P. K. Sen and C. R. Rao, eds., Handbook of Statistics, Vol. 18 © 2000 Elsevier Science B.V. All rights reserved. k.y Hierarchical and Empirical Bayes Methods for Environmental Risk Assessment Gauri Datta, Malay Ghosh and Lance A. Waller The study of geographical variability in mortality rates for diseases such as cancer has received a great deal of attention from statisticians in recent years. A con- siderable amount of effort has been devoted to the production of disease atlases, i.e., collections of maps in which the geographical distribution of incidence of a specific disease is presented across a country or region. Typically, the study area is partitioned into smaller spatial districts, and the disease map displays local rates (incidence or mortality) associated with each district. The main problem faced in production of these atlases is the choice of a suitable measure of local mortality (or incidence of disease) to be mapped. This paper reviews hierarchical and empirical Bayes methods for the estima- tion of mortality rates. Unlike standardized mortality ratios, these methods are particularly well-suited to account for spatially structured heterogeneity in disease risks. The inference is based on certain spatial generalized linear models. Spatio- temporal models are also discussed. The papers concludes with a data example. 1. Introduction The study of geographical variability in mortality rates for diseases such as cancer has received a great deal of attention from statisticians in recent years. A con- siderable amount of effort has been devoted to the production of disease atlases, i.e., collections of maps in which the geographical distribution of incidence of a specific disease is presented across a country or region. Typically, the study area is partitioned into smaller spatial districts, and the disease map displays local rates (incidence or mortality) associated with each district. The main problem faced in production of these atlases is the choice of a suitable measure of local mortality (or incidence of disease) to be mapped. Until recently, the two widely used measures were ratios of standardized rates (such as standardized mortality ratios) and P-values of statistical significance of local deviations of risk from the overall risk averaged across the entire study region. 223