Impact of Likelihoods on Class Enumeration in Bayesian Growth Mixture Modeling

Bayesian methods have been widely used to estimate models with complex structures. To assess model fit and compare different models, researchers typically use model selection criteria such as Deviance Information Criteria (DIC), Watanabe-Akaike Information Criteria (WAIC) and leave-one-out cross validation (LOO-CV), the calculation of which is based on the likelihoods of the models. When models contain latent variables, the likelihood is often specified as conditional on the latent variables in popular Bayesian software (e.g., BUGS, JAGS, and Stan). Although this practice reduces computation work and does not affect model estimation, the previous literature has shown that model comparisons based on the conditional likelihood could be misleading. In contrast, marginal likelihoods can be obtained by integrating out the latent variables and be used to calculate model selection criteria. In this study, we evaluate the effect of using conditional likelihoods and marginal likelihoods in model selection for growth mixture models. Simulation results suggest that marginal likelihoods are much more reliable and should be generally used for growth mixture modeling.