Generalized multilevel structural equation modeling

A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.gllamm can be downloaded from http://www.gllamm.org. The paper was written while Sophia Rabe-Hesketh was employed at and Anders Skrondal was visiting the Department of Biostatistics and Computing, Institute of Psychiatry, King's College London.     

Comparing standardized coefficients in structural equation modeling: a model reparameterization approach

We propose a two-stage method for comparing standardized coefficients in structural equation modeling (SEM). At stage 1, we transform the original model of interest into the standardized model by model reparameterization, so that the model parameters appearing in the standardized model are equivalent to the standardized parameters of the original model. At stage 2, we impose appropriate linear equality constraints on the standardized model and use a likelihood ratio test to make statistical inferences about the equality of standardized coefficients. Unlike other existing methods for comparing standardized coefficients, the proposed method does not require specific modeling features (e.g., specification of nonlinear constraints), which are available only in certain SEM software programs. Moreover, this method allows researchers to compare two or more standardized coefficients simultaneously in a standard and convenient way. Three real examples are given to illustrate the proposed method, using EQS, a popular SEM software program. Results show that the proposed method performs satisfactorily for testing the equality of standardized coefficients.     

On specifying the null model for incremental fit indices in structural equation modeling

In structural equation modeling, incremental fit indices are based on the comparison of the fit of a substantive model to that of a null model. The standard null model yields unconstrained estimates of the variance (and mean, if included) of each manifest variable. For many models, however, the standard null model is an improper comparison model. In these cases, incremental fit index values reported automatically by structural modeling software have no interpretation and should be disregarded. The authors explain how to formulate an acceptable, modified null model, predict changes in fit index values accompanying its use, provide examples illustrating effects on fit index values when using such a model, and discuss implications for theory and practice of structural equation modeling.     

Constrained versus unconstrained estimation in structural equation modeling

Recently, R. D. Stoel, F. G. Garre, C. Dolan, and G. van den Wittenboer (2006) reviewed approaches for obtaining reference mixture distributions for difference tests when a parameter is on the boundary. The authors of the present study argue that this methodology is incomplete without a discussion of when the mixtures are needed and show that they only become relevant when constrained difference tests are conducted. Because constrained difference tests can hide important model misspecification, a reliable way to assess global model fit under constrained estimation would be needed. Examination of the options for assessing model fit under constrained estimation reveals that no perfect solutions exist, although the conditional approach of releasing a degree of freedom for each active constraint appears to be the most methodologically sound one. The authors discuss pros and cons of constrained and unconstrained estimation and their implementation in 5 popular structural equation modeling packages and argue that unconstrained estimation is a simpler method that is also more informative about sources of misfit. In practice, researchers will have trouble conducting constrained difference tests appropriately, as this requires a commitment to ignore Heywood cases. Consequently, mixture distributions for difference tests are rarely appropriate.     

Missing data techniques for structural equation modeling

As with other statistical methods, missing data often create major problems for the estimation of structural equation models (SEMs). Conventional methods such as listwise or pairwise deletion generally do a poor job of using all the available information. However, structural equation modelers are fortunate that many programs for estimating SEMs now have maximum likelihood methods for handling missing data in an optimal fashion. In addition to maximum likelihood, this article also discusses multiple imputation. This method has statistical properties that are almost as good as those for maximum likelihood and can be applied to a much wider array of models and estimation methods.     

Hopes and cautions in implementing Bayesian structural equation modeling

Muthén and Asparouhov (2012) have proposed and demonstrated an approach to model specification and estimation in structural equation modeling (SEM) using Bayesian methods. Their contribution builds on previous work in this area by (a) focusing on the translation of conventional SEM models into a Bayesian framework wherein parameters fixed at zero in a conventional model can be respecified using small-variance priors and (b) implementing their approach in software that is widely accessible. We recognize potential benefits for applied researchers as discussed by Muthén and Asparouhov, and we also see a tradeoff in that effective use of the proposed approach introduces increased demands in terms of expertise of users to navigate new complexities in model specification, parameter estimation, and evaluation of results. We also raise cautions regarding the issues of model modification and model fit. Although we see significant potential value in the use of Bayesian SEM, we also believe that effective use will require an awareness of these complexities. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Meta-analytic structural equation modeling: a two-stage approach

To synthesize studies that use structural equation modeling (SEM), researchers usually use Pearson correlations (univariate r), Fisher z scores (univariate z), or generalized least squares (GLS) to combine the correlation matrices. The pooled correlation matrix is then analyzed by the use of SEM. Questionable inferences may occur for these ad hoc procedures. A 2-stage structural equation modeling (TSSEM) method is proposed to incorporate meta-analytic techniques and SEM into a unified framework. Simulation results reveal that the univariate-r, univariate-z, and TSSEM methods perform well in testing the homogeneity of correlation matrices and estimating the pooled correlation matrix. When fitting SEM, only TSSEM works well. The GLS method performed poorly in small to medium samples.     

Bayesian model selection for mixtures of structural equation models with an unknown number of components

This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea is to construct a continuous path linking the competing models; then the Bayes factor can be estimated efficiently via grids in [0, 1] and simulated observations that are generated by the Gibbs sampler from the posterior distribution. Bayesian estimates of the structural parameters, latent variables, as well as other statistics can be produced as by‐products. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.     

Asymptotic biases in exploratory factor analysis and structural equation modeling

Formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the Wishart maximum likelihood estimation for normally and nonnormally distributed variables. When multivariate normality is satisfied, considerable simplification is obtained for the models of unstandardized variables. Formulas for the models of standardized variables are also provided. Numerical examples with Monte Carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data. Some relationships between the asymptotic biases and other asymptotic values are discussed.The author is indebted to the editor and anonymous reviewers for their comments, corrections, and suggestions on this paper, and to Yutaka Kano for discussion on biases.     

A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling

Meta-analysis and structural equation modeling (SEM) are two important statistical methods in the behavioral, social, and medical sciences. They are generally treated as two unrelated topics in the literature. The present article proposes a model to integrate fixed-, random-, and mixed-effects meta-analyses into the SEM framework. By applying an appropriate transformation on the data, studies in a meta-analysis can be analyzed as subjects in a structural equation model. This article also highlights some practical benefits of using the SEM approach to conduct a meta-analysis. Specifically, the SEM-based meta-analysis can be used to handle missing covariates, to quantify the heterogeneity of effect sizes, and to address the heterogeneity of effect sizes with mixture models. Examples are used to illustrate the equivalence between the conventional meta-analysis and the SEM-based meta-analysis. Future directions on and issues related to the SEM-based meta-analysis are discussed.     

Introduction to the special section on structural equation modeling

Structural equation modeling (SEM) is a frequently used data-analytic technique in psychopathology research. This popularity is due to the unique capabilities and broad applicability of SEM and to recent advances in model and software development. Unfortunately, the popularity and accessibility of SEM is matched by its complexities and ambiguities. Thus, users are often faced with difficult decisions regarding a variety of issues. This special section is designed to increase the effective use of SEM by reviewing recently developed modeling capabilities, identifying common problems in application, and recommending appropriate strategies for analysis and evaluation.     

Standard errors of fit indices using residuals in structural equation modeling

The asymptotic standard errors of the correlation residuals and Bentler's standardized residuals in covariance structures are derived based on the asymptotic covariance matrix of raw covariance residuals. Using these results, approximations of the asymptotic standard errors of the root mean square residuals for unstandardized or standardized residuals are derived by the delta method. Further, in mean structures, approximations of the asymptotic standard errors of residuals, standardized residuals and their summary statistics are derived in a similar manner. Simulations are carried out, which show that the asymptotic standard errors of the various types of residuals and the root mean square residuals in covariance, correlation and mean structures are close to actual ones.The author is indebted to the reviewers for their comments and suggestions which have led to an improvement of this work.     

Fitting multivariage normal finite mixtures subject to structural equation modeling

This paper is about fitting multivariate normal mixture distributions subject to structural equation modeling. The general model comprises common factor and structural regression models. The introduction of covariance and mean structure models reduces the number of parameters to be estimated in fitting the mixture and enables one to investigate a variety of substantive hypotheses concerning the differences between the components in the mixture. Within the general model, individual parameters can be subjected to equality, nonlinear and simple bounds constraints. Confidence intervals are based on the inverse of the Hessian and on the likelihood profile. Several illustrations are given and results of a simulation study concerning the confidence intervals are reported.     

Thought-action fusion: a comprehensive analysis using structural equation modeling

Thought-action fusion (TAF), the phenomenon whereby one has difficulty separating cognitions from corresponding behaviors, has implications in a wide variety of disturbances, including eating disorders, obsessive-compulsive disorder, generalized anxiety disorder, and panic disorder. Numerous constructs believed to contribute to the etiology or maintenance of TAF have been identified in the literature, but to date, no study has empirically integrated these findings into a comprehensive model. In this study, we examined simultaneously an array of variables thought to be related to TAF, and subsequently developed a model that elucidates the role of those variables that seem most involved in this phenomenon using a structural equation modeling approach. Results indicated that religiosity, as predicted by ethnic identity, was a significant predictor of TAF. Additionally, the relation between ethnic identity and TAF was partially mediated by an inflated sense of responsibility. Both TAF and obsessive-compulsive symptoms were found to be significant predictors of engagement in neutralization activities. Clinical and theoretical implications are discussed.     

The structure of personality disorders: comparing the DSM-IV-TR Axis II classification with the five-factor model framework using structural equation modeling

Earlier factor analytical studies on the empirical validity of the DSM-IV-TR (American Psychological Association, 2000) Axis II classification have offered little support for the current three-cluster structure. In his large-scale meta-analysis of previously published personality disorder correlation matrices, O'Connor (2005) found four factors, corresponding to the neuroticism, extraversion, agreeableness, and conscientiousness domains of the five-factor model of personality. In the present study, this dimensional four-factor model and the categorical DSM three-cluster structure were fitted to the Assessment of DSM-IV Personality Disorders questionnaire (ADP-IV; Schotte & De Doncker, 1994) scale scores using structural equation modelling. The results strongly favored the dimensional model, which also resembled other well-founded four-factor proposals (Livesley, Jang, & Vernon, 1998; Widiger & Simonsen, 2005). Moreover, a multigroup confirmatory factor analysis showed that this model was highly invariant and thus generalizable across two large clinical (n = 1,029) and general population (n = 659) samples.     

Testing parameters in structural equation modeling: every "one" matters

A problem with standard errors estimated by many structural equation modeling programs is described. In such programs, a parameter's standard error is sensitive to how the model is identified (i.e., how scale is set). Alternative but equivalent ways to identify a model may yield different standard errors, and hence different Z tests for a parameter, even though the identifications produce the same overall model fit. This lack of invariance due to model identification creates the possibility that different analysts may reach different conclusions about a parameter's significance level even though they test equivalent models on the same data. The authors suggest that parameters be tested for statistical significance through the likelihood ratio test, which is invariant to the identification choice.