A constrained empirical Bayes estimator for incidence rates in areas with small populations

Maps that show the geographic distribution of incidence rates can be useful tools for analysing spatial variation in mortality and morbidity. To attain the necessary geographic resolution, however, production of such maps often requires estimation of incidence in areas with small populations where the observed rates may be highly unstable. Manton et al. have presented an empirical Bayes stabilization procedure in which the observed rate is combined with an area-specific estimate of the underlying incidence. The approach allows for the mapping of outcomes with varied and possibly unknown etiologies without necessitating covariate dependent modelling of the expected rate. The empirical distribution of a collection of these estimates, however, may not provide an adequate description of the dispersion among the true rates. As a result, decisions based on the histogram of the empirical Bayes estimates may be suspect. We propose a modified version of the approach in which the mean and sample variance of the ensemble of estimates are constrained to equal the appropriate moments of the posterior distribution. The resulting collection of constrained empirical Bayes estimators has nearly the stability of the unconstrained approach and provides an improved estimator of the true rate distribution. We illustrate use of the estimator by producing stabilized county-level maps of U.S. fire- and burn-related mortality rates and validate the analytic results using a simulation analysis.