| Disease mapping plays an enormously vital role in spatial epidemiology, allowing us to assess geographic patterns in disease risk and perhaps more importantly, to uncover risk factors associated with disease. Moreover, since maps by their very nature display spatial information that varies with resolution, the patterns that emerge are directly influenced by the particular scale at which the disease maps are presented. In other words, a change in resolution results in a potential change in the structure of the spatial object; therefore, it is of importance to study the effects of scale that arise as a consequence of moving between scales. While the issue of scale in the analysis of general spatial data is long-standing, there has been very little substantive work focused on a formal statistical framework for analyzing such data simultaneously at multiple scales. A recent exception is the framework of Kolaczyk and Huang (2001), based on multiscale likelihood factorizations, which extends the now-classical wavelet-based paradigm for the special case of image data to general spatial data structures. A key consequence of this framework is the ability to carry out estimation, hypothesis testing, and characterization of uncertainty in a scale-sensitive manner.; We extend and formalize the framework of Kolaczyk and Huang (2001) yielding a class of multiscale spatial process models (MSSpP's). Focus is on the modeling of the parameters that capture the effects of scale and the spatial processes that are induced as a result of specifying models on these "multiscale" parameters. Furthermore, we characterize the spatial variation of MSSpP's through scale recursive formulas for the spatial covariance structure, calculations for the eigenvalues of the covariance structures under certain constraints, and a scale-indexed analogue of the variogram. The existence of a Poisson multiscale likelihood factorization is ideal for analyzing disease maps due to the discrete nature of the data; therefore we extend the MSSpP's to measures of relative risk for tract count (aggregate) data, providing frameworks for characterizing uncertainty and scale/location-specific testing of regions with elevated incidence. This research has relevance in public health policy and epidemiology. |
