Simulation Based Model Checking for Hierarchical Models

Recent computational advances have made it feasible to t hierarchical models in a wide range of serious applications. If one entertains a collection of such models for a given data set, the problems of model adequacy and model choice arise. We focus on the former. While model checking usually addresses the entire model speciication, model failures can occur at each hierarchical stage. Such failures include outliers, mean structure errors, dispersion misspeciication, and inappropriate exchangeabilities. We propose another approach which is entirely simulation based. It only requires the model speciication and that, for a given data set, one be able to simulate draws from the posterior under the model. By replicating a posterior of interest using data obtained under the model we can \see" the extent of variability in such a posterior. Then, we can compare the posterior obtained under the observed data with this medley of posterior replicates to ascertain whether the former is in agreement with them and accordingly, whether it is plausible that the observed data came from the proposed model. This suggests the large scale use of Monte Carlo tests, each focusing on a potential model failure. It thus suggests the possibility of examining not only the overall adequacy of the hierarchical model but, using suitable posteriors, the adequacy of each stage. 1 This raises the question of when individual stages are separable and checkable which we explore in some detail. Finally, we develop this strategy in the context of generalized linear mixed models and ooer a simulation study to demonstrate its capabilities.