A Bayesian Hierarchical Selection Model for Academic Growth With Missing Data

Using a sample of schools testing annually in grades 9–11 with a vertically linked series of assessments, a latent growth curve model is used to model test scores with student intercepts and slopes nested within school. Missed assessments can occur because of student mobility, student dropout, absenteeism, and other reasons. Missing data indicators are modeled using logistic regression, with grade 9 and potentially unobserved growth scores used as covariates. Under a hierarchical selection model, estimates of school effects on academic growth and missingness are obtained. The results from the selection model are compared to a model that ignores the missing data process.     

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.     

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 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 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.     

Evaluation of Test Statistics for Robust Structural Equation Modeling With Nonnormal Missing Data

A 2-stage robust procedure as well as an R package, rsem, were recently developed for structural equation modeling with nonnormal missing data by Yuan and Zhang (2012). Several test statistics that have been used for complete data analysis are employed to evaluate model fit in the 2-stage robust method. However, properties of these statistics under robust procedures for incomplete nonnormal data analysis have never been studied. This study aims to systematically evaluate and compare 5 test statistics, including a test statistic derived from normal-distribution-based maximum likelihood, a rescaled chi-square statistic, an adjusted chi-square statistic, a corrected residual-based asymptotical distribution-free chi-square statistic, and a residual-based F statistic. These statistics are evaluated under a linear growth curve model by varying 8 factors: population distribution, missing data mechanism, missing data rate, sample size, number of measurement occasions, covariance between the latent intercept and slope, variance of measurement errors, and downweighting rate of the 2-stage robust method. The performance of the test statistics varies and the one derived from the 2-stage normal-distribution-based maximum likelihood performs much worse than the other four. Application of the 2-stage robust method and of the test statistics is illustrated through growth curve analysis of mathematical ability development, using data on the Peabody Individual Achievement Test mathematics assessment from the National Longitudinal Survey of Youth 1997 Cohort.     

Effects of Missing Data Methods in Structural Equation Modeling With Nonnormal Longitudinal Data

The purpose of this study is to investigate the effects of missing data techniques in longitudinal studies under diverse conditions. A Monte Carlo simulation examined the performance of 3 missing data methods in latent growth modeling: listwise deletion (LD), maximum likelihood estimation using the expectation and maximization algorithm with a nonnormality correction (robust ML), and the pairwise asymptotically distribution-free method (pairwise ADF). The effects of 3 independent variables (sample size, missing data mechanism, and distribution shape) were investigated on convergence rate, parameter and standard error estimation, and model fit. The results favored robust ML over LD and pairwise ADF in almost all respects. The exceptions included convergence rates under the most severe nonnormality in the missing not at random (MNAR) condition and recovery of standard error estimates across sample sizes. The results also indicate that nonnormality, small sample size, MNAR, and multicollinearity might adversely affect convergence rate and the validity of statistical inferences concerning parameter estimates and model fit statistics.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

Structural Equation Models and Mixture Models With Continuous Nonnormal Skewed Distributions

In this article we describe a structural equation modeling (SEM) framework that allows nonnormal skewed distributions for the continuous observed and latent variables. This framework is based on the multivariate restricted skew t distribution. We demonstrate the advantages of skewed SEM over standard SEM modeling and challenge the notion that structural equation models should be based only on sample means and covariances. The skewed continuous distributions are also very useful in finite mixture modeling as they prevent the formation of spurious classes formed purely to compensate for deviations in the distributions from the standard bell curve distribution. This framework is implemented in Mplus Version 7.2.     

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.     

Evaluating Interventions with Multimethod Data: A Structural Equation Modeling Approach

In many intervention and evaluation studies, outcome variables are assessed using a multimethod approach comparing multiple groups over time. In this article, we show how evaluation data obtained from a complex multitrait–multimethod–multioccasion–multigroup design can be analyzed with structural equation models. In particular, we show how the structural equation modeling approach can be used to (a) handle ordinal items as indicators, (b) test measurement invariance, and (c) test the means of the latent variables to examine treatment effects. We present an application to data from an evaluation study of an early childhood prevention program. A total of 659 children in intervention and control groups were rated by their parents and teachers on prosocial behavior and relational aggression before and after the program implementation. No mean change in relational aggression was found in either group, whereas an increase in prosocial behavior was found in both groups. Advantages and limitations of the proposed approach are highlighted.