Collaborative double robustness using the e-score
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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 high-dimensional pre-treatment covariate, but positivity is less likely in analyses with more predictors of treatment. 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 recent 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.
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