Simultaneous classification and multidimensional scaling with external information

For the exploratory analysis of a matrix of proximities or (dis)similarities between objects, one often uses cluster analysis (CA) or multidimensional scaling (MDS). Solutions resulting from such analyses are sometimes interpreted using external information on the objects. Usually the procedures of CA, MDS and using external information are carried out independently and sequentially, although combinations of two of the three procedures (CA and MDS, or multidimensional scaling and using external information) have been proposed in the literature. The present paper offers a procedure that combines all three procedures in one analysis, using a model that describes a partition of objects with cluster centroids represented in a low-dimensional space, which in turn is related to the information in the external variables. A simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure. Also, to illustrate the method, it is applied to two empirical data sets.     

A comparison of factor scores under conditions of factor obliquity

Six different methods of computing factor scores were investigated in a simulation study. Population scores created from oblique factor patterns selected from the psychological literature served as the bases for the simulations, and the stability of the different methods was assessed through cross-validation in a subject-sampling model. Results from 5 evaluative criteria indicated that a simplified, unit-weighting procedure based on the factor score coefficients was generally superior to several unit-weighting procedures based on the pattern or structure coefficients. This simplified method of computing factor scores also compared favorably with an exact-weighting scheme based on the full factor score coefficient matrix. Results are discussed with regard to their potential impact on current practice, and several recommendations are offered.     

Rotation to achieve factorial invariance

Under certain conditions it is reasonable to assume that the same factor pattern matrix will describe the regression of observed on factor scores in different populations. However, ordinary factoring procedures will not reveal in general the existence of such a factor pattern matrix. Two procedures for rotating any number of factor pattern matrices based on different populations to conform to a single best fitting factor pattern matrix are developed in this paper. It is assumed that the same number of factors have been determined for each population. Both procedures will yield oblique results in the various populations. The procedures are illustrated with data taken from the 1939 Holzinger-Swineford monograph. Four groups of individuals are utilized.     

The relationship between external variables and common factors

A theorem is presented which gives the range of possible correlations between a common factor and an external variable (i.e., a variable not included in the test battery factor analyzed). Analogous expressions for component (and regression component) theory are also derived. Some situations involving external correlations are then discussed which dramatize the theoretical differences between components and common factors.Support by the National Research Council of Canada (NRC Grant No. A4640) and the University of British Columbia (UBC Humanities and Social Sciences Grant 26-9718) is gratefully acknowledged.This paper is based in part on the author's Ph.D. dissertation. I am particularly grateful to my dissertation advisor, Dr. Peter H. Schönemann. Thanks also to the editor and the anonymous reviewers, who contributed many helpful comments and suggestions.     

Influence analysis for the factor analysis model with ranking data

Influence analysis is an important component of data analysis, and the local influence approach has been widely applied to many statistical models to identify influential observations and assess minor model perturbations since the pioneering work of Cook (1986) . The approach is often adopted to develop influence analysis procedures for factor analysis models with ranking data. However, as this well‐known approach is based on the observed data likelihood, which involves multidimensional integrals, directly applying it to develop influence analysis procedures for the factor analysis models with ranking data is difficult. To address this difficulty, a Monte Carlo expectation and maximization algorithm (MCEM) is used to obtain the maximum‐likelihood estimate of the model parameters, and measures for influence analysis on the basis of the conditional expectation of the complete data log likelihood at the E‐step of the MCEM algorithm are then obtained. Very little additional computation is needed to compute the influence measures, because it is possible to make use of the by‐products of the estimation procedure. Influence measures that are based on several typical perturbation schemes are discussed in detail, and the proposed method is illustrated with two real examples and an artificial example.     

Regression among factor scores

Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.     

The factor structure and dimensional scoring of the generalized anxiety disorder questionnaire for DSM-IV

Despite favorable psychometric properties, the Generalized Anxiety Disorder Questionnaire for the Diagnostic and Statistical Manual of Mental Disorders (4th ed.) (GAD-Q-IV) does not have a known factor structure, which calls into question use of its original weighted scoring system (usually referred to as the dimensional score). Analyses appropriate to categorical item responses in a large sample of undergraduates were used to establish the scale's factor structure. Analyses indicated that a one-factor structure resulted in good fit. A scoring method based on this one-factor structure was compared with a variety of alternative scoring procedures, and it was found that a method based on factor scores did relatively well but that the previously suggested dimensional score failed to perform better than a simple sum of items. Results support the general unity of the measure but raise doubts regarding its scoring and response options.     

IMINCE: An unrestricted factor-analysis-based program for assessing measurement invariance

In this article, a Windows program for analyzing measurement invariance in two different populations is described. Factor analysis is a common way of assessing measurement invariance, and restricted factor analysis is now the most popular method. However, applied researchers have usually found that the theoretical advantages of restricted factor analysis do not always apply in practical situations. For example, when the participant sample is large, as is the case in Internet-based questionnaires, the available software for restricted factor analysis might fail to converge on a solution. Our program is based on unrestricted factor analysis and considers the three parameters that define factor invariance: difficulties, discriminations, and residual variances. The statistical significance of the tests for evaluating invariance is obtained using Bootstrap resampling procedures. A real-life example demonstrates the usefulness of the program.     

Dynamical system approach to factor analysis parameter estimation

A new unified approach to solving and studying the factor analysis parameter estimation problem is proposed. The maximum likelihood and least squares formulations of factor analysis are considered. The approach leads to globally convergent procedures for simultaneous estimation of the factor analysis parameters. The method presented necessarily leads to proper factor analysis estimations.     

Testing the assumptions underlying tetrachoric correlations

A method is proposed for empirically testing the appropriateness of using tetrachoric correlations for a set of dichotomous variables. Trivariate marginal information is used to get a set of one-degree of freedom chi-square tests of the underlying normality. It is argued that such tests should preferrably preceed further modeling of tetrachorics, for example, modeling by factor analysis. The assumptions are tested in some real and simulated data.Presented at the Psychometric Society meeting in Santa Barbara, California, June 25–26, 1984. The research of the first author was supported by Grant No. SES-8312583 from the National Science Foundation.     

A non-graphical method for transforming an arbitrary factor matrix into a simple structure factor matrix

The most commonly used method of factoring a matrix of intercorrelations is the centroid method developed by L. L. Thurstone. It is, however, necessary to transform the centroid matrix of factor loadings into a simple structure matrix in order to facilitate the interpretation of the factor loadings. Current methods for effecting this transformation are chiefly graphical and require considerable experience and personal judgment. This paper presents a new method for transforming an arbitrary factor matrix into a simple structure matrix by methods almost completely objective. The theory underlying the method is developed and approximation procedures are derived. The method is applied to a matrix of factor loadings previously analyzed by Thurstone.     

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Indefinite symmetric matrices that are estimates of positive-definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based methods for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.     

The effect of additional variables on factor indeterminacy in models with a single common factor

Determinate solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.     

A Proposed Number Correct Scoring Procedure Based on Classical True-Score Theory and Multidimensional Item Response Theory

A hybrid procedure for number correct scoring is proposed. The proposed scoring procedure is based on both classical true-score theory (CTT) and multidimensional item response theory (MIRT). Specifically, the hybrid scoring procedure uses test item weights based on MIRT and the total test scores are computed based on CTT. Thus, what makes the hybrid scoring method attractive is that this method accounts for the dimensionality of the test items while test scores remain easy to compute. Further, the hybrid scoring does not require large sample sizes once the item parameters are known. Monte Carlo techniques were used to compare and contrast the proposed hybrid scoring method with three other scoring procedures. Results indicated that all scoring methods in this study generated estimated and true scores that were highly correlated. However, the hybrid scoring procedure had significantly smaller error variances between the estimated and true scores relative to the other procedures.     

FACTOR: A computer program to fit the exploratory factor analysis model

Exploratory factor analysis (EFA) is one of the most widely used statistical procedures in psychological research. It is a classic technique, but statistical research into EFA is still quite active, and various new developments and methods have been presented in recent years. The authors of the most popular statistical packages, however, do not seem very interested in incorporating these new advances. We present the program FACTOR, which was designed as a general, user-friendly program for computing EFA. It implements traditional procedures and indices and incorporates the benefits of some more recent developments. Two of the traditional procedures implemented are polychoric correlations and parallel analysis, the latter of which is considered to be one of the best methods for determining the number of factors or components to be retained. Good examples of the most recent developments implemented in our program are (1) minimum rank factor analysis, which is the only factor method that allows one to compute the proportion of variance explained by each factor, and (2) the simplimax rotation method, which has proved to be the most powerful rotation method available. Of these methods, only polychoric correlations are available in some commercial programs. A copy of the software, a demo, and a short manual can be obtained free of charge from the first author.     

Distinguishing Constructs from Variables in Designing Research

Current research practices often conflate theoretical constructs and explanatory hypotheses with variables and predicted effects, to the detriment of research progress. This has led to the use of procedures such as manipulation checks, mediation analysis, and boundary conditions predicated on the idea that matching constructs to variables is necessary to validate that a theory corresponds to an effect . An alternative perspective, Inference to the Best Explanation (IBE), calls for designing research to exploit the power of distinguishing constructs from variables, hypotheses from predictions, and theory from effects. IBE calls for stating hypotheses (Hs) about construct‐to‐construct relationships and, separately, the predictions (Ps) about variable‐to‐variable effects that are explained by the hypotheses. In addition, articles should include disparate effects, a single explanation covering all studies, and a discussion of the use of the research in specific problem contexts. The application of IBE is illustrated with research investigating when judgments are based on a feeling about the ease of information retrieval versus the information content itself.     

On local influence analysis of full information item factor models

The full information item factor (FIIF) model is very useful for analyzing relations of dichotomous variables. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the FIIF model. The development is based on a Q-displacement function which is closely related with the Monte Carlo EM algorithm in the ML estimation. In the E-step of this algorithm, the conditional expectations are approximated by sample means of observations simulated by the Gibbs sampler from the appropriate conditional distributions. It turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures. The diagnoses are based on the conformal normal curvature that can be computed easily. A number of interesting perturbation schemes are considered. The methodology is illustrated with two real examples.The research is fully supported by a grant (CUHK 4356/00H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the Editor, Associate Editor, anonymous reviewers, and W.Y. Poon for valuable comments for improving the paper, and to ICPSR and the relevant founding agency for allowing us to use of their data. The assistance of Michael Leung and Esther Tam is gratefully acknowledged.     

Abstract: A Comparison of the Factor Mixture Model and Taxometric Procedures

Taxometric procedures and the Factor Mixture Model (FMM) have a complimentary set of strengths and weaknesses. Both approaches purport to detect evidence of a latent class structure. Taxometric procedures, popular in psychiatric and psychopathology literature, make no assumptions beyond those needed to compute means and covariances. However, Taxometric procedures assume that observed items are uncorrelated within a class or taxon. This assumption is violated when there are individual differences in the trait underlying the items (i.e., severity differences within class). FMMs can model within-class covariance structures ranging from local independence to multidimensional within-class factor models and permits the specification of more than two classes. FMMs typically rely on normality assumptions for within-class factors and error terms. FMMs are highly parameterized and susceptible to misspecifications of the within-class covariance structure.

The current study compared the Taxometric procedures MAXEIG and the Base-Rate Classification Technique to the FMM in their respective abilities to (1) correctly detect the two-class structure in simulated data, and to (2) correctly assign subjects to classes. Two class data were simulated under conditions of balanced and imbalanced relative class size, high and low class separation, and 1-factor and 2-factor within-class covariance structures. For the 2-factor data, simple and cross-loaded factor loading structures, and positive and negative factor correlations were considered. For the FMM, both correct and incorrect within-class factor structures were fit to the data.

FMMs generally outperformed Taxometric procedures in terms of both class detection and in assigning subjects to classes. Imbalanced relative class size (e.g., a small minority class and a large majority class) negatively impacted both FMM and Taxometric performance while low class separation was much more problematic for Taxometric procedures than the FMM. Comparisons of alterative FMMs based on information criteria generally resulted in correct model choice but deteriorated when small class separation was combined with imbalanced relative class size.     

Psychometric properties of beliefs about relationship violence against women and gender stereotypes scale

This article describes psychometric properties of the Gender-Based Relationship Violence Beliefs Scale (BEREVIWOS) and the Gender Stereotypes and Beliefs (GESTABE) scale in a Nigerian setting. Analysis was based on a convenience sample of 202 respondents. Confirmatory factor analysis (CFA) was used to determine construct validity of the three-factor structure of each scale and measurement invariance procedures were utilised to determine whether the factor structure was equivalent across gender. Convergent and discriminant validity of the scales, and predictive and incremental validity were examined for consistency with theory on gender beliefs/stereotypes and violence in patriarchal society. A single-factor CFA model approach was used to examine common method bias. Results suggest the BEREVIWOS and GESTABE are multidimensional scales with an intercorrelated three-factor structure. The fit of the models and composite reliability were adequate. Scores from the BEREVIWOS and GESTABE appear reliable and valid measures of underlying beliefs and stereotypes associated with violence against women in Nigeria.     

The Early History of Multivariate Techniques in Psychological Research

The origins of multivariate techniques are traced to Galtun, and the circumstances and precedents for Pearson's development of the method of principal components are described. Some early forms of factor extraction are reviewed with emphasis upon the assumptions involved. The different purposes of general psychology and of individual psychology are shown to entail differences in factoring technique. Burt's adoption of the hypothetico-disjunctive method in connection with factor analysis is explained and illustrated. Some fundamental similarities between analysis of variance and correlational analysis are discussed; and arguments are given for the joint application of these and other mathematical procedures now available for handling multivariate problem.