  Common causal estimands include the average treatment effect (ATE), the
average treatment effect of the treated (ATT), and the average treatment effect
on the controls (ATC). Using augmented inverse probability weighting methods,
parametric models are judiciously leveraged to yield doubly robust estimators,
i.e., estimators that are consistent when at least one the parametric models is
correctly specified. Three sources of uncertainty are associated when we
evaluate these estimators and their variances, i.e., when we estimate the
treatment and outcome regression models as well as the desired treatment
effect.
  In this paper, we propose methods to calculate the variance of the
normalized, doubly robust ATT and ATC estimators and investigate their finite
sample properties. We consider both the asymptotic sandwich variance
estimation, the standard bootstrap as well as two wild bootstrap methods. For
the asymptotic approximations, we incorporate the aforementioned uncertainties
via estimating equations. Moreover, unlike the standard bootstrap procedures,
the proposed wild bootstrap methods use perturbations of the influence
functions of the estimators through independently distributed random variables.
We conduct an extensive simulation study where we vary the heterogeneity of the
treatment effect as well as the proportion of participants assigned to the
active treatment group. We illustrate the methods using an observational study
of critical ill patients on the use of right heart catherization.
