Spatial Statistics and Its Applications in Biostatistics and Environmental Statistics

This dissertation presents some topics in spatial statistics and their application in biostatistics and environmental statistics. The field of spatial statistics is an energetic area in statistics. In Chapter 2 and Chapter 3, the goal is to build subregion models under the assumption that the responses or the parameters are spatially correlated. For regression models, considering spatially varying coecients is a reasonable way to build subregion models. There are two different techniques for exploring spatially varying coecients. One is geographically weighted regression (Brunsdon et al. 1998). The other is a spatially varying coecients model which assumes a stationary Gaussian process for the regression coecients (Gelfand et al. 2003). Based on the ideas of these two techniques, we introduce techniques for exploring subregion models in survival analysis which is an important area of biostatistics. In Chapter 2, we introduce modied versions of the Kaplan-Meier and Nelson-Aalen estimators which incorporate geographical weighting. We use ideas from counting process theory to obtain these modied estimators, to derive variance estimates, and to develop associated hypothesis tests. In Chapter 3, we introduce a Bayesian parametric accelerated failure time model with spatially varying coefficients. These two techniques can explore subregion models in survival analysis using both nonparametric and parametric approaches. In Chapter 4, we introduce Bayesian parametric covariance regression analysis for a response vector. The proposed method denes a regression model between the covariance matrix of a p-dimensional response vector and auxiliary variables. We propose a constrained Metropolis-Hastings algorithm to get the estimates. Simulation results are presented to show performance of both regression and covariance matrix estimates. Furthermore, we have a more realistic simulation experiment in which our Bayesian approach has better performance than the MLE. Finally, we illustrate the usefulness of our model by applying it to the Google Flu data. In Chapter 5, we give a brief summary of future work.