A matrix formulation of Kaiser's varimax criterion

Kaiser has given the varimax criterion for the solution of the rotation problem in factor analysis as well as a practical computational procedure for maximizing this criterion. In the present paper, the maximization condition is shown as a matrix equation involving only the unknown orthogonal rotation matrix. This matrix equation can be solved iteratively as a sequence of symmetric eigenproblems.This investigation was supported by Public Health Service grant number MH 07285-03 from the National Institute of Mental Health.     

A joint treatment of varimax rotation and the problem of diagonalizing symmetric matrices simultaneously in the least-squares sense

The present paper contains a lemma which implies that varimax rotation can be interpreted as a special case of diagonalizing symmetric matrices as discussed in multidimensional scaling. It is shown that the solution by De Leeuw and Pruzansky is essentially equivalent to the solution by Kaiser. Necessary and sufficient conditions for maxima and minima are derived from first and second order partial derivatives. A counter-example by Gebhardt is reformulated and examined in terms of these conditions. It is concluded that Kaiser's method or, equivalently, the method by De Leeuw and Pruzansky is the most attractive method currently available for the problem at hand.The author is obliged to Dirk Knol for computational assistance and to Dirk Knol, Klaas Nevels and Frits Zegers for critically reviewing an earlier draft of this paper.     

Varimax solution for primary mental abilities

The varimax solution for Thurstone's classic Primary Mental Abilities study is presented. Comparisons between the factors of Thurstone's original subjectively rotated factor pattern, Zimmerman's subjectively revised solution, Wrigley, Saunders, and Neuhaus' quartimax results, and the present varimax factor matrix are made by finding correlations between factors defined by these four solutions. It is pointed out that any possible ultimate merit of the varimax solution should be based on its psychological meaningfulness and on the rationale of the varimax criterion—not on its relationship to the other studies.The computations for this paper were done on Illiac, an electronic computer of the Digital Computer Laboratory of the University of Illinois. Mr. S. M. Hunka assisted in these computations.     

Comparison Of Oblique And Orthogonal Simple Structure Solutions For Personality And Interest Factors

The study was directed to the need to structure in a few variables the domain measured by personality and interest measures commonly employed in educational counseling: Strong, Kuder, EPPS and the Study of Values. Despite initial uncertainty regarding number of factors to be employed, effects of ipsative scores and of mixing test formats, both an oblique and orthogonal rotation yielded nearly identical results. Of the twenty factors identified by both the biquartimin and varimax solutions, Seven linked vocational interest clusters with personality. Two of the remaining factors had only interest loadings, while of the eleven personality factors, only four were scale specific. Definition of the 16 common factors required that extraction proceed beyond the unit latent root criterion. The results offer evidence that over- extracting factors does not confuse the results of rotation. Further, psycho- metric differences between tests had essentially no effect on the factors found. Of three oblimin rotations attempted, only the biquartimin was successful, yielding results essentially like those of the varimax solution. Because of the vast difference in computation time for these two solutions (computer time 20 times greater for biquartimin), however, the orthogonal varimax remains the method of choice.     

The varimax criterion for analytic rotation in factor analysis

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that thenormal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure—although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.Part of the material in this paper is from the writer's Ph.D. thesis. I am indebted to my committee, Professors F. T. Tyler, R. C. Tryon, and H. D. Carter, chairman, for many helpful suggestions and criticisms. Dr. John Caffrey suggested the namevarimax, and wrote the original IBM 602A computer program for this criterion.I am also indebted to the staff of the University of California Computer Center for help in programming the procedures described in the paper for their IBM 701 electronic computer. Since their installation is partially supported by a grant from the National Science Foundation, the assistance of this agency is acknowledged.     

A general rotation criterion and its use in orthogonal rotation

Measures of test parsimony and factor parsimony are defined. Minimizing their weighted sum produces a general rotation criterion for either oblique or orthogonal rotation. The quartimax, varimax and equamax criteria are special cases of the expression. Two new criteria are developed. One of these, the parsimax criterion, apparently gives excellent results. It is argued that one of the most important factors bearing on the choice of a rotation criterion for a particular problem is the amount of information available on the number of factors that should be rotated. This research was supported by the National Research Council of Canada research grant 291-13 to Dr. G. A. Ferguson.     

Tandem criteria for analytic rotation in factor analysis

Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.     

Factor structure of the Beck Depression Inventory in a university sample

Factor analysis was performed on the Beck Depression Inventory with a university sample to examine its potential multidimensionality. A principal components analysis with an oblimin and varimax rotation produced a two-factor solution. These factors were labeled Cognitive-Affective and Physiological and accounted for approximately 39% of the common variance. This finding is consistent with multidimensionality of the inventory and with a similar study of college students. The commonalities of the two studies suggest the reliability (internal consistency) of the Cognitive-Affective and Physiological constructs among "minimally" depressed university samples.     

A factor analysis of the most frequently used Korean personality trait adjectives

We tested the cross‐cultural generalizability of personality structure by factor‐analysing self‐ratings of 435 Korean university students on the 406 most frequently used Korean personality trait adjectives. A plot of eigenvalues and a test of factor replicability both suggested a four‐factor solution. The four varimax‐rotated factors showed strong correlations with the first four factors of the Big Five (Extraversion, Agreeableness, Conscientiousness, and Emotional Stability), as measured by markers selected a priori from the pool of 406 adjectives. We also investigated a five‐factor solution, in an attempt to recover an Intellect factor. The five varimax‐rotated factors corresponded closely to the classic Big Five, but with a minor difference in the rotation of the Conscientiousness and Intellect factors. Solutions involving six and seven factors were also investigated, and these solutions produced a Truthfulness factor similar to some previously discovered lexical factors. The results of the study were discussed in relation to the lexical hypothesis and to previous studies of personality structure in East Asian languages. Copyright © 1999 John Wiley & Sons, Ltd.     

Interlest Checklist

The purpose of this study was to investigate the empirical structure of the five a priori theoretical categories of the Interest Checklist (Matsutsu:yu, 1969). Factor analysis with varimax rotation was utilized to assess the underlying structure of the 80 activity items. Sixty-seven adult psychiatric patients comprised the study sample. Results suggest a four factor solution rather than fivc: Factor I - Sports and physical tasks; Factor I1 -Intellectual and musical; Factor 111-Sociall; and Factor IV-Fine manual tasks and homemaking. A revised Activity Checklist that can be used for current clinical purposes is provided.     

Relationship of numbing to alexithymia, apathy, and depression

The present study assessed the relationship between numbing and three associated conditions of alexithymia, apathy, and depression, utilizing data collected on 353 Vietnam combat veterans diagnosed with Posttraumatic Stress Disorder from in- and out-patient settings and an outreach center at various Department of Veterans Affairs Medical centers. All subjects completed four self-report measures: the Glover Numbing Scale, the Beck Depression Inventory, the Apathy Evaluation Scale, and the Toronto Alexithymia Scale-20. The correlation matrix indicated that scores on the four measures were moderately to highly correlated. Principal components analysis with a varimax rotation indicated a five-factor solution that provided evidence for the factorial validity of each of the constructs assessed. Results of the factor analysis of items from the four measures were consistent with numbing being a separate and distinct construct from alexithymia, apathy, and depression. In general, results indicated that all constructs measured were separate and distinct from one another.     

Factor structure of the Depression Self-Rating Scale in an Iranian adolescent sample

Previous research showed concurrent validity for the Depression Self-rating Scale for Iranian adolescents, so the present purpose was to evaluate construct validity. The scale was distributed to 2012 school students in Grades 5 to 10. Factor analysis using varimax rotation yielded four factors interpretable as depressed mood, hopelessness, lack of pleasurable activities, and physiological symptoms. These factors accounted for 44% of the variance and provide some support for construct validity of the scale in Iranian adolescents.     

Item format and the structure of the Eysenck Personality Inventory: a replication

A seven-choice response format for personality inventories allows finer distinctions by subjects than the traditional two-choice format. The Eysenck Personality Inventory was employed in the present study to test the hypothesis that use of the expanded format would result in a clearer indication of test structure. A principal components analysis with a varimax rotation retained two components for the two-choice format, clearly identifiable as Eysenck's "Neuroticism" and "Extraversion" which accounted for 16% of the variance. The expanded format retained four components accounting for 28% of the variance.     

Standard errors for rotated factor loadings

Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.This research was supported in part by NIH Grant RR-3. The authors are grateful to Dorothy T. Thayer who implemented the algorithms discussed here as well as those of Lawley and Maxwell. We are particularly indebted to Michael Browne for convincing us of the significance of this work and for helping to guide its development and to Harry H. Harman who many years ago pointed out the need for standard errors of estimate.     

On meredith's solution for weighted procrustes rotation

Meredith developed a criterion for weighted procrustes rotation. The solution was given using Lagrange multipliers. We show that the solution can be obtained more simply, and that our approach enables us to transform the orthogonalization problem, also considered by Meredith, to that of minimizing a quadratic form on the unit sphere.     

Is the state-trait anxiety inventory multidimensional?

The factor structure of the State-Trait Anxiety Inventory (STAI) was investigated to determine whether the STAI State and Trait Anxiety scales were multidimensional. The STAI (Form Y) was administered to a heterogeneous sample of Air Force recruits (N = 1728); the scores for all 40 STAI items were factor-analyzed together, using the principal axis method and varimax rotation. Two- and four-factor solutions were found to be equally good in simple structure and psychological meaningfulness. Clearly defined state and trait anxiety factors were found in the more parsimonious two-factor solution. In the four-factor solution, the factors were: State Anxiety-Present, State Anxiety-Absent, Trait Anxiety-Present and Trait Anxiety-Absent. Using Cattell's (1966) confactor method, these factors were found to be congruent with similar factors previously identified for high school students by Spielberger et al. (1980). The anxiety-present and anxiety-absent factors were interpreted as reflecting either ‘item method’ variance or ‘item-intensity specificity’. There was no evidence in the present study that the STAI scales were multidimensional in terms of item content.     

Development of a measure of work motivation for a meta-theory of motivation

This study presents a measure of work motivation designed to assess the motivational concepts of the meta-theory of motivation. These concepts include intrinsic process motivation, goal internalization motivation, instrumental motivation, external self-concept motivation, and internal self-concept motivation. Following a process of statement development and identification, six statements for each concept were presented to a sample of working professionals (N = 330) via a paper-and-pencil questionnaire. Parallel analysis supported a 5-factor solution, with a varimax rotation identifying 5 factors accounting for 48.9% of total variance. All 5 scales had Cronbach alpha coefficients above .70. Limitations of the newly proposed questionnaire and suggestions for its further development and use are discussed.     

Revising the Competitiveness Index using factor analysis

The Competitiveness Index is a 20-item true-false measure designed to assess the desire to win in interpersonal situations. To develop a more psychometrically sound form of the scale, 213 undergraduates were administered the original form and a modified version containing a 5-point Likert-type scale. An initial principal component analysis using a varimax rotation of the modified version yielded a four-factor solution accounting for 54.5% of the explained variance. Based on a subsequent reliability analysis. six items were dropped from the modified scale. A second analysis produced a two-factor solution accounting for 54.1% of the explained variance. Both factors (Enjoyment of Competition and Contentiousness) formed reliable subscales. The 14-item Revised Competitiveness Index had high internal consistency and was positively correlated with the original Competitiveness Index, the competitiveness subscales of the Work and Family Orientation Questionnaire, the Sports Orientation Questionnaire, and the Nach Naff measure of Need for Achievement.     

A Brief Parent Rating Scale for Assessing Young Children's Adjustment

This report describes the development of the Parent Rating Scale (PRS), a 20-item scale for use by parents in rating the adjustment of young normal children. Based on the responses of 416 parents of urban and suburban 2nd and 3rd grade children, a principal components factor analysis with varimax rotation yielded a 4-factor solution accounting for 48.3% in total variance. Low to moderate relationshps were found between PRS scores and teacher, self, and sociometric ratings fo children. PRS total and factor and total scores discriminated clearly between referred and nonreferred children. Suburban children had higher PRS scores than urban children and girls had higher scores than boys.