Hierarchical Bayesian Spatial Models for Disease Mortality Rates

The spatial epidemiology is the study of the occurrences of a disease in spatial locations. In spatial epidemiology, the disease to be examined usually occurs within a map that needs spatial statistical methods to model the observed data. The methods used should be appropriate and catered for the variation of the disease. The classical approach, which used to estimate the risk associated with the spread of the disease, did not seem to give a good estimation when there were different factors expected to influence the spread of the disease. In this research, the relative risk heterogeneity was investigated, while the hierarchical Bayesian models with different sources of heterogeneity were proposed using the Bayesian approach within the Markov Chain Monte Carlo (MCMC) method. The Bayesian models were developed in such a way that they allowed several factors, classified as fixed and random effects, to be included in the models. The effects were the covariate effects, interregional variability and the spatial variability, which were all investigated in three different hierarchical Bayesian models. These factors showed substantial effects in the relative risk estimation. The Bayesian approach, within the MCMC method, produced stable estimates for each individual (e.g. county) in the spatially arranged regions. It also allowed for unexplained heterogeneity to be investigated in the disease maps. The disease maps were employed to exploratory investigate the spread of the disease and to clean the maps off the extra noise via the Bayesian approach to expose the underlying structure. Using the MCMC method, particular sets of prior densities over the space of possible relative risks parameters and hyper-parameters were adopted for each model. The products of the likelihood and the prior densities produced the joint and conditional posterior densities of the parameters, from which all statistical inferences can be made for each model. Convergence of the MCMC simulation to the stationary posterior distributions was assessed. This was achieved by monitoring the samples of the history graphs for posterior means of the parameters, applying statistical diagnostic test and conducting sensitivity analysis for several trials of different choices of priors. The hierarchical models and the classical approach were applied on a spatial set of lip cancer data. The spatial correlation among the counties was examined and found to be spatially correlated. The results of the estimated relative risk for each county were compared with the result of the maximum likelihood estimation using the disease maps. The final model selection was accomplished by applying the deviance information criterion. The performance of each model was investigated using the posterior predictive simulations. The predictive simulation for each model was carried out using the Bayesian analysis results of the real data. The graphical and numerical posterior predictive checks were used as the assessment tests for each model. The numerical results showed a good agreement with the graphical results, in which the full model with both fixed and random effects was appropriate since it was found to be capable of providing the most similar values of the original and predicted samples compared to the other models. This model was also found to be flexible since it can be reduced or extended according to the nature of the data. Nevertheless, great care must be considered in the choice of prior densities.

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