Robustness and Efficiency of Rosenbaum's Rank-based Estimator in Randomized Trials: A Design-based Perspective

TL;DR

This paper develops a theoretical framework for Rosenbaum's rank-based estimator in randomized trials, demonstrating its robustness to outliers and comparing its efficiency to mean-based methods, especially with heavy-tailed residuals.

Contribution

It introduces a design-based asymptotic theory for Rosenbaum's estimator, analyzing its robustness, efficiency, and behavior under covariate adjustment and potential treatment effect heterogeneity.

Findings

Rosenbaum's estimator is highly robust against outliers.

Regression adjustment with Rosenbaum's estimator is often more efficient with heavy-tailed residuals.

The estimator's efficiency is within 13.6% of mean-based methods under mild conditions.

Abstract

Mean-based estimators of causal effects in randomized experiments may behave poorly if the potential outcomes have a heavy tail or contain outliers. An alternative estimator proposed by Rosenbaum (1993) estimates a constant additive treatment effect by inverting a randomization test using ranks. We develop a design-based asymptotic theory for this rank-based estimator and study its robustness and efficiency properties. We show that Rosenbaum's estimator is robust against outliers with a breakdown point that uniformly dominates that of any weighted quantile estimator. When pretreatment covariates are available, a regression-adjusted version of Rosenbaum's estimator uses an agnostic linear regression on the covariates and bases inference on the ranks of residuals. Under mild integrability conditions, we show that this estimator is at most 13.6% less efficient, in the worst case, than the commonly used mean-based regression adjustment method proposed by Lin (2013); often outperforming it when the residuals have heavy tails. Moreover, under suitable assumptions, Rosenbaum's regression-adjusted estimator is at least as efficient as the unadjusted one. Finally, we initiate the study of Rosenbaum's estimator when the constant treatment effect assumption may be violated. To analyze the regression-adjusted estimator, we develop local asymptotics of rank statistics under the design-based framework, which may be of independent interest.