The simultaneous maximization of congruence for two or more matrices under orthogonal rotation

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.     

Procrustes matching by congruence coefficients

Matching by Procrustes methods involves the transformation of one matrix to match with another. A special least squares criterion, the congruence coefficient, has advantages as a criterion for some factor analytic interpretations. A Procrustes method maximizing the congruence coefficient is given. This solution is identical to Mosier's [1939] approximate solution, but is an exact solution for maximum congruence.     

Coefficients alpha and reliabilities of unrotated and rotated components

It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.Comments by Henk A.L. Kiers and by anonymous referees are gratefully acknowledged.     

A weighted procrustes criterion

The Procrustes criterion is a common measure for the distance between two matricesX andY, and can be interpreted as the sum of squares of the Euclidean distances between their respective column vectors. Often a weighted Procrustes criterion, using, for example, a weighted sum of the squared distances between the column vectors, is called for. This paper describes and analyzes the performance of an algorithm for rotating a matrixX such that the column-weighted Procrustes distance toY is minimized. The problem of rotatingX intoY such that an aggregate measure of Tucker's coefficient of congruence is maximized is also discussed.We wish to thank Richard A. Harshman and C. F. Jeff Wu for valuable discussions in the early stages of this work. We would also like to thank Jos ten Berge, John Gower, and the Editor, Associate Editor, and referees whose comments and suggestions greatly improved this paper.     

Matched Determiners Vs. Factor Invariance: A Reply To Korth

Korth (1978) does well to describe factor matching as vital to personality research but seriously underestimates the extent of successful matching both within and between cultures. His evaluation of matching of a set of factors is an advance, but the achievement of "diagonalization" of r[SUBc] coefficients in a matching matrix probably has a higher significance than his method would indicate.

Regarding Monte Carlo determinations of r[SUBc] distributions, the writer maintains that treating loadings as random normal deviates is incorrect and that a special distribution (here presented) is required. Further, it is argued that "factor invariance," as commonly defined, is not the required proof of identify of determiners. Instead, the principles of real base factor analysis need to be applied to demonstrate degree of matching of determiners. A numerical illustration shows that when congruence is actually perfect for the real base factor patterns, it is not so for ordinary factor analysis patterns. Even in this framework the congruence, coefficient has weaknesses, and it is suggested that decisions be based on the joint outcome of r[SUBc] and 8, the salient variable similarity index.     

Measures of invariance and comparability in factor analysis for fixed variables

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.     

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.     

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.     

How the 2SLS/IV estimator can handle equality constraints in structural equation models: A system‐of‐equations approach

Parameters in structural equation models are typically estimated using the maximum likelihood (ML) approach. Bollen (1996) proposed an alternative non‐iterative, equation‐by‐equation estimator that uses instrumental variables. Although this two‐stage least squares/instrumental variables (2SLS/IV) estimator has good statistical properties, one problem with its application is that parameter equality constraints cannot be imposed. This paper presents a mathematical solution to this problem that is based on an extension of the 2SLS/IV approach to a system of equations. We present an example in which our approach was used to examine strong longitudinal measurement invariance. We also investigated the new approach in a simulation study that compared it with ML in the examination of the equality of two latent regression coefficients and strong measurement invariance. Overall, the results show that the suggested approach is a useful extension of the original 2SLS/IV estimator and allows for the effective handling of equality constraints in structural equation models.     

Tests of statistical hypotheses in the case of unequal or disproportionate numbers of observations in the subclasses

General solutions of the analysis of variance in the case of unequal numbers of observations in the subclasses are presented. If we havek criteria for the classification, there will bek! solutions in making a complete analysis and 2^k–1 answers, bearing different meanings, for the sum of squares between subclasses of each criterion. The sum of squares for the interaction of any order, however, will be identical in different solutions of same problem.     

USING A META-ANALYTIC PERSPECTIVE TO ENHANCE JOB COMPONENT VALIDATION

This paper develops synthetic validity estimates based on a meta-analytic-weighted least squares (WLS) approach to job component validity (JCV), using position analysis questionnaire (PAQ) estimates of job characteristics, and the Data, People, & Things ratings from the  Dictionary of Occupational Titles  as indices of job complexity. For the general aptitude test battery database of 40,487 employees, nine validity coefficients were estimated for 192 positions. The predicted validities from the WLS approach had lower estimated variability than would be obtained from either the classic JCV approach or local criterion-related validity studies. Data, People, & Things summary ratings did not consistently moderate validity coefficients, whereas the PAQ data did moderate validity coefficients. In sum, these results suggest that synthetic validity procedures should incorporate a WLS regression approach. Moreover, researchers should consider a comprehensive set of job characteristics when considering job complexity rather than a single aggregated index.     

Generalized structured component analysis

We propose an alternative method to partial least squares for path analysis with components, called generalized structured component analysis. The proposed method replaces factors by exact linear combinations of observed variables. It employs a well-defined least squares criterion to estimate model parameters. As a result, the proposed method avoids the principal limitation of partial least squares (i.e., the lack of a global optimization procedure) while fully retaining all the advantages of partial least squares (e.g., less restricted distributional assumptions and no improper solutions). The method is also versatile enough to capture complex relationships among variables, including higher-order components and multi-group comparisons. A straightforward estimation algorithm is developed to minimize the criterion.The work reported in this paper was supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the second author. We wish to thank Richard Bagozzi for permitting us to use his organizational identification data and Wynne Chin for providing PLS-Graph 3.0.     

Prevalence and structure of behavior symptoms among children in special education and regular classroom settings

Studied the factor structure of the Behavior Problem Checklist using children from a middle to upper-middle-class suburban school system, determined the extent to which factors generated from children in regular classrooms approximated those generated from children in special education, and examined the effects of class placement, sex, and the interactions of these variables on the average rating children receive on the generated factors. One hundred children from 12 classes for the emotionally disturbed or learning disabled and 527 children from regular forst-through seventh-grade classes were rated by their teachers on the Behaviors Problem Checklist. Following a varimax factor rotation, coefficients of factor congruence and analysis of variance were conducted to test the hypotheses. Results indicated that although teacher ratings of special education and normal children are structurally similar, special education children scored higher on all three factors than did their regular class counterparts. Furthermore, almost all percentages of occurance for individual items were higher for special education children. Further research is suggested to deal with a possible expectancy effect in teacher ratings of problem behaviors.     

Enhancing Interpretability in Factor Analysis by Means of Mathematical Optimization

Abstract

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors’ interpretability by building a sparse loading matrix. In this paper, we propose an optimization-based procedure to give meaning to the factors arising in EFA by means of an additional set of variables, called explanatory variables, which may include in particular the set of observed variables. A goodness-of-fit criterion is introduced which quantifies the quality of the interpretation given this way. Our methodology also exploits the rotational invariance of EFA to obtain the best orthogonal rotation of the factors, in terms of the goodness-of-fit, but making them match to some of the explanatory variables, thus going beyond traditional rotation methods. Therefore, our approach allows the analyst to interpret the factors not only in terms of the observed variables, but in terms of a broader set of variables. Our experimental results demonstrate how our approach enhances interpretability in EFA, first in an empirical dataset, concerning volumes of reservoirs in California, and second in a synthetic data example.     

Longitudinal factor analysis

A model is presented for factor analysing scores on a set of psychological tests administered as both pre- and postmeasures in a study of change. The model assumes that the same factors underlie the tests on each occasion, but that factor scores as well as factor loadings may change between occasions. Factors are defined to be orthogonal between as well as within occasions. A two-stage least squares procedure for fitting the model is described, and generally provides a unique rotation solution for the factors on each occasion.We thank M. W. Browne, R. P. McDonald, R. Pruzek, and the Managing Editor for their constructive comments.     

NFACOM: A new program for relating solutions in exploratory factor analysis

A new version of the program FACOM (comparison of exploratory factor analysis solutions) is presented. The new version (NFACOM) greatly extends the possibilities of the previous versions. It is now possible to use congruence as well as least squares comparisons in both orthogonal and oblique situations. What is more, n solutions, not just two, can be compared. Recently developed algorithms have been implemented; others, which have proved not to be theoretically well founded, have been omitted.     

On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.     

Cross‐cultural Generalizability of the Alternative Five‐factor Model Using the Zuckerman–Kuhlman–Aluja Personality Questionnaire

Several personality models are known for being replicable across cultures, such as the Five‐Factor Model (FFM) or Eysenck's Psychoticism–Extraversion–Neuroticism (PEN) model, and are for this reason considered universal. The aim of the current study was to evaluate the cross‐cultural replicability of the recently revised Alternative FFM (AFFM). A total of 15 048 participants from 23 cultures completed the Zuckerman–Kuhlman–Aluja Personality Questionnaire (ZKA‐PQ) aimed at assessing personality according to this revised AFFM. Internal consistencies, gender differences and correlations with age were similar across cultures for all five factors and facet scales. The AFFM structure was very similar across samples and can be considered as highly replicable with total congruence coefficients ranging from .94 to .99. Measurement invariance across cultures was assessed using multi‐group confirmatory factor analyses, and each higher‐order personality factor did reach configural and metric invariance. Scalar invariance was never reached, which implies that culture‐specific norms should be considered. The underlying structure of the ZKA‐PQ replicates well across cultures, suggesting that this questionnaire can be used in a large diversity of cultures and that the AFFM might be as universal as the FFM or the PEN model. This suggests that more research is needed to identify and define an integrative framework underlying these personality models. Copyright © 2016 European Association of Personality Psychology     

Negative congruence effects in letter and pseudo-letter recognition: the role of similarity and response conflict

Letters and pseudo-letters were presented in three experiments using a sequential same-different task. While first items were always presented in isolation, the second item was either presented in isolation or surrounded by geometrical non-target shapes that could be congruent or incongruent to the target. In two experiments, a physical sameness criterion was used. In Experiment 1, in one condition, different pairs were always distinct in shape, in another they were similar in shape. Negative congruence effects were obtained for different pairs that are similar. In Experiment 2, this effect is replicated within participants. In this experiment, similar and dissimilar stimuli were mixed. The results were explained in terms of the difficulty of responding different to stimuli that are similar in shape: when the second item is surrounded by a congruent shape, the similarity is emphasized, making this response even more difficult. In Experiment 3, the same stimuli were presented using a categorical sameness criterion. This reduces the role of physical similarity and thus eliminates the response conflict. As a result, negative congruence effects were no longer observed. Taken together, the three experiments identify another source of negative congruence effects besides the ones recently reported in the literature.     

Measurement Invariance,Predictive Invariance,and the Duality Paradox

The statistical literature on bias in psychological testing distinguishes at least two forms of bias: measurement bias and predictive bias. Measurement bias concerns group differences in the relationship between a test and the latent variable to be measured. Predictive bias concerns group differences in the relationship between a test and an external criterion. How are these two forms of bias related? For example. if a test is unbiased in the predictive sense, does this fact support the hypothesis that the test is unbiased in the measurement sense? A theorem is given that describes the conditions under which measurement invariance (lack of bias) is consistent with predictive invariance for the linear case. Paradoxically, these two forms of invariance are shown to be inconsistent under realistic conditions. This duality or inconsistency is illustrated in simulated data. The implications of the duality for group differences research are illustrated in real data involving gender and ethnic differences on the SAT. The phenomenon of duality may force a reinterpretation of common empirical findings of test criterion regression slope invariance. and of invariance in test validities. Other implications are discussed.