An empirical evaluation of various priors in the empirical Bayes estimation of small area disease risks.

Empirical and fully Bayes estimation of small area disease risks places a prior distribution on area-specific risks. Several forms of priors have been used for this purpose including gamma, log-normal and non-parametric priors. Spatial correlation among area-specific risks can be incorporated in log-normal priors using Gaussian Markov random fields or other models of spatial dependence. However, the criterion for choosing one prior over others has been mostly logical reasoning. In this paper, we evaluate empirically the various priors used in the empirical Bayes estimation of small area disease risks. We utilize a Spanish mortality data set of a 12-year period to give the underlying true risks, and estimate the true risks using only a 3-year portion of the data set. Empirical Bayes estimates are shown to have substantially smaller mean squared errors than Poisson likelihood-based estimates. However, relative performances of various priors differ across a variety of mortality outcomes considered. In general, the non-parametric prior provides good estimates for lower-risk areas, while spatial priors provide good estimates for higher-risk areas. Ad hoc composite estimates averaging the estimates from the non-parametric prior and those from a spatial log-normal prior appear to perform well overall. This suggests that an empirical Bayes prior that strikes a balance between these two priors, if one can construct such a prior, may prove to be useful for the estimation of small area disease risks.