When inference is desired regarding some attribute of a particular geographic region, it often happens that data are not directly available for that region. However, it may be that data are available over the same general area, but reported according to a different set of regional boundaries. Recently, powerful computer programs called geographic information systems (GIS's) have enabled the simultaneous display of such "misaligned" datasets, but these systems address only the descriptive needs of the user, leaving the inferential goal unmet. In this article we describe a hierarchical Bayes approach, implemented via Markov chain Monte Carlo methods, which provides a natural solution to this problem through its ability to sensibly combine information from several sources of data and available prior information. After presenting a simple, idealized example to illustrate the method, we apply it to a dataset on leukemia rates in Tompkins County, New York, wherein we use block group-level covariate information to interpolate disease counts given only aggregate (census tract-level) summaries. We display our results graphically, using both statistical (S-plus) and GIS (ARC/INFO, MapInfo) software packages. The approach emerges as flexible, accurate, and suggestive of promising related methods for spatial smoothing of underlying relative risks.

Key Words
Bayesian methods; Markov chain Monte Carlo; Misaligned data; Spatial statistics.

Andrew S. Mugglin is Assistant Professor, Northwestern College, St. Paul, MN 55113 (E-mail: andy@muskie.biostat.umn.edu). Bradley P. Carlin is Associate Professor, Division of Biostatistics, School of Public Health, University of Minnesota, Box 303, Mayo Memorial Building, Minneapolis, MN 55455–0392.(E-mail: brad@muskie.biostat.umn.edu).