Random Voronoi tesselations for mapping residual spatial variation in disease risk based upon an 'individual-level' matched case-control design Seminar

Abstract
Methods for exploring geographical heterogeneity in health outcomes continue to motivate much research. Sub-dynamics of any atypical regions can be used to guide resource allocation, generate aetiological hypotheses and motivate further research. In general, variation arising as a consequence of 'known' spatially varying risk factors, such as measures of socio-economic status and age, is not of intrinsic interest and the ability to accommodate known confounders is paramount. In addition, models which are flexible and make few assumptions about the underlying functional form are sought after. Assuming individual-level geo-referenced health data (cases and controls) confounding can be handled at the analysis stage, by means of covariate adjustment, or alternatively, at the design stage by using the confounders as stratifying factors for selecting case-matched controls. A consequence of this design-based approach is that the matching needs to be accommodated in the analysis which in turn results in consistency issues since the number of parameters increases with the sample size. In general, analysis thus proceeds on the basis of the matched, conditional likelihood (Breslow and Day, 1980) in which the 'nuisance' parameters are eliminated. This work presents a Bayesian partition model formulation for the analysis of matched case-control data. The model assumes that the region of interest can be decomposed in to a number of disjoint regions in which the residual disease risk has a constant mean. Random Voronoi tessellations are used to decompose the region of interest and the Mulitinomial-Poisson model formulation, which yiels a likelihood which is proportional to the matched conditional likelihood, is used to ease the computational burden and establish the modelling as a Bayesian generalised linear modelling problem. Key features of the approach include a relaxation of the customary assumption of a stationary, isotropic, covariance structure, and moreover, the ability to detect spatial discontinuity. Reversible Jump MCMC (Green, 1995) is used to sample from the posterior distribution; Bayesian least squares (Gamerman, 1996) and re-parameterisation is performed to provide efficient means of candidate generation. The Poisson model formulation maintains the conditional independence structure underpinning the partitioning idea and negates the need to monitor the locations of, and compute covariate differences between, the case and the matched controls in each group, at each iterative step. The methodology is also generalised to handle additional 'non-matched-for' covariate information. Simulations demonstrated the capacity to recover known smooth and discontinuous risk surfaces and the estimated covariate effects were consistent with the underlying truth. The methodology is demonstrated on matched perinatal mortality data derived in the North-West Thames region of the UK. Infants were matched on the basis of sex and date-of-birth. A measure of social deprivation, Carstairs' index, known to be associated with infant death, was available and was included in the analysis. As anticipated, social deprivation was found to be a highly significant predictor of perinatal mortality. The unadjusted analysis (not accounting for the Carstairs' index) resulted in surface estimates which exhibited regions of atypical risk. When adjusted for Carstairs' index, however, there was no evidence of residual spatial variation in perinatal risk. Breslow, N.E. and Day, N.E. (1980). Statistical methods in cancer research. Vol 1. The analysis of case-control studies IARC, Scientific publications No. 32. Gamerman, D. (1996). Sampling from the posterior distribution in generalised linear mixed models. Statistics and Computing, 7, 57-68. Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82,4, pp. 711-731.