Methods for the analysis of spatio-temporal multi-state processes

Studies of recurring infection or chronic disease often collect longitudinal data on the disease status of subjects. Multi-state transitional models are commonly used for describing the development of such longitudinal data. In this setting, we model a stochastic process, which at any point in time will occupy one of a discrete set of states and interest centers on the transition process between states. For example, states may refer to the number of recurrences of an event or the stage of a disease. Geographic referencing of data collected in longitudinal studies is progressively more common as scientific databases are being linked with GIs systems. This has created a need for statistical methods addressing the resulting spatial-longitudinal structure of the data. In this thesis, we develop hierarchical mixed multi-state models for the analysis of such longitudinal data when the processes corresponding to different subjects may be correlated spatially over a region. Methodological developments have been strongly driven by studies in forestry and spatial epidemiology. Motivated by an application in forest ecology studying pine weevil infestations, the second chapter develops methods for handling mixtures of populations for spatial discrete-time twestate processes. The two-state discrete-time transitional model, often used for studying chronic conditions in human populations, is extended to settings where subjects are spatially arranged. A mixed spatially correlated mover-stayer model is developed. Here, clustering of infection is modelled by a spatially correlated random effect reflecting the density or closeness of the individuals under study. Analysis is carried out using maximum likelihood with a Monte Carlo EM algorithm for implementation and also using a fully Bayesian analysis. The third chapter presents continuous-time spatial multi-state models. Here, joint modelling of both the spatial correlation as well as correlation between different transition rates is required and a multivariate spatial approach is employed. A proportional intensities frailty model is developed where baseline intensity functions are modelled using both parametric Weibull forms as well as flexible representations based on cubic B-splines. The methodology is applied to a study of invasive cardiac procedure in Quebec examining readmission and mortality rates over a four-year period. Finally, in the fourth chapter we return to the two-state discrete-time setting. An extension of the mixed mover-stayer model is motivated and developed within the Bayesian framework. Here, a multivariate conditional autoregressive (MCAR) model is incorporated providing flexible joint correlation structures. We also consider a test for the number of mixture components, quantifying the existence of a hidden subgroup of 'stayers' within the population. Posterior summarization is based on a Metropolis-Hastings sampler and methods for assessing the model goodness-of-fit are based on posterior predictive comparisons.