Posterior Predictive Model Checking: An Application to Multivariate Normal Heterogeneity

Analysts fitting a hierarchical Bayesian model must specify the distribution of heterogeneity. There are several distributions to choose from, including the multivariate normal, mixture of normals, Dirichlet processes priors, and so forth. Although significant progress has been made, estimating the models and obtaining measures for model selection remain ongoing areas of research for more flexible distributions of heterogeneity. As a result, the multivariate normal remains the default choice for many researchers and software packages. This article proposes model-checking statistics that signal the adequacy of the multivariate normal assumption for the distribution of heterogeneity; these methods do not require the analyst to fit alternative models. The authors use posterior predictive model checking to determine whether a discrepancy exists between the individual-level parameters and those implied by the assumed distribution of heterogeneity. In simulated and real data sets, the results show that these statistics are useful for identifying when the multivariate normal distribution is adequate, when there is a departure in the tails of the distribution, and when a multimodal distribution of heterogeneity may be more appropriate.