Bayesian Spatial and Spatio-temporal Models for Skewed Areal Count Data

Abstract

Disease mapping models have found wide range of applications to epidemiology and public health. These models typically extend from generalized linear models (GLM) and are usually implemented using a Bayesian approach. Most of the disease mapping models incorporate random effects that assume either a Gaussian exchangeable prior for the spatially unstructured heterogeneity or the popular Gaussian CAR priors for the spatially structured variability. However, this Gaussian assumption is often violated since random effects can be skewed. This thesis proposed models that relax the usual normality assumption on the spatially unstructured random effect by using skew normal and skew-t distributions. In the analysis of 2016 HIV and AID data in Kenya, it was found out that models whose unstructured random effects follow asymmetric skewed distributions perform better than models with corresponding symmetric distributed unstructured random effects. Classical random-effects models for count data includes the Poisson-gamma model, that utilizes the conjugate feature between the Poisson and Gamma distributions to attain closed-form posterior distribution but accounts only for overdispersion or extra variation, and the Gaussian conditional autoregressive (CAR) models, that model spatial correlation but does not have a closed-form posterior distribution. This thesis also considers an alternative model that combines a Poisson-gamma model with a spatially structured skew-t random effect in the same model thus accounting for the extra variability, spatial correlation and skewness in the data. In the analysis of 2016 Kenya HIV and AIDS data, the skew-t spatial combined random effects model was found to provide a better alternative to the classical disease mapping models. Simulation studies also show that the proposed models perform better than the classical disease mapping models. To model spatio-temporal variation, this thesis considered Leroux CAR (LCAR) prior for spatial random effect and implemented Bayesian analysis using integrated nested Laplace approximations (INLA). In the analysis of spatio-temporal variation of HIV and AIDS in Kenya for the period 2013–2016, it was found out that counties located in theWestern region of Kenya show significantly higher HIV and AIDS risks as compared to the other counties.