Smaller Confidence Intervals From IPW Estimators via Data-Dependent Coarsening

TL;DR

This paper introduces Coarse IPW (CIPW) estimators that produce smaller, more robust confidence intervals in causal inference, even with inaccurate propensity scores, by coarsening covariate spaces.

Contribution

The paper proposes a data-dependent coarsening approach to create robust IPW estimators with confidence intervals that scale favorably with sample size and score inaccuracies.

Findings

CIPW estimators achieve confidence interval sizes of O(ε + 1/√n).

Existing estimators' confidence intervals remain large regardless of data accuracy.

The proposed method is efficient and robust under mild assumptions.

Abstract

Inverse propensity-score weighted (IPW) estimators are prevalent in causal inference for estimating average treatment effects in observational studies. Under unconfoundedness, given accurate propensity scores and $n$ samples, the size of confidence intervals of IPW estimators scales down with $n$, and, several of their variants improve the rate of scaling. However, neither IPW estimators nor their variants are robust to inaccuracies: even if a single covariate has an $\varepsilon>0$ additive error in the propensity score, the size of confidence intervals of these estimators can increase arbitrarily. Moreover, even without errors, the rate with which the confidence intervals of these estimators go to zero with $n$ can be arbitrarily slow in the presence of extreme propensity scores (those close to 0 or 1). We introduce a family of Coarse IPW (CIPW) estimators that captures existing IPW estimators and their variants. Each CIPW estimator is an IPW estimator on a coarsened covariate space, where certain covariates are merged. Under mild assumptions, e.g., Lipschitzness in expected outcomes and sparsity of extreme propensity scores, we give an efficient algorithm to find a robust estimator: given $\varepsilon$-inaccurate propensity scores and $n$ samples, its confidence interval size scales with $\varepsilon+1/\sqrt{n}$. In contrast, under the same assumptions, existing estimators' confidence interval sizes are $\Omega(1)$ irrespective of $\varepsilon$ and $n$. Crucially, our estimator is data-dependent and we show that no data-independent CIPW estimator can be robust to inaccuracies.