Bayesian non-parametric models for spatially indexed data of mixed type

Abstract

We develop Bayesian non-parametric models for spatially indexed data of mixed type. Our work is motivated by challenges that occur in environmental epidemiology, where the usual presence of several confounding variables that exhibit complex interactions and high correlations makes it difficult to estimate and understand the effects of risk factors on health outcomes of interest. The modelling approach that we adopt assumes that responses and confounding variables are manifestations of continuous latent variables and uses multivariate Gaussian distributions to model these jointly. Responses and confounding variables are not treated equally as relevant parameters of the distributions of the responses only are modelled in terms of explanatory variables or risk factors. Spatial dependence is introduced by allowing the weights of the non-parametric process priors to be location specific, obtained as probit transformations of Gaussian Markov random fields. Confounding variables and spatial configuration have a similar role in the model, in that they influence, along with the responses, only the allocation probabilities of the areas into the mixture components, thereby allowing for flexible adjustment of the effects of observed confounders, while allowing for the possibility of residual spatial structure, possibly occurring because of unmeasured or undiscovered spatially varying factors. Aspects of the model are illustrated in simulation studies and an application to a real data set.