On variance of the treatment effect in the treated using inverse probability weighting

In the analysis of observational studies, inverse probability weighting (IPW) is commonly used to consistently estimate the average treatment effect (ATE) or the average treatment effect in the treated (ATT). The variance of the IPW ATE estimator is often estimated by assuming the weights are known and then using the so-called "robust" (Huber-White) sandwich estimator, which results in conservative standard error (SE) estimation. Here it is shown that using such an approach when estimating the variance of the IPW ATT estimator does not necessarily result in conservative SE estimates. That is, assuming the weights are known, the robust sandwich estimator may be conservative or anti-conservative. Thus confidence intervals of the ATT using the robust SE estimate will not be valid in general. Instead, stacked estimating equations which account for the weight estimation can be used to compute a consistent, closed-form variance estimator for the IPW ATT estimator. The two variance estimators are compared via simulation studies and in a data analysis of the effect of smoking on gene expression.