Disease Mapping With Generative Models

Disease mapping focuses on learning about areal units presenting high relative risk. Disease mapping models assume that the disease counts are distributed as Poisson random variables with the respective means typically specified as the product of the relative risk and the expected count. These models usually incorporate spatial random effects to accomplish spatial smoothing of the relative risks. Fitting of these models often computes expected disease counts via internal standardization. This places the data on both sides of the model, that is, the counts are on the left side but they are also used to obtain the expected counts on the right side. As a result, these internally standardized models are incoherent and not generative; probabilistically, they could not produce the data we observe. Here, we argue for adopting the direct generative model for disease counts, modeling disease incidence rates instead of relative risks, using a generalized logistic regression. Then, the relative risks are then extracted post model fitting. We first demonstrate the benefit of the generative model without incorporating spatial smoothing using simulation. Then, spatial smoothing is introduced using the customary conditionally autoregressive model. We also extend the generative model to dynamic settings. The generative models are compared with internally standardized models, again through simulated datasets but also through a well-examined lung cancer morbidity dataset in Ohio. Both models are spatial and both smooth the data similarly with regard to relative risks. However, the generative coherent models tend to provide tighter credible intervals. Since the generative specification is coherent, is at least as good inferentially, and is no more difficult to fit, we suggest that it should be the model of choice for spatial disease mapping.