Bayesian Subgroup Analysis with Hierarchical Models

Abstract

Estimates of treatment effects within subgroups tend to have too much variation relative to the true treatment effects, hampering clinical interpretation. To address this unwanted variation, we consider Bayesian hierarchical modeling of subgroups. When considered exchangeable in a one-way hierarchical structure, the subgroup treatment effects have posterior means that shrink the sample estimates toward the overall estimate. The amount of shrinkage increases with decreasing evidence of treatment effect heterogeneity, as measured by the variation between relative to within the subgroups. Because the shrinkage estimates borrow strength from the overall estimate, Bayesian subgroup analysis is more precise than separate analyses of the sample treatment effects. To illustrate, we re-analyze race subgroups in the LIFE study comparing hypertension treatments losartan and atenolol on time to first major adverse cardiac event. We also present exchangeability structures for two or more subgrouping factors. Aims of the chapter are to provide a basic understanding of the behavior of Bayesian hierarchical models for subgroup analysis and how to implement them. To facilitate understanding, formulas for Bayesian posterior means, interval estimates and hypothesis tests are presented and interpreted when the parameters of a hierarchical model are known. For implementation, we provide an Appendix listing code for analysis of the LIFE study with empirical Bayes and more fully Bayes approaches. The discussion section includes selected bibliographic notes on Bayesian hierarchical subgroup analysis and computation.