Hierarchical Multiresolution Approaches for Dense Point-Level Breast Cancer Treatment Data

The analysis of point-level (geostatistical) data has historically been plagued by computational difficulties, owing to the high dimension of the nondiagonal spatial covariance matrices that need to be inverted. This problem is greatly compounded in hierarchical Bayesian settings, since these inversions need to take place at every iteration of the associated Markov chain Monte Carlo (MCMC) algorithm. This paper offers an approach for modeling the spatial correlation at two separate scales. This reduces the computational problem to a collection of lower-dimensional inversions that remain feasible within the MCMC framework. The approach yields full posterior inference for the model parameters of interest, as well as the fitted spatial response surface itself. We illustrate the importance and applicability of our methods using a collection of dense point-referenced breast cancer data collected over the mostly rural northern part of the state of Minnesota. Substantively, we wish to discover whether women who live more than a 60-mile drive from the nearest radiation treatment facility tend to opt for mastectomy over breast conserving surgery (BCS, or "lumpectomy"), which is less disfiguring but requires 6 weeks of follow-up radiation therapy. Our hierarchical multiresolution approach resolves this question while still properly accounting for all sources of spatial association in the data.