Spatially correlated allocation models for count data

Peter J. Green (Bristol) & Sylvia Richardson (Imperial College, London)
Spatial heterogeneity of count data on a rare phenomenon occurs commonly in many domains of application, in particularly in disease mapping. We present new methodology to analyse such data, based on a hierarchical allocation model. We assume that the counts follow a Poisson model at the lowest level of the hierarchy, and introduce a finite mixture model for the Poisson rates at the next level. The novelty lies in the allocation model to the mixture components, which follows a spatially correlated process, the Potts model, and in treating the number of components of the spatial mixture as unknown. Inference is performed in a Bayesian framework using reversible jump MCMC. The model introduced can be viewed as a Bayesian semiparametric approach to specifying flexible spatial distribution in hierarchical models. It could also be used in contexts where the spatial mixture subgroups are themselves of interest, as in health care monitoring.

Performance of the model and comparison with an alternative well-known Markov random field model specification for the Poisson rates are demonstrated on synthetic data sets. We found that our allocation model avoids the problem of oversmoothing in cases where the underlying rates exhibit discontinuities, while giving equally good results in cases of smooth gradient-like or highly autocorrelated rates. The methodology is illustrated on epidemiological applications to data on rare disease and health outcome in France.

Some key words: Allocation, Bayesian hierarchical model, Disease mapping, Finite mixture distributions, Heterogeneity, Markov chain Monte Carlo, Poisson mixtures, Potts model, Reversible jump algorithms, Semiparametric model, Split/merge moves.
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