 065 0. 072 0. 188 0. 126
0. 056 0. 097 0. 151 0. 109
0. 080 0. 105 0. 117 0. 131
0. 071 0. 168 0. 174 0. 144
0. 095 0. 131 0. 119 0. 124
0. 064 0. 114 0. 199 0. 162
0. 075 0. 098 0. 118 0. 182
0. 086 0. 118 0. 130 0. 098
0. 076 0. 125 0. 141 0. 124
1 2 3 4
0. 052 0. 068 0. 028 0. 041
0. 091 0. 031 0. 066 0. 050
0. 040 0. 052 0. 058 0. 065
0. 027 0. 074 0. 038 0. 038
0. 034 0. 058 0. 059 0. 047
0. 036 0. 070 0. 064 0. 065
0. 026 0. 031 0. 240 0. 034
0. 036 0. 044 0. 042 0. 063
0. 052 0. 058 0. 063 0. 066
0. 054 0. 064 0. 055 0. 083
0. 038 0.
059 0. 053 0. 045
0. 036 0. 065 0. 032 0. 141
r o
ro
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Asset Metadata
Creator
Hamburger, Charles David
(author)
Core Title
Factorial Stability As A Function Of Analytical Rotational Method, Type Of Simple Structure, And Size Of Sample
Degree
Doctor of Philosophy
Degree Program
Psychology
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
OAI-PMH Harvest,psychology, general
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
Guilford, Joy P. (
committee chair
), Cliff, Norman (
committee member
), Whiteman, Albert Leon (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c18-359128
Unique identifier
UC11359034
Identifier
6503106.pdf (filename),usctheses-c18-359128 (legacy record id)
Legacy Identifier
6503106.pdf
Dmrecord
359128
Document Type
Dissertation
Rights
Hamburger, Charles David
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA
Tags
psychology, general