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.     

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).     

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.We thank David Spiegelhalter for suggesting applying the Gibbs sampler to structural equation models to the first author at a 1994 workshop in Wiesbaden. We thank Ulf Böckenholt, Chris Meek, Marijtje van Duijn, Clark Glymour, Ivo Molenaar, Steve Klepper, Thomas Richardson, Teddy Seidenfeld, and Tom Snijders for helpful discussions, mathematical advice, and critiques of earlier drafts of this paper.     

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.     

贝叶斯结构方程模型及其研究现状

在心理学研究中结构方程模型(Structural Equation Modeling, SEM)被广泛用于检验潜变量间的因果效应, 其估计方法有频率学方法(如, 极大似然估计)和贝叶斯方法两类。近年来由于贝叶斯统计的流行及其在结构方程建模中易于处理小样本、缺失数据及复杂模型等方面的优势, 贝叶斯结构方程模型发展迅速, 但其在国内心理学领域的应用不足。主要介绍了贝叶斯结构方程模型的方法基础和优良特性, 及几类常用的贝叶斯结构方程模型及其应用现状, 旨在为应用研究者介绍新的研究工具。     

Accuracy in parameter estimation for targeted effects in structural equation modeling: sample size planning for narrow confidence intervals

In addition to evaluating a structural equation model (SEM) as a whole, often the model parameters are of interest and confidence intervals for those parameters are formed. Given a model with a good overall fit, it is entirely possible for the targeted effects of interest to have very wide confidence intervals, thus giving little information about the magnitude of the population targeted effects. With the goal of obtaining sufficiently narrow confidence intervals for the model parameters of interest, sample size planning methods for SEM are developed from the accuracy in parameter estimation approach. One method plans for the sample size so that the expected confidence interval width is sufficiently narrow. An extended procedure ensures that the obtained confidence interval will be no wider than desired, with some specified degree of assurance. A Monte Carlo simulation study was conducted that verified the effectiveness of the procedures in realistic situations. The methods developed have been implemented in the MBESS package in R so that they can be easily applied by researchers.     

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.     

结构方程模型统计检验力分析：原理与方法

结构方程模型是心理学、管理学、社会学等学科中重要的统计工具之一。然而, 大量使用结构方程模型的研究忽视了对该方法的统计检验力进行必要的分析和报告, 在一定程度上降低了这些研究的结果的证明效力。结构方程模型的统计检验力分析方法主要有Satorra-Saris法、MacCallum法与Monte Carlo法三类。其中Satorra-Saris法适用于备择模型清晰、检验对象相对简单、检验方法基于χ²分布的情形; MacCallum法适用于基于χ²分布的模型拟合检验且备择模型不明的情形; Monte Carlo法适用于检验对象相对复杂、采用模拟或重抽样方法进行检验的情形。在实际应用中, 研究者应当首先判断检验的目的、方法以及是否有明确的备择模型, 并根据这些信息选择具体的分析方法。     

The problem of model selection uncertainty in structural equation modeling

Model selection in structural equation modeling (SEM) involves using selection criteria to declare one model superior and treating it as a best working hypothesis until a better model is proposed. A limitation of this approach is that sampling variability in selection criteria usually is not considered, leading to assertions of model superiority that may not withstand replication. We illustrate that selection decisions using information criteria can be highly unstable over repeated sampling and that this uncertainty does not necessarily decrease with increases in sample size. Methods for addressing model selection uncertainty in SEM are evaluated, and implications for practice 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.     

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.     

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.     

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.