Evaluating the informativeness of the Besag-York-Mollié CAR model

The use of the conditional autoregressive framework proposed by Besag, York, and Mollié (1991; BYM) is ubiquitous in Bayesian disease mapping and spatial epidemiology. While it is understood that Bayesian inference is based on a combination of the information contained in the data and the information contributed by the model, quantifying the contribution of the model relative to the information in the data is often non-trivial. Here, we provide a measure of the contribution of the BYM framework by first considering the simple Poisson-gamma setting in which quantifying the prior's contribution is quite clear. We then propose a relationship between gamma and lognormal priors that we then extend to cover the framework proposed by BYM. Following a brief simulation study in which we illustrate the accuracy of our lognormal approximation of the gamma prior, we analyze a dataset comprised of county-level heart disease-related death data across the United States. In addition to demonstrating the potential for the BYM framework to correspond to a highly informative prior specification, we also illustrate the sensitivity of death rate estimates to changes in the informativeness of the BYM framework.