Uncertainty in Propensity Score Estimation: Bayesian Methods for Variable Selection and Model-Averaged Causal Effects

Abstract

Causal inference with observational data frequently relies on the notion of the propensity score (PS) to adjust treatment comparisons for observed confounding factors. As decisions in the era of “big data” are increasingly reliant on large and complex collections of digital data, researchers are frequently confronted with decisions regarding which of a high-dimensional covariate set to include in the PS model to satisfy the assumptions necessary for estimating average causal effects. Typically, simple or ad hoc methods are employed to arrive at a single PS model, without acknowledging the uncertainty associated with the model selection. We propose three Bayesian methods for PS variable selection and model averaging that (a) select relevant variables from a set of candidate variables to include in the PS model and (b) estimate causal treatment effects as weighted averages of estimates under different PS models. The associated weight for each PS model reflects the data-driven support for that model’s ability to adjust for the necessary variables. We illustrate features of our proposed approaches with a simulation study, and ultimately use our methods to compare the effectiveness of surgical versus nonsurgical treatment for brain tumors among 2606 Medicare beneficiaries. Supplementary materials for this article are available online.

KEY WORDS:

• Bayesian statistics
• Causal inference
• Comparative effectiveness
• Model averaging

SUPPLEMENTARY MATERIALS

• Web Appendix: A supplementary appendix with additional simulation studies, a complement to the analysis of Medicare data of Section 5, and a detailed account of the MCMC algorithm for the SSVS approach of Section 3.3.