Variance reduction in the inverse probability weighted estimators for the average treatment effect using the propensity score

The propensity methodology is widely used in medical research to compare different treatments in designs with a nonrandomized treatment allocation. The inverse probability weighted (IPW) estimators are a primary tool for estimating the average treatment effect but the large variance of these estimators is often a significant concern for their reliable use in practice. Inspired by Rao-Blackwellization, this paper proposes a method to smooth an IPW estimator by replacing the weights in the original estimator by their mean over a distribution of the potential treatment assignment. In our simulation study, the smoothed IPW estimator achieves a substantial variance reduction over its original version with only a small increased bias, for example two-to-sevenfold variance reduction for the three IPW estimators in Lunceford and Davidian [Statistics in Medicine, 23(19), 2937-2960]. In addition, our proposed smoothing can also be applied to the locally efficient and doubly robust estimator for added protection against model misspecification. An implementation in R is provided.