Doubly robust nonparametric inference on the average treatment effect

Abstract

Doubly robust estimators are widely used to draw inference about the average effect of a treatment. Such estimators are consistent for the effect of interest if either one of two nuisance parameters is consistently estimated. However, if flexible, data-adaptive estimators of these nuisance parameters are used, double robustness does not readily extend to inference. We present a general theoretical study of the behaviour of doubly robust estimators of an average treatment effect when one of the nuisance parameters is inconsistently estimated. We contrast different methods for constructing such estimators and investigate the extent to which they may be modified to also allow doubly robust inference. We find that while targeted minimum loss-based estimation can be used to solve this problem very naturally, common alternative frameworks appear to be inappropriate for this purpose. We provide a theoretical study and a numerical evaluation of the alternatives considered. Our simulations highlight the need for and usefulness of these approaches in practice, while our theoretical developments have broad implications for the construction of estimators that permit doubly robust inference in other problems.

Keywords: Adaptive estimation; Doubly robust estimation; Efficient influence function; Targeted minimum loss-based estimation.

Fig. 1.

Simulation results when only the outcome regression is consistently estimated, with the following performance measures plotted against the sample size : (a) bias; (b) bias; (c) coverage of 95% confidence intervals; (d) accuracy of the standard error estimator. Squares represent estimators that do not account for inconsistent nuisance parameter estimation, circles represent estimators using the bivariate correction of van der Laan (2014), and triangles represent estimators using the proposed univariate corrections.

Fig. 2.

Simulation results when only the propensity score is consistently estimated: (a) bias, (b) bias, (c) coverage of 95% confidence intervals, and (d) accuracy of the standard error estimator plotted against . Squares represent estimators that do not account for inconsistent nuisance parameter estimation, circles represent estimators using the bivariate correction of van der Laan (2014), and triangles represent estimators using the proposed univariate corrections.

Fig. 3.

Simulation results when both the outcome regression and the propensity score are consistently estimated: (a) bias, (b) bias, (c) coverage of 95% confidence intervals, and (d) accuracy of the standard error estimator plotted against . Squares represent estimators that do not account for inconsistent nuisance parameter estimation, circles represent estimators using the bivariate correction of van der Laan (2014), and triangles represent estimators using the proposed univariate corrections.