Marginal and conditional posterior predictive p-values in Bayesian SEM

Abstract

The posterior predictive p-value (ppp-value) is currently the primary measure of fit for Bayesian SEM. It is a measure of discrepancy between observed data and a posited model, comparing an observed likelihood ratio test (LRT) statistic to the posterior distribution of LRT statistics under a fitted model. However, the LRT statistic requires a likelihood, and multiple likelihoods are available for a given SEM: we can use a marginal likelihood that integrates out the latent variable(s), or we can use a conditional likelihood that conditions on the latent variable(s). A ppp-value based on conditional likelihoods is unexplored in the SEM literature, so the goal of this project is to study its performance alongside the marginal ppp-value. We present comparisons of the marginal and conditional ppp-values using real and simulated data, leading to recommendations on uses of the metrics in practice.