Empirical bayes estimators for spatially correlated incidence rates

Abstract Assessments of the potential health impacts of contaminants and other environmental risk factors are often based on comparisons of disease rates among collections of spatially aligned areas. These comparisons are valid only if the observed rates adequately reflect the true underlying area‐specific risk. In areas with small populations, observed incidence values can be highly unstable and true risk differences among areas can be masked by spurious fluctuations in the observed rates. We examine the use of Bayes and empirical Bayes methods for stabilizing incidence rates observed in geographically aligned areas. While these methods improve stability, both the Bayes and empirical Bayes approaches produce a histogram of the estimates that is too narrow when compared to the true distribution of risk. Constrained empirical Bayes estimators have been developed that provide improved estimation of the variance of the true rates. We use simulations to compare the performance of Bayes, empirical Bayes, and constrained empirical Bayes approaches for estimating incidence rates in a variety of multivariate Gaussian scenarios with differing levels of spatial dependence. The mean squared error of estimation associated with the simulated observed rates was, on average, five times greater than that of the Bayes and empirical Bayes estimates. The sample variance of the standard Bayes and empirical Bayes estimates was consistently smaller than the variance of the simulated rates. The constrained estimators produced collections of rate estimates that dramatically improved estimation of the true dispersion of risk. In addition, the mean square error of the constrained empirical Bayes estimates was only slightly greater than that of the unconstrained rate estimates. We illustrate the use of empirical and constrained empirical Bayes estimators in an analysis of lung cancer mortality rates in Ohio.