A note on Bayesian residuals as a hierarchical model diagnostic technique

When one wants to check a tentatively proposed model for departures that are not well specified, looking at residuals is the most common diagnostic technique. Here, we investigate the use of Bayesian standardized residuals to detect unknown hierarchical structure. Asymptotic theory, also supported by simulations, shows that the use of Bayesian standardized residuals is effective when the within group correlation, ρ, is large. However, we show that standardized residuals may not detect hierarchical structure when ρ is small. Thus, if it is important to detect modest hierarchical structure (i.e., ρ small) one should use other diagnostic techniques in addition to the standardized residuals. We use “quality of care” data from the Patterns of Care Study, a two-stage cluster sample of patients undergoing radiation therapy for cervix cancer, to illustrate the potential use of these residuals to detect missing hierarchical structure.