Bayesian Analysis of Structural Equation Model With Fixed Covariates

This article extends the LISREL model to incorporate fixed covariates at both the measurement and the structural equations of the model. A Bayesian procedure with conjugate type prior distributions is established. The joint Bayesian estimates of the latent variables and the structural parameters that involve the regression coefficients of the covariates, the variances, covariances and causations among the manifest and latent variables are obtained via the Gibbs sampler algorithm. It is shown that the conditional distributions required in the Gibbs sampler are familiar distributions, hence the algorithm is very efficient. A goodness of fit statistic for assessing the proposed model is presented. An illustrative example with some real data is presented.     

Bayesian Analysis of Semiparametric Hidden Markov Models With Latent Variables

In psychological, social, behavioral, and medical studies, hidden Markov models (HMMs) have been extensively applied to the simultaneous modeling of heterogeneous observation and hidden transition in the analysis of longitudinal data. However, the majority of the existing HMMs are developed in a parametric framework without latent variables. This study considers a novel semiparametric HMM, which comprises a semiparametric latent variable model to investigate the complex interrelationships among latent variables and a nonparametric transition model to examine the linear and nonlinear effects of potential predictors on hidden transition. The Bayesian P-splines approach and Markov chain Monte Carlo methods are developed to estimate the unknown, a Bayesian model comparison statistic, is employed to conduct model comparison. The empirical performance of the proposed methodology is evaluated through simulation studies. An application to a data set derived from the National Longitudinal Survey of Youth is presented.     

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.     

Analyzing Mixed-Dyadic Data Using Structural Equation Models

Mixed-dyadic data, collected from distinguishable (nonexchangeable) or indistinguishable (exchangeable) dyads, require statistical analysis techniques that model the variation within dyads and between dyads appropriately. The purpose of this article is to provide a tutorial for performing structural equation modeling analyses of cross-sectional and longitudinal models for mixed independent variable dyadic data, and to clarify questions regarding various dyadic data analysis specifications that have not been addressed elsewhere. Artificially generated data similar to the Newlywed Project and the Swedish Adoption Twin Study on Aging were used to illustrate analysis models for distinguishable and indistinguishable dyads, respectively. Due to their widespread use among applied researchers, the AMOS and Mplus statistical analysis software packages were used to analyze the dyadic data structural equation models illustrated here. These analysis models are presented in sufficient detail to allow researchers to perform these analyses using their preferred statistical analysis software package.     

Bayesian Analysis of Multivariate Latent Curve Models With Nonlinear Longitudinal Latent Effects

In longitudinal studies, investigators often measure multiple variables at multiple time points and are interested in investigating individual differences in patterns of change on those variables. Furthermore, in behavioral, social, psychological, and medical research, investigators often deal with latent variables that cannot be observed directly and should be measured by 2 or more manifest variables. Longitudinal latent variables occur when the corresponding manifest variables are measured at multiple time points. Our primary interests are in studying the dynamic change of longitudinal latent variables and exploring the possible interactive effect among the latent variables.

Much of the existing research in longitudinal studies focuses on studying change in a single observed variable at different time points. In this article, we propose a novel latent curve model (LCM) for studying the dynamic change of multivariate manifest and latent variables and their linear and interaction relationships. The proposed LCM has the following useful features: First, it can handle multivariate variables for exploring the dynamic change of their relationships, whereas conventional LCMs usually consider change in a univariate variable. Second, it accommodates both first- and second-order latent variables and their interactions to explore how changes in latent attributes interact to produce a joint effect on the growth of an outcome variable. Third, it accommodates both continuous and ordered categorical data, and missing data.     

Comparing Exploratory Structural Equation Modeling and Existing Approaches for Multiple Regression with Latent Variables

Exploratory structural equation modeling (ESEM) is an approach for analysis of latent variables using exploratory factor analysis to evaluate the measurement model. This study compared ESEM with two dominant approaches for multiple regression with latent variables, structural equation modeling (SEM) and manifest regression analysis (MRA). Main findings included: (1) ESEM in general provided the least biased estimation of the regression coefficients; SEM was more biased than MRA given large cross-factor loadings. (2) MRA produced the most precise estimation, followed by ESEM and then SEM. (3) SEM was the least powerful in the significance tests; statistical power was lower for ESEM than MRA with relatively small target-factor loadings, but higher for ESEM than MRA with relatively large target-factor loadings. (4) ESEM showed difficulties in convergence and occasionally created an inflated type I error rate under some conditions. ESEM is recommended when non-ignorable cross-factor loadings exist.     

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.     

Constructing Approximate Confidence Intervals for Parameters With Structural Equation Models

Confidence intervals (CIs) for parameters are usually constructed based on the estimated standard errors. These are known as Wald CIs. This article argues that likelihood-based CIs (CIs based on likelihood ratio statistics) are often preferred to Wald CIs. It shows how the likelihood-based CIs and the Wald CIs for many statistics and psychometric indexes can be constructed with the use of phantom variables (Rindskopf, 1984 Rindskopf, D. 1984. Using phantom and imaginary latent variables to parameterize constraints in linear structural models. Psychometrika, 49: 37–47. [Crossref], [Web of Science ®] , [Google Scholar]) in some of the current structural equation modeling (SEM) packages. The procedures to form CIs for the differences in correlation coefficients, squared multiple correlations, indirect effects, coefficient alphas, and reliability estimates are illustrated. A simulation study on the Pearson correlation is used to demonstrate the advantages of the likelihood-based CI over the Wald CI. Issues arising from this SEM approach and extensions of this approach are discussed.     

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.     

一个重要的二元二次不定方程——佩尔方程

1.从变三角队为方队谈起 一所学校运动会的团体操表演,有这样一个场面,运动员站成了第1排1人,第2排2人,第3排3人,……第n排n人的正二三角形队型,忽然,又变成了每排m人的正方形队型．     

Unconstrained Structural Equation Models of Latent Interactions: Contrasting Residual- and Mean-Centered Approaches

Little, Bovaird and Widaman (2006) proposed an unconstrained approach with residual centering for estimating latent interaction effects as an alternative to the mean-centered approach proposed by Marsh, Wen, and Hau (2004, 2006). Little et al. also differed from Marsh et al. in the number of indicators used to infer the latent interaction factor and how they were represented, but this issue is separate from the mean versus residual centering distinction that was their primary focus. However, their implementation of the Marsh et al. mean-centered approach failed to incorporate the mean structure that Marsh et al. argued was necessary to obtain unbiased estimates. One might suppose that their new approach would suffer this same problem, an issue not addressed by Little et al. However, we demonstrate here why the Little et al. approach obviates this requirement that heretofore was thought to be necessary for all constrained, partially constrained, and unconstrained approaches. Both the Marsh et al. and Little et al. unconstrained approaches typically result in similar results and are much easier to implement than traditional constrained approaches. They differ primarily in that the Little et al. approach is a 2-step approach involving a potentially large number of separate analyses prior to estimating the structural equation model that apparently does not require the estimation of a mean structure, whereas the Marsh et al. approach is a 1-step approach that includes a mean structure.     

Transformation Structural Equation Models With Highly Nonnormal and Incomplete Data

As useful multivariate techniques, structural equation models have attracted significant attention from various fields. Most existing statistical methods and software for analyzing structural equation models have been developed based on the assumption that the response variables are normally distributed. Several recently developed methods can partially address violations of this assumption, but still encounter difficulties in analyzing highly nonnormal data. Moreover, the presence of missing data is a practical issue in substantive research. Simply ignoring missing data or improperly treating nonignorable missingness as ignorable could seriously distort statistical influence results. The main objective of this article is to develop a Bayesian approach for analyzing transformation structural equation models with highly nonnormal and missing data. Different types of missingness are discussed and selected via the deviance information criterion. The empirical performance of our method is examined via simulation studies. Application to a study concerning people’s job satisfaction, home life, and work attitude is presented.     

Structural Equation Models of Latent Interactions: Clarification of Orthogonalizing and Double-Mean-Centering Strategies

The purpose of this investigation is to compare a new (double-mean-centering) strategy to estimating latent interactions in structural equation models with the (single) mean-centering strategy (Marsh, Wen, & Hau, 2004 Marsh, H. W., Wen, Z. and Hau, K. T. 2004. Structural equation models of latent interactions: Evaluation of alternative estimation strategies and indicator construction.. Psychological Methods, 9: 275–300. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 2006 Marsh, H. W., Wen, Z. and Hau, K. T. 2006. “Structural equation models of latent interaction and quadratic effects”. In A second course in structural equation modeling Edited by: Hancock, G. and Mueller, R. 225–265. Greenwich, CT: Information Age.  [Google Scholar]) and the orthogonalizing strategy (Little, Bovaird, & Widaman, 2006 Little, T. D., Bovaird, J. A. and Widaman, K. F. 2006. On the merits of orthogonalizing powered and product term: Implications for modeling interactions among latent variables.. Structural Equation Modeling, 13: 497–519. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Marsh et al., 2007 Marsh, H. W., Wen, Z., Hau, K. T., Little, T. D., Bovaird, J. A. and Widaman, K. F. 2007. Unconstrained structural equation models of latent interactions: Contrasting residual- and mean-centered approaches.. Structural Equation Modeling, 14: 570–580. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A key benefit of the orthogonalizing strategy is that it eliminated the need to estimate a mean structure as required by the mean-centering strategy, but required a potentially cumbersome 2-step estimation procedure. In contrast, the double-mean-centering strategy eliminates both the need for the mean structure and the cumbersome 2-stage estimation procedure. Furthermore, although the orthogonalizing and double-mean-centering strategies are equivalent when all indicators are normally distributed, the double-mean-centering strategy is superior when this normality assumption is violated. In summary, we recommend that applied researchers wanting to estimate latent interaction effects use the double-mean-centering strategy instead of either the single-mean-centering or orthogonalizing strategies, thus allowing them to ignore the cumbersome mean structure.     

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.     

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 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.     

Two Rules of Identification for Structural Equation Models

Identification of structural equation models remains a challenge to many researchers. Although empirical tests of identification are readily available in structural equation modeling software, these examine local identification and rely on sample estimates of parameters. Rules of identification are available, but do not include all models encountered in practice. In this article we provide 2 rules of identification: the 2+ emitted paths rule and the exogenous X rule. The former is a necessary condition of identification and the latter is a sufficient condition. We explain and prove each of these rules and provide illustrations of their application. These rules extend the coverage of structural equation models that we can check for identification. We also explain how they can be part of a piecewise identification strategy that extends their use even further.     

Modeling Latent Growth Curves With Incomplete Data Using Different Types of Structural Equation Modeling and Multilevel Software

This article offers different examples of how to fit latent growth curve (LGC) models to longitudinal data using a variety of different software programs (i.e., LISREL, Mx, Mplus, AMOS, SAS). The article shows how the same model can be fitted using both structural equation modeling and multilevel software, with nearly identical results, even in the case of models of latent growth fitted to incomplete data. The general purpose of this article is to provide a demonstration that integrates programming features from different software. The most immediate goal is to help researchers implement these LGC models as a useful way to test hypotheses of growth.     

Multiple Imputation Strategies for Multiple Group Structural Equation Models

Although structural equation modeling software packages use maximum likelihood estimation by default, there are situations where one might prefer to use multiple imputation to handle missing data rather than maximum likelihood estimation (e.g., when incorporating auxiliary variables). The selection of variables is one of the nuances associated with implementing multiple imputation, because the imputer must take special care to preserve any associations or special features of the data that will be modeled in the subsequent analysis. For example, this article deals with multiple group models that are commonly used to examine moderation effects in psychology and the behavioral sciences. Special care must be exercised when using multiple imputation with multiple group models, as failing to preserve the interactive effects during the imputation phase can produce biased parameter estimates in the subsequent analysis phase, even when the data are missing completely at random or missing at random. This study investigates two imputation strategies that have been proposed in the literature, product term imputation and separate group imputation. A series of simulation studies shows that separate group imputation adequately preserves the multiple group data structure and produces accurate parameter estimates.     

Variance Estimation Using Replication Methods in Structural Equation Modeling With Complex Sample Data

This article discusses replication sampling variance estimation techniques that are often applied in analyses using data from complex sampling designs: jackknife repeated replication, balanced repeated replication, and bootstrapping. These techniques are used with traditional analyses such as regression, but are currently not used with structural equation modeling (SEM) analyses. This article provides an extension of these methods to SEM analyses, including a proposed adjustment to the likelihood ratio test, and presents the results from a simulation study suggesting replication estimates are robust. Finally, a demonstration of the application of these methods using data from the Early Childhood Longitudinal Study is included. Secondary analysts can undertake these more robust methods of sampling variance estimation if they have access to certain SEM software packages and data management packages such as SAS, as shown in the article.