Fixed Size Blocks of Varying Correlation Random Effects Model Auto-Regressive Model We select the “ice” model from among the examples provided with WinBUGS as an auto-regressive (AR; Harvey, 1993) example, which is also analyzed in Breslow and Clayton (1993). The data contains 77 incident counts of breast cancer occurring in Iceland, which are modeled as Poisson counts. Explanatory variables include age group, year of birth (represented using 11 cohorts ranging between 1840 and 1949), and the total person-years for the subjects in each group. The model uses second-order AR smoothing of birth cohort effects. We select the “litters” model from among the original example models provided with the MCMC package WinBUGS. This random effects model contains two groups of 16 binomial observations. Within each group, the binomial probabilities are modeled as random effects arising from a beta distribution. The particular parameterization of the beta distributions (in terms of α and β , rather than µ and σ ) results in strong correlations between each α$_{i}$, β$_{i}$ pair. The WinBUGS manual comments upfront that this model exhibits slow mixing. We consider an informed MCMC algorithm, which blocks each α$_{i}$, β$_{i}$ pair. In addition, the beta-binomial conjugacy relationships permit use of cross-level sampling, where we jointly sample top-level parameters and conjugate latent states, as used by Rue and Held (2005, p.141-143). The next model structure exhibits fixed size groupings of parameters, with posterior correlations ranging between 0 and 0.9. Each such model contains N = 10 n parameters. Again employing multivariate normal distributions, we induce nine disjoint groupings of n parameters each, having intra-group pairwise correlations of 0.1, 0.2, . . . , and 0.9. The remaining n parameters are fully uncorrelated. Three such models of this structure are constructed, using the values n = 2, 5, and 10. As in the previous models, these do not include any likelihood term. 19