NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET Modelling spatial variation in disease risk using Gaussian Markov random field proxies for Gaussian random fields

Analyses of spatial variation in disease risk based on area-level summaries of disease counts are most often based on the assumption that the relative risk is uniform across each region. Such approaches introduce an artificial piecewise-constant relative risk-surface with discontinuities at regional boundaries. A more natural approach is to assume that the spatial variation in risk can be represented by an underlying smooth relative risk-surface over the area of interest. This approach was taken by Kelsall and Wakefield (2002), who used an underlying Gaussian random field (GRF) to derive a multivariate log-Normal distribution of the risk at the regional level. The derivation rely on the approximation , which is frequently used in similar contexts in the geostatistics literature, but the different sizes and shapes of the regions typically found in disease mapping applications indicate that the validity of the approximation is questionable. We propose an approach to the modelling of a smoothly varying risk surface based on aggregated data avoiding this approximation. We also derive computationally efficient block MCMC-algorithms using a re-formulation of the geostatistical GRF model using Gaussian Markov random fields (GMRFs). We make extensive use of recent developments for GMRFs, including a method for fitting GMRFs to Gaussian random fields and computationally efficient algorithms for GMRFs based on numerical methods for sparse matrices. We demonstrate our approach on simulated data as well as a set of German oral cavity cancer mortality data from the period – , which have been previously analysed in the literature.