  Disease mapping focuses on learning about areal units presenting high
relative risk. Disease mapping models for disease counts specify Poisson
regressions in relative risks compared with the expected counts. These models
typically incorporate spatial random effects to accomplish spatial smoothing.
Fitting of these models customarily computes expected disease counts via
internal standardization. This places the data on both sides of the model,
i.e., 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 observed data. Here, we argue for adopting the direct generative
model for disease counts. We model disease incidence instead of relative risks,
using a generalized logistic regression. We extract relative risks post model
fitting. We also extend the generative model to dynamic settings. We compare
the generative models with internally standardized models through simulated
datasets and a well-examined lung cancer morbidity data in Ohio. Each model is
a spatial smoother and they smooth the data similarly with regard to relative
risks. However, the generative models tend to provide tighter credible
intervals. Since the generative specification is no more difficult to fit, is
coherent, and is at least as good inferentially, we suggest it should be the
model of choice for spatial disease mapping.
