Presenter: Kori Khan, Assistant Professor, Department of Statistics, Iowa State University

Title: On Sum-to-Zero Constraints in Intrinsic Conditional Autoregressions 

Abstract: Intrinsic conditional autoregressions (ICARs) are widely used in myriad spatial statistic applications. In this presentation, I focus particularly on one specific model, often referred to as the “BYM” model. This model is often used when the primary inferential goal is to explore the relationship between a spatially aggregated (count) response variable and a spatially aggregated exposure (aka explanatory variable). Thus, the focus is typically on regression coefficients of the fixed effects in the model. For Bayesian applications, this means the focus is on the marginal posterior distribution of such regression coefficients. 

In Bayesian applications, ICAR models, and the BYM model specifically, incorporate spatial dependence by including a random effect in a regression model. This random effect is assumed to have the improper ICAR prior, which accounts for spatial dependence with a “precision” matrix. When intercepts are included in fixed effect, models with the ICAR prior will have identifiability issues. This is a direct consequence of the kernel of the precision matrix in the ICAR prior. Existing software handles this issue by enforcing sum-to-zero constraints when sampling the components of the random effect in the MCMC. There are two popular ways to do this: 1) “centering on the fly”, and 2) a sparsity-influenced approach. While the approaches are often used interchangeably, there is some (not rigorous) evidence that the two approaches may not result in the same marginal posterior distribution for regression coefficients. 

 This presentation is on a work very much in progress exploring 1) whether the marginal posterior distributions for regression coefficients are invariant to the choice of method of sum-to-zero constraints, 2) if not, whether the difference “matters” in any practical settings.