Double robust variance estimation with parametric working models

Abstract

Doubly robust estimators have gained popularity in the field of causal inference due to their ability to provide consistent point estimates when either an outcome or an exposure model is correctly specified. However, for nonrandomized exposures, the influence function based variance estimator frequently used with doubly robust estimators of the average causal effect is only consistent when both working models (ie, outcome and exposure models) are correctly specified. Here, the empirical sandwich variance estimator and the nonparametric bootstrap are demonstrated to be doubly robust variance estimators. That is, they are expected to provide valid estimates of the variance leading to nominal confidence interval coverage when only 1 working model is correctly specified. Simulation studies illustrate the properties of the influence function based, empirical sandwich, and nonparametric bootstrap variance estimators in the setting where parametric working models are assumed. Estimators are applied to data from the Improving Pregnancy Outcomes with Progesterone (IPOP) study to estimate the effect of maternal anemia on birth weight among women with HIV.

Keywords: M-estimation; augmented inverse probability weighting; causal inference; double robustness; empirical sandwich variance.

FIGURE 1

95% confidence interval (CI) half-widths by model specification and estimator. FC = full covariate set; NO = naive outcome model; NP = naive propensity model.

FIGURE 2

Ratio between each simulation’s estimated standard error and the empirical standard error by estimator and model specification, continuous outcome, n = 800, , 5000 simulations. was approximately −60. Black squares denote the mean variance ratio (=SER). Results exclude 1 simulation where models failed to converge. The 0.33% of correct model specification simulations, 4.02% of misspecified outcome model simulations, and 0.004% of misspecified propensity model simulations where the ratio was above 1.2 or below 0.8 are not displayed.

FIGURE 3

Ratio between each simulation’s estimated standard error and the empirical standard error by estimator and model specification, continuous outcome, n = 800, , 5000 simulations under the null. Black squares denote the mean variance ratio (=SER). The 0.01% of correct model specification simulations and 1.72% of misspecified outcome model simulations where the ratio was above 2.75 or below 0.5 are not displayed.