Posterior distribution of hierarchical models using CAR(1) distributions

Abstract

We examine properties of the CAR1 model, which is commonly used to represent regional e ects in Bayesian analyses of mortality rates. We consider a Bayesian hierarchical linear mixed model where the xed e ects have a v ague prior such a s a constant prior and the random e ect follows a class of CAR1 models including those whose joint prior distribution of the regional e ects is improper. We give su cient conditions for the existence of the posterior distribution of the xed and random e ects and variance components. We then prove the necessity of the conditions and give a one-way analysis of variance example where the posterior may o r m a y not exist. Finally, w e extend the result to the generalised linear mixed model, which includes as a special case the Poisson log-linear model commonly used in disease mapping.

Bayesian Factor Analysis for Spatially Correlated Data, With Application to Summarizing Area-Level Material Deprivation From Census Data

“…The matrix 9 of spatial correlations can be parameterized either marginally (Cressie 1993;Conlon and Waller 1998) or conditionally (Besag 1974;Sun, Tsutakawa, and Speckman 1999). We use both approaches in our application.…”

“…Sun et al (1999) described a rather general structure for Gaussian CAR models. Let R i denote the set of indices for areas that are neighbors of area i; typically, but not necessarily, neighbors are de ned by adjacency.…”

“…then the joint marginal distributionof ± D .± 1 ; : : : ; ± N / T follows N.0; ºB ¡1 /, where B is an N £ N matrix with f® 1 ; : : : ; ® N g along the diagonal and ¡® i¯ij on the off-diagonal, provided that B is symmetric and positive de nite (Besag 1974;Sun et al 1999). A simple and common parameterization of the CAR model gives some insight about parameter interpretation.…”

Modeling Area-Level Health Rankings

“…We use the conditional autoregressive specification (Besag, 1974;Sun et al, 1999), which produces a tractable relationship between the conditional and the marginal specifications. Hogan and Tchernis (2004) show that this specification performs well relative to several alternatives.…”

Phylogenetic Mapping of Recombination Hotspots in Human Immunodeficiency Virus via Spatially Smoothed Change-Point Processes

“…In the context of small area estimation, Ghosh et al (1998) and Sun et al (1999) show that despite the singularity of matrix Q such autoregressive priors lead to a proper posterior distribution under mild conditions on the model likelihood function. Our ''pseudolikelihood'' (7) does not satisfy these conditions when C s ¼ 0 for all s, or when C s ¼ T s for all s. Although very unlikely, such values of recombination counts do have strictly positive probability mass a posteriori.…”