Variance Estimation for Weighted Average Treatment Effects.

Common variance estimation methods for weighted average treatment effects (WATEs) in observational studies include nonparametric bootstrap and model-based, closed-form sandwich variance estimation. However, the computational cost of bootstrap increases with the size of the data at hand. Besides, some replicates may exhibit random violations of the positivity assumption even when the original data do not. Sandwich variance estimation relies on regularity conditions that may be structurally violated. Moreover, the sandwich variance estimation is model-dependent on the propensity score model, the outcome model, or both; thus it does not have a unified closed-form expression. Recent studies have explored the use of wild bootstrap to estimate the variance of the average treatment effect on the treated (ATT). This technique adopts a one-dimensional, nonparametric, and computationally efficient resampling strategy. In this article, we propose a "post-weighting" bootstrap approach as an alternative to the conventional bootstrap, which helps avoid random positivity violations in replicates and improves computational efficiency. We also generalize the wild bootstrap algorithm from ATT to the broader class of WATEs by providing new justification for correctly accounting for sampling variability from multiple sources under different weighting functions. We evaluate the performance of all four methods through extensive simulation studies and demonstrate their application using data from the National Health and Nutrition Examination Survey (NHANES). Our findings offer several practical recommendations for the variance estimation of WATE estimators.