When is a correlation matrix appropriate for factor analysis? Some decision rules.

Discusses 3 techniques for assessing the psychometric adequacy of correlation matrices; (a) computation of M. S. Bartlett's test of sphericity, (b) inspection of the off-diagonal elements of the anti-image covariance matrix, and (c) computation of the Kaiser-Meyer-Olkin 1970 measure of sampling adequacy. The advantages and disadvantages of each are compared with respect to assessment of correlation matrices prior to factor analysis. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reexamining the circumplex model of affect.

The circumplex model of affect has been among the most widely studied representations of affect. Despite the considerable evidence cited in support of it, methods typically used to evaluate the model have substantial limitations. In this article, the authors attempt to correct past limitations by using a covariance structure model specifically designed to assess circumplex structure. This model was fit to 47 individual correlation matrices from published data sets. Analyses revealed that model fit was typically acceptable and that opposing affective states usually demonstrated strong negative correlations with one another. However, analyses also indicated substantial variability in both model fit and correlations among opposing affective states and suggested several characteristics of studies that partially accounted for this variability. Detailed examination of the locations of affective states for 10 of the correlation matrices with relatively optimal characteristics provided mixed support for the model. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Evaluating model fit for growth curve models: Integration of fit indices from SEM and MLM frameworks.

Evaluating overall model fit for growth curve models involves 3 challenging issues. (a) Three types of longitudinal data with different implications for model fit may be distinguished: balanced on time with complete data, balanced on time with data missing at random, and unbalanced on time. (b) Traditional work on fit from the structural equation modeling (SEM) perspective has focused only on the covariance structure, but growth curve models have four potential sources of misspecification: within-individual covariance matrix, between-individuals covariance matrix, marginal mean structure, and conditional mean structure. (c) Growth curve models can be estimated in both the SEM and multilevel modeling (MLM) frameworks; these have different emphases for the evaluation of model fit. In this article, the authors discuss the challenges presented by these 3 issues in the calculation and interpretation of SEM- and MLM-based fit indices for growth curve models and conclude by identifying some lines for future research. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Applications of covariance structure modeling in psychology: Cause for concern?

Methods of covariance structure modeling are frequently applied in psychological research. These methods merge the logic of confirmatory factor analysis, multiple regression, and path analysis within a single data analytic framework. Among the many applications are estimation of disattenuated correlation and regression coefficients, evaluation of multitrait–multimethod matrices, and assessment of hypothesized causal structures. Shortcomings of these methods are commonly acknowledged in the mathematical literature and in textbooks. Nevertheless, serious flaws remain in many published applications. For example, it is rarely noted that the fit of a favored model is identical for a potentially large number of equivalent models. A review of the personality and social psychology literature illustrates the nature of this and other problems in reported applications of covariance structure models. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Examining the structural validity of the Strengths and Difficulties Questionnaire (SDQ) in a U.S. sample of custodial grandmothers.

The authors examined the structural validity of the parent informant version of the Strengths and Difficulties Questionnaire (SDQ) with a sample of 733 custodial grandparents. Three models of the SDQ's factor structure were evaluated with confirmatory factor analysis based on the item covariance matrix. Although indices of fit were good across all 3 models, a model that included a newly hypothesized positive construal method factor in addition to the 4 symptom factors (Emotional Symptoms, Conduct Problems, Hyperactivity-Inattention, Peer Problems) and the single Prosocial Behavior factor originally intended for the SDQ provided the best representation of this instrument's latent structure. Structural validity was further evidenced by measurement invariance across grandparent race and grandchild gender and age, a conceptually meaningful pattern of cross-scale correlations, and the acceptable internal reliability estimates found for each subscale. Measurement and clinical implications of the results are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Linear models for the analysis and construction of instruments in a facet design.

Describes linear models for the situation wherein a measurement instrument is constructed for all elements of the Cartesian product of several facets when the elements of each facet are not ordered. The structure of the covariance matrix of the instrument is derived from the models. By using covariance structure analysis, the models can be tested, and estimates of the parameters can be obtained. Models for 20 tests were formulated and tested and were constructed from a design with a behavioral and a situational facet measuring social anxiety in children; for 15 tests a model proved to fit the data. It is concluded that covariance structure analysis is useful for the analysis and construction of measurement instruments. (34 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Commentary on Steinley and Brusco (2011): Recommendations and cautions.

I discuss the recommendations and cautions in Steinley and Brusco's (2011) article on the use of finite models to cluster a data set. In their article, much use is made of comparison with the K-means procedure. As noted by researchers for over 30 years, the K-means procedure can be viewed as a special case of finite mixture modeling in which the components are in equal (fixed) proportions and are taken to be normal with a common spherical covariance matrix. In this commentary, I pay particular attention to this link and to the use of normal mixture models with arbitrary component-covariance matrices. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Gender and smoking cessation: A factor structure comparison of processes of change.

This study examined gender differences in the pattern of process use for smoking cessation using the Processes of Change Questionnaire (J. O. Prochaska et al; see record 1989-03620-001). The goals were (a) to determine the degree to which the covariance structure of the Processes of Change Questionnaire is invariant across gender, (b) to test the existence of the theoretical 2-factor process model using confirmatory factor analysis, and (c) to explore mean differences, if found, in the use of the 10 processes of change across 4 stages of change (precontemplation, contemplation, preparation, and action). The sample (N?=?516) had an equal distribution of men and women across the stages of change. Results demonstrated that the structure of the measure for men and women was invariant at the level of the variance–covariance matrices and that the hypothesized 2-factor model fit the data. Only stage of change predicted the experiential and behavioral process factors. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On the dimensionality of the California Psychological Inventory Socialization scale.

The California Psychological Inventory Socialization (So) scale is assumed to measure a dimension of socialized behaviors based on role-taking ability. Some studies have reported that the internal structure of the So scale may be complex. The present study tested its dimensionality on 3 groups: (a) 154 female undergraduates, (b) 38 delinquent and 146 nondelinquent 18–31 yr old males, and (c) 64 18–49 yr old males who were undergoing court-ordered psychiatric examination prior to sentencing. All So items except one were used, and these items were divided into 6 subscales. The number of factors producing the correlations of subscales was tested in maximum likelihood factor analyses made separately for the groups. The results made a single factor seem reasonable. The variance–covariance matrices differed significantly for the groups. The fit of a 1-factor model for 3 groups was therefore tested in a simultaneous factor analysis for several populations, and good fit was obtained. (46 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comment on Dziuban and Shirkey's decision rules for factor analysis.

C. D. Dziuban and E. C. Shirkey (see record 1974-28961-001) recommended some rules for deciding whether to fit a factor-analytic model to a correlation matrix. This comment argues that they belabored a necessary but far from sufficient condition for the factor-analytic model to be appropriate. It is also argued that models suggested by the theory of analysis of covariance structures should be considered alternatives to the factor model under certain circumstances. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comments on the Meehl-Waller (2002) procedure for appraisal of path analysis models.

P. E. Meehl and N. G. Waller (see record 2002-18342-001) proposed an innovative method for assessing path analysis models wherein they subjected a given model, along with a set of alternatives, to risky tests using selected elements of a sample correlation matrix. Although the authors find much common ground with the perspective underlying the Meehl-Waller approach, they suggest that there are aspects of the proposed procedure that require close examination and further development. These include the selection of only one subset of correlations to estimate parameters when multiple solutions are generally available, the fact that the risky tests may test only a subset of parameters rather than the full model of interest, and the potential for different results to be obtained from analysis of equivalent models. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

No more than skin deep: Ethnic and racial similarity in developmental process.

Many studies adduce evidence of ethnic or racial dissimilarities in developmental outcomes (e.g., delinquency and achievement). Many researchers fail to distinguish between group average levels and developmental processes (correlations). Evidence is reviewed that developmental processes are nearly identical for US Black, Hispanic, White, and Asian ethnic and racial groups. Using diverse and representative data sources, covariance matrices were computed for these ethnic groups and then compared by using a LISREL goodness-of-fit test. Not only were these matrices nearly identical but they also were no less alike than covariance matrices computed from random halves within 1 ethnic or racial group. This article documents the importance of accepting ethnic and racial similarity of developmental processes. Thus, group average level differences may result from different levels of developmental antecedents working through common developmental pathways. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Using covariance structure analysis to detect correlates and predictors of individual change over time.

Recently, methodologists have shown how 2 disparate conceptual arenas (individual growth modeling and covariance structure analysis) can be integrated. The integration brings the flexibility of covariance analysis to bear on the investigation of systematic interindividual differences in change and provides another powerful data-analytic tool for answering questions about the relationship between individual true change and potential predictors of that change. The individual growth modeling framework uses a pair of hierarchical statistical models to represent (1) within-person true status as a function of time and (2) between-person differences in true change as a function of predictors. This article explains how these models can be reformatted to correspond, respectively, to the measurement and structural components of the general LISREL model with mean structures and illustrates, by means of worked example, how the new method can be applied to a sample of longitudinal panel data. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Allowing for correlations between correlations in random-effects meta-analysis of correlation matrices.

Practical meta-analysis of correlation matrices generally ignores covariances (and hence correlations) between correlation estimates. The authors consider various methods for allowing for covariances, including generalized least squares, maximum marginal likelihood, and Bayesian approaches, illustrated using a 6-dimensional response in a series of psychological studies concerning prediction of exercise behavior change. Quantities of interest include the overall population mean correlation matrix, the contrast between the mean correlations, the predicted correlation matrix in a new study, and the conflict between the existing studies and a new correlation matrix. The authors conclude that accounting for correlations between correlations is unnecessary when interested in individual correlations but potentially important if concerned with a composite measure involving 2 or more correlations. A simulation study indicates the asymptotic normal assumption appears reasonable. Because of potential instability in the generalized least squares methods, they recommend a model-based approach, either the maximum marginal likelihood approach or a full Bayesian analysis. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A model for interpreting genetic correlations with estimates of parameters.

Presents a general additive model for kinship correlational structures which includes genetic, environmental, covariance, and measurement error components. The properties and assumptions of the model and other models which are viewed as special cases are discussed. A procedure is described for estimating the model parameters from sample correlation coefficients. The procedure results in numerical estimates of the ratio of genetic to environmental variance, estimates of the correlation between genetic and environmental factors, and estimates of the correlation between environments. A simple procedure for partitioning a correlation coefficient into its variance components is suggested. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Measurement of psychological disorders using cognitive diagnosis models.

Cognitive diagnosis models are constrained (multiple classification) latent class models that characterize the relationship of questionnaire responses to a set of dichotomous latent variables. Having emanated from educational measurement, several aspects of such models seem well suited to use in psychological assessment and diagnosis. This article presents the development of a new cognitive diagnosis model for use in psychological assessment--the DINO (deterministic input; noisy "or" gate) model--which, as an illustrative example, is applied to evaluate and diagnose pathological gamblers. As part of this example, a demonstration of the estimates obtained by cognitive diagnosis models is provided. Such estimates include the probability an individual meets each of a set of dichotomous Diagnostic and Statistical Manual of Mental Disorders (text revision [DSM-IV-TR]; American Psychiatric Association, 2000) criteria, resulting in an estimate of the probability an individual meets the DSM-IV-TR definition for being a pathological gambler. Furthermore, a demonstration of how the hypothesized underlying factors contributing to pathological gambling can be measured with the DINO model is presented, through use of a covariance structure model for the tetrachoric correlation matrix of the dichotomous latent variables representing DSM-IV-TR criteria. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Stability of skilled performance across time: Some generalizations and limitations on utilities.

We review two closely related phenomena in the study of ability and performance: (a) decreasing predictive validity in ability–performance relations when studied longitudinally and (b) the superdiagonal stability matrices (i.e., simplex-like matrices) found in the correlations between repeated trials on a variety of tasks. We discuss briefly two models that explain these phenomena, a changing-task model and a changing-subject model, and report an empirical study in which (a) the generality of the simplex phenomenon in an area of human performance not previously studied and (b) whether the decrease in correlations across time would eventually stabilize (a hypothesis of the changing-task model) were investigated. Intercorrelation matrices of major league baseball performance across 10 years (four measures) indicate that the correlations systematically decrease over time, with no evidence of stabilizing. We discuss the implications of these empirical findings and conceptual developments for utilities of selection programs. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Two-sample T–2 procedure and the assumption of homogeneous covariance matrices.

Reports on an empirical investigation of the robustness of H. Hotelling's (1931) 2-sample T–2 test with respect to violation of the assumption of homogeneity of covariance matrices. In a Monte Carlo study, empirical sampling distributions of the T–2 statistic were obtained from a large number of sets, each consisting of 2,000 samples drawn from multivariate normal parent populations. Average sample size (n), extent of inequality of sample sizes, number of variables (p), and degree of inequality of covariance matrices were combined to 108 different conditions. Actual proportions of values that exceeded nominal α levels are presented. For equal ns, the procedure was found to be generally robust. With unequal ns, the procedure is shown to become increasingly less robust as covariance matrix heterogeneity and p increase. The results are related to earlier findings, and implications for the proper use of the T–2 procedure are noted. (19 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The meta-analysis of factor analyses: An illustration based on the cumulation of correlation matrices.

This article introduces and illustrates a procedure for aggregating data from several factor analyses in a meta-analysis. The central idea is to combine the correlation matrices first and then perform a new factor analysis. Special attention is given to the handling of general factors in data sets and to the use of correlations corrected for attenuation. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A note on analysis of covariance.

S. H. Evans and E. J. Anastasio (see record 1968-09688-001) constructed 2 sets of hypothetical data to show ways in which the analysis of covariance can yield erroneous results. It is argued that analysis of covariance does not give incorrect results for either set of their data. Analysis of covariance correctly (within rounding error) recovered the form of the models used to construct their data and also provided accurate estimates of parameters for both sets of data. Although analysis of covariance produces accurate results, it must be used cautiously because it may or may not answer the substantive question a researcher asks. (PsycINFO Database Record (c) 2010 APA, all rights reserved)