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-population areas while maintaining overall geographic trends and patterns. Recent work extends these methods to the analysis of areal data which are spatially misaligned, that is, involving variables (typically counts or rates) which are aggregated over differing sets of regional boundaries. The addition of a temporal aspect complicates matters further, since now the misalignment can arise either within a given time point, or across time points (as when the regional boundaries themselves evolve over time). Hierarchical Bayesian methods (implemented via modern Markov chain Monte Carlo computing methods) enable the fitting of such models, but a formal comparison of their fit 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 information criterion (AIC) designed for complex hierarchical model settings like ours. We investigate the use of the delta method for obtaining an approximate variance estimate for DIC, in order to attach significance to apparent differences between models. We illustrate our approach using a spatially misaligned data set relating a measure of traffic density to paediatric asthma hospitalizations in San Diego County, California.