Application of the quartimax method of rotation to Thurstone's primary mental abilities study

This study compares a quartimax rotation of the centroid factor loadings for Thurstone's Primary Mental Abilities Test Battery with factorings of the same correlation matrix by Thurstone (simple structure), Zimmerman (revised simple structure), Holzinger and Harman (bi-factor analysis), and Eysenck (group factor analysis). The quartimax results agree very closely with the solutions of Holzinger and Harman and of Eysenck, and reasonably well with the two simple structure analyses. The principal difference is the general factor provided by the quartimax solution. Reproduction of the factorial structure is sufficiently good to justify its use at least as the first stage of rotation. More extensive trial of the method will be needed with more varied data before it will be possible to decide whether quartimax factors meet psychological requirements sufficiently well without further rotation.We wish to thank Professor L. G. Henyey and the University of California Computer Center for making the IBM 701 electronic computer available for this study, and the National Science Foundation for its support of the work of the Computer Center. Professor H. F. Kaiser of the University of Illinois has made helpful criticisms of the paper, and Mr. Louis S. Davis of the University of California has assisted with preparation of the tables. The research was supported in part by the United States Air Force under Contract No. AF 33 (038)-25726 monitored by the Air Force Personnel and Training Research Center. Permission is granted for reproduction, translation, publication, use and disposal in whole and in part by or for the United States Government.A 701 program for calculation of the quartimax and varimax loadings, prepared by Professor H. F. Kaiser, is available in the library of computer programs held by the Computer Center at the University of California (Program No. 464). Mr. J. O. Neuhaus and Mr. K. W. Dickman have prepared a quartimax program for Illiac at the University of Illinois. This Illiac program will be usable on three other computers recently built or under construction: Mistic (Michigan State University), Silliac (University of Sydney), and the machine being constructed by Iowa State College.     

Recurrent Personality Factors Based on Trait Ratings

Intercorrelations among ratings on 35 personality traits, selected as representative of the personality domain, were obtained for eight samples. These samples differed in length of acquaintanceship from 3 days to more than a year; in kind of acquaintanceship from assessment programs in a military training course to a fraternity house situation; in type of subject from airmen with only a high-school education to male and female undergraduate students to first-year graduate students; and in type of rater from very naive persons to clinical psychologists and psychiatrists with years of experience in the evaluation of personality. Centroid or multiple-group factors were extracted and rotated orthogonally to simple structure. For one study, an independent solution was obtained in which analytic rotations were accomplished on an IBM 650 computer using Kaiser's normal varimax criterion. Five fairly strong and recurrent factors emerged from each analysis, labeled as (a) Surgency, (b) Agreeableness, (c) Dependability, (d) Emotional Stability, and (e) Culture.     

Orthogonal rotation algorithms

The quartimax and varimax algorithms for orthogonal rotation attempt to maximize particular simplicity criteria by a sequence of two-factor rotations. Derivations of these algorithms have been fairly complex. A simple general theory for obtaining two factor at a time algorithms for any polynomial simplicity criteria satisfying a natural symmetry condition is presented. It is shown that the degree of any symmetric criterion must be a multiple of four. A basic fourth degree algorithm, which is applicable to all symmetric fourth degree criteria, is derived and applied using a variety of criteria. When used with the quartimax and varimax criteria the algorithm is mathematically identical to the standard algorithms for these criteria. A basic eighth degree algorithm is also obtained and applied using a variety of eighth degree criteria. In general the problem of writing a basic algorithm for all symmetric criteria of any specified degree reduces to the problem of maximizing a trigonometric polynomial of degree one-fourth that of the criteria.This research was supported by the Bell Telephone Laboratories, Murray Hill, New Jersey and NIH Grant FR-3.     

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.     

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.     

Factor analysis of the social performance survey schedule

There are several studies suggesting that the Social Performance Survey Schedule (SPSS) is a reliable and valid measure of social skill. However, the factorial structure of the SPSS has never been examined. In the present study, SPSSs taken by 652 college students were factor analyzed by the method of principal components and the resulting factors were rotated according to the varimax criterion. Separate factor analyses were conducted for men and women. Seven factors, which accounted for a little over a third of the total variance, were retained for each sex. A large general factor labeled Prosocial Competence emerged for both sexes. Beyond this, the factorial solutions for the two sexes tended to diverge, with males showing a large negative factor (Social Offensiveness) and three smaller negative factors and females showing five relatively small negative factors. Suggestions were made concerning additional factor analytic studies of the SPSS and the ways in which derived factors might be used in research and clinical work.This research was supported by funds from the Rutgers University Research Council. Thanks are due to Leona Aiken for her statistical consultation.     

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.     

Group Differences in Regression Intercepts: Implications for Factorial Invariance

Studies of differential prediction typically examine group differences in linear regression slopes or intercepts for predicting criterion scores from one or more test scores. When there are no group differences in slopes, what are the implications of differences in regression intercepts for the measurement equivalence of the tests or criterion across groups? Measurement equivalence is here defined as factorial invariance under a single-factor model for the tests and criterion. Two theorems are given that describe conditions under which intercept differences can exist under factorial invariance. In such cases, intercept differences do not result from measurement bias in either the tests or criterion. The conditions of the theorems are testable using multiple-group confirmatory factor analysis. These test procedures are illustrated in real data. The implications of the theorems and the test procedures for studies of differential prediction are discussed.     

Orthomax rotation and perfect simple structure

A loading matrix has perfect simple structure if each row has at most one nonzero element. It is shown that if there is an orthogonal rotation of an initial loading matrix that has perfect simple structure, then orthomax rotation with 0 1 of the initial loading matrix will produce the perfect simple structure. In particular, varimax and quartimax will produce rotations with perfect simple structure whenever they exist.     

Rotation to simple loadings using component loss functions: The orthogonal case

Component loss functions (CLFs) are used to generalize the quartimax criterion for orthogonal rotation in factor analysis. These replace the fourth powers of the factor loadings by an arbitrary function of the second powers. Criteria of this form were introduced by a number of authors, primarily Katz and Rohlf (1974) and Rozeboom (1991), but there has been essentially no follow-up to this work. A method so simple, natural, and general deserves to be investigated more completely. A number of theoretical results are derived including the fact that any method using a concave CLF will recover perfect simple structure whenever it exists, and there are methods that will recover Thurstone simple structure whenever it exists. Specific CLFs are identified and it is shown how to compare these using standardized plots. Numerical examples are used to illustrate and compare CLF and other methods. Sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them.The author is very indebted to a reviewer for pointing him to the generalized hyperplane count literature and to all the reviewers for valuable comments and suggestions.     

Factorial invariance and significance

It is shown that invariance requirements remove the indeterminacy in factor determination and lead to an integration of factorial studies with promise of considerable reduction in computational labor. The selection of significant primary factors is discussed, with special reference to Thurstone's simple structure criterion.     

A test of the hypothesis that Cronbach's alpha reliability coefficient is the same for two tests administered to the same sample

In measurement studies the researcher may wish to test the hypothesis that Cronbach's alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.The author is indebted to Jerry S. Gilmer for development of the computer programs used in this study.     

Factorial invariance in multilevel confirmatory factor analysis

This paper presents a procedure to test factorial invariance in multilevel confirmatory factor analysis. When the group membership is at level 2, multilevel factorial invariance can be tested by a simple extension of the standard procedure. However level‐1 group membership raises problems which cannot be appropriately handled by the standard procedure, because the dependency between members of different level‐1 groups is not appropriately taken into account. The procedure presented in this article provides a solution to this problem. This paper also shows Muthén's maximum likelihood (MUML) estimation for testing multilevel factorial invariance across level‐1 groups as a viable alternative to maximum likelihood estimation. Testing multilevel factorial invariance across level‐2 groups and testing multilevel factorial invariance across level‐1 groups are illustrated using empirical examples. SAS macro and Mplus syntax are provided.     

Replicability and invariance of dimensions of parental rearing behaviour: Further Dutch experiences with the EMBU

In a recent article, Ross, Clayer and Campbell (1983) argued and showed by means of factor analysis that the number of child-rearing dimensions found previously with the EMBU appears to be dependent on the type of rotation used in factoring. Based on the assumption and empirical evidence that the dimensions found earlier are significantly correlated, Ross et al. (1983) argued that Oblique rotation would appear to give rise to a larger number of distinguishable dimensions than previous analyses (e.g. Arrindell, Emmelkamp, Brilman and Monsma, 1983). We contend that their conclusion was based on an inappropriate analysis of the data with a neglect of relevant psychometric principles. Employing objective techniques of factorial invariance the factors obtained in our previous study were shown here to be congruent across rotational procedures (Oblique vs Varimax). Supporting the stability and factorial and construct validity of the EMBU Rejection, Emotional Warmth, Overprotection and Favouring Subject dimensions, they were shown in further analyses to be: (a) replicable across split samples of phobics (Ns: 421 vs 420); and (b) invariant across distinct populations (phobics, N = 841 vs normals, N = 277).     

Simple Structure Detection Through Bayesian Exploratory Multidimensional IRT Models

In modern validity theory, a major concern is the construct validity of a test, which is commonly assessed through confirmatory or exploratory factor analysis. In the framework of Bayesian exploratory Multidimensional Item Response Theory (MIRT) models, we discuss two methods aimed at investigating the underlying structure of a test, in order to verify if the latent model adheres to a chosen simple factorial structure. This purpose is achieved without imposing hard constraints on the discrimination parameter matrix to address the rotational indeterminacy. The first approach prescribes a 2-step procedure. The parameter estimates are obtained through an unconstrained MCMC sampler. The simple structure is, then, inspected with a post-processing step based on the Consensus Simple Target Rotation technique. In the second approach, both rotational invariance and simple structure retrieval are addressed within the MCMC sampling scheme, by introducing a sparsity-inducing prior on the discrimination parameters. Through simulation as well as real-world studies, we demonstrate that the proposed methods are able to correctly infer the underlying sparse structure and to retrieve interpretable solutions.     

Orthogonal procrustes rotation maximizing congruence

Procedures for assessing the invariance of factors found in data sets using different subjects and the same variables are often using the least squares criterion, which appears to be too restrictive for comparing factors.Tucker's coefficient of congruence, on the other hand, is more closely related to the human interpretation of factorial invariance than the least squares criterion. A method maximizing simultaneously the sum of coefficients of congruence between two matrices of factor loadings, using orthogonal rotation of one matrix is presented. As shown in examples, the sum of coefficients of congruence obtained using the presented rotation procedure is slightly higher than the sum of coefficients of congruence using Orthogonal Procrustes Rotation based on the least squares criterion.The author is obliged to Lewis R. Goldberg for critically reviewing the first draft of this paper.     

Simple structure in component analysis techniques for mixtures of qualitative and quantitative variables

Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.The Netherlands organization for scientific research (NWO) is gratefully acknowledged for funding this project. This research was conducted while the author was supported by a PSYCHON-grant (560-267-011) from this organization. The author is obliged to Jos ten Berge for his comments on an earlier version.     

Belief attribution in science: Folk psychology under theoretical stress

Some eliminativists have predicted that a developed neuroscience will eradicate the principles and theoretical kinds (belief, desire, etc.) implicit in our ordinary practices of mental state attribution. Prevailing defenses of common-sense psychology infer its basic integrity from its familiarity and instrumental success in everyday social commerce. Such common-sense defenses charge that eliminativist arguments are self-defeating in their folk psychological appeal to the belief that eliminativism is true. I argue that eliminativism is untouched by this simple charge of inconsistency, and introduce a different dialectical strategy for arguing against the eliminativist. In keeping with the naturalistic trend in the sociology and philosophy of science, I show that neuroscientists routinely rely on folk psychological procedures of intentional state attribution in applying epistemically reliable standards of scientific evaluation. These scientific contexts place ordinary procedures of attribution under greater stress, producing evidence of folk psychological success that is less equivocal than the evidence in mundane settings. Therefore, the dependence of science on folk psychology, when combined with an independently plausible explanatory constraint on reduction and an independently motivated notion of theoretical stress, allows us to reconstitute the charge of (neurophilic) eliminativist inconsistency in a more sophisticated form.This research was supported by a National Science Foundation Predoctoral Fellowship (RCD87-58409). I completed the paper during an enjoyable year as a Mellon Postdoctoral Fellow at Bryn Mawr College. I am indebted to Richard Boyd, Sydney Shoemaker, and Robert Stalnaker for their detailed appraisals of the ideas presented here. I am especially grateful to Richard Boyd for many long conversations and specific suggestions. Anthony Appiah, Paul Churchland, Phil Gasper, Frank Keil, Dick Moran, Dave Reichling, Frank Wilson, and Rob Wilson improved the final product either through comments, conversation, or both. Finally, two anonymous referees for Synthese provided very useful recommendations for improvement.     

A comparative evaluation of several prominent methods of oblique factor transformation

The oblimax, promax, maxplane, and Harris-Kaiser techniques are compared. For five data sets, of varying reliability and factorial complexity, each having a graphic oblique solution (used as criterion), solutions obtained using the four methods are evaluated on (1) hyperplane-counts, (2) agreement of obtained with graphic within-method primary factor correlations and angular separations, (3) angular separations between obtained and corresponding graphic primary axes. The methods are discussed and ranked (descending order): Harris-Kaiser, promax, oblimax, maxplane. The Harris-Kaiser procedure—independent cluster version for factorially simple data,P'P proportional to , with equamax rotations, for complex—is recommended.This paper is based upon part of the author's doctoral dissertation [Hakstian, 1969] completed at the University of Colorado. The author is greatly indebted to Dr. Gene V Glass, who, as thesis advisor, generously contributed his time, erudition, and encouragement.