Comparing hierarchical models for spatio-temporally misaligned data using the deviance information criterion

Bayes and empirical Bayes methods have proven effective in smoothing crude maps of disease risk, eliminating the instability of estimates in low-popul ation areas while maintaining overall geographic trends and patterns. Recen t work extends these methods to the analysis of areal data which are spatia lly misaligned, that is, involving variables (typically counts or rates) wh ich are aggregated over differing sets of regional boundaries. The addition of a temporal aspect complicates matters further, since now the misalignme nt can arise either within a given time point, or across time points (as wh en the regional boundaries themselves evolve over time). Hierarchical Bayes ian methods (implemented via modern Markov chain Monte Carlo computing meth ods) enable the fitting of such models, but a formal comparison of their fi t is hampered by their large size and often improper prior specifications. In this paper, we accomplish this comparison using the deviance information criterion (DIC), a recently proposed generalization of the Akaike informat ion criterion (AIC) designed for complex hierarchical model settings like o urs. We investigate the use of the delta method for obtaining an approximat e variance estimate for DIC, in order to attach significance to apparent di fferences between models. We illustrate our approach using a spatially misa ligned data set relating a measure of traffic density to paediatric asthma hospitalizations in San Diego County, California. Copyright (C) 2000 John W iley & Sons, Ltd.