Bayesian Methods for Analyzing Structural Equation Models With Covariates,Interaction, and Quadratic Latent Variables

The analysis of interaction among latent variables has received much attention. This article introduces a Bayesian approach to analyze a general structural equation model that accommodates the general nonlinear terms of latent variables and covariates. This approach produces a Bayesian estimate that has the same statistical optimal properties as a maximum likelihood estimate. Other advantages over the traditional approaches are discussed. More important, we demonstrate through examples how to use the freely available software WinBUGS to obtain Bayesian results for estimation and model comparison. Simulation studies are conducted to assess the empirical performances of the approach for situations with various sample sizes and prior inputs.     

Bayesian Analysis of Mixture Structural Equation Models With an Unknown Number of Components

Multivariate heterogenous data with latent variables are common in many fields such as biological, medical, behavioral, and social-psychological sciences. Mixture structural equation models are multivariate techniques used to examine heterogeneous interrelationships among latent variables. In the analysis of mixture models, determination of the number of mixture components is always an important and challenging issue. This article aims to develop a full Bayesian approach with the use of reversible jump Markov chain Monte Carlo method to analyze mixture structural equation models with an unknown number of components. The proposed procedure can simultaneously and efficiently select the number of mixture components and conduct parameter estimation. Simulation studies show the satisfactory empirical performance of the method. The proposed method is applied to study risk factors of osteoporotic fractures in older people.     

Model Comparison of Bayesian Semiparametric and Parametric Structural Equation Models

Structural equation models have wide applications. One of the most important issues in analyzing structural equation models is model comparison. This article proposes a Bayesian model comparison statistic, namely the L _ν-measure for both semiparametric and parametric structural equation models. For illustration purposes, we consider a Bayesian semiparametric approach for estimation and model comparison in the context of structural equation models with fixed covariates. A finite dimensional Dirichlet process is used to model the crucial latent variables, and a blocked Gibbs sampler is implemented for estimation. Empirical performance of the L _ν-measure is evaluated through a simulation study. Results obtained indicate that the L _ν-measure, which additionally requires very minor computational effort, gives satisfactory performance. Moreover, the methodologies are demonstrated through an example with a real data set on kidney disease. Finally, the application of the L _ν-measure to Bayesian semiparametric nonlinear structural equation models is outlined.     

Bayesian Model Averaging Over Directed Acyclic Graphs With Implications for the Predictive Performance of Structural Equation Models

This article examines Bayesian model averaging as a means of addressing predictive performance in Bayesian structural equation models. The current approach to addressing the problem of model uncertainty lies in the method of Bayesian model averaging. We expand the work of Madigan and his colleagues by considering a structural equation model as a special case of a directed acyclic graph. We then provide an algorithm that searches the model space for submodels and obtains a weighted average of the submodels using posterior model probabilities as weights. Our simulation study provides a frequentist evaluation of our Bayesian model averaging approach and indicates that when the true model is known, Bayesian model averaging does not yield necessarily better predictive performance compared to nonaveraged models. However, our case study using data from an international large-scale assessment reveals that the model-averaged submodels provide better posterior predictive performance compared to the initially specified model.     

Bayesian Quantile Structural Equation Models

Structural equation modeling is a common multivariate technique for the assessment of the interrelationships among latent variables. Structural equation models have been extensively applied to behavioral, medical, and social sciences. Basic structural equation models consist of a measurement equation for characterizing latent variables through multiple observed variables and a mean regression-type structural equation for investigating how explanatory latent variables influence outcomes of interest. However, the conventional structural equation does not provide a comprehensive analysis of the relationship between latent variables. In this article, we introduce the quantile regression method into structural equation models to assess the conditional quantile of the outcome latent variable given the explanatory latent variables and covariates. The estimation is conducted in a Bayesian framework with Markov Chain Monte Carlo algorithm. The posterior inference is performed with the help of asymmetric Laplace distribution. A simulation shows that the proposed method performs satisfactorily. An application to a study of chronic kidney disease is presented.     

A Bayesian Approach to Multilevel Structural Equation Modeling With Continuous and Dichotomous Outcomes

Multilevel Structural equation models are most often estimated from a frequentist framework via maximum likelihood. However, as shown in this article, frequentist results are not always accurate. Alternatively, one can apply a Bayesian approach using Markov chain Monte Carlo estimation methods. This simulation study compared estimation quality using Bayesian and frequentist approaches in the context of a multilevel latent covariate model. Continuous and dichotomous variables were examined because it is not yet known how different types of outcomes—most notably categorical—affect parameter recovery in this modeling context. Within the Bayesian estimation framework, the impact of diffuse, weakly informative, and informative prior distributions were compared. Findings indicated that Bayesian estimation may be used to overcome convergence problems and improve parameter estimate bias. Results highlight the differences in estimation quality between dichotomous and continuous variable models and the importance of prior distribution choice for cluster-level random effects.     

A Bayesian Approach for Analyzing Hierarchical Data With Missing Outcomes Through Structural Equation Models

Structural equation models are widely appreciated in behavioral, social, and psychological research to model relations between latent constructs and manifest variables, and to control for measurement errors. Most applications of structural equation models are based on fully observed data that are independently distributed. However, hierarchical data with a correlated structure are common in behavioral research, and very often, missing data are encountered. In this article, we propose a 2-level structural equation model for analyzing hierarchical data with missing entries, and describe a Bayesian approach for estimation and model comparison. We show how to use WinBUGS software to get the solution conveniently. The proposed methodologies are illustrated through a simulation study, and a real application in relation to organizational and management research concerning the study of the interrelationships of the latent constructs about job satisfaction, job responsibility, and life satisfaction for citizens in 43 countries.     

Multiple-Group Analysis for Structural Equation Modeling With Dependent Samples

Multigroup structural equation modeling (SEM) plays a key role in studying measurement invariance and in group comparison. However, existing methods for multigroup SEM assume that different samples are independent. This article develops a method for multigroup SEM with correlated samples. Parallel to that for independent samples, the focus here is on the cross-group stability of the within-group structure and parameters. In particular, the method does not require the specification of any between-group relationship. Rescaled and adjusted statistics as well as sandwich-type covariance matrices make the developed method work for possibly nonnormal variables with finite 4th-order moments. The method is applied to a longitudinal data set on the development of entrepreneurial teams across 4 phases. Detailed analysis is provided regarding the stability of the effect of psychological compatibility on team performance, as it is mediated by fairness perception and team cohesion.     

Model Selection in Structural Equation Models With Continuous and Polytomous Data

Recently, analysis of structural equation models with polytomous and continuous variables has received a lot of attention. However, contributions to the selection of good models are limited. The main objective of this article is to investigate the maximum likelihood estimation of unknown parameters in a general LISREL-type model with mixed polytomous and continuous data and propose a model selection procedure for obtaining good models for the underlying substantive theory. The maximum likelihood estimate is obtained by a Monte Carlo Expectation Maximization algorithm, in which the E step is evaluated via the Gibbs sampler and the M step is completed via the method of conditional maximization. The convergence of the Monte Carlo Expectation Maximization algorithm is monitored by the bridge sampling. A model selection procedure based on Bayes factor and Occam's window search strategy is proposed. The effectiveness of the procedure in accounting for the model uncertainty and in picking good models is discussed. The proposed methodology is illustrated with a real example.     

Bayesian Data-Model Fit Assessment for Structural Equation Modeling

Bayesian approaches to modeling are receiving an increasing amount of attention in the areas of model construction and estimation in factor analysis, structural equation modeling (SEM), and related latent variable models. However, model diagnostics and model criticism remain relatively understudied aspects of Bayesian SEM. This article describes and illustrates key features of Bayesian approaches to model diagnostics and assessing data–model fit of structural equation models, discussing their merits relative to traditional procedures.     

A Systematic Evaluation and Comparison Between Exploratory Structural Equation Modeling and Bayesian Structural Equation Modeling

In this study, we contrast two competing approaches, not previously compared, that balance the rigor of CFA/SEM with the flexibility to fit realistically complex data. Exploratory SEM (ESEM) is claimed to provide an optimal compromise between EFA and CFA/SEM. Alternatively, a family of three Bayesian SEMs (BSEMs) replace fixed-zero estimates with informative, small-variance priors for different subsets of parameters: cross-loadings (CL), residual covariances (RC), or CLs and RCs (CLRC). In Study 1, using three simulation studies, results showed that (1) BSEM-CL performed more closely to ESEM; (2) BSEM-CLRC did not provide more accurate model estimation compared with BSEM-CL; (3) BSEM-RC provided unstable estimation; and (4) different specifications of targeted values in ESEM and informative priors in BSEM have significant impacts on model estimation. The real data analysis (Study 2) showed that the differences in estimation between different models were largely consistent with those in Study1 but somewhat smaller.     

Using Fixed Thresholds with Grouped Data in Structural Equation Modeling

Valuable methods have been developed for incorporating ordinal variables into structural equation models using a latent response variable formulation. However, some model parameters, such as the means and variances of latent factors, can be quite difficult to interpret because the latent response variables have an arbitrary metric. This limitation can be particularly problematic in growth models, where the means and variances of the latent growth parameters typically have important substantive meaning when continuous measures are used. However, these methods are often applied to grouped data, where the ordered categories actually represent an interval-level variable that has been measured on an ordinal scale for convenience. The method illustrated in this article shows how category threshold values can be incorporated into the model so that interpretation is more meaningful, with particular emphasis given to the application of this technique with latent growth models.     

Comparison of Frequentist and Bayesian Regularization in Structural Equation Modeling

Research in regularization, as applied to structural equation modeling (SEM), remains in its infancy. Specifically, very little work has compared regularization approaches across both frequentist and Bayesian estimation. The purpose of this study was to address just that, demonstrating both similarity and distinction across estimation frameworks, while specifically highlighting more recent developments in Bayesian regularization. This is accomplished through the use of two empirical examples that demonstrate both ridge and lasso approaches across both frequentist and Bayesian estimation, along with detail regarding software implementation. We conclude with a discussion of future research, advocating for increased evaluation and synthesis across both Bayesian and frequentist frameworks.     

大学教师组织认同的结构方程模型分析

大学教师对学校的认同结构是测量大学教师对学校认同的一个重要基础。以H省11所大学664名教师为调查对象,探讨大学教师组织认同的结构,问卷显示：大学教师的组织认同是由组织认同感和组织公民行为两方面组成,两方面存在一定的相关,但各自具有独立的结构因素。其中,大学教师组织认同感主要由组织适合性和组织归属性两部分组成,大学教师组织公民行为主要由热爱学校、帮助同事和自我发展3个维度组成。     

Testing Complex Correlational Hypotheses With Structural Equation Models

It is often of interest to estimate partial or semipartial correlation coefficients as indexes of the linear association between 2 variables after partialing one or both for the influence of covariates. Squaring these coefficients expresses the proportion of variance in 1 variable explained by the other variable after controlling for covariates. Methods exist for testing hypotheses about the equality of these coefficients across 2 or more groups, but they are difficult to conduct by hand, prone to error, and limited to simple cases. A unified framework is provided for estimating bivariate, partial, and semipartial correlation coefficients using structural equation modeling (SEM). Within the SEM framework, it is straightforward to test hypotheses of the equality of various correlation coefficients with any number of covariates across multiple groups. LISREL syntax is provided, along with 4 examples.     

A Second-Order Conditionally Linear Mixed Effects Model With Observed and Latent Variable Covariates

A conditionally linear mixed effects model is an appropriate framework for investigating nonlinear change in a continuous latent variable that is repeatedly measured over time. The efficacy of the model is that it allows parameters that enter the specified nonlinear time-response function to be stochastic, whereas those parameters that enter in a nonlinear manner are common to all subjects. In this article we describe how a variant of the Michaelis-Menten (M-M) function can be fit within this modeling framework using Mplus 6.0. We demonstrate how observed and latent covariates can be incorporated to help explain individual differences in growth characteristics. Features of the model including an explication of key analytic decision points are illustrated using longitudinal reading data. To aid in making this class of models accessible, annotated Mplus code is provided.     

Modeling of Nonrecursive Structural Equation Models With Categorical Indicators

Nonrecursive structural equation models generally take the form of feedback loops, involving 2 latent variables that are connected by 2 unidirectional paths, 1 starting with each variable and terminating in the other variable. Nonrecursive models belong to a larger class of path models that require the use of instrumental variables (IVs) to achieve model identification. Prior research has focused on SEM parameter estimation with IVs when indicators were continuous and normally distributed. Much less is known about how estimators function in the presence of categorical indicators, which are commonly used in the social sciences, such as with cognitive and affective instruments. In this study, there was specific interest in comparing the 2-stage least squares (2SLS) estimator and its categorical variant to other recommended estimators. This study compares the performance of several estimation approaches for fitting structural equation models with categorical indicator variables when IVs are necessary to obtain proper model estimates. Across conditions, 1 extension of the nonlinear 2SLS (N2SLS) approach, the nonlinear 3-stage least squares (N3SLS), which accounts for correlated errors among regressors within each model (as does the N2SLS), as well as correlations of errors across models, which N2SLS does not, appears to work the best among methods compared.     

Assessing Model Similarity in Structural Equation Modeling

Two models can be nonequivalent, but fit very similarly across a wide range of data sets. These near-equivalent models, like equivalent models, should be considered rival explanations for results of a study if they represent plausible explanations for the phenomenon of interest. Prior to conducting a study, researchers should evaluate plausible models that are alternatives to those hypothesized to evaluate whether they are near-equivalent or equivalent and, in so doing, address the adequacy of the study’s methodology. To assess the extent to which alternative models for a study are empirically distinguishable, we propose 5 indexes that quantify the degree of similarity in fit between 2 models across a specified universe of data sets. These indexes compare either the maximum likelihood fit function values or the residual covariance matrices of models. Illustrations are provided to support interpretations of these similarity indexes.     

A Second-Order Conditionally Linear Mixed Effects Model With Observed and Latent Variable Covariates

A conditionally linear mixed effects model is an appropriate framework for investigating nonlinear change in a continuous latent variable that is repeatedly measured over time. The efficacy of the model is that it allows parameters that enter the specified nonlinear time-response function to be stochastic, whereas those parameters that enter in a nonlinear manner are common to all subjects. In this article we describe how a variant of the Michaelis–Menten (M–M) function can be fit within this modeling framework using Mplus 6.0. We demonstrate how observed and latent covariates can be incorporated to help explain individual differences in growth characteristics. Features of the model including an explication of key analytic decision points are illustrated using longitudinal reading data. To aid in making this class of models accessible, annotated Mplus code is provided.     

Identifying Aberrant Data in Structural Equation Models With IRLS-ADF

In structural equation models, outliers could result in inaccurate parameter estimates and misleading fit statistics when using traditional methods. To robustly estimate structural equation models, iteratively reweighted least squares (IRLS; Yuan & Bentler, 2000) has been proposed, but not thoroughly examined. We explore the large-sample properties of IRLS and its effect on parameter recovery, model fit, and aberrant data identification. A parametric bootstrap technique is proposed to determine the tuning parameters of IRLS, which results in improved Type I error rates in aberrant data identification, for data sets generated from homogenous populations. Scenarios concerning (a) simulated data, (b) contaminated data, and (c) a real data set are studied. Results indicate good parameter recovery, model fit, and aberrant data identification when noisy observations are drawn from a real data set, but lackluster parameter recovery and identification of aberrant data when the noise is parametrically structured. Practical implications and further research are discussed.