Modelling multivariate disease rates with a latent structure mixture model

There has been considerable recent interest in multivariate modelling of the geographical distribution of morbidity or mortality rates for potentially related diseases. The motivations for this include investigation of similarities or dissimilarities in the risk distribution for the different diseases, as well as ‘borrowing strength’ across disease rates to shrink the uncertainty in geographical risk assessment for any particular disease. A number of approaches to such multivariate modelling have been suggested and this paper proposes an extension to these which may provide a richer range of dependency structures than those encompassed so far. We develop a model which incorporates a discrete mixture of latent structures and argue that this provides potential to represent an enhanced range of correlation structures between diseases at the same time as implicitly allowing for less restrictive spatial correlation structures between geographical units. We compare and contrast our approach to other commonly used multivariate disease models and demonstrate comparative results using data taken from cancer registries on four carcinomas in some 300 geographical units in England, Scotland and Wales.