DISTORTION OF FACTOR LOADINGS AS A FUNCTION OF
THE NUMBER OF FACTORS ROTATED UNDER
VARYING LEVELS OF COMMON VARIANCE AND ERROR
By
AZZA SHOIWKET GUERTIN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
ACKNOWLEDGMENTS
The writer wishes to thank her supervisory Committee members for
their guidance and support. To my Chairman, Dr. Vynce A. Hines, I owe
a debt of gratitude for his encouragement and support throughout this
enterprise.
For the diligent guidance and editorial assistance my deepest
appreciation goss to the cochairman of my Committee, Dr. William B.
Ware. I will be forever grateful.
Sincere thanks go to Dr. Lewis Berner and Dr. Robert S. Soar for
serving patiently and supportively as Committee members.
Special thanks go to Dr. Wilson H. Guertin for the technical
assistance that only he could have supplied.
To my friend and former advisor Dr. Ned E. Bingham, many thanks
for all the faith you have had in me.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS. . . . . . . . . . . . i
LIST OF TABLES . . . . . . . . . . . . . v
LIST OF FIGURES. . . . . . . . . ... ....... .viii
ABSTRACT . . . . . . .... .. .. . . . . .. ix
CHAPTER I Introduction. . . . . . . . ... . 1
The Purpos . . . . . . . . . 2
The Procedure: An Overview . . . . . . 3
Some Limitations. . . . . ... . .. 5
Significance of the Study . . . . . . 6
Organization of the Study . . . . . . 6
CHAPTER II Related Literature. . . . . . ... .. 7
Procedural Rules and Criteria for Rotation. . . 7
The statistical approach. . . . . . . 8
The psychometric approach . . . . . . 9
Alternative procedures for determining the
number of factors . . . . . . . 12
Studies on the Effects of Under and Overrotation. 15
Relevant Factor Analytic Methods and Procedures .20
Common factors, principal axes and
communalities . . . . . . . . 20
Simple structure and the Varimax method .... .22
The use of the rootmeansquares (RMS). .... .23
Summary . . . . . . . . . . . 24
CHAPTER III Methodology . . . . . . . . . . 26
The Statistical Hypotheses. . . . . . ... 26
Selection of the Matrices . . . . . . 27
The Cormmion Variance Adjustment. . . . . ... 28
The Choice and Generation of Eiro;. . . ... 29
Procedures. .. . . . . . . . . 30
Problem One . . . . ... .. . . . 30
Problem Tw . . . . . . . . .. 40
Problem Three . . . .... . . . . 40
Problem Four. .... .. . . . . . . 42
Summary . ... . . . . . . . . 45
TABLE OF CONTENTS continued
Page
CHAPTER IV Results . . . . . . . . ... .... . 47
Problem One . . . . . . . .... . 47
Problem Two . . . . . . . . . 55
Problem Three. . . . . . . . .57
Problem Four. . . . . . . . . ... 72
Summary . . . . ... . . . . . 72
CHAPTER V Discussion and Summary. . . . . . . ... 81
Discussion ................... . 82
The findings in relation to the literature. .. 82
Comparison of the Problem Matrices. . . .. 84
The effects of common variance and error on
the number of factors rotated . . . ... 87
A direction for future research . . . . 89
Summary . . . . . . . .... .. .. 90
APPENDIX A . . . . . .. . . . . . . . 92
APPENDIX B . . . . . . . . ... . . .... 95
BIBLIOGRAPHY . . . . . . . . . . . . 114
BIOGRAPHICAL SKETCH. . . . . . . . . ... ...... 119
LIST OF TABLES
Page
1 The Input Factor Matrix Taken from Fruchter . . . ... 31
2 Fruchter's Matrix Adjusted to Account for
Three Levels of Common Variance . . . . . . .... 32
3 intercorrelation Matrices R' and R" for
Replication One with 30% of Common Variance . . . ... 33
4 The Criterion Matrices with Three Levels
of Common Variance. . . . . . .. .. . . . 35
5 Error Values for Replication One
Pseudcrandomly Generated Error Values Added to the Adjusted
Intercorrelation Matrix R' and Reflected in R''. The
Standard Error is .09 and the Mean is .01 . . . . . 36
6 Total RootMeanSquares for 30% Common Variance
and Three Levels of Error . . . . . . . ... 38
7 The Input Factor Matrix Taken from Harman . . . .... 41
8 The Input Factor Matrix Taken from Mulaik . . . .... 43
9 The Input Factor Matrix Taken from Whimbey and Denenberg . 44
10 Means (X) and Standard Deviations (S.D.) for Each Ten
Replications Under Three Levels of Common Variance, Three
Levels of Error and Five Rotations. . . . . . . ... 48
11 ANOVA Summary Table for RMS Mean Values for
Five Different Rotations. . . . ... . . . . 52
12 ANOVA Summary Tables for Linear, Quadratic and Cubic Trends .54
13 Trend Components, Observed aid Predicted RMS Means
for Five Different Rotations (B). . . . . . . . 56
14 Means (X) and Standard Deviations (S.D.) for Each Ten
Replications Under Three Levels of Common Variance, Three
Levels of Error and Five Rotations. . . . . . . ... 58
15 ANOVA Summary for RMS Mean Values for Five
Different Rotations . . .. . . .. . . . . 62
LIST OF TABLES continued
Pag_e
15 ANOVA Summary Table for Linear, Quadratic and Cubic Trends. .. 63
17 Trend Components, Observed and Predicted RMS Means for
Five Different Rotations (B). . . . . . . . . 64
18 Means (X) and Standard Deviations (S.D.) for Each Ten
Replications Under Three Levels of Common Variance, Three
Levels of Error and Five Rotations. . . . . . . ... 65
19 ANOVA Summary for RMS Mean Values for
Five Different Rotations. . . . . . . . . ... 69
20 ANOVA Summary Tables for Linear, Quadratic and Cubic Trends .70
21 Trend Components, Observed and Predicted RMS Means for
Five Different Rotations (B). . . . . . . . . 71
22 Means (X) and Standard Deviations (S.D.) for Each Ten
Replications Under Three Levels of Common Variance, Three
Levels of Error and Seven Rotations . . . . . .... 73
23 ANOVA Suzmmary for ':IS Mean Values for
Seven Different Rotations . . . . . . . . . 77
24 ANOVA Summary Table for Linear, Quadratic, and Cubic Trends .78
25 Trend Components, Observed and Predicted RMS Means for
Seven Different Rotations (B) . . . . . . . .. 79
APPENDICES
Al Means of Random Error Under Three Levels of Standard
Error and Three Levels of Common Variance (Expected
Mean Value = 0.0) . .. ... . . . . . . . . 93
A2 Standard Deviations for Three Levels of Random Error
Under Three Levels of Common Variance . . . . .... 94
BI The Matrix Adjusted for Three Levels of Common Variance . . 96
B2 The Criterion Matrices with Three Levels of Common Variance .99
B3 The Matrix Adjusted to Account for Three Levels
of Common Variance. .... . . . . . . . . 102
B4 The Criterion Matrices with Three Levels of Common Variance . 105
LIST OF TABLES continued
B5 The Matrix Adjusted to Account for Three Levels
of Common Variance . . . . . . . . . . . 108
B6 The Criterion Matrices with Three Levels of Common Variance . 111
LIST OF FIGURES
Page
1 RMS means for the five different rotations for Problem One. 49
2 RMS means for the interaction of the five different
rotations with the three levels of common variance
for Problem One . . . . ... .. . . . . . . 50
3 RMS means for the interaction of the five different
rotations with the three levels of error for
Problem One . . . . . . . . . . . . . 51
4 IMS means for the five different rotations for Problem Two. . 59
5 RMS means for the interaction of the five different
rotations with the three levels of common variance
for Problem Two . . . . . .. . . . . . . 60
6 RMS means for the interaction of the five different
rotations with the three levels of error
for Problem Two . . . . . . ... . . . . . 61
7 RMS means for the five different rotations for Problem Three. 66
8 RMS means for the interaction of the five different
rotations with the three levels of common variance
for Problem Three . . . . . . . . . . . 67
9 RMS means for the interaction of the five different
rotations with the three levels of error
for Problem Three . . . ... .. . . . . . . 68
10 RMS means for the seven different rotations for Problem Four. 74
11 RMS means for the interaction of the seven different
rotations with the three levels of common variance
for Problem Four. . . . ... . . . . . . . 75
12 RMS means for the interaction of the seven different
rotations with the three levels of error
for Problem Four. ..... . . . . . ...... 76
13 Comparison of the RMS mean values for the different
rotations for the Four Problems . . . . . . . 83
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor oF Philosophy
DISTORTION OF FACTOR LOADINGS AS A FUNCTION OF
THE NUMBER OF FACTORS ROTATED UNDER
VARYING LEVELS OF COMMON VARIANCE AND ERROR
By
Azza Showket Guertin
June 1977
Chairman: Vynce A. Hines
Major Department: Foundations of Education
This study examined factor loading stability as a function of the
number of factors rotated for four problem matrices under three levels of
common variance: 30%, 45%,and 60%; and three levels of sample size: 100,
200,and 500. The sample sizes correspond to standard error values of .10,
.07,and .04 respectively.
Four representative problem factor matrices were selected from the
literature. Each was treated in the following manner. The matrix was
adjusted to account for each of the three specified levels of common
variance. The intercorrelation matrix was obtained for each adjusted
matrix; the latter was factor analyzed by the principal axes method and
a criterion commonfactor matrix obtained. For each problem, the three
criterion matrices, adjusted to the three levels of common variance, had
the same number of factors as that of the original problem matrix.
Computergenerated pseudorandom error at each of the three levels
specified was added to the intercorrelation matrices mentioned above,
and the errorladen matrices were factor analyzed, principal axes extracted,
and several factor rotations performed. The factor rotations involved a
series of successive under and overrotations below and above the correct
number of factors for a given problem matrix. Rootmeansquare (RMS)
deviation values were calculated between the factor loadings of each
criterion matrix and the corresponding factor loadings in each of the
successively rotated factor matrices. The RMS values were computed for
only the initial two or three rotated factors for each problem. The
procedures of the addition of random error to the intercorrelation matrix,
the Factor extraction, the successive rotations, and the calculation of
the RMS discrepancies were replicated ten times under each of the nine
conditions of common variance by error.
The obtained RMS mean values were plotted and tested for significance
using a multifactor repeated measures ANOVA design. Linear, quadratic,
and cubic trend analyses were performed. Goodness of fit of the plotted
curves of the RMS means for the four problems was examined by computing
predicted PIMS means and comparing them with the observed RMS means.
For all four problems at the .05 level, the ANOVA results were
significant for the number of factors rotated; this was also true for
the rotation x common variance and rotation x error interactions. The
three trend analyses were also found significant at the .05 level.
The polynomial cubic equation
Y = k0 + I1X1 + 2Xl 2 + 03X3 + E
gave the best approximation for the trend of the data for all four
selected problem matrices.
This study provided support for the literature's position on under
rotation; namely, it was not recommended. The view on overrotation,
which advocated overrotation by one or two extra factors, could be neither
supported nor rejected by the findings of the study.
There did not seem to be a clear relationship between the number of
variables and/or factors for a given matrix and its factor loading
stability. Factors with large amounts of common variance and low levels
of error were found to be the most stable.
Chairman
CHAPTER I
Introduction
The number of factors to rotate has been long recognized as a problem
in factor analysis, since to a certain extent the decision involves the
skill and subjective judgment of the analyst (Fruchter, 1954). With the
adveit of computers most decisions are made automatically, but the decision
on the number of factors to rotate and interpret ultimately rests upon the
investigator (Guertin & Bailey, 1970).
Even for the classic Holzinger and Harman 24 psychological tests
problem, "Harman answers. .unequivocally that the best number of factors
for the problem is either four or five but refuses to co,'rriit himself as to
which of these two answers is better" (Kaiser, 1970, p. 412413). Regarding
factor analytic methodology, it was Kaiser's (1970) position that "the
most important future work, as I see it, should continue to concentrate on
the numberoffactors question" (p. 414).
A survey cf the literature revealed Kaiser's concern was not unique.
Many studies have focused upon the numberoffactors problem. Suggestions,
rationales,and solutions abound in the literature (e.g., Cattell, 1966b;
Horn, 1965; Hump'ireys & Ilgen, 1969; Linn, 1965, .'58; Mosier, 1939).
The issue of the number of factors to rotate is directly related to
tnatof.th etiumber of factors to extract from the intercorrelation matrix.
However, widely available computer programs for the principal axes
factoring procedure extract simultaneous factors that account for all the
variance of the intercorrelation matrix (Guertin & Bailey, 1970). The
question then centers upon the number of extracted principal axes factors
which must be carried into rotation to yield the final interpretable
factors.
This task would be relatively simple if the earlier extracted large
factors contained only commonfactor variance and the later small ones
contained nothing but error. Unfortunately this is not the case, for "in
the extraction process one does not begin to extract only substantive
factors until one suddenly gets to 'error' factors, but that some degree
of error variance is present from the beginning" (Cattell, 1966a,p. 201).
It is the process of rotation that enables the researcher to separate
out substantive 'real' factors from those of error. The aim is to rotate
so that the maximum number of 'real' factors is retained "while cutting
off as much as possible of the error variance as will not simultaneously
carry away too much real variance" (Cattell, 1966a,p. 204). The goal,
then, is to rotate substantive factors and ignore the errorladen ones.
The Purpose
While several methods and criteria have been suggested to determine
the correct number of factors, none of these seems to have gained
unanimous acceptance. As Guilford (1974) has pointed out, "The need for
rotation of axes in factor analysis is. .the most serious weakness of
this very useful method of reduction of numerical data. . ." (p. 498).
In light of this weakness, it would be desirable to employ an empirical
approach in examining this aspect of factor analysis. The goal was to
gain Further insight into this troublesome area.
This study examined factor loading stability for four different
factor matrices of known solutions under the following conditions: (a)
variation of the number of factor rotations, below and above the known
number for the criterion matrices; (b) three levels of common variance
accounted for by the factor matrices; (c) three levels of random error
added to the matrices, i.e., for three different sample sizes.
This stability would be reflected, in part at least, by the
characteristics of the shape of plotted curves of the means of rootmean
square discrepancies between each criterion matrix and its corresponding
manipulated matrix under the experimental conditions identified above.
The Procedure: An Overview
Four matrices of known factor solutions were selected. These matrices
were based upon different numbers of variables. Each factor matrix was
adjusted to account for three proportions of common variance; the inter
correlation matrix for each factor matrix was obtained, factored, and
rotated orthogonally by the Varimax method (Kaiser, 1958). Thus for the
four selected problem matrices, a total of twelve adjusted criterion
matrices was obtained  three per problem.
Error representing three levels of sample size was added to each of the
four intercorrelation matrices. These errorladen matrices were subse
quently factor analyzed by the principal axes method and several rotations
to the Varimax criterion were tried (Kaiser, 1958). The number of factors
rotated ranged from two or three factors less to two or three factors more
than that of the original problem matrix.
The rootmeansquare (RHS) discrepancies betw ,n the first two or
three factors of each criterion matrix and their corresponding factors in
each trial rotation of the erroradded factor matrices were calculated.
I'T R:S statistic is an appropriate, common statistical measure used to
make direct co; :parisons of corresponding factor loadings (Harman, 1976,
p. 2917).
Ten replications were performed under each condition of common
variance by level of error, so that, for example, for a fourfactor prob
ler. with five trial rotations, 450 RtlS values were obtained for later
analysis.
The means of the total RMS's for the trial rotations for each prob
lem were plotted. Null hypotheses about these means were tested for
significance at the .05 level by the analysis of variance multifactor
repeated measures design (Winer, 1971).
Further analyses were completed to examine the trends in the plotted
curves of the means of the RMS's, and, finally, orthogonal polynomial
coefficients were used to solve for the predicted values of these means.
These procedures were conducted with the aid of several "software"
options. The errors were produced by the use of the Fortran Subroutine
NDIST which generates pseudorandom error as specified. The factor
analyses were performed by using a modified version of factor analytic
program ED 501, (Guertin & Bailey, 1970), available through the University
of Florida Educational Evaluation Library, as adapted for the IBM 360.
The multifactor repeated measures analysis of variance was completed by
using computer program BED 80V (Dixon, 1974).
For each representative problem examined the research questions to
be answered were:
1. 1, hat effect does under end overratation have on the loadings of
a known factor matrix given three levels of sample size and
three levels of cormon variance? Since the RMS's are measures
of deviation, a test of differences about their means should be
an indicator of this effect.
2. If the F test, at the .05 level of significance indicates a trend
in the data, what is the nature of this trend? What degree
equation best fits the trend of the data, i.e., what is the shape
obtained when the means of the RMS's are plotted?
Some Limitations
This study had several limitations. Only one problem (matrix) of
each size was used. To do otherwise would have been impractical in terms
of cost and presentation.
When each intercorrelation matrix was factored, only commonfactor
analysis was employed where communalities were known, i.e., neither
principal components analysis nor image analysis was used. All rotations
of the principal axes were performed to the Varimax criterion regardless
cf the original rotation of the input matrices.
Another limiLation to the study was that only the first two or three
factors in each criterion matrix were matched by the trial rotation factors
and their RMiS's calculated. Only a limited number of the factors con
tributing to the shape of the line or curve of discrepancies were investi
gated. Specifically, underrotation and overrotation were examined as they
affect only the first two or three factors of the criterion matrix. The
overall effects on all the factors were not assessed. The reason for this
iimitation was the logistics of being unable to calculate the RMS dis
crepancies when only two or three trial factors are rotated. This is per
haps the most important limitation to the study.
Because of the practical limitations for the number of trial rotations
to be examined and reported, only two or three factors below and two or
three factors above the ideal number in the criterion matrices were
exami ned.
For each problem, the variable of sample size was limited to three
levels judged to be fairly representative of those used in factor analytic
studies. The proportion of common variance accounted for was confined to
only three levels, also for reasons of representativeness. Matrices
intermediate to, or outside these ranges might give different results.
Significance oF the Study
One focus of commonfactor analytic methodology has been the number
offactors problem, i.e., the optimum number of common factors that should
be carried into rotation to yield a meaningful solution. A recurrent,
though not unanimous, theme has been that it is best to rotate one or two
additional factors than to underrotate. It seemed, therefore, that an
empirical examination of this issue was appropriate. New information
might be gained that could aid the researcher in deciding on the ideal
number of factors to retain and interpret.
Organization of the Study
Chapter I has dealt with the purpose of and the background to the
study; an overview of the procedure; the research questions; the limita
tions and the significance of the study. A review of the related
literature is presented in Chapter II. The complete procedure and a
detailed description of the problem matrices are discussed in Chapter III.
Results are presented in Chapter IV. A discussion of the results, con
clusions, and the summary appear in Chapter V.
CHAPTER II
Related Literature
The purpose of this study was to examine factor loading stability as
a function of under and overrotation of common factors, under three levels
of common variance and three levels of error. In this Chapter is reviewed
the literature concerned with the issue of the number of factors to rotate
as it affects factor stability in commonfactor analysis.
For the purpose of presentation, the Chapter is divided into three
major sections: (a) the procedural rules and criteria for determining
the number of factors; (b) the research findings and conclusions on the
effects of under and overrotation; (c) the factor analytic methods and
procedures relevant to this investigation.
Procedural Rules and Criteria for Rotation
The issue of the number of factors to rotate and interpret is fairly
straightfoyward. The goal in factor analysis is to arrive at a small
number of common factors which maximally account for the common variance
of aD intercorrelation matrix (Linn, 1968). Procedural rules have been
developed to determine this number. These rules generally fall into two
categories: statistical and psychometric (Cliff & Hamburger, 1967; Linn,
1968). The former attempt to generalize from the data to a population
of subjects, while the latter seek to generalize from the data to a domain
of interest, i.e., a universe of measures. Both approaches have their
vicgrous proponents. Obviously, rules and procedures espoused by one
camp do not necessarily yield factor solutions identical to those ob
tained from the other's (Hakstian & Muller, 1973).
The statistical approach. A number of rigorously derived statistical
procedures have been developed. Among the workers in this area were
Bartiett (1950); Joreskog (1963); Lawley (1951); and Rao (1955). Tests
of significance have been developed for the hypothesis that a given number
of factors is necessary to account for a set of data. Unfortunately,
these methods are of narrow applicability. For example, "Barlett's X
test is limited to the principal components model with unities in the
diagonal and thus is not applicable to the usual communality model" (Linn,
1968, p. 38).
The status of the statistical approach was described by Cliff and
Hamburger (1967):
The results available from statistical theory, while
useful, leave a larger area where the needs of the
investigator are unsatisfied. The statistical tests
for the number of factors are all tied, naturally
enough, to the respective methods for estimating
factors. Moreover, these methods of estimation are
either computationally arduous, as in the case of
Lawley's (1953) or Rao's (1955) method, or unfamiliar,
as in the case of Joreskog's1 (1963) method. Con
sequently, they are rarely used and so the statistical
tests are rarely applied. More important than this
is the fact that the number of factors in a given
matrix is only one of many concerns of the investi
gator. He is interested in a wide variety of sta
tistical questions, and he is interested in them
as they arise in the methods of factor analysis
currently in use (p. 431).
Hakstian and Muller (1973) expressed some reservations about the use
1Joreskog's K statistic has been extensively investigated by Monte
Carlo techniques and seems to give good results except when the N's are
rather low (100) (Cliff & Hamburger, 1967).
of inferential procedure in factor analysis. One concern was the "lack
of rigorous control over Type I error" (p. 465). Their major objection,
however, centered around the "dependence of this approach upon the total
sample size, with the resulting problem of 'statistical but not practical
significance' with particularly large N" (p. 465).
Linn (1968) pointed to another problem regarding the statistical
approach, namely, the "almost complete lack of knowledge concerning the
distribution and standard error of individual factor loadings and their
differences. The mathematical difficulty of developing the necessary
distribution theory has proven to be exceedingly great" (p. 37). Never
theless, investigations of sampling error continue to be made. Since the
1960's several of these studies have employed Monte Carlo techniques. A
review of these appears in Cliff and Hamburger (1967).
Just as in the statistical case, the psychometric approach has led
to several procedural rules to establish the number of interpretable
factors that must be retained. Generally, it is the latter methodological
approach,used in this study, that is frequently encountered in factor
analytic literature. This seems to be related to the previously mentioned
limitations and objections to the available statistical tests.
The psychometric approach. A number of psychometric rules of thumb
have been developed: (a) the KaiserGuttman (Guttman, 1954; Kaiser, 1960)
latent root greater than one criterion; (b) Cattell's (1958) rule for
computing the percentage of common variance, which led to (c) Cattell's
(166b)scree test. The last two will be considered together since they
are interrelated.
The KaiserGuttman criterion (Guttman, 1954; Kaiser, 1960) has been
adopted by the psychometric approach, although it clearly applies only to
ccrrelations of a population and not to those of a universe of measures
(Cliff & Hanhurger, 1967; Linn, 1968). The rule states that, with unities
in the diagonal of a correlation matrix, the number of common Factors is
equal to the number of latent roots greater than one (Harman, 1976).
Some factor analysts take exception to the application of this rule
to obtain rotated common factors (Cattell, 1966b;Gorsuch, 1974; Guertin
& Bailey, 1970). Their position is that when unities are used in the
diagonal, the principal components obtained should not be rotated; com
poner : retain unique as well as common variance and therefore cannot be
expected to yield interpretable common factors. Hakstian and Muller (1973)
stated that the application of the KaiserGuttman rule, with its "procedural
implications. .for only the component model, is seen as theoretically
inappropriate when a commonfactor. .analysis is being performed" (pp.
470471).
Linn (1968) used a Monte Carlo approach to develop criteria for the
number of factors. He augmented observed intercorrelation matrices with
generated random normal deviates and factor analyzed the augmented matrices.
The factoring method, sample size, number of variables,and the estimates
of communalities were varied. The latent root criterion correctly esti
mated the number of factors in only six cases, underestimated in five
cases, and grossly overestimated the correct number of factors in four
instances. He concluded that the application of this rule in deciding
upon the correct number of common factors "cannot be recommended on the
basis of the results of the present study" (p. 67).
Hulphreys (1964) analyzed intercorrelations of the 21 variables of
the 1944 Air Cre Classification Battery based on 8158 cases. He obtained
ten rotated interpr table factors corresponding to those obtained by
previous studies on the same variables. Had he used the KaiserGuttman
rule, he would have had to retain only five factors, since, with unities
in the diagonals, only five factors had latent roots larger than one.
Humphreys (1964) concluded that "The Kaiser [Guttman] criterion, when
N is very large is clearly too conservative with respect to the number of
factors" (p. 466).
It would seem, then, that the KaiserGuttman rule, statistically
sound as it is, has been applied in an inappropriate manner in common
factor analysis. As Gorsuch (1974) said, "The major criticism of the
root >1 criterion lies more in its use than in its concept" (p. 149).
Computing the percentage of variance extracted as a basis for the
number of factors to rotate stipulates that rotation should not be
terrinated until 95 to 98% of the complete principal axes variance is
accounted for (Cattell, 1966b;Guertin & Bailey, 1970). This principle
subsequently led to the development of the scree test.
Cattell's (1966b)scree test probably best exemplifies the psycho
metric approach in factor analysis. He stated that ". .it should be
left to rotation to separate substantive [real] and error of measurement
factors" (p. 246). Cattell's (1966b)main reservations about the use of
statistical tests was that factor extraction may be terminated too soon,
resulting in the rejection of substantive variance that may be needed
for subsequent rotation.
In the scree test, the latent roots of the principal axes are
plotted. At first, the roots fall off rapidly because common variance
is extracted early. Subsequently, the roots level off in a linear
fashion when almost nothing but measurement error is extracted. The
cutoff point that indicates the number of factors is just before the
linear descent (Cattell and Jaspers, 1967). Thus the scree test gives
the minimum number of factors for the maximum amount of variance
(Gorsuch, 1974).
Several empirical studies have evaluated the scree test. Linn's
(1968) Nonte Carlo study, mentioned earlier, found that the scree
technique identified the correct number of factors in seven instances,
underestimated in two cases, and overestimated in one. In the remaining
six cases, the results of the scree were not clear cut. Tucker, Koopnmn,
and Linn (1969) reported that the scree technique correctly identified
the number of factors in 12 out of 18 instances. Similar findings made
by others led Corsuch (1974) to conclude that "the scree test is in the
correct general area" (p. 155).
In a study that compared several statistical and psychometric factor
analytic rules for determining the number of interpretable factors,
Hakstian and Muller (1973) reanalyzed 17 published correlation matrices.
Their results suggested "that the appropriate number of factors . .
depends, in part, upon the view held regarding factors and factor analysis
and the consequent linear model employed in the analysis" (p. 470).
The scree test, for example, was found to yield too few factors in
many cases, while the latent root >1 criterion was seen theoretically in
appropriate when either commonfactor or image analysis is performed.
They recommended for the commonfactor case, at least, that the number of
factors be found for rotation "so that an optimally clear solution results"
(p. 473). Regarding the number of common factors to rotate, they suggested
rotating more factors than will ultimately be interpreted.
Alternative procedures for determining the number of factors. In
addition to the KaiserGuttman rule (Kaiser, 1960; Guttman, 1954) and
Cattell's (1966b) scree test, several investigators have recommended
alternative procedures for arriving at the correct number of interpretable
factors.
Horn (1965) developed a procedure as a correction for the latent root
>1 criterion for determining the number of factors. His rationale for the
necessity for this correction was that the criterion overestimates the
number of factors. The technique he presented was designed to determine
the number of nonerror latent roots.
Horn (1965) used a Monte Carlo procedure to generate random normal
deviates for the same number of subjects and variables as ones in an
observed 297 x 65 raw score matrix. The latent roots were calculated
for the raw score intercorrelation matrix and the randomlygenerated
data. Horn (1965) stated that the correct number of factors is equal to
the number of latent roots of the real data that are larger than their
counterparts in the random data. He proposed that this procedure be
routinely incorporated in computer programs.
By counterexample, Cliff and Hamburger (1966, p. 433) showed that
Horn's (1965) method can underestimate the number of common factors.
Linn (1968) found Horn's results "while interesting, can only be taken
as suggestive, due to the fact that they consist of only one example"
(p. 39).
In a generalization of Horn's (1965) procedure to the commonfactor
model, Humphreys and Ilgen (1969) recommended what seems to be a promising
technique for determining the number of factors to rotate and interpret.
Using procedures by Horn (1965) and Linn (1968) as points of departure,
the authors employed parallel analysis on matrices of real and random
data. Latent roots for the intercorrelation matrices of the real and
the random data were plotted and the point at which the latent root curves
crossed were assumed to indicate the number of common factors.
Their results compared favorably with maximum likelihood statistical
solutions for the same matrices. In fact, with squared multiples in the
diagonals, parallel analysis seemed to give a bit more accurate results
than maximum likelihood. (Neither unities nor the highest r adjusted
in the diagonal gave equally satisfactory results.) The writers recom
mended the routine use of their technique since it can be used concurrently
with latent root inspection for breaks, and as incorporated in Cattell's
scree test (1966b).
Humphreys and Ilgen's (1969) findings were confirmed by sampling
studies conducted by Humphreys and Montanelli (1975). They concluded
that when the commonfactor model provided a good fit to the data,
parallel analysis was more accurate than maximum likelihood in determining
the number of cornon factors.
Howard and Gordon (1963) presented an illustration and extension of
a method proposed by Wrigley (1960) for identifying common factors.
Wrigley (1960) recommended overfactoring, then rotating successive numbers
of factors either by Varimax or Quartimax. Next, the rotated factor
matrix is searched for specific factors. These are factors that have high
loadings of only oie variable. IF such a specific factor is found, the
last principal axes Factor is dropped and Varimax or Quartimax reapplied.
This procedure is repeated until a factor solution is obtained where each
factor has high loadings of at least two variables. By this method, only
common factors are obtained.
Howard and Gordon (1963) offer an illustration and a refinement of
W'rigley's (1960) method since ". .there may still be certain ambiguities
associated with some of these factors" (p. 245). They analyzed 37
activities variables taken on 598 streetcorner gang boys. The 37
variables were intercorrelated, communality estimates were inserted in
the diagonal; factoring was done by the principal axes method and eleven
factors extracted. "Varimax rotations were performed using the first two,
three, four, five, and so on, up to eleven of the principal axes factors"
(p. 248). Howard and Gordon (1963) report that, for this illustration,
only five common factors are meaningful. But had they followed Wrigley's
(1960) criterion, six factors would have had to be retained and interpreted.
They recommend that Wrigley's (1960) procedure be followed up to the point
when one specific factor emerges. At that time, "an evaluation is made
of the stability of the loadings of the remaining common factors" (p. 250),
and only the maximum number of stable common factors is retained.
While several approaches and procedural rules have been suggested,
there does not seem to be a unanimous agreement on any single way to
determine the number of factors to rotate. As a consequence several
researchers have investigated the stability of the rotated factors as
more factors are carried into rotation. Their findings are now examined.
Studies on the Effects of Under and Overrotation
An empirical investigation of special interest to this study was
Mosier's (1939)1. The procedure followed in the present study parallels
to some extent that of Mosier's. The purpose of the Mosier (1939) study
was to assess the influence of chance error and communality estimates on
IMosier's (1939) study is considered the "earliest paper on the
subject of procrustes rotation" (Harman, 1976, p. 336).
simple structure. Nosier constructed a representative hypothetical
factor matrix for 20 variables and four orthogonal factors, "satisfying
the criterion of simple structure (p. 34). The intercorrelation matrix
was obtained by postmultiplying the factor matrix by its transpose.
Just as was done in the present study, normally distributed "chance
error" (p. 35) assuming N = 100, was added to each off diagonal coefficient
in the matrix. "This matrix with unknown diagonal entries, represents
the situation met in. .factor analysis, where the individual rjk's are
subject to error and the communalities must be estimated" (p. 36).
The intercorrelation matrix was factored using two different esti
mates of the communality. Four factors were rotated and rootmeansquare
discrepancies calculated between the hypothetical factor loadings and
those of the erroradded matrix. Mosier (1939) concluded that neither
the added error nor the estimated communalities prevent accurate deter
mination of a Factor solution "provided that the rank of the centroid
natrix is equal to or greater than that of the underlying primary trait
matrix" (p. 43).
In addition Hosier (1939) investigated several criteria for "comple
tion of the analysis" (p. 39) under the two conditions of estimated
communalities and added error. Using the same hypothetical fourfactor
matrix, he took out three, four, then six factors. None of the criteria
tested was considered wholly satisfactory in determining the correct
number of factors, although he recommended that "It is safer to have too
many than too few factors" (p. 43).
Kiel and Wrigley (1960) used analytical rotational procedures to
compare solutions from successive factor rotations. They found that,
initially, the existing factors will subdivide when another factor is
carried into rotation, and an interp;retable factor emerges. A point of
stability is assumed to be reach?.. when no further a eptable factors re
sult with further rotation. An acceptable factor, according to Kiel and
Wrigley,is one on which at least two variables have their highest loadings.
They recommnded that the point of stability be used as a criterion for
terminating factor rotation.
Ding;an, Miller, and Eyman (1964) studied the effect of rotating
too many factors for both the orthogonal and the oblique case. Only the
former is pertinent to this study. The data were based on three aptitude
factors, each with three levels of difficulty. Tests representing the
aptitude levels were administered to 479 male college students. Factor
extraction was by the centroid method and communality estimates were
iterated until stability was reached. Varimax was used for rotating
first the three "ideal" Factors; this was followed by four, five,and six
factors carried into rotation.
Dinginan et al. (1964) reported that in the orthogonal case, "as the
number of factors rotated was increased over the optimum number of 3;
si;rple structure progressively got worse and more factors tended to appear
in the overall dimension of common factor space up to and including the
5factor solution" (p. 78). Nevertheless, the authors maintained that
meaningful factors can be obtained even when there is overrotation. They
admitted, however, that this conclusion may be peculiar to their highly
structured data; the three optimum factors remained fairly recognizable
in spite of overrotations.
Levonian and Comrey (1966) stated that rotating too few factors can
result in a distortion of the rotated i .trix. They pointed out that when
fdcioring is stopped too soon, the extracted variance will be crowded
into a lesser number of factors than are necessary to represent the
underlying factor structure, nor will rotation of these factors clarify
the structure of their matrix. It is possible, they stated, that none
of the real factors of the matrix will emerge, and those that appear
will be severely distorted by "foreign" variance (p. 101).
Levonian and Comrey (1966) pointed out that the effect of rotating
too few factors "would seem to become more serious as the degree of under
extraction increases" (p. 401). They also suggested that ". .the con
sequences of rotating too many factors is less clear" (p. 401); a possible
consequence may be an instability in the common factor loadings.
The authors investigated factorial stability as related to the number
of orthogonally rotated factors for two separate problems. The problems
were treated differently in terms of the correlations computed and the
method of factor extraction. However, both sets of factors were rotated
to tlhe Varimax criterion.
The number of rotations was varied for each problem. For example,
for the first study, of the first 25 centroid factors extracted, the
first 6 were rotated, then the first 10, 14, 18,and finally all 25.
Levonian and Comrey (1966) concluded that, though generalization
was not possible from only two studies, "stability considerations suggest
the rotation of many, rather than few, factors" (p. 404). Further, that
if the number of variables is not small, reasonable stability may be
achieved; the ratio of factors to variables should approach 1/3 or
possibly larger.
In a study to determine the number of principal axes factors to carry
into rotation, Veldman (1974) used the Varii::x criterion to rotate suc
cessively greater numbers of factors for nine published problems. The
Varimax criterion value, C, was considered by Veldman to be "an index of
the degree to which the rotation process has approximated 'simple
structure'  the goal of analytic rotation" (p. 193).
Veldman found that the Varimax criterion value C appears to be useful
in identifying the rank of a factor matrix. The C values were found to
be unimodel and peaked at the correct number of factors. Another finding
was that overrotation was not necessarily disastrous, when the principal
axes were rotated. Moreover, overrotation when image analysis was used
did not disturb the major factors. However, the criterion values C
fluctuated erratically when the latent structure of a matrix was weak.
A general theme in the literature seems to be that retaining one or
two additional factors for rotation does little harm and is advocated by
some investigators (e.g., Gorsuch, 1974: Mosier, 1939). Underrotation is
discouraged since it forces commonfactor variance to be compressed into
too few factors, thus distorting commonfactor space (Guertin & Bailey,
1970).
On the other hand, "factor fission" as Cattell (1952) refers to it,
can result when too many factors beyond the scree point are rotated (p.
334). As the number of rotated factors is increased the common variance
is redistributed across too many factors, causing some factors to split.
The resulting factor matrix degenerates into an uninterpretable, psycho
logically meaningless solution (Guertin & Bailey, 1970).
Because of the lack of agreement on any one approach or method in
determining the correct number of factors to rotate, several kinds of
factor solutions have been developed. The differences among these solu
tions "correspond to the different mathematical theories in the explana
tion of a particular scientific problem" (Harman, 1976, p. 10).
Comprehensive presentations of the various rationales and
procedures may be found in a number of available books (Comrey, 1973;
Gorsuch, 1974; Harman, 1976; 1lulaik, 1972). In a 1972 investigation,
Dielman, Cattell and Wagner included a summary of comparative studies
of rotational procedures since 1954. Hakstian and Muller (1973) pre
sented a tabular summary of the views, models, bases for inference, the
rationales, and procedures that have been traditionally employed in factor
analytic studies. Of the various methods and procedures recommended,
the following we'e selected as most appropriate for this investigation.
Relevant Factor Analytic Methods and Procedures
Because this study was concerned with only commonfactor analysis,
the principal axes method was used, with squared multiple correlations
inserted as communality estimates. Iterations and refactoring were per
formed until satisfactory convergence was achieved. The resulting
principal axes were rotated to approximate simple structure by the use
of the Varimax method. The literature pertaining to each of these
phases of the analysis will be examined. The use of the RMS mean
deviations in factor analytic studies will be presented.
Common factors, principal axes and communalities. The principal
axes method developed by Thurstone (1932) has remained popular because
it extracts the maximum amount of commonfactor variance from a reduced
intercorrelation matrix. Additionally, it has the virtue of producing
"a lowervalued final residual matrix" (Guertin & Bailey, 1970, p. 62).
In a comparison with Harman's (1967) Minres and Lawley's (1951) Maximum
likelihood factor extraction procedures, with different communality
ertiiates, the principal axes method gave very similar results after
Lhe factor matrices were rotated to Varimax (Guertin, 1971).
A reduced intercorrelation matrix is one where the values in the
diagonal are estimated (cosinunalities) prior to factor extraction.
Several values for the initial estimates have been proposed. Wrigley
(1956) performed an empirical iterationbyrefactoring study in which he
compared fifteen different methods of initial communality estimates. He
concluded that, with the use of computers, the squared multiple correla
tion of each variable with the remaining ones is the best initial esti
mate of the communality. Wrigley (1957) pointed out that "Various ob
jections raised against communalities can be met. .by the use of the
S.M.C.'s" since "the S.M.C. measure variance common to a test [variable]
and the remaining pl tests in the selection" (p. 94). Guttman (1956)
viewed the squared multiple correlations not only as the best possible
estimates of communalities, but also as the lower bound for these
estimates.
Other workers in the field share similar views on the merits of the
squared multiple correlations (Gorsuch, 1974; Harman, 1976). Humphreys
and Ilgen (1969) found the use of the squared multiple correlations an
"objective useful way of estimating communalities," with the additional
advantage of remaining "stable from sample to sample since they depend
upon all the data" (p. 572).
A concluding word on the use of communality estimates might be
Harman's (1976), "It has been argued, and substantiated by empirical
evidence, that it matters little what values are placed in the principal
diagonal of the correlation matrix when the number of variables is large
(say, n>20)" (p. 86).
The iterationbyrefactoring procedure used in this study has much
to recr:..',end it. (Gorsuch, 1974). Haian (1976) viewed it as one method
for es:tiating communalities "which has the semblance of" objectivity
(p. 65).
In performing commonfactor analysis, then, the chief emphasis is
upon obtainir.g the maximum amount of commonfactor variance. The use of
the cr.munalities in the diagonal prior to factor extraction makes this
possible; a definition of the communality is that it is "the amount of
variance a test [variable] shares with all others in commonfactor space"
(Guertin & Bailey, 1970, p. 165).
/ Simple structure and the Varimax method. The aim of rotating the
extracted principal axes factors is to gain the clearest view of common
faccor space. Since the principal axes factors extract the maximum
possible common variance from the intercorrelation matrix, the question
now bccoi;es that of the number of those factors that must be carried into
rotation and to what criterion.
The universally accepted criterion that is followed is Thurstone's
(1947, p. 335) principle of simple structure which yields factors that
are relatively invariant across studies (Guertin & Bailey, 1970).
This criterion is doubly parsimonious: in rotating factors in
commonfactor space, simple structure dictates that both variables and
factors should be described by a minimum number of sizable loadings.
Although other criteria have been proposed, none has become as widely
used (Gorsuch, i974). To approximate the ideal of simple structure for
a given factor matrix, the factors may be rotated in either an obliquely
or an orthogonal fashion.
1Sinco only orthogonal rotation was employed in this study, oblique
snlutions will not be discussed. See larman (1976) for a comprehensive
treati ?ent.
Several analytical orthogonal rotation methods have been developed,
all of w,,hich were referred to collectively by Harman (1976) as Quartimax.
With Kaiser's (1958) development of the Varimax method for orthogonal ro
tation (used in this study), the Quartimax approach was abandoned (Comrey,
1973).
The VariOid: A method is the best known and most popular rotational
procedure used today (Butler, 1969; Comrey, 1973; Guertin & Bailey, 1970).
It is available at most computer centers, and is included in Dixon's
(1974) BMO package of computer programs. It has become so important that
special sections are devoted to it in several texts (Comrey, 1973; Gorsuch,
1974; Harman, 1976).
Several studies have compared different rotational procedures (Dielman
et al., 1972; Gorsuch, 1970; Guertin & Bailey, 1970). The findings were
fairly consistent. The Varimax method was found to satisfy the principle
of simple structure and that of factorial invariance. These two criteria
are considered fundamental to a successful rotational method (Harman, 1976).
In their review of analytic methodology, Glass and Taylor (1966)
concluded that "The search for an acceptable analytical orthogonal rota
tion procedure for attaining simple structure was effectively ended in
19S8 with the publication of Kaiser's Varimax procedure" (p. 570). In
Gl.ss and Toylor's (1966) view, future interest in improving on Varimax
is not expected since "Those who apply factor analysis appear to be con
tent with Varimax" (p. 570).
The use of the rootmeansouares (RSf_. Of the various methods
aviiable for comparing the factor loadings of one matrix with those of
another, the RMS deviations method was the most appropriate one to use
in this study (Idarnan, 1976). As stated previously (Chapter 1, p. 3)
the RMS deviation is a common statistical iteasure for comparing pairs of
corresponding factor loading in two studies "since the variables are the
same" (Harran, 1976, p. 343). In comparing Varimax, Quartimax, and sub
jective solutions for the same factor matrix, Harman (1976) used the RMS
index of deviation.
In a Monte Carlo study, Hamburger (1965) computed the RMS deviations
between corresponding loadings of rotated factors from sample and popula
tion matrices. A similar use of the RMS is in an empirical study by
Joreskog (1963) who compared unrotated commonfactor loadings from samples
and populations.
In an empirical investigation, emulated somewhat by this study,
Mosier (1939) calculated the RMS deviations to compare errorfree hypo
thetical factor solutions with erroradded ones.
Bailey (1969) computed the RMS discrepancies to compare variable
dependence in several oblique solutions.
Because of its simplicity, ease in calculation, and its common use
in factor analytic methodology, the RMS index seemed an appropriate
measure of deviation to use in this study. A rootmeansquare value of
zero would mean perfect agreement between two corresponding values
(Harman, 1976). Successive increases in the RMS values away from zero
should indicate greater degrees of disagreement. The data thus obtained
become suitable for further analysis, e.g., as a dependent variable in
an analysis of variance design.
Sumia y
A variety of rationales, rules,and procedures were found in the
1 ierature as to the correct number of factors that muist be rotated. A
small number of studies was noted that examined the effects of under and
overrotation on factor structure. The general results were fairly con
sistent. It is better to overrotate by one or two factors but not much
more, otherwise factor fission occurs (Cattell, 1966a). Underrotation
was not recommended.
The survey revealed no specific empirical study that dealt with
factor loading stability as a function of under and overrotation under
all the conditions proposed for the present study. Different investi
gators dealt with certain aspects of the problem, with no attempt at a
boead empirical examination. There also appeared to be few investiga
tions in this troublesome area of factor analysis. In addition, within
the investigations noted, the number of intercorrelation matrices
examined was quite small. It seemed, therefore, appropriate to employ
an empirical approach, with a large number of replicated matrices, under
varying experimental conditions representative of observed data. The
results may contribute some insights to this aspect of factor analysis.
CHAPTER III
Methodology
Because of the length and complexity of the procedure in this study,
Chapter III has been divided into five main sections: (a) the statistical
hypotheses generated by the research questions; (b) the selection of the
problem matrices; (c) the selection of and the matrices' adjustments to
the specified levels of common variance; (d) the selection and generation
of the chosen random error levels; (e) the application of the procedures
to the problem matrices.
Two major research questions were to be answered by this study.
First, what effect does under and overrotation of factors have on the
loadings of a factor matrix under the specified levels of common variance
and error? A test of the differences in the obtained RMS means should
be an indicator of this effect. Second, what are the trends in the data,
i.e., what is the form of the eauation(s) that best describes) the
plotted RMS mean values? These questions led to the formulation of
several statistical hypotheses.
The Statistical Hypotheses
The research questions generated the following statistical hypotheses,
tested at a = .05:
a. For each selected problem, there are no differences among the
RPMS mean values under the various levels of the number of factors rotated,
i .e. :
Ho: PI = 2 = 13 .. . k
H1: some pj's are unequal.
Any significant differences that exist among the selected levels of
cormon variance and among the selected levels of error were not a focus
of concern for this study. Those levels were chosen to allow for the
gereralizability of the findings. However, the effect of these levels
on the number of factors to rotate is of importance to the investigation.
b. For each problem, there is no linear, quadratic, or cubic com
ponent of the model:
2 3
Y = 0 + B1X1 + B2X + B3X1 + E,
That is:
1. H : 1 = 0
H1: B1 0
2. Ho: 2 = 0
H1: B2 0
3. Ho: 3 = 0
H1: B3 / 0.
Selection of the Matrices
Four factor matrices with different numbers of variables and factors
weie chosen. The matrices were selected to be fairly representative of
ones reported in the literature.
Though hypothetical matrices could have been constructed, it was
decided that, in keeping with a common practice in factor analytic
studies, only published matrices be included (Gorsuch, 1974). Logistical
considerations prevented the selection of matrices of extreme size. The
ones chosen were Fruchter's (1954) 11x5 matrix, Harman's (1976) 24x4
natrix, Mulaik's (1972) 36x5 matrix, and Whimbey and Denenberg's (1966)
23x6 matrix. The Fruchter and Harman matrices have been used extensively
in factor analytic methodological investigations. Harman's matrix, in
particular, has become a classic in factor analytic demonstrations.
The Common Variance Adjustment
Since each of the four selected problem matrices accounted for a
different amount of common variance, it was necessary to adjust this
variance so as to allow comparisons among the matrices. Three levels
of the proportion of common variance were chosen. The rationale for
choosing .30, .45, and .60 (30%, 45%,and 60% of the common variance
accounted for) was to make the study as generalizable as possible by
including proportions that were representative of those found in pub
lishe'd works. A proportion of .30 is fairly low but not infrequently
ercoi:.tered; .45 is rather commonly and typically reported; and the .60
proportion is somewhat high but also found.
For each factor matrix chosen, several operations were performed.
Regardless of the original amount of common variance for which the input
matrix accounted, it was adjusted to conform to each of the three pre
specified levels. Each factor matrix was "stretched" or "compressed" as
follows: every value (factor loading) in the matrix was squared and
then the values were summed across the rows to give the communalities;
all the communalities thus calculated were then summed and the total
divided by the number of variables for the particular matrix.
The value thus obtained was then divided into each of the three
p.;aspecified proportions of common variance, yielding constants by which
every squared value in the original factor matrix was multiplied. Then
the square root values were obtained. The adjusted matrices would now
conform to th2 amount of common variance specified.
The Choice and Generation of Error
Because external validity was a concern, three levels of pseudo
randomlygenerated error were chosen, each of which represented a
different sample size (N). The N's were 100, 200, and 500. The
corresponding standard error (S.E.) for each N was computed by the use
of /N when the correlation is zero. This is an acceptable formula
for determining the S.E. when samples are fairly large (Guilford and
Fruchter, 1973).
For an N = 100, the S.E. is .10 with a mean of zero; N = 200, the
S.E. is .07 with a mean of zero; and For N = 500, the S.E. is .04 with
a mean of zero. The errors were computergenerated from a specified
normal distribution for each level with a mean of zero, and the above
specified standard deviations.
To assess the effectiveness of Fortran routine NDIST for generating
the specified random error, the means and standard deviations of the
requested error values were examined. The number of error values
generated per intercorrelation matrix was 10 m(m1) where 10 is the
2
number of replications per condition, and m is the number of variables.
For example, for Fruchter's (1954) 11x5 matrix, for each level of error,
550 random values were generated and their means and standard deviations
computed. Comparisons of the means and standard deviations of the random
error values requested and those actually generated for the three levels
of error under the three levels of common variance for the four problem
matrices are shown in Tables Al and A2 in Appendix A.
Procedures
All four problem matrices were analyzed in a similar fashion.
Differences in the analyses, when they existed, were related to the
number of factors that were carried into rotation. This number depended
not only on the original rank of each of the problem matrices, but also
on the logistical limitations to the analyses.
Because the analytical procedures were complex and lengthy, the
first problem matrix will be presented in detail to illustrate completely
the technique used in the analyses. A less detailed description of the
procedures is given for the other three selected matrices.
Problem One. Problem One was chosen from Fruchter (1954, p. 147),
and is the smallest matrix used in this study. It is a fivefactor
problem based upon eleven variables. The latter were eleven tests, part
of a larger battery used by the U.S. Army Air Force during World War II.
The fie oblique referencefactors, which accounted for 21.33% of the
common variance were rotated by Harris's (1948) direct method. This
11x5 factor matrix is shown in Table 1.
Fruchter's (1954) matrix was adjusted by the procedure described
earlier to account for 30%, 45%,and 60% of the common variance; the
three resulting matrices are shown in Table 2. Each of these adjusted
matrices was postmultiplied by its transpose to yield an intercorrela
tion matrix R.
R', the intercorrelation matrix based on 30% common variance, is
shown above the principal diagonal in Table 3. The matrix, R', was
1See Guilford, 1947, for a complete description of these tests,
including reliability and validity statistics.
31
TABLE 1
PROBLEM ONE
The Input Factor Matrix Taken
from Fruchtera
Factor
I II III
1 .07 .32
2 .43 .04
3 .03 .12
4 .03 .03
5 .03 .02
6 .45 .00
7 .00 .63
8 .09 .68
9 .01 .01
10 .00 .00
11 .00 .00
.13 .16 .21
.06 .00 .05
.05 .42 .05
.03 .08 .30
.35 .03 .09
.00 .00 .00
.00 .00 .00
.11 .02 .13
.39 .01 .05
.00 .41 .00
.00 .00 .33
aFruchter (1954, p. 147)
Variable
IV V
TABLE 2
PROBLEM ONE
Fruchter's Matrix Adjusted to Account
for Three Levels of Common Variance
Factor
Variance Variable I II III IV V
1 .0a .38 .15 .19 .25
2 .51 .05 .07 .00 .06
3 .04 .14 .06 .50 .06
4 .04 .04 .04 .09 .36
5 .04 .02 .42 .04 .11
30% 6 .53 .00 .00 .00 .00
7 .00 .75 .00 .00 .00
8 .11 .81 .13 .02 .15
9 .01 .01 .46 .01 .06
10 .00 .00 .00 .49 .00
11 .00 .00 .00 .00 .39
1 .10 .46 .19 .23 .31
2 .62 .06 .09 .00 .07
3 .04 .17 .07 .61 .07
4 .04 .04 .04 .12 .44
5 .04 .03 .51 .04 .13
45% 6 .65 .00 .00 .00 .00
7 .00 .92 .00 .00 .00
8 .13 .99 .16 .03 .19
9 .01 .01 .57 .01 .07
10 .00 .00 .00 .60 .00
11 .00 .00 .00 .00 .48
1 .12 .54 .22 .27 .35
2 .72 .07 .10 .00 .08
3 .05 .20 .08 .70 .08
4 .05 .05 .05 .13 .50
5 .05 .03 .59 .05 .15
60% 6 .75 .00 .00 .00 .00
7 .00 .99 .00 .00 .00
8 .15 .99 .18 .03 .22
9 .02 .02 .65 .02 .08
10 .00 .00 .00 .69 .00
11 .00 .00 .00 .00 .55
aValues only to second decimal place accuracy were retained to
facilitate inspection.
CM1 C\ CM Z COC3 CD cO LnC
C 000c o CD ooo
0
SCM Mc OOMLni 0
u 0 C"OCMC D0LO CDM M
C OODCJOOOOOCVO
. . .CMCM.* * 0)
S1 0
5
coo( D _c M) 0 1,cr)oo 4
Sn 01 o Co c r Mo 
o 01 OI' ^rfMOl OO C)CO T *I
0 I U
S4 *e
Q )C co mL LfZD CO 4 
r 't CMC Mo) 0 0)
= m 0
4) 4' 0
u, a0
*s memor oo cm i o
S0 0
CDM'COM CMLCOt 'O MCD00 0 0)
0 0. CMcMCoO' fcMLc r2 S
*i UCC CM 0)
SI I
o 0 fO
03 c)o o0 0,
r S) 'S CD 8C0 S
.' CM .9c. ,. . 0C
S O i I i 41
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factored by the principal axes method and the extracted factors were
rotated to the Varimax criterion. The obtained rotated fivefactor
solution was now the criterion matrix. The latter is shown in the
upper third of Table 4. To create the first condition of error, the
intercorrelation matrix, R', was subjected to the addition of the pseudo
randomlygenerated error level of .10 which represents a sample size of
100. No error was added to the principal diagonal (the exact
communalities) so as not to alter the rank of the matrix. The adjusted,
erroradded intercorrelation matrix R" for Fruchter's (1954) original
factor matrix appears in Table 3 below the principal diagonal. The
coimputergenerated error added to the first replication is shown in
Table 5.
The R" matrix was factor analyzed by the principal axes method
yielding seven factors. The first three principal axes were rotated to
Varimax and the differences between their loadings and those of their
counterparts in the criterion factor matrix (Table 4) were calculated
by the RMS method. By adding the three RMS's for the differences between
paired loadings on the three factors, a single value, the total RMS
was obtained.
A fourth principal axis was carried into rotation and, in a similar
fashion, the total RMS's obtained. The same was done with five, six,
then seven factor rotations,and total RMS's calculated. Therefore, for
the first replication, five values were obtained each of which repre
sented the total RMS discrepancy between the first three factors of the
criterion matrix and the first three factors of the five tri;:l rotation
matrices.
Beginning again with the errorfree intercorrelation matrix, R',
TABLE 4
PROBLEM ONE
The Criterion Matrices with Three
Levels of Common Variance
Factor
Variance Variable I II III IV V
1 .08 .39 .16 .19 .23
2 .51 .06 .07 .01 .05
3 .03 .13 .06 .50 .07
4 .03 .05 .06 .08 .35
5 .04 .04 .42 .03 .08
30% 6 .53 .01 .00 .01 .00
7 .01 .75 .03 .02 .03
8 .09 .82 .11 .05 .11
9 .02 .03 .46 .01 .03
10 .01 .01 .00 .49 .02
11 .00 .02 .03 .01 .39
1 .09 .48 .19 .24 .28
2 .62 .07 .08 .01 .06
3 .03 .16 .07 .61 .08
4 .04 .06 .07 .10 .43
5 .05 .05 .51 .04 .09
45% 6 .65 .01 .01 .01 .00
7 .02 .91 .04 .03 .04
8 .12 .99 .13 .06 .13
9 .02 .04 .57 .01 .03
10 .01 .02 .00 .59 .02
11 .00 .02 .03 .02 .48
1 .11 .55 .22 .28 .32
2 .72 .09 .10 .01 .07
3 .04 .19 .08 .71 .10
4 .05 .07 .08 .12 .50
5 .05 .06 .59 .05 .11
60% 6 .75 .01 .01 .01 .00
7 .02 ,99 .04 .03 .05
8 .13 .99 .15 .07 .15
9 .02 .04 .66 .02 .04
10 .01 .02 .00 .69 .03
11 .00 .03 .04 .02 .55
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the whole process of adding newlygenerated error, Factoring, rotating,
and calculating the total RMIS's was completed nine more times for a
total of ten replications. The total RMS's for the ten replications
for Problem One, where the common variance is 30%, the error is .10,
for the five rotations tried, are shown in the upper portion of Table 6.
Fruchter's (1954) matrix was examined under the second level of
error, .07, with the common variance remaining at 30%. The same number
of rotations was performed and the total RMS's similarly computed. The
procedure was replicated ten times. The same operations were repeated
with the third error level of .04. Therefore, for one level of common
variance and three levels of error, with ten replications each, 150
total RMS values were obtained for Problem One. These values are shown
in Table 6.
F,'chter's (1954) original matrix was similarly examined under the
condition of 45% of the common variance accounted for and a second
criterion factor matrix was obtained. This matrix is shown in the
middle portion of Table 4. Again, error, generated at the three levels,
was added to the intercorrelation matrix in each instance and the same
procedure followed in obtaining total RMS's for five different trial
rotations. Ten replications were performed each time and another 150
total R'.iS values were thus obtained.
The last 150 total RMS's for Problem One were the result of adjusting
Fruchter's (1954) matrix to account for 60% of the common variance (see
the lower portion of Table 2) and examining it under the three levels of
error. The criterion matrix for this experimental condition appears in
the lower third of Table 4.
The overall means of the RMS values for the five different rotations
TABLE 6
PROBLEM ONE
Total RootMeanSquares for 30O
Common Variance and Three Levels of Error
Replication
Number of Factors Rotated
3 4 5 6 7
Note. Only this sample of the raw RMS data is included in this study.
All the data are available from the author upon request.
aValues only to second decimal place retained.
Error
were calculated and plotted. Plotted also were the RMS mean values
under the nine conditions of common variance/error level. To test the
differences in the RMS means under the conditions of the five different
rotations and the nine common variance/error levels, an analysis of
variance was completed for a multifactor repeated measures design (Winer,
1971). In this type of design the experimental unit is observed under
more than one treatment. As a consequence these repeated observations
(measures) will be correlated, i.e., dependent (Winer, 1971).
The element of dependence which necessitated analysis by the re
peated measures design stems from the fact that for each replication,
the factors rotated were drawn from the same principal axes factor matrix.
For Problem One there were five rotations per replication. At every
rotation, each additional factor rotated was dependent for its loadings
upon the ones preceding it. Therefore, the obtained RMS's for Problem
One were analyzed by means of the repeated measures multifactor design,
where there were three levels of common variance (A), within each of
which there were three levels of error (C); there were ten "subjects"
replicationss) (S) per experimental condition and five "measures" (rota
tions) (B) on each replication. This represented a 3x3x5 factorial
design with ten subjects per cell. The number of trial rotations, i.e.,
measures B, for each problem, was considered a quantitative variable
with an underlying continuum, having equal treatment levels and equal
N's (Winer, 1971; Kirk, 1968). Hence, where there were significant main
effects and interactions, trend analyses were performed.
Linear, quadratic,and cubic orthogonal polynomial coefficients
were used to calculate the predicted values for the RMS means. Goodness
of fit of the polynomial equations was determined by comparing the
predicted and obtained values for these means (Kirk, 1968).
Problem Two. The second problem was Harman's (1976, p. 296),
twentyfour psychological tests fourfactor matrix, rotated to the
Varimax criterion. The input matrix accounted for 47.50% of the common
variance. The same procedures described for Problem One were followed
for this problem. The original matrix appears in Table 7. For this
problem, two factors were rotated and RMS's calculated, then similarly
three, four, five,and finally six rotations tried. This was done under
each combination of the three levels of common variance adjustment and
three levels of error. The three matrices reflecting the variance adjust
ments and the three criterion matrices used for the RMS calculations are
shown in Tables B1 and B2 in Appendix B.
As in Problem One,five trial rotations were performed for Problem
Two under the nine combinations of common variance and levels nf error.
The obtained RMS values were plotted, then analyzed by means of a 3x3x5
multifactor repeated measures design as was the case for Problem One.
Trend analyses were performed, and goodness of fit of the equations to
the data determined.
Problem Three. Problem Three was a modification of a matrix
appearing in Mulaik (1972, p. 395). It is a Varimax factor matrix of 35
tests of the Language Modalities Test for Aphasia. The original authors,
Jones and Wepman (1961), included two more variables, age and education,
in the analysis and rotated six factors. Since the variable of education
had a single high loading of .59 on factor six, and loadings of .16 or
less on the remaining five factors, it was deleted from the matrix.
Factor six, in turn, was a very weak factor, except for its educa
tion variable loading, and it, also, was excluded from the analysis. The
TABLE 7
PROBLEM TWO
The Input Factor Matrix Taken
from Harmana
Factor
Variable I II III IV
aHarman (1976, p. 296)
matrix finally used for Problem Three was based upon a 36variable five
factor matrix that accounted for 79.75% of the common variance, the
highest amount for all four problems. The input matrix appears in Table
8.
The procedures outlined for Problems One and Two were followed for
Problem Three. The number of factors that were rotated was a minimum of
three rotations to a maximum of seven inclusive. Therefore, the number
of "measures" on each principal axes matrix was five. The obtained RMS's
were analyzed by means of a 3x3x5 repeated measures model as was done
in the two previous problems. The adjusted and criterion matrices for
Problem Three are shown in Appendix B in Tables B3 and B4.
Problem Four. Problem Four, the last one examined, was based upon
a sixfactor orthogonal solution taken from Whimbey and Denenberg (1966,
p. 284). The variables for the matrix were 23 behavioral tests administered
to a group of PurdueWistar rats. The input matrix, which accounted for
72.74% of the common variance, is shown in Table 9.
Although this 23x6 matrix was smaller than the 36x5 matrix of
Problem Three, the number of the total RMS values obtained was much
larger. This was because more rotations were tried for this matrix than
for any of the other three problems. A total of seven trial rotations
was performed: three to nine rotations inclusive. Because the
rank of this matrix was the highest of all the matrices selected, it
was logistically possible to examine a wider range of the effects of
successive rotations.
Therefore, for Problem Four, seven RMS "measures" were computed per
replication under each experimental condition. The RMS means were plotted,
then analyzed by a repeated measures 3x3x7 factorial design. The
TABLE 8
PROBLEM THREE
The Input Factor Matrix Taken from Mulaika
_Factor
II III IV V
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
.16
.83
.89
.88
.88
.86
.88
.83
.85
.82
.81
.84
.24
.15
.11
.10
.25
.30
.28
.40
.34
.46
.47
.36
.43
.48
.63
.61
.50
.47
.54
.55
.26
.28
.16
.17
aMulaik (1972, p. 395)
.25
.35
.26
.27
.29
.28
.25
.32
.27
.26
.31
.27
.12
.07
.18
.10
.05
.01
.13
.10
.08
.77
.77
.76
.76
.78
.60
.63
.13
.20
.26
.19
.18
.12
.36
.12
Variable
.26 .02
.18 .04
.10 .11
.12 .13
.10 .14
.13 .16
.10 .18
.12 .16
.14 .06
.17 .19
.11 .26
.14 .19
.81 .10
.83 .13
.81 .14
.77 .08
.28 .04
.25 .11
.22 .20
.18 .18
.11 .18
.18 .15
.15 .12
.17 .09
.14 .06
.08 .01
.03 .23
.01 .24
.30 .59
.23 .60
.21 .65
.21 .62
.10 .27
.36 .09
.25 .18
.31 .02
.49
.28
.19
.25
.20
.24
.20
.25
.26
.25
.23
.23
.33
.36
.30
.38
.78
.70
.58
.69
.71
.21
.13
.24
.11
.16
.08
.12
.42
.42
.28
.30
.60
.64
.68
.57

44
TABLE 9
PROBLEM FOUR
The Input Factor Matrix Taken from
Whimbey and Denenberga
Factor
Variable I II III IV V VI
1 .697 .004 .192 .210 .309 .057
2 .028 .178 .096 .040 .840 .242
3 .394 .280 .051 .461 .436 .001
4 .046 .671 .341 .103 .061 .102
5 .117 .083 .013 .728 .021 .080
6 .134 .047 .073 .562 .220 .235
7 .529 .060 .435 .431 .188 .046
8 .418 .276 .371 .057 .636 .229
9 .051 .033 .115 .059 .234 .819
10 .228 .324 .201 .132 .777 .186
11 .037 .233 .940 .007 .142 .056
12 .175 .070 .783 .165 .005 .312
13 .391 .376 .072 .641 .084 .196
14 .743 .157 .015 .043 .039 .406
15 .007 .006 .547 .062 .124 .135
16 .096 .253 .341 .315 .044 .787
17 .905 .081 .012 .036 .123 .120
18 .493 .098 .028 .060 .012 .711
19 .067 .573 .675 .191 .020 .194
20 .090 .073 .250 .769 .129 .374
21 .081 .414 .256 .617 .335 .175
22 .504 .120 .083 .005 .424 .275
23 .330 .335 .053 .105 .651 .366
aWhimbey and Denenberg (1966, p. 284)
remainder of the analyses followed procedures identical to those used in
examining the other three problems. The adjusted and criterion matrices
for Problem Four are shown in Appendix B in Tables B5 and B6.
Because four F statistics were computed for the hypotheses of the
four problems, it was necessary to determine the probability of obtaining
four significant F's, all by chance. This was done by following the
procedure outlined in Jones and Fiske (1953) and Levitt (1961). Both
sources stated the rationale and assumptions for the use of Wilkinson's
(1951) table. Wilkinson (1951) provided probability tables based upon
the expansion of the binomial distribution (p + q)n, where p is the
specified level of significance, q = (1 p), and n is the total number
of tests of significance. "The fundamental assumption for the binomial
model is that the several experimental results are independent, that the
probability value for any one result in no way influences the value for
any other result" (Jones & Fiske, 1953, p. 376). Certainly, the assump
tion of statistical independence was met by this study by using four
totally unrelated input matrices.
In applying Wilkinson's (1951) method to this study, the probability
of making four Type I errors in four tests at the .05 level was found to
be .00000625.
Summary
The research questions generated several statistical hypotheses
which were tested at a = .05. Four representative factor matrices,
selected from the literature were analyzed in a similar fashion. Each
was adjusted to three selected levels of common variance and subsequently
three criterion matrices were obtained. Three levels of randomlygenerated
error were selected. Random error from each level was added to inter
correlation matrices of the adjusted Factor matrices.
For each problem, the intercorrelation matrices were factor analyzed,
principal axes extracted,and several orthogonal factor rotations tried.
These rotations were a series of successive under and overrotations of
the factors of each selected matrix.
Ten replications were performed for each trial rotation, under each
of the nine conditions of common variance/level of error. Root mean
square (RMS) deviation values were obtained at each replication. The
RMS's represented differences between factor loadings for the initial
factors in each criterion matrix and their counterpart loadings on the
corresponding factors in the trial rotations.
The means of the RMS values thus obtained were plotted, then
analyzed by a multifactor repeated measures design. Trend analyses were
performed where indicated. Orthogonal polynomial coefficients were used
to compute predicted RMS mean values so as to compare them to the obtained
RMS means. This was done to test the goodness of fit of the trends to
the data.
Wilkinson's (1951) method was used to calculate the probability of
making four Type I errors in four tests at the .05 level.
CHAPTER IV
Results
The aim of this investigation was to examine the stability of factor
loadings as a function of the number of factors rotated under specified
levels of common variance and error. To this end, four problem matrices
were selected and examined under the specified experimental conditions.
Because of the nature and length of the analyses performed, the obtained
results for each of the four selected problems are presented separately.
For each problem the following will be given: descriptive data
including the plotted curves of the RMS mean values; ANOVA summary table
and the hypothesis tested; results of the trend analyses; comparisons
between the observed and the computed RMS mean values.
Problem One
The means and the standard deviations for the RMS values for each
ten replications under the stated experimental conditions of common
variance, error, and rotations are presented in Table 10. The overall
RMS means for the five rotations are plotted in Figure 1. The mean
values of the RMS's under the three levels of common variance and the
three levels of error are plotted in Figures 2 and 3, respectively.
Results of the ANOVA test for the hypothesis of equality of the RMS
means are summarized in Table 11. It can be seen that at the .05 level
0
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49
3 4 5 6 7
Number of Factors Rotated
Figure 1. RMS means for the five different rotations
for Problem One.
u
CJ
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ro
"4
k0
E
o
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45% Common Variance
Number of Factors Rotated
V
different rotations with the three levels
of common variance for Problem One.
5 
z
30% Coninon Variance
d iffren 45% Commion Variance
of. c o r cm rO
60% Comion Variance
3 4 5 6 7
Number of Factors Rotated
Figure 2. RFlS rmcans for the interaction of the five
different rotations wsith the three levels
of common variance for Problem One.
_ _~~U__
S
0
s
in
5
(D \
M2
521 
E S.E. =.10
SS.E. =.07
u
3 4 5 6 7
E S.E. = .10
Number of Factors Rotated
Figure 3. MI1S means for the interaction of the five
different rotations with the three levels
of error for Problem One.
of error for Problem One.
TABLE 11
PROBLEM ONE
ANOVA Summary Table for RMS iean Values for
Five Different Rotations
Source of Variation
SS df MS
Between rotations
Common variance (A)
Error (C)
Common variance x error (AC)
Replications within common variance
and error (S/AC)
Within rotations
Rotations (B)
Common variance x rotations (AB)
Error x rotations (CB)
Common variance x rotations x
error (ABC)
Rotations x replications within
common variance and error (BS/AC)
.2022 89
.0142 2
.1453 2
.0087 4
.0071
.0727
.0022
.0340 81 .0004
.4140 360
.3566 4
.0137 8
.0050 8
.0892
.0017
.0006
.0023 16 .0001
.0364 324
.0001
Total .6162 449
aValues rounded to four places for presentation which accounts for
apparent inconsistencies in the F ratios.
*p < .05.
18.94*
193.50*
5.78*
793.45*
15.23*
5.55*
1.27
the main effect of the different rotations B is significant. The
hypothesis of no difference is therefore rejected in favor of the
alternative hypothesis.
The effect of the three levels of common variance on the number
of factors rotated, i.e., the variance and rotations interaction AB is
found significant. This is also true of the interaction between the
different factor rotations and the three levels of error BC. The over
all interaction between common variance, rotations and error ABC is not
found significant.
The other three significant F's in the ANOVA summary table are
those for the common variance A, error levels C, and their interaction
AC. It should be noted that the significance of the latter three terms
is not of primary concern to this investigation. The three levels of
common variance and the three levels of error were selected as independent
variables only to allow the generalization of the findings. Therefore,
future references to the levels of common variance and error will not be
made in the remainder of this Chapter. However, these terms can be
found in each ANOVA summary table as part of the overall analysis.
Results of the analysis of variance of the linear, quadratic, and
cubic trends of the RMS mean values are shown in Table 12. The results
of the trend analyses indicate that the means of the RMS deviation
values are a curvilinear function of the number of factors rotated. The
three trend components are included in the polynomial cubic equation:
Y = +0 + 1X1 + F2Xl2 + +3X13 + E.
It seems that an equation of this form represents the simplest description
of the curve connecting the five RMS means in Figure 1.
A comparison of the observed RMS mean values for the five rotations
TABLE 12
PROBLEM ONE
ANOVA Summary Tables for Linear, Quadratic and Cubic Trends
Source of variation
SS df
MS F
Analysis of Linear Trend
Within rotations linear
Rotations (B) linear
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within common
variance x error (BS/AC)
.2749 90
.2534 1 .2435
.0063 2 .0032
.0003 2 .0002
.0012 4 .0057
.0137 81 .0002
Analysis of Quadratic Trend
Within rotations quadratic
Rotations (B) quadratic
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within common
variance x error (BS/AC)
.1092
.0931
.0049
.0004
.0931
.0025
.0002
.0019 4 .0005
.0089 81 .0001
Analysis of Cubic Trend
Within rotations cubic
Rotations (B) cubic
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within common
variance x error (BS/AC)
.0240 90
.0111 1 .0111
.0019 2 .0010
.0018 2 .0009
.0004 4 .0001
.0088 81 .0001
1499.41*
18.64*
<1.00
33.58*
847.31*
25.00*
2.00
4.75*
111.00*
10.00*
9.00*
1.0
_ nr
and the predicted ones are in Table 13. The three trend components
used in computing the predicted RMS mean values are also included. (The
computations were done by the use of orthogonal polynomial coefficients
and solving for Y.) It can be seen from Table 13 that adequate approxi
mations of the RMS mean values as a function of the number of different
factor rotations are those given by the cubic components. It should be
noted, however, that the plotted curve of the RMS mean values in Figure
1 shows no visible inflection corresponding to the significant cubic
trend.
Problem Two
The means and standard deviations for the RMS mean values for each
ten replications under the prespecified experimental conditions are in
Table 14. The RMS mean values B for the five different rotations are
plotted in Figure 4; the interactions of common variance with rotation
AB and error with rotations CB are shown in Figure 5 and 6, respectively.
In Table 15 is shown the summary of the ANOVA completed with the
RMS mean values for the five different rotations for Problem Two. The
main effect of rotations B is significant at the .05 level, as are the
AB and BC interaction terms. The rest of the significant terms are
indicated in the table. Therefore, the hypothesis of the equality of
the B's is rejected in favor of the alternative hypothesis.
The three trend analyses performed are presented in Table 16. The
results of the analyses indicate the presence of a cubic trend in the
data. Therefore, a polynomial cubic equation of the form:
Y = 40 + IX1 ]+ 2X 2 + 3X3 + F
is accepted as best representing the curvilinear nature of the RMS mean
TABLE 13
PROBLEM ONE
Trend Components, Observed and Predicted RMS
Means for Five Different Rotations (B)
Trend Components
Linear
Quadratic
Cubic
1.68 x
.85 x
.35 x
Number of Factors Predicted B's Observed B's
Rotated Linear Quadratic Cubic
3 (Bl) .1053 .1223 .1258 .1259
4 (B2) .0885 .0800 .0730 .0725
5 (B3) .0552 .0717 .0547 .0552
6 (B4) .0530 .0549 .0464 .0530
7 (B5) .0518 .0381 .0551 .0518
deviation values for the five different rotations.
In Table 17 are found the three trend components and comparisons
between the observed RMS mean values for the five different rotations
and the RMS values predicted. The best approximation of the data is
that of the cubic trend, although, in Figure 4, no indication of this
cubic trend is observed.
Problem Three
The means and standard deviations for each ten replications under
the experimental conditions of common variance and error are shown in
Table 18. In Figures 7, 8,and 9 are the plotted curves of the RMS mean
values.
Results of the ANOVA test for the hypothesis of equality of the RIS
means are summarized in Table 19. At the .05 level, the main effect of
rotations B is significant, as are the interaction terms AB and BC. The
null hypothesis is rejected in favor of the alternative hypothesis.
The results of the linear, quadratic,and cubic trend analyses,
given in Table 20, indicate the presence of a significant cubic trend in
the data. Therefore, a polynomial cubic equation of the form:
Y = B0 + BlXl + B2XI2 + B3X3 + E
is accepted as the best representative of the curvilinear trend of the
RMS mean values under the condition of five different rotations.
The three trend components and a comparison between the observed
an.d the predicted RMS mean values are presented in Table 21. It can be
seen that the observed B's compare bet with the predicted B's when the
cubic components are included in the equations. Figure 7, however, shows
no inflection in the curve to indicate the cubic trend, significant though
it is.
o0
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i oo ooo ooo
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V)
2 3 4 5 6
Number of Factors Rotated
Figure 4. RMS means for the five different rotations
for Problem Two.
60
0E
\
U
I 
"45% Common Variance
60% Common Variance
S 3 4 5 6
Number of Factors Rotated
Figure 5. RMS means for the interaction of the five different
5 
I
5 
rotations with the three levels of common variance
fo45 Common VarianceTwo.
''*...~00~'
60% Common Variance
Number of Factors Rotated
Figure 5. RMS means for the interaction of the five different
rotations with the three levels of common variance
for Prohler Trio.
s
o 
G
oL
G i\
.5
S.E \.0
S.E. = .04
2 3 4 5 6
Number of Factors Rotated
Figure 6. RMS means for the interaction of the five
different rotations with the three levels
of error for Problem Two.
TABLE 15
PROBLEM TWO
ANOVA Summary for RMS Mean Values for
Five Different Rotations
Source of variation SS df MS F
Between rotations .1593 89
Common variance (A) .0077 2 .0038 15.67*
Error (C) .1265 2 .0633 259.52*
Common variance x error (AC) .0053 4 .0013 5.61*
Replications within common variance
and error (S/AC) .0197 81 .0002
Within rotations .9917 360
Rotations (B) .8614 4 .2154 1538.21*
Common variance x rotations (AB) .0659 8 .0082 58.52*
Error x rotations (CB) .0169 8 .0021 15.11*
Common variance x rotations x error
(ABC) .0016 16 .0001 <1.00
Rotations x replications within
common variance and error (BS/AC) .0459 324 .0001
Total 1.1510 449
*P < .05.
TABLE 16
PROBLEM TWO
ANOVA Summary Table for Linear, Quadratic and Cubic Trends
Source of variation SS df MS F
Analysis of Linear Trend
Within rotations linear .5994 90
Rotations (B) linear .5341i 
Rotations x common variance (BA) .0367 2
Rotations x error (BC) .0115 2
Rotations x common variance x error
(BAC) .0007 4
Rotations x replications within
common variance x error (BS/AC) .0164 81
Analysis of Quadratic Trend
Within rotations uadratic .3218 90
Rotations (B) quadratic .2795 1
Rotations x common variance (BA) .0230 2
Rotations x error (BC) .0048 2
Rotations x common variance x error
(BAC) .0005 4
Rotations x replications within common
variance x error (BS/AC) .0140 81
Analysis of Cubic Trend
Within rotations cubic .0639 90
Rotations (B) cubic .0478 1
Rotations x common variance (BA) .0054 2
Rotations x error (BC) .0010 2
Rotations x common variance x error
(BAC) .0001 4
Rotations x replications within common
variance x error (BS/AC) .0096 81
.5341
.0184
.0058
.0002
.0002
.2795
.0115
.0024
.0001
.0017x10l
2670.50*
92.00*
29.00*
1.00
1615.60*
66.47*
13.87*
<1.00
.0478 402.05*
.0027 22.67*
.0005 4.21*
.0003x101 <1.00
.0012x101
*p< .05.
TABLE 17
PROBLEM TWO
Trend Components, Observed and Predicted RMS
Means for Five Different Rotations (B)
Trend Components
Linear
Quadratic
Cubic
2.44 x 102
1.49 x 102
.72 x 102
Number of Factors Predicted B's Observed B's
Rotated Linear Quadratic Cubic
2 (B1) .1312 .1610 .1682 .1685
3 (B2) .1068 .0919 .0775 .0758
4 (B3) .0824 .0526 .0526 .0552
5 (B4) .0580 .0431 .0575 .0559
6 (B5) .0336 .0634 .0562 .0567
s
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w
3 4 5 6 7
Number of Factors Rotated
Figure 7. RMS means for the five different rotations
for Problem Three.
0)
0
S,
4
0
30% Common Variance
60% Common Variance
3 45
3 4 5 6 7
Number of Factors Rotated
Figure 8. RMS means for the interaction of the five
different rotations with the three levels
of common variance for Problem Three.
ai A
0
C)
s S.E. .10
4
0
0
S.E. .04
........... .........
3 4 5 6 7
Number of Factors Rotated
Figure 9. RMS means for the interaction of the five
different rotations with the three levels
of error for Problem Three.
~~_______I
TABLE 19
PROBLEM THREE
ANOVA Summary for RMS Mean Values for
Five Different Rotations
Source of variation SS df MS F
Between rotations .4227 89
Common variance (A) .0265 2 .0132 24.63*
Error (C) .3469 2 .1734 322.57*
Common variance x error (AC) .0057 4 .0014 2.67
Replications within common variance
and error (S/AC) .0436 81 .0005
Within rotations .1648 360
Rotations (B) .0948 4 .0237 190.90*
Common variance x rotations (AB) .0132 8 .0017 13.34*
Error x rotations (CB) .0145 8 .0018 14.65*
Common variance x rotations x error
(BAC) .0020 16 .0001 1.024
Rotations x replicatio.is within
common variance and error (BS/AC) .0402 324 .0001
Total .5874 449
*P < .05.
TABLE 20
PROBLEM THREE
ANOVA Summary Tables for Linear, Quadratic and Cubic Trends
Source of variance SS df MS F
Analysis oF Linear Trend
Within rotations linear
Rotations (B) linear
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within
common variance x error (BS/AC)
.1060
.0733
.0096
.0050
.0733
.0048
.0025
366.50*
24.00*
12.50*
.0007 4 .0018x101<1.00
.0174 81 .0002
Analysis of Quadratic Trend
Within rotations quadratic
Rotations (B) quadratic
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within common
variance x error (BS/AC)
.0418
.0203
.0031
.0069
.0203
.0016
.0035
145.00*
11.43*
25.00*
.0002 4 .0005x10<1.00
.0113 81 .0014x101
Analysis of Cubic Trends
Within rotations cubic
Rotations (B) cubic
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within
common variance x error (BS/AC)
*P < .05.
.0113
.0012
.0002
.0024
.0012
.0001
.0012
.0008 4 .0002
.0067 81 .0008x101
14.99*
1,?5
15.13*
2.38
TABLE 21
PROBLEM THREE
Trend Components, Observed and Predicted RMS
Means for Five Different Rotations (B)
Trend Components
Linear
Quadratic
Cubic
x 102
x 102
x 102
Number of Factors
Rotated
3 (BI)
4 (B2)
5 (B3)
6 (B4)
7 (B5)
Predicted B's
Linear Quadratic
.1006
.0916
.0826
.0735
.0645
.1086
.0876
.0745
.0695
.0725
Cubic
.1098
.0853
.0745
.0718
.0714
Observed B's
.1098
.0850
.0749
.0716
.0714
_~_~
~
~I_
Problem Four
The means and standard deviations of the RMS mean values for each
ten replications under the specified three levels of common variance and
error and the seven different rotations tried are presented in Table 22.
In Figure 9 is the curve of the RMS means for the seven different rota
tions. The profiles of the interactions of the three levels of common
variance with rotations are seen in Figure 10. In Figure 11 are depicted
the three interaction terms of error with rotations.
It can be seen from the ANOVA summary in Table 19 that, at the .05
level, rotations B, interaction terms AB,and CB are significant. The
null hypothesis is, therefore, rejected and the alternative one is not
rejected.
The three trend analyses performed to test the trend in the overall
RMS mean values under the seven rotations are shown in Table 24. At the
.05 level, the tests for trends are found significant. A polynomial
cubic equation of the form Y = B0 + BIX1 + 2X12 + B3X13 + E is accepted
as the best representative of the curvilinear trend of the RMS mean
values under the seven rotations.
In Table 25 are shown the three trend components and the observed
and predicted RMS mean values B. The best approximation of the observed
B's is given by the cubic component. However, the significant cubic
trend is not reflected in the plotted curve shown in Figure 10.
Summary
This study investigated factor stability as a function of the
number of factors carried into rotation under specified levels of common
73
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3 4 5 6 7 8
Number of Factors Rotated
Figure 10. RMS means for the seven different rotations
for Problem Four.
43
o \
30% Common V,
[ .
E '*
................. .................
60% Common V,
3 4 5 6 7 8
Number of Factors Rotated
Figure 11. RMS means for the interaction of the seven
different rotations with the three levels
of common variance for Problem Four.
U) \
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s
0
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(1
'S
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S.E.
S\ \
S.S.E. \
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% a S .E .
Number of Factors Rotated
Figure 12. RMS means for the interaction of the seven
different rotations v.with the three levels of
error for Problem Four.
\ 0, ^ o3_ o 
error for Problem Four.
TABLE 23
PROBLEM FOUR
ANOVA Summary for RMS Mean Values for
Seven Different Rotations
Source of variation SS df MS F
Between rotations .3232 89
Common variance (A) .0177 2 .0088 24.54*
Error (C) .2686 2 .1343 372.53*
Common variance x error (AC) .0077 4 .0019 5.32*
Replications within common variance
and error (S/AC) .0292 81 .0003
Within rotations .5674 540
Rotations (B) .4721 6 .0787 1183.16*
Common variance x rotations (AB) .0415 12 .0035 51.94*
Error x rotations (CB) .0183 12 .0015 22.87*
Common variance x rotations x error
(ABC) .0032 24 .0013x10 2.00
Rotations x replications within
common variance and error (BS/AC) .0323 486
Total .8906 629
*p< .05.
TABLE 24
PROBLEM FOUR
ANOVA Summary Table for Linear, Quadratic, and Cubic Trends
Source of variation SS df MS F
Analysis of Linear Trend
Within rotations linear
Rotations (B) linear
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x cor:on variance x error
(BAC)
Rotations x replications within
common variance x error (BS/AC)
.2997
.2493
.0254
.0115
.2493
.0127
.0058
.0017 4 .0004
.0118 81 .0015x101
Analysis of Ouadratic Trends
Within rotations quadratic
Rotations (B) quadratic
Rotations x common variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within
common variance x error (BS/AC)
.2094
.1838
.0132
.0053
.1838
.0066
.0027
.0007 4 .0002
2326.58*
83.54*
33.54*
2.22
.0064 81 .0008x101
Analysis of Cubic Trend
Within rotations cu :ic
Rotations (B) cubic
Rotations x conron variance (BA)
Rotations x error (BC)
Rotations x common variance x error
(BAC)
Rotations x replications within
common variance x error (BS/AC)
.0447 90
.0377 1
.0019 2
.0013 2
.0068x102 4
.0377
.0009
.0006
838.60*
22.00*
13.98*
.0017x102 <1.00
.0037 81 .0045x102
*P < .05.
712.23*
87.23*
39.49*
2.92
1
TABLE 25
PROBLEM FOUR
Trend Components, Observed and Predicted RMS
Means for Seven Different Rotations (B)
Trend Components
Linear
Quadratic
Cubic
x 102
x 102
x 10
Number of Factors
Rotated
3 (B1)
4 (B2)
5 (B3)
6 (B4)
7 (B5)
8 (B6)
9 (B7)
Predicted B's
Linear Quadratic Cubic
.1104
.1005
.0906
.0807
.0708
.0609
.0510
.1349
.1005
.0759
.0611
.0561
.0609
.0755
.1432
.0921
.0675
.0611
.0644
.0692
.0671
Observed B's
.1440
.0916
.0657
.0638
.0646
.0668
.0681
~~~
~~
variance and error. Te RMIS deviation measures obtained were used as a
dependent variable in a series of statistical tests. The RMS mean values
were plotted and the trr'nds of the obtained curves were analyzed. The
shape of these curve; was considered to be an index of the stability of
factor loadings when the number of factor rotations was varied. It was
assumed that the lowest RMS values would be those associated with the
correct number of rotated factors for a given matrix. The lowest point
in a plotted curve should, therefore, graphically illustrate this
assumption.
The findings of this study neither support nor reject this assumption.
In this respect the results are inconclusive. In only Prcblems Two and
Four are the lowest RMS mean values obtained associated with the correct
number of factors.
The results indicate, however, that factor loading stability is a
curvilinear function of the number of factors rotated. There is also a
significant interaction between the amount of common variance for which
the matrices account and the number of factors that are carried into
rotation. The interaction between the error added to the matrices and
the number of rotations is also found significant. For all four problems,
a form of the cubic polynomial equation represents the best approximation
of the plotted curves of the RMS means.
CHAPTER V
Discussion and Summary
This study investigated factor loading stability as a function of
the number of common factors carried into rotation under specified levels
of common variance and error. A survey of the literature revealed fairly
consistent recommendations as to the correct number of factors that must
be rotated to obtain interpretable common factors. It is best to over
rotate by one or two factors, but no more, otherwise factor fission may
result. Underrotation was not recommended because it results in the
compression of common variance into too few factors, thus distorting the
factor structure.
The results of this investigation provided support for the literature's
position on underrotation but were inconclusive in relation to the view on
overrotation. Factor loading stability was found to be significantly
affected by the number of factors that are rotated. Furthermore a
statistically significant interaction was found between the number of
factors that are rotated and the amount of common variance for which a
matrix accounts. The interaction between the number of rotations and
the amount of error added to a matrix was found significant also. The
RMS mean deviation values, used as indicators of the effects of the number
of rotation on factor loading stability, were found to be a curvilinear
function of the number of factors rotated.
Discussion
Each of these findings warrants further discussion. This is divided
into four main sections: (a) interpretation of the results in relation
to the existing literature; (b) comparisons among the problem matrices;
(c) examination of the effects of common variance and error on the number
of factors rotated; (d) suggestion for future research.
A summary concludes this chapter.
The findings in relation to the literature. It should be noted that
a major limitation to this study was that only the first few factors in
the selected matrices were examined for the effects of under and over
rotation under the experimental conditions of common variance and error.
Also, only four representative matrices were selected for examination
from the large number of matrices available in the literature. Within
the confines of this and the limitations stated previously in Chapter I,
the results of this investigation seem to confirm some, but not all, of
the positions in the literature regarding under and overrotation.
For all four problem matrices, the highest RMS values are those
associated with underrotation. An examination of Figure 13 clearly shows
this result. This finding is as it should be. When only a few of the
total number of factors are rotated, the common variance is compressed
and the factors are overdetermined. This results in underdimensioned
distorted commonfactor space. Because of this distorting effect on the
factor loadings, the general position in the literature neither supports
nor recommends underrotation. In this respect the study confirms this
position.
As the number of factors rotated approaches the correct number of
factors for each of the four problem matrices, the RMIS values begin to
Three (36x5)
o Four (23x6)
^d. _____L _UI_ LL ^J__ J
2 3 4 5 6 7 8 9
Number of Factors Rotated
Figure 13. Comparison of the RMS mean values for the
different rotations for the Four Problems.
decrease. In this study it has been assumed tnat the lowest RMS mean
deviation values obtained would be those identified with the correct
number of factors for each of the selected matrices. This assumption is
confirmed in only two of the four problem matrices.
In Problems Two and Four the lowest RMS values obtained are the ones
corresponding to the correct number of factors. This is not true, how
ever, of Problems One and Three. In this case, the iRS values continue
to decrease as the number of successive factor rotation increases. Be
cause of these conflicting results each two similar matrices are now com
pared.
Comparison of the Problem Matrices. The results of Problems Two
and Four tend to lend support to the literature's position on overrotation.
It is obvious from Figure 13 that the lowest points in the plotted RMS
mean value curves are those associated with the known number of factors
for these two matrices. The upward inclination at the right terminus of
the two curves, corresponding to a successive increase in the number of
factors rotated, indicates an increase in the RMS deviation values. This
increase is interpreted as the beginning of distortion effects on the
loadings of the factors examined. It is probable that factor fission may
result eventually with an increase in the number of successive over
rotations.
Problems Two and Four have a comparable number of variables, 24 and
23 respectively. The original matrix of the former accounts for 47.50%
of the common variance and it is a fourfactor solution; the latter
accounts for 72.74% of the common variance and is a sixfactor matrix.
However, this Iaount of common variance each matrix accounts for originally
should have no bearing on the results because these matrices were adjusted
prior to the analyses. The number of factor rotations tried for each of
the two matrices differs drastically, as can be seen from Figure 13.
Nevertheless the overall profile of the two matrices is very similar.
Problem Two, the 24x4 matrix, is of special interest since it is the
only matrix where two to six successive factor rotations were tried. (The
number of factors that were carried into rotation was determined by
logistical considerations related to the rank of each matrix.) The
plotted curves for Problem Two, found in Figures 4, 5, 6,and 13, show
an interesting pattern. When only two factors are rotated, the RMS mean
deviation value that is plotted begins at a much higher point on the Y
axis than was the case for the other three problems. This is as expected:
when all the common variance is compressed in only two factors the RMS
deviation value is extremely high. It could be higher still if all the
variance was compressed in one factor only.
The factors of Problem Four, the 23x6 matrix, were treated by three
under and three overrotations. Yet the shape of the plotted RMS mean
values curve for this problem is very similar to that for the 24x4 problem.
The profiles of these two problems agree with expectations. One can
speculate that as an increasing number of successive overrotations is
performed, there would be a corresponding increase in the RMS deviation
values obtained.
Just as there are notable resemblances between Problems Two and Four,
there are striking similarities between the curves for Problems One and
Three. An overall comparison of these matrices shows that they are both
fivefactor matrices; the number of trial rotations performed is the same
for both; the two problems have the most shallow curves of all the four
problems. This is seen in the RMS mean deviation curves in Figure 13.
For both these problems, the curves are shown to begin at lower levels
than is the case for Problems Two and Four. The curves continue to descend
with each successive overrotation, indicating a continued corresponding
decrease in the RMS mean deviation values.
The pattern of the curves of the RMS values for these two problems,
therefore, does not seem to support the findings in the literature re
garding overrotation. Since it is assumed that the ideal factor number
is that associated with the lowest RMS mean deviation values obtained,
then Problems One and Three violate this assumption. It would be of
interest if future studies were to examine matrices such as these when
additional successive overrotation can be performed. (Because of the
limited number of principal axes obtained in this study, further over
rotation was not possible.)
An examination of the differences between the matrices of Problems
One and Three, indicates that they differ greatly in terms of the number
of variables upon which each is based. The original amount of common
variance for which each accounts is also dissimilar. However, the amount
of original common variance should have no effect on the findings because
of the adjustment of the matrices prior to the analyses.
The inconsistencies in the results of the effects of overrotation on
factor stability makes it difficult to reach any general conclusions on
this aspect of the study. There does not seem to be any reasonable com
mon denominator relating each two of the problem matrices having similar
RMS profiles. The resemblances in profiles do not appear to be a function
of either the number of variables or the number of factors of the matrices.
Since both the amount of common variance and error are controlled in this
study, the effect of these two variables on the results is discounted.
Although this study has strived for ,eneralizability through its
empirical approach to the analyses, it is probable that the results ob
tained are peculiar to the matrices selected and to the limitations
imposed.
The effects of common variance and error on the number of factors
rotated. There is a remarkable but not unexpected resemblance across the
four problems in terms of the profiles of the curves representing the
interactions of the number of factors rotated and the amount of common
variance to which each matrix was adjusted. A consistent pattern is seen
in Figures 2, 5, 8,and 11. The most extreme initial RMS discrepancies
are those associated with the 60% common variance level and the fewest
number of factors rotated. The curves are steepest and descend the lowest
at the 60% level than they do for the other two levels. This is reasonable
and consistent with the expected compression that occurs when a compara
tively large amount of variance is confined to a smaller number of factors
than is ideal. As a successively increasing number of factors is rotated,
the variance redistributes itself accordingly across the factors.
The curves for the 45% and the 30% common variance levels begin at
correspondingly lower points on the Y axis than was the case for the 60%
curve, then level off as expected. The less the initial amount of common
variance for which a matrix accounts, the more shallow the curve repre
senting the RMS mean deviation values is expected to be. Only in Problem
Three is there a reversal of the points of origin on the Y axis for the
curves representing the 30% and 45% common variance. The reason for this
reversal is not obvious. (It may be related to properties unique to this
particular matrix.)
These results indicate the positive relationship that appears to
exist between factor loading stability and the amount of common variance
for which a matrix accounts. The larger this amount, the more stable the
factor loadings tend to be, and the less subject are they to the vagaries
of overrotation.
The implications of these findings for future research are obvious.
Special attention needs to be given in factor analytic investigations to
the proportion of common variance for which a given matrix accounts. To
assure factor loading stability, this proportion must be large.
There is also a resemblance in the RMS mean deviation curves repre
senting the interaction between the levels of the added error and the
number of factors rotated. This is seen in Figures 3, 6, 9,and 12. Ob
viously, the larger the amount of the error added to a matrix, the higher
are the RMS discrepancy values. This pattern is observable in all four
problems. The highest RMS deviation values are those associated with
the .10 added error, and the lowest are those for the .04 level. This
finding simply reinforces the importance of the size of the sample of
subjects in factor analytic research.
It is of interest to note that the RMS curves for the successive
factor rotations is found to be best represented by third degree poly
nomial equations in all four matrices. Although the significant cubic
components give better approximations of the observed RMS mean values
throughout, there is no observable indication of this cubic trend in any
of the plotted means. The cubic trend can be considered to have statis
tical but not practical significance.
In conclusion, within the stated limitations, the results of this
study confirm the literature's position regarding underrotation of common
factors. The evidence indicates that distortion occurs in the factor
loadings when the common variance is compressed and commonfactor space
is underdimensioned.
The literature's view on overrotation is neither clearly supported
nor obviously rejected. In two of the problems, the results confirm the
position; in the other two instances they do not.
The more common variance for which a factor matrix accounts, the
more stable its factor loadings appear to be. The reverse seems to be
also true. The less error added to a matrix the more it is stable and
vice versa. Ther does not appear to be a clear relationship between
the number of variables and/or factors in a given matrix and its factor
loading stability.
The shape of the RMS mean value deviations curves under successive
rotations can be best represented by cubic polynomial equations. However,
this trend is not demonstrated by the plotted curves. Although the study
has attempted generalizability, it is acknowledged that the conclusions
drawn must be considered only within the confines of the stated limitations.
A direction for future research. Several questions remain unanswered.
The results regarding overrotation are inconclusive and warrant further
investigation. Ihis study examined only the first few factors in each
matrix. Future work needs to examine the effects of rotation on a
different number of factors in selected factor matrices. The RMS deviation
or other appropriate measures for factor congruence can be used for this
type of investigation (Harman, 1976).
The effects of alternative factor analytic techniques on the number
of factors to rotate merit further examination. It would be of interest
to investigate the stability of factor loadings, e.g., when image analysis
is used as compared to the use of the principal axes method. The results
