Title: Distortion of factor loadings as a function of the number of factors
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Title: Distortion of factor loadings as a function of the number of factors rotated under varying levels of common variance and error
Physical Description: xi, 119 leaves : ill. ; 28 cm.
Language: English
Creator: Guertin, Azza Showket, 1935-
Publication Date: 1977
Copyright Date: 1977
 Subjects
Subject: Factor analysis   ( lcsh )
Mathematical statistics   ( lcsh )
Educational Administration and Supervision thesis Ph. D   ( lcsh )
Dissertations, Academic -- Educational Administration and Supervision -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 114-118.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Azza Showket Guertin.
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Bibliographic ID: UF00098857
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000197911
oclc - 03759542
notis - AAW4601

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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 root-mean-squares (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 Root-Mean-Squares 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 common-factor 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.

Computer-generated pseudorandom error at each of the three levels

specified was added to the intercorrelation matrices mentioned above,

and the error-laden 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. Root-mean-square (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. 412-413). 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 number-of-factors question" (p. 414).

A survey cf the literature revealed Kaiser's concern was not unique.

Many studies have focused upon the number-of-factors 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

tnat-of.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 common-factor 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

o-ut 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 error-laden 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 root-mean-

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 error-laden 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 root-mean-square (RHS) discrepancies betw ,-n the first two or

three factors of each criterion matrix and their corresponding factors in

each trial rotation of the error-added 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 four-factor 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 pseudo-random 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 common-factor

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 common-factor analytic methodology has been the number-

of-factors 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 common-factor 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

straight-foyward. 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 Kaiser-Guttman (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 Kaiser-Guttman 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 Kaiser-Guttman rule, with its "procedural

implications. .for only the component model, is seen as theoretically

inappropriate when a common-factor. .analysis is being performed" (pp.

470-471).

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 Kaiser-Guttman

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 Kaiser-Guttman 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

cut-off 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 common-factor or image analysis is performed.

They recommended for the common-factor 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 Kaiser-Guttman 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 non-error 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 randomly-generated

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 counter-example, 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 common-factor

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 common-factor 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 street-corner 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 root-mean-square

discrepancies calculated between the hypothetical factor loadings and

those of the error-added 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 four-factor

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 recomm-nded 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 over-all dimension of common factor space up to and including the

5-factor 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 common-factor variance to be compressed into

too few factors, thus distorting common-factor 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 common-factor 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 common-factor variance from a reduced

intercorrelation matrix. Additionally, it has the virtue of producing

"a lower-valued 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 iteration-by-refactoring 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 p-l 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 iteration-by-refactoring procedure used in this study has much








to recr:..',end it. (Gorsuch, 1974). Hai-an (1976) viewed it as one method

for es:tiating communalities "which has the semblance of" objectivity

(p. 65).

In performing common-factor analysis, then, the chief emphasis is

upon obtainir.g the maximum amount of common-factor 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 common-factor 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

common-factor 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 root-mean-souares (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 common-factor loadings from samples

and populations.

In an empirical investigation, emulated somewhat by this study,

Mosier (1939) calculated the RMS deviations to compare error-free hypo-

thetical factor solutions with error-added 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 root-mean-square 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-

randomly-generated 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 computer-generated 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(m-1) 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 five-factor

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 reference-factors, 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.

















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factored by the principal axes method and the extracted factors were

rotated to the Varimax criterion. The obtained rotated five-factor

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-

randomly-generated 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,

error-added intercorrelation matrix R" for Fruchter's (1954) original

factor matrix appears in Table 3 below the principal diagonal. The

coimputer-generated 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 error-free 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 newly-generated 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 Root-Mean-Squares 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),

twenty-four psychological tests four-factor 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 36-variable 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 six-factor orthogonal solution taken from Whimbey and Denenberg (1966,

p. 284). The variables for the matrix were 23 behavioral tests administered

to a group of Purdue-Wistar 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 randomly-generated








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
































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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
U
ro
"4-


k0
E
o

o






















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 c---m- 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.


























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2 3 4 5 6
Number of Factors Rotated

Figure 4. RMS means for the five different rotations
for Problem Two.






60




















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"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-
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rotations with the three levels of common variance
fo45 Common VarianceTwo.
''*--.---.-.~---0------0~'
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 -
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S.E \.0













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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*

.0003x10-1 <1.00

.0012x10-1


*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 10-2

1.49 x 10-2

-.72 x 10-2


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






























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3 4 5 6 7
Number of Factors Rotated

Figure 7. RMS means for the five different rotations
for Problem Three.






























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S,
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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

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s -S.E. .10
4-


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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 .0018x10-1<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 .0008x10-1


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 10-2

x 10-2

x 10-2


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









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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.
































4-3
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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) \
ci'\45 oinnV
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s-
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S.S.E. \
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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 F-our.










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 .0015x10-1


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 .0008x10-1


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

.0068x10-2 4


.0377
.0009
.0006


838.60*
22.00*
13.98*


.0017x10-2 <1.00


.0037 81 .0045x10-2


*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 10-2

x 10-2

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 common-factor 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 four-factor solution; the latter

accounts for 72.74% of the common variance and is a six-factor 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

five-factor 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 common-factor 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




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