Simultaneous factor analysis in several populations

This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large samplex ² of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable.This research was supported by grant NSF-GB-12959 from National Science Foundation. My thanks are due to Michael Browne for his comments on an earlier draft of this paper and to Marielle van Thillo who checked the mathematical derivations and wrote and debugged the computer program SIFASP.Now at Statistics Department, University of Uppsala, Sweden.     

Simultaneous factor analysis of dichotomous variables in several groups

A new method is proposed for a simultaneous factor analysis of dichotomous responses from several groups of individuals. The method makes it possible to compare factor loading pattern, factor variances and covariances, and factor means over groups. The method uses information from first and second order proportions and estimates the model by generalized least-squares. Hypotheses regarding different degrees of invariance over groups may be evaluated by a large-sample chi-square test.The research of the first author was supported partly by the Bank of Sweden Tercentenary Foundation under project Structural Equation Models in the Social Sciences, project director Karl G. Jöreskog, and partly by the Swedish Research Council in the Humanistic and Social Sciences under project Analysis of Contingency Tables, project director Bengt Muthén.     

Simultaneous analysis of coupled data matrices subject to different amounts of noise

In many areas of science, research questions imply the analysis of a set of coupled data blocks, with, for instance, each block being an experimental unit by variable matrix, and the variables being the same in all matrices. To obtain an overall picture of the mechanisms that play a role in the different data matrices, the information in these matrices needs to be integrated. This may be achieved by applying a data‐analytic strategy in which a global model is fitted to all data matrices simultaneously, as in some forms of simultaneous component analysis (SCA). Since such a strategy implies that all data entries, regardless the matrix they belong to, contribute equally to the analysis, it may obfuscate the overall picture of the mechanisms underlying the data when the different data matrices are subject to different amounts of noise. One way out is to downweight entries from noisy data matrices in favour of entries from less noisy matrices. Information regarding the amount of noise that is present in each matrix, however, is, in most cases, not available. To deal with these problems, in this paper a novel maximum‐likelihood‐based simultaneous component analysis method, referred to as MxLSCA, is proposed. Being a stochastic extension of SCA, in MxLSCA the amount of noise in each data matrix is estimated and entries from noisy data matrices are downweighted. Both in an extensive simulation study and in an application to data stemming from cross‐cultural emotion psychology, it is shown that the novel MxLSCA strategy outperforms the SCA strategy with respect to disclosing the mechanisms underlying the coupled data.     

A note on the factor analysis of partial covariance matrices

A relationship is given between the joint common factor structure of two sets of variables, and the factor structure of the partial covariance matrix of one of the sets with the other partialled out.     

Sensitivity analysis in factor analysis: Difference between using covariance and correlation matrices

Influence curves of some parameters under various methods of factor analysis have been given in the literature. These influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis. The differences between the influence curves based on the covariance and the correlation matrices are derived in this paper. Simple formulas for the differences of the influence curves, based on the two matrices, for the unique variance matrix, factor loadings and some other parameter are obtained under scale-invariant estimation methods, though the influence curves themselves are in complex forms.The authors are most grateful to the referees, the Associate Editor, the Editor and Raymond Lam for helpful suggestions for improving the clarity of the paper.     

Distinguishing personality and biodata items using confirmatory factor analysis of multitrait-multimethod matrices

Hard biodata, soft biodata, and personality items were developed to measure three personal constructs (Dependability, Demeanor, and Ambition). This resulted in nine sets of items forming a three-method (personality, soft biodata, and hard biodata) by three-trait (Dependability, Demeanor, and Ambition) Multitrait-Multimethod (MTMM) matrix. Four models were proposed to represent the nine scales and each was tested on a sample of 477 job applicants and subsequently cross-validated on a sample of 958 job applicants, using confirmatory factor analysis. The model where the personality and soft biodata items represented one factor and the hard biodata items represented another, distinct factor, was both the best fitting and most parsimonious model, thus confirming Asher's (1972) taxonomy.This research was based in part on a doctoral dissertation submitted to Wayne State University. I would like to thank committee members Lois Tetrick (Chair), Alan Bass, Sebastiano Fisicaro, and Thomas Naughton. An earlier version of this paper was presented at the 8th annual conference of the Society for Industrial and Organizational Psychology (SIOP), San Francisco, CA: April, 1993.     

Simultaneous testing of McNemar's problem for several populations

McNemar's problem concerns the hypothesis of equal probabilities for the unlike pairs of correlated binary variables. We consider four different extensions to this problem, each for testing simultaneous equality of proportions of unlike pairs inc independent populations of correlated binary variables, but each under different assumptions and/or additional hypotheses. For each extension both the likelihood ratio test and the goodness-of-fit chi-square test are given. Whenc=1, all cases reduce to McNemar's problem. Forc ≥ 2, however, the tests are quite different, depending on exactly how the hypothesis and alternatives of McNemar are extended. An example illustrates how widely the results may differ, depending on which extended framework is appropriate.     

Identification with deficient rank loading matrices in confirmatory factor analysis: Multitrait-multimethod models

This paper presents some results on identification in multitrait-multimethod (MTMM) confirmatory factor analysis (CFA) models. Some MTMM models are not identified when the (factorial-patterned) loadings matrix is of deficient column rank. For at least one other MTMM model, identification does exist despite such deficiency. It is also shown that for some MTMM CFA models, Howe's (1955) conditions sufficient for rotational uniqueness can fail, yet the model may well be identified and rotationally unique. Implications of these results for CFA models in general are discussed.     

Rotational equivalence of factor loading matrices with specified values

Under mild assumptions, when appropriate elements of a factor loading matrix are specified to be zero, all orthogonally equivalent matrices differ at most by column sign changes. Here a variety of results are given for the more complex case when the specified values are not necessarily zero. A method is given for constructing reflections to preserve specified rows and columns. When the appropriatek(k – 1)/2 elements have been specified, sufficient conditions are stated for the existence of 2 ^k orthogonally equivalent matrices.This research was supported in part by the National Institute of Health Grant RR-3.     

Simultaneous optimization of factor assessibility and representativeness

P. M. Bentler has shown that Rao's canonical factor analysis is in effect a psychometric analysis, leading to factors that are maximally assessible from the data. He contrasts this with Kaiser and Caffrey's alpha factor analysis that leads to factors that maximally represent the true factors in the content domain. Noting the problems associated with factors that may be highly assessible, but not very representative, or vice versa, Bentler suggests the need for a technique that would, insofar as possible, be optimal with respect to both criteria. Such a technique is presented here, and is shown to resolve into a traditional scaling method, which in turn acquires a richer psychometric interpretation.The author would like to gratefully acknowledge support for this research from the Workshop in Political Theory and Policy Analysis (Indiana University) and Esco Research Associates. The paper benefitted substantially from the criticisms and suggestions of an anonymous referee.     

The analysis of multitrait-multimethod matrices via constrained components analysis

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to three anonymous reviewers and an associate editor for constructive suggestions on the first version of this paper.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

A note on the identification of restricted factor loading matrices

It is shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions. A necessary (under a regularity assumption) and sufficient condition for local uniqueness is given and a counterexample is provided to a theorem by J. Algina concerning necessary and sufficient conditions for global uniqueness.     

Procrustean analysis of matrices with different row order

Procrustean analysis is extended here to the case where matrices have different row order. Using a definition system in the sense of facet theory, fitting transformations are determined for the set of centroids that represent common substantive equivalence classes of all or some matrices. The resulting operators are used to transform the original matrices. An index, δ, which is consistent with this approach expresses the goodness of fit.     

The extension of component analysis to four-mode matrices

A model for four-mode component analysis is developed and presented. The developed model, which is an extension of Tucker's three-mode factor analytic model, allows for the simultaneous analysis of all modes of a four-mode data matrix and the consideration of relationships among the modes. An empirical example based upon viewer perceptions of repetitive advertising shows the four-mode model applicable to real data.This research was supported by the University of Kansas School of Business Research Fund provided by the Fourth National Bank & Trust Company, Wichita. The ideas and opinions expressed herein are solely those of the author.     

Covariance structure analysis in several populations

This paper is concerned with the study of covariance structural models in several populations. Estimation theory of the parameters that are subject to general functional restraints is developed based on the generalized least squares approach. Asymptotic properties of the constrained estimator are studied; and asymptotic chi-square tests are presented to evaluate appropriate model comparisons. The method of multipliers and the standard reparametrization technique are discussed in obtaining the estimates. The methodology is demonstrated by a set of real data.Computer facilities were provided by the Computer Services Center, The Chinese University of Hong Kong. The authors are indebted to several anonymous reviewers for suggestions for improvement of this paper.     

Evaluation of factor analytic research procedures by means of simulated correlation matrices

In order to study the effectiveness of factor analytic methods, a procedure was developed for computing simulated correlation matrices which are more similar to real data correlation matrices than are those matrices computed from the factor analysis structural model. In the present investigation, three methods of factor extraction were studied as applied to 54 simulated correlation matrices which varied in proportion of variance derived from a major factor domain, number of factors in the major domain, and closeness of the simulation procedure to the factor analysis structural model. While the factor extraction methods differed little from one another in quality of results for matrices more dissimilar to the factor analytic model, major differences in quality of results were associated with fewer factors in the major domain, higher proportion of variance from the major domain, and closeness of the simulation procedure to the factor analysis structural model.The research was jointly supported by the University of Illinois and the office of Naval Research under contracts Nonr 1834(39) and N00014-67-A-0305-0003.