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Title: Conditional autoregressive models for disconnected graph Authors:  Anna Freni Sterrantino - Imperial College London (United Kingdom) [presenting]
Haavard Rue - KAUST (Saudi Arabia)
Massimo Ventrucci - University of Bologna (Italy)
Abstract: Conditional autoregressive (CAR) distributions are widely used in disease mapping to created smoothed relative risk maps. The underlying maps are defined using an adjacency matrix based on the spatial neighborhood of areal units, de facto defining connected graphs, but in presence of islands or discontinue geographical regions disconnected graphs are created. Currently, if there are islands in the map (singletons) usually are connected to the closest areal unit or a constant prior is assigned for the singleton making it difficult for the singleton random effect to shrink to the global mean. Additionally, if the map has several connected the marginal deviation from its (component) mean, depends on the graph, and will, in general, be different for each connected component. To solve the issues we scale the precision matrix of the CAR, in detail we scale each connected component of size larger than one, independently and for connected components of size one, instead of setting the singleton random effect equal to zero, we replace it with a standard Gaussian with precision. This scaling gives a well-defined interpretation of the precision and the same typical (conditional) marginal variance within each connected component, no matter the size of the map.