Doubly robust estimators for the average treatment effect under positivity violations: introducing the $e$-score

Abstract: Estimation of causal parameters from observational data requires complete confounder adjustment, as well as positivity of the propensity score for each treatment arm. There is often a trade-off between these two assumptions: confounding bias may be reduced through adjustment for a large number of pre-treatment covariates, but positivity is less likely in analyses with irrelevant predictors of treatment such as instrumental variables. Under empirical positivity violations, propensity score weights are highly variable, and doubly robust estimators suffer from high variance and large finite sample bias. To solve this problem, we introduce the $e$-score, which is defined through a dimension reduction for the propensity score. This dimension reduction is based on a result known as collaborative double robustness, which roughly states that a propensity score conditioning only on the bias of the outcome regression estimator is sufficient to attain double robustness. We propose methods to construct doubly robust estimators based on the $e$-score, and discuss their properties such as consistency, efficiency, and asymptotic distribution. This allows the construction of asymptotically valid Wald-type confidence intervals and hypothesis tests. We present an illustrative application on estimating the effect of smoking on bone mineral content in adolescent girls well as a synthetic data simulation illustrating the bias and variance reduction and asymptotic normality achieved by our proposed estimators.