We review common situations in Bayesian latent variable models where the prior distribution that a researcher specifies differs from the prior distribution used during estimation. These situations can arise from the positive definite requirement on correlation matrices, from sign indeterminacy of factor loadings, and from order constraints on…

Dynamic structural equation modeling (DSEM) is a useful technique for analyzing intensive longitudinal data. A challenge of applying DSEM is the missing data problem. The impact of missing data on DSEM, especially on widely applied DSEM such as the two-level vector autoregressive (VAR) cross-lagged models, however, is understudied. To fill the…

Model evaluation is a crucial step in SEM, consisting of two broad areas: global and local fit, where local fit indices are use to modify the original model. In the modification process, the modification index (MI) and the standardized expected parameter change (SEPC) are used to select the parameters that can be added to improve the fit. The…

With the prevalence of missing data in social science research, it is necessary to use methods for handling missing data. One framework in which data with missing values can still be used for parameter estimation is the Bayesian framework. In this tutorial, different missing data mechanisms including Missing Completely at Random, Missing at…

Bayesian inference for structural equation models (SEMs) is increasingly popular in social and psychological sciences owing to its flexibility to adapt to more complex models and the ability to include prior information if available. However, there are two major hurdles in using the traditional Bayesian SEM in practice: (1) the information nested…

Structural equation models comprise a large class of popular statistical models, including factor analysis models, certain mixed models, and extensions thereof. Model estimation is complicated by the fact that we typically have multiple interdependent response variables and multiple latent variables (which may also be called random effects or…

Research questions regarding how, for whom, and where a treatment achieves its effect on an outcome have become increasingly valued in substantive research. Such questions can be answered by causal moderated mediation analysis, which assesses the heterogeneity of the mediation mechanism underlying the treatment effect across individual and…

Despite its importance to structural equation modeling, model evaluation remains underdeveloped in the Bayesian SEM framework. Posterior predictive p-values (PPP) and deviance information criteria (DIC) are now available in popular software for Bayesian model evaluation, but they remain under-utilized. This is largely due to the lack of…

Posterior Predictive Model Checking (PPMC) is a Bayesian model checking method that compares the observed data to (plausible) future observations from the posterior predictive distribution. We propose an alternative to PPMC in the context of structural equation modeling, which we term the Poor Persons PPMC (PP-PPMC), for the situation wherein one…