Bayesian model diagnostics based on artificial autoregressive errors

Abstract

Hierarchical Bayes models provide a natural way of incorporating covariate information into the inferential process through the elaboration of regression equations for one or more of the model parameters, with errors that are often assumed to be i.i.d. Gaussian. Unfortunately, building adequate regression models is a complicated art form that requires the practitioner to make numerous decisions along the way. Assessing the validity of the modeling decisions is often difficult. In this article I develop a simple and effective device for ascertaining the quality of the modeling choices and detecting lack-of-fit. I specify an artificial autoregressive structure (AAR) in the probability model for the errors that incorporates the i.i.d. model as a special case. Lack-of-fit can be detected by examining the posterior distribution of AAR parameters. In general, posterior distributions that assign considerable mass to a region of the AAR parameter space away from zero provide evidence that apparent dependencies in the errors are compensating for misspecifications of some other aspects (typically conditional means) of the model. I illustrate the methodology through several examples including its application to the analysis of data on brain and body weights of mammalian species and response time data.