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.     

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 SEM Under Conditions of Incomplete and Nonnormal Data

Using Monte Carlo simulations, this research examined the performance of four missing data methods in SEM under different multivariate distributional conditions. The effects of four independent variables (sample size, missing proportion, distribution shape, and factor loading magnitude) were investigated on six outcome variables: convergence rate, parameter estimate bias, MSE of parameter estimates, standard error coverage, model rejection rate, and model goodness of fit—RMSEA. A three-factor CFA model was used. Findings indicated that FIML outperformed the other methods in MCAR, and MI should be used to increase the plausibility of MAR. SRPI was not comparable to the other three methods in either MCAR or MAR.     

Using Monte Carlo Normal Distributions to Evaluate Structural Models With Nonnormal Data

Statistical theories of goodness-of-fit tests in structural equation modeling are based on asymptotic distributions of test statistics. When the model includes a large number of variables or the population is not from a multivariate normal distribution, the asymptotic distributions do not approximate the distribution of the test statistics very well at small sample sizes. A variety of methods have been developed to improve the accuracy of hypothesis testing at small sample sizes. However, all these methods have their limitations, specially for nonnormal distributed data. We propose a Monte Carlo test that is able to control Type I error with more accuracy compared to existing approaches in both normal and nonnormally distributed data at small sample sizes. Extensive simulation studies show that the suggested Monte Carlo test has a more accurate observed significance level as compared to other tests with a reasonable power to reject misspecified models.     

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.     

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 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.     

semPlot: Unified Visualizations of Structural Equation Models

Structural equation modeling (SEM) has a long history of representing models graphically as path diagrams. This article presents the freely available semPlot package for R, which fills the gap between advanced, but time-consuming, graphical software and the limited graphics produced automatically by SEM software. In addition, semPlot offers more functionality than drawing path diagrams: It can act as a common ground for importing SEM results into R. Any result usable as input to semPlot can also be represented in any of the 3 popular SEM frameworks, as well as translated to input syntax for the R packages sem (Fox, Nie, & Byrnes, 2013) and lavaan (Rosseel, 2012). Special considerations are made in the package for the automatic placement of variables, using 3 novel algorithms that extend the earlier work of Boker, McArdle, and Neale (2002). The article concludes with detailed instructions on these node-placement algorithms.     

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.     

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.     

Automated Selection of Robust Individual-Level Structural Equation Models for Time Series Data

In order to analyze intensive longitudinal data collected across multiple individuals, researchers frequently have to decide between aggregating all individuals or analyzing each individual separately. This paper presents an R package, gimme, which allows for the automatic specification of individual-level structural equation models that combine group-, subgroup-, and individual-level information. This R package is a complement of the GIMME program currently available via a combination of MATLAB and LISREL. By capitalizing on the flexibility of R and the capabilities of the existing structural equation modeling package lavaan, gimme allows for the automated specification and estimation of group-, subgroup-, and individual-level relations in time series data from within a structural equation modeling framework. Applications include daily diary data as well as functional magnetic resonance imaging data.     

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.     

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 Nonlinear Dynamic Latent Class Structural Equation Model

In this article, we propose a nonlinear dynamic latent class structural equation modeling (NDLC-SEM). It can be used to examine intra-individual processes of observed or latent variables. These processes are decomposed into parts which include individual- and time-specific components. Unobserved heterogeneity of the intra-individual processes are modeled via a latent Markov process that can be predicted by individual- and time-specific variables as random effects. We discuss examples of sub-models which are special cases of the more general NDLC-SEM framework. Furthermore, we provide empirical examples and illustrate how to estimate this model in a Bayesian framework. Finally, we discuss essential properties of the proposed framework, give recommendations for applications, and highlight some general problems in the estimation of parameters in comprehensive frameworks for intensive longitudinal data.     

Examining Measure Correlations With Incomplete Data Sets

A 2-stage procedure for estimation and testing of observed measure correlations in the presence of missing data is discussed. The approach uses maximum likelihood for estimation and the false discovery rate concept for correlation testing. The method can be used in initial exploration-oriented empirical studies with missing data, where it is of interest to estimate manifest variable interrelationship indexes and test hypotheses about their population values. The procedure is applicable also with violations of the underlying missing at random assumption, via inclusion of auxiliary variables. The outlined approach is illustrated with data from an aging research study.     

The Performance of Multilevel Models When Outcome Data Are Incomplete

When data for multiple outcomes are collected in a multilevel design, researchers can select a univariate or multivariate analysis to examine group-mean differences. When correlated outcomes are incomplete, a multivariate multilevel model (MVMM) may provide greater power than univariate multilevel models (MLMs). For a two-group multilevel design with two correlated outcomes, a simulation study was conducted to compare the performance of MVMM to MLMs. The results showed that MVMM and MLM performed similarly when data were complete or missing completely at random. However, when outcome data were missing at random, MVMM continued to provide unbiased estimates, whereas MLM produced grossly biased estimates and severely inflated Type I error rates. As such, this study provides further support for using MVMM rather than univariate analyses, particularly when outcome data are incomplete.     

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.     

Fitting Structural Equation Model Trees and Latent Growth Curve Mixture Models in Longitudinal Designs: The Influence of Model Misspecification

When conducting longitudinal research, the investigation of between-individual differences in patterns of within-individual change can provide important insights. In this article, we use simulation methods to investigate the performance of a model-based exploratory data mining technique—structural equation model trees (SEM trees; Brandmaier, Oertzen, McArdle, & Lindenberger, 2013)—as a tool for detecting population heterogeneity. We use a latent-change score model as a data generation model and manipulate the precision of the information provided by a covariate about the true latent profile as well as other factors, including sample size, under the possible influences of model misspecifications. Simulation results show that, compared with latent growth curve mixture models, SEM trees might be very sensitive to model misspecification in estimating the number of classes. This can be attributed to the lower statistical power in identifying classes, resulting from smaller differences of parameters prescribed by the template model between classes.     

Multivariate Meta-Analysis as Structural Equation Models

Multivariate meta-analysis has become increasingly popular in the educational, social, and medical sciences. It is because the outcome measures in a meta-analysis can involve more than one effect size. This article proposes 2 mathematically equivalent models to implement multivariate meta-analysis in structural equation modeling (SEM). Specifically, this article shows how multivariate fixed-, random- and mixed-effects meta-analyses can be formulated as structural equation models. metaSEM (a free R package based on OpenMx) and Mplus are used to implement the proposed procedures. A real data set is used to illustrate the procedures. Formulating multivariate meta-analysis as structural equation models provides many new research opportunities for methodological development in both meta-analysis and SEM. Issues related to and extensions on the SEM-based meta-analysis are discussed.