  Rapid developments in geographical information systems (GIS) continue to
generate interest in analyzing complex spatial datasets. One area of activity
is in creating smoothed disease maps to describe the geographic variation of
disease and generate hypotheses for apparent differences in risk. With multiple
diseases, a multivariate conditionally autoregressive (MCAR) model is often
used to smooth across space while accounting for associations between the
diseases. The MCAR, however, imposes complex covariance structures that are
difficult to interpret and estimate. This article develops a much simpler
alternative approach building upon the techniques of smoothed ANOVA (SANOVA).
Instead of simply shrinking effects without any structure, here we use SANOVA
to smooth spatial random effects by taking advantage of the spatial structure.
We extend SANOVA to cases in which one factor is a spatial lattice, which is
smoothed using a CAR model, and a second factor is, for example, type of
cancer. Datasets routinely lack enough information to identify the additional
structure of MCAR. SANOVA offers a simpler and more intelligible structure than
the MCAR while performing as well. We demonstrate our approach with simulation
studies designed to compare SANOVA with different design matrices versus MCAR
with different priors. Subsequently a cancer-surveillance dataset, describing
incidence of 3-cancers in Minnesota's 87 counties, is analyzed using both
approaches, showing the competitiveness of the SANOVA approach.
