Cutting Feedback in Bayesian Regression Adjustment for the Propensity Score

McCandless, Gustafson and Austin (2009) describe a Bayesian approach to regression adjustment for the propensity score to reduce confounding. A unique property of the method is that the treatment and outcome models are combined via Bayes theorem. However, this estimation procedure can be problematic if the outcome model is misspecified. We observe feedback that can bias propensity score estimates. Building on new innovation in Bayesian computation, we propose a technique for cutting feedback in a Bayesian propensity analysis. We use the posterior distribution of the propensity scores as an input in the regression model for the outcome. The method is approximately Bayesian in the sense that it does not use the full likelihood for estimation. Nonetheless, it severs feedback between the treatment and outcome giving propensity score estimates that are free from bias but modeled with uncertainty. We illustrate the method in a matched cohort study investigating the effect of statins on primary stroke prevention.