A Distributional Approach for Causal Inference Using Propensity Scores

Drawing inferences about the effects of treatments and actions is a common challenge in economics, epidemiology, and other fields. We adopt Rubin’s potential outcomes framework for causal inference and propose two methods serving complementary purposes. One can be used to estimate average causal effects, assuming no confounding given measured covariates. The other can be used to assess how the estimates might change under various departures from no confounding. Both methods are developed from a nonparametric likelihood perspective. The propensity score plays a central role and is estimated through a parametric model. Under the assumption of no confounding, the joint distribution of covariates and each potential outcome is estimated as a weighted empirical distribution. Expectations from the joint distribution are estimated as weighted averages or, equivalently to first order, regression estimates. The likelihood estimator is at least as efficient and the regression estimator is at least as efficient and robust as existing estimators. Regardless of the no-confounding assumption, the marginal distribution of covariates times the conditional distribution of observed outcome given each treatment assignment and covariates is estimated. For a fixed bound on unmeasured confounding, the marginal distribution of covariates times the conditional distribution of counterfactual outcome given each treatment assignment and covariates is explored to the extreme and then compared with the composite distribution corresponding to observed outcome given the same treatment assignment and covariates. We illustrate the methods by analyzing the data from an observational study on right heart catheterization.