Multivariate areal modeling with application to mapping multiple disease data

With the ready availability of spatial databases and Geographical Information System (GIS) software, statisticians are increasingly encountering multivariate modeling settings featuring associations of more than one type: spatial associations between data locations, and associations between the variables within the locations. A multivariate areal model is needed for modeling such associations.; In this dissertation, we first include the existing multivariate conditionally autoregressive (MCAR) distributions of Carlin and Banerjee (2003) into spatial and spatiotemporal survival models, where the frailty and regression coefficients (if any) are allowed both to vary spatially and be correlated with each other. We then apply these models to a data set of county-level breast cancer survival rates from women living in the state of Iowa, as recorded by Surveillance, Epidemiology, and End Results (SEER) program.; Second, we propose a flexible new class of generalized MCAR (GMCAR) models for areal data, and show how it enriches the existing MCAR class. Our approach differs from earlier ones in that it directly specifies the joint distribution for a multivariate Markov random field (MRF) through specification of simpler conditional and marginal models. We compare our approach with existing MCAR models in the literature via simulation, and also offer some applications of our proposed GMCAR approach.; While flexible modeling of multivariate point-referenced data have recently been addressed using a linear model of coregionalization (LMC; see e.g. Gelfand et al., 2004), existing methods for multivariate areal data (e.g. Kim et al., 2001; Gelfand and Vounatsou, 2003) typically suffer from unnecessary restrictions on the covariance structure, and there is undesirable dependence on the conditioning order of the variables present in the GMCAR models. As such, the third contribution of this dissertation is to propose a multivariate areal model that avoids these restrictions, permitting flexible and order-free modeling of correlations both between variables and across areal units. We illustrate the strengths of our approach over existing models using simulation studies, and also offer another cancer-related application.