Conditional Autoregressive models for disconnected graph and applications

Bayesian Methods in disease mapping are widely used to estimate and smooth relative risk. Usually, the underlying maps are defined using an adjacency matrix based on the spatial neighbourhood of areal units, de facto defining connected graphs. In the presence of islands or discontinuous geographical regions, disconnected graphs are created. Currently, these issues are solved by assigning the singletons to the closest areal unit and then the usual smoothing techniques are carried out to produce the relative risk maps. This leads to incorrect relative risk estimates and results in overfitting. But we define a scaled version for the intrinsic Conditional autoregressive model on a map with islands and provide a clear and unambiguous definition of the parameters and hyperparameters. To account for the islands, we use a scaling option so the precision has the same interpretation regardless of the particular structure of the map. This immediately suggests a fair prior for random effects associated with the islands in a disconnected graph in terms of a normal distribution. These improvements have been implemented in R-INLA.