# Efficient and Robust Causal Inference: A Distributional Approach

#### Abstract

Drawing inferences about the effects of exposures or treatments is a common challenge in many scientific fields. We propose two methods serving complementary purposes in causal inference. One can be used to estimate average causal effects, assuming ``no confounding" given measured covariates. The other can be used to assess the sensitivity of the estimates to possible departures from "no confounding." We establish asymptotic results for the methods, and also address practical issues in planning data analysis, checking propensity score models, and interpreting sensitivity parameters. Both methods are developed from a nonparametric likelihood perspective. The propensity score plays a central role, and is estimated through a parametric model. Under "no confounding," the joint distribution of the 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, as regression-adjusted estimates. The likelihood and regression estimators can be more efficient and more robust than those in the literature. Without "no confounding," the marginal distribution of the covariates times the conditional distribution of the observed outcome given each treatment assignment and the covariates is estimated. For a fixed bound on unmeasured confounding, the marginal distribution of the covariates times the conditional distribution of the counterfactual outcome given each treatment assignment and the covariates is explored to its extreme, and is compared with the corresponding distribution that involves the observed outcome given the same treatment assignment. We illustrate the methods by analyzing the data from an observational study on right heart catheterization.

*This paper has been withdrawn.*