In the analysis of spatially referenced data, interest often focuses not on prediction of the spatially indexed variable itself, but on 'boundary analysis', i.e., the determination of boundaries on the map that separate areas of higher and lower values. Existing boundary analysis methods are sometimes generically referred to as 'wombling', after a foundational paper by Womble (1951). In this paper we propose wombling methods for areal data (i.e., data which consist only of sums or averages over geopolitical regions). Such methods are valuable in determining boundaries for data sets that, perhaps due to confidentiality concerns, are available only in ecological (aggregated) format, or are only collected this way (e.g., delivery of health care or cost information). We propose a fully model-based framework for areal wombling, using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo (MCMC) methods. While our approaches parallel similar developments in statistical image restoration using Markov random fields, important differences arise due to the irregular nature of our lattices, ]the sparseness and high variability of our data, the existence of important covariate information, and most importantly, our desire for full posterior inference on the boundary. We use our methods to determine the service area of two particular cancer hospice systems in northeastern Minnesota based only on death counts abstracted from Medicare billing records.
Group for Research in Decision Analysis