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.     

Evaluating Model Fit and Structural Coefficient Bias: A Bayesian Approach to Multilevel Bifactor Model Misspecification

Appropriate model specification is fundamental to unbiased parameter estimates and accurate model interpretations in structural equation modeling. Thus detecting potential model misspecification has drawn the attention of many researchers. This simulation study evaluates the efficacy of the Bayesian approach (the posterior predictive checking, or PPC procedure) under multilevel bifactor model misspecification (i.e., ignoring a specific factor at the within level). The impact of model misspecification on structural coefficients was also examined in terms of bias and power. Results showed that the PPC procedure performed better in detecting multilevel bifactor model misspecification, when the misspecification became more severe and sample size was larger. Structural coefficients were increasingly negatively biased at the within level, as model misspecification became more severe. Model misspecification at the within level affected the between-level structural coefficient estimates more when data dependency was lower and the number of clusters was smaller. Implications for researchers are discussed.     

On the Correspondence between the Latent Growth Curve and Latent Change Score Models

There has been a great deal of work in the literature on the equivalence between the mixed-effects modeling and structural equation modeling (SEM) frameworks in specifying growth models (Willett &; Sayer, 1994). However, there has been little work on the correspondence between the latent growth curve model (LGM) and the latent change score model (see Grimm, Zhang, Hamagami, &; Mazzocco, 2013 Grimm, K. J., Zhang, Z., Hamagami, F., &; Mazzocco, M. M. (2013). Modeling nonlinear change via latent change and latent acceleration frameworks: Examining velocity and acceleration of growth trajectories. Multivariate Behavioral Research, 48, 117–143.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We demonstrate that four popular variants of the latent change score model – the no change, constant change, proportional change, and dual change models – have LGM equivalents. We provide equations that allow the translation of parameters from one approach to the other and vice versa. We then illustrate this equivalence using mathematics achievement data from the National Longitudinal Survey of Youth.     

Bayesian Meta-Analytic SEM: A One-Stage Approach to Modeling Between-Studies Heterogeneity in Structural Parameters

Meta-analytic structural equation modeling (MASEM) refers to a set of meta-analysis techniques for combining and comparing structural equation modeling (SEM) results from multiple studies. Existing approaches to MASEM cannot appropriately model between-studies heterogeneity in structural parameters because of missing correlations, lack model fit assessment, and suffer from several theoretical limitations. In this study, we address the major shortcomings of existing approaches by proposing a novel Bayesian multilevel SEM approach. Simulation results showed that the proposed approach performed satisfactorily in terms of parameter estimation and model fit evaluation when the number of studies and the within-study sample size were sufficiently large and when correlations were missing completely at random. An empirical example about the structure of personality based on a subset of data was provided. Results favored the third factor structure over the hierarchical structure. We end the article with discussions and future directions.     

Inference and Interval Estimation Methods for Indirect Effects With Latent Variable Models

Although much is known about the performance of recent methods for inference and interval estimation for indirect or mediated effects with observed variables, little is known about their performance in latent variable models. This article presents an extensive Monte Carlo study of 11 different leading or popular methods adapted to structural equation models with latent variables. Manipulated variables included sample size, number of indicators per latent variable, internal consistency per set of indicators, and 16 different path combinations between latent variables. Results indicate that some popular or previously recommended methods, such as the bias-corrected bootstrap and asymptotic standard errors had poorly calibrated Type I error and coverage rates in some conditions. Likelihood-based confidence intervals, the distribution of the product method, and the percentile bootstrap emerged as leading methods for both interval estimation and inference, whereas joint significance tests and the partial posterior method performed well for inference.     

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.     

Time-Varying Effect Sizes for Quadratic Growth Models in Multilevel and Latent Growth Modeling

Multilevel and latent growth modeling analysis (GMA) is often used to compare independent groups in linear random slopes of outcomes over time, particularly in randomized controlled trials. The unstandardized coefficient for the effect of group on the slope from a linear GMA can be transformed into a model-estimated effect size for the group difference at the end of a study. Because effect sizes vary nonlinearly in quadratic GMA, the effect size at the end of a study using quadratic GMA cannot be derived from a single coefficient, and cannot be used to estimate effect sizes at intermediate time points with backward extrapolation. This article formulates equations and associated input commands in Mplus for time-varying effect sizes for quadratic GMA. Illustrative analyses that produced these time-varying effect sizes were presented, and a Monte Carlo study found that bias in the effect sizes and their confidence intervals was ignorable.     

Bayesian Regularized Multivariate Generalized Latent Variable Models

We consider a multivariate generalized latent variable model to investigate the effects of observable and latent explanatory variables on multiple responses of interest. Various types of correlated responses, such as continuous, count, ordinal, and nominal variables, are considered in the regression. A generalized confirmatory factor analysis model that is capable of managing mixed-type data is proposed to characterize latent variables via correlated observed indicators. In addressing the complicated structure of the proposed model, we introduce continuous underlying measurements to provide a unified model framework for mixed-type data. We develop a multivariate version of the Bayesian adaptive least absolute shrinkage and selection operator procedure, which is implemented with a Markov chain Monte Carlo (MCMC) algorithm in a full Bayesian context, to simultaneously conduct estimation and model selection. The empirical performance of the proposed methodology is demonstrated through a simulation study. An application of the proposed method to a study of adolescent substance abuse based on the National Longitudinal Survey of Youth is presented.     

Software Packages for Bayesian Multilevel Modeling

Multilevel modeling is a statistical approach to analyze hierarchical data that consist of individual observations nested within clusters. Bayesian method is a well-known, sometimes better, alternative of Maximum likelihood method for fitting multilevel models. Lack of user friendly and computationally efficient software packages or programs was a main obstacle in applying Bayesian multilevel modeling. In recent years, the development of software packages for multilevel modeling with improved Bayesian algorithms and faster speed has been growing. This article aims to update the knowledge of software packages for Bayesian multilevel modeling and therefore to promote the use of these packages. Three categories of software packages capable of Bayesian multilevel modeling including brms, MCMCglmm, glmmBUGS, Bambi, R2BayesX, BayesReg, R2MLwiN and others are introduced and compared in terms of computational efficiency, modeling capability and flexibility, as well as user-friendliness. Recommendations to practical users and suggestions for future development are also discussed.     

Alternative Methods for Assessing Mediation in Multilevel Data: The Advantages of Multilevel SEM

Multilevel modeling (MLM) is a popular way of assessing mediation effects with clustered data. Two important limitations of this approach have been identified in prior research and a theoretical rationale has been provided for why multilevel structural equation modeling (MSEM) should be preferred. However, to date, no empirical evidence of MSEM's advantages relative to MLM approaches for multilevel mediation analysis has been provided. Nor has it been demonstrated that MSEM performs adequately for mediation analysis in an absolute sense. This study addresses these gaps and finds that the MSEM method outperforms 2 MLM-based techniques in 2-level models in terms of bias and confidence interval coverage while displaying adequate efficiency, convergence rates, and power under a variety of conditions. Simulation results support prior theoretical work regarding the advantages of MSEM over MLM for mediation in clustered data.     

Estimating Latent Variable Interactions with the Unconstrained Approach: A Comparison of Methods to Form Product Indicators for Large,Unequal Numbers of Items

This Monte Carlo simulation study investigated methods of forming product indicators for the unconstrained approach for latent variable interaction estimation when the exogenous factors are measured by large and unequal numbers of indicators. Product indicators were created based on multiplying parcels of the larger scale by indicators of the smaller scale, multiplying the three most reliable indicators of each scale matched by reliability, and matching items by reliability to create as many product indicators as the number of indicators of the smallest scale. The unconstrained approach was compared with the latent moderated structural equations (LMS) approach. All methods considered provided unbiased parameter estimates. Unbiased standard errors were obtained in all conditions with the LMS approach and when the sample size was large with the unconstrained approach. Power levels to test the latent interaction and Type I error rates were similar for all methods but slightly better for the LMS approach.     

Evaluation of Structural Relationships in Autoregressive Cross-Lagged Models Under Longitudinal Approximate Invariance:A Bayesian Analysis

To infer longitudinal relationships among latent factors, traditional analyses assume that the measurement model is invariant across measurement occasions. Alternative to placing cross-occasion equality constraints on parameters, approximate measurement invariance (MI) can be analyzed by specifying informative priors on parameter differences between occasions. This study evaluated the estimation of structural coefficients in multiple-indicator autoregressive cross-lagged models under various conditions of approximate MI using Bayesian structural equation modeling. Design factors included factor structures, conditions of non-invariance, sizes of structural coefficients, and sample sizes. Models were analyzed using two sets of small-variance priors on select model parameters. Results showed that autoregressive coefficient estimates were more accurate for the mixed pattern than the decreasing pattern of non-invariance. When a model included cross-loadings, an interaction was found between the cross-lagged estimates and the non-invariance conditions. Implications of findings and future research directions are discussed.     

Multilevel Factorial Invariance in n-Level Structural Equation Modeling (nSEM)

We present a multigroup multilevel confirmatory factor analysis (CFA) model and a procedure for testing multilevel factorial invariance in n-level structural equation modeling (nSEM). Multigroup multilevel CFA introduces a complexity when the group membership at the lower level intersects the clustered structure, because the observations in different groups but in the same cluster are not independent of one another. nSEM provides a framework in which the multigroup multilevel data structure is represented with the dependency between groups at the lower level properly taken into account. The procedure for testing multilevel factorial invariance is illustrated with an empirical example using an R package xxm2.     

Mixture Simultaneous Factor Analysis for Capturing Differences in Latent Variables Between Higher Level Units of Multilevel Data

Given multivariate data, many research questions pertain to the covariance structure: whether and how the variables (e.g., personality measures) covary. Exploratory factor analysis (EFA) is often used to look for latent variables that might explain the covariances among variables; for example, the Big Five personality structure. In the case of multilevel data, one might wonder whether or not the same covariance (factor) structure holds for each so-called data block (containing data of 1 higher level unit). For instance, is the Big Five personality structure found in each country or do cross-cultural differences exist? The well-known multigroup EFA framework falls short in answering such questions, especially for numerous groups or blocks. We introduce mixture simultaneous factor analysis (MSFA), performing a mixture model clustering of data blocks, based on their factor structure. A simulation study shows excellent results with respect to parameter recovery and an empirical example is included to illustrate the value of MSFA.     

Latent Differential Equation Models for Binary and Ordinal Data

This study evaluates latent differential equation models on binary and ordinal data. Binary and ordinal data are widely used in psychology research and many statistical models have been developed, such as the probit model and the logit model. We combine the latent differential equation model with the probit model through a threshold approach, and then compare the threshold model with a naive model, which blindly treats binary and ordinal data as continuous. Simulation results suggest that the naive model leads to bias on binary data and on ordinal data with fewer than 5 levels, whereas the threshold model is unbiased and efficient for binary and ordinal data. Two example analyses on empirical binary data and ordinal data show that the threshold model also has better external validity. The R code for the threshold model is provided.     

A Comparison of Strategies for Forming Product Indicators for Unequal Numbers of Items in Structural Equation Models of Latent Interactions

This Monte Carlo simulation study investigated different strategies for forming product indicators for the unconstrained approach in analyzing latent interaction models when the exogenous factors are measured by unequal numbers of indicators under both normal and nonnormal conditions. Product indicators were created by (a) multiplying parcels of the larger scale by items of the smaller scale, and (b) matching items according to reliability to create several product indicators, ignoring those items with lower reliability. Two scaling approaches were compared where parceling was not involved: (a) fixing the factor variances, and (b) fixing 1 loading to 1 for each factor. The unconstrained approach was compared with the latent moderated structural equations (LMS) approach. Results showed that under normal conditions, the LMS approach was preferred because the biases of its interaction estimates and associated standard errors were generally smaller, and its power was higher than that of the unconstrained approach. Under nonnormal conditions, however, the unconstrained approach was generally more robust than the LMS approach. It is recommended to form product indicators by using items with higher reliability (rather than parceling) in the matching and then to specify the model by fixing 1 loading of each factor to unity when adopting the unconstrained approach.     

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.     

Brief Research Report: Bayesian Versus REML Estimations With Noninformative Priors in Multilevel Single-Case Data

Abstract

Recently, researchers have used multilevel models for estimating intervention effects in single-case experiments that include replications across participants (e.g., multiple baseline designs) or for combining results across multiple single-case studies. Researchers estimating these multilevel models have primarily relied on restricted maximum likelihood (REML) techniques, but Bayesian approaches have also been suggested. The purpose of this Monte Carlo simulation study was to examine the impact of estimation method (REML versus Bayesian with noninformative priors) on the estimation of treatment effects (relative bias, root mean square error) and on the inferences about those effects (interval coverage) for autocorrelated multiple-baseline data. Simulated conditions varied with regard to the number of participants, series length, and distribution of the variance within and across participants. REML and Bayesian estimation led to estimates of the fixed effects that showed little to no bias but that differentially impacted the inferences about the fixed effects and the estimates of the variances. Implications for applied researchers and methodologists are discussed.     

Implementing Restricted Maximum Likelihood Estimation in Structural Equation Models

Structural equation modeling (SEM) is now a generic modeling framework for many multivariate techniques applied in the social and behavioral sciences. Many statistical models can be considered either as special cases of SEM or as part of the latent variable modeling framework. One popular extension is the use of SEM to conduct linear mixed-effects modeling (LMM) such as cross-sectional multilevel modeling and latent growth modeling. It is well known that LMM can be formulated as structural equation models. However, one main difference between the implementations in SEM and LMM is that maximum likelihood (ML) estimation is usually used in SEM, whereas restricted (or residual) maximum likelihood (REML) estimation is the default method in most LMM packages. This article shows how REML estimation can be implemented in SEM. Two empirical examples on latent growth model and meta-analysis are used to illustrate the procedures implemented in OpenMx. Issues related to implementing REML in SEM are discussed.