AbstractsMathematics

Hierarchical spatial modelling is useful for modelling complex spatially correlated data in a variety of settings. Due to the complexity of spatial analyses, hierarchical spatial models for disease mapping studies have not generally found application at Vital Statistics agencies. Chapter 2 compares penalized quasi-likelihood relative risk estimates to target values based on Bayesian Markov Chain Monte Carlo methods. Results show penalized quasi-likelihood to be a simple, reasonably accurate method of inference for exploratory studies of small-area relative risks and ranks of risks. Often the identification of extreme risk areas is of interest. Isolated ‘hot spots’/‘low spots’ which are distinct from those of neighbouring sites are not accommodated by standard hierarchical spatial models. In Chapter 3, spatial methods are developed which allow extreme risk areas to arise in proximity to one another in a smooth spatial surface, or in isolated ‘hot spots’/‘low spots’. The former is modelled by a spatially smooth surface using a conditional autoregressive model; the latter is addressed with the addition of a discrete clustering component, which accommodates extreme isolated risks and is not limited by spatial smoothness. A Bayesian approach is employed, graphical techniques for isolating extremes are illustrated, and model assessment is conducted via cross-validation posterior predictive checks. Zero-inflated data are not uncommon yet they are not handled well by standard models. These values may be of particular interest in species abundance studies where such zeros may provide clues to physical characteristics associated with habitat suitability or individual immunity. In Chapter 4 we review the overdispersion and zero-inflation literature and develop a series of zero-inflated spatial models. Each model highlights different features of white pine weevil infestation data. The spatial models use a variety of structures for the probability of belonging to the zero component, thus allowing the probability of ‘resistance’ to differ across models. One model focuses on individually resistant trees which are located among infested trees while another focuses on clusters of resistant trees which are likely located within protective habitats. The final chapter discusses future research ideas which have been motivated by this thesis.