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

Aditya Ghosh, Nabarun Deb, Bikram Karmakar, Bodhisattva Sen

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.

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

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.
Paper Structure (60 sections, 47 theorems, 402 equations, 1 figure, 12 tables)

This paper contains 60 sections, 47 theorems, 402 equations, 1 figure, 12 tables.

Table of Contents

  1. Introduction
  2. Summary of our contributions
  3. Inference without regression adjustment
  4. Rosenbaum's estimator based on the WRS statistic
  5. Robustness of Rosenbaum's estimator
  6. Asymptotic distribution of Rosenbaum's estimator
  7. Efficiency of Rosenbaum's estimator
  8. Inference with regression adjustment
  9. Asymptotic distribution of Rosenbaum's regression-adjusted estimator
  10. Efficiency gain by regression adjustment
  11. Numerical experiments
  12. Rosenbaum's estimator under treatment heterogeneity
  13. Discussion
  14. Appendices
  15. Additional results
  16. ...and 45 more sections

Key Result

proposition 1

Let $t_N:=t(\vec{Z}_N, \vec{Y}_N - \tau_0\vec{Z}_N)$ be the WRS statistic for a sample of size $N$, with $t(\cdot,\cdot)$ as defined in WRS_N. Suppose that cterandomization hold, and that none of the ranks in null-upranks dominates all the others, that is, where $\overline{q}_N^{(\tau_0)} := N^{-1}\sum_{j=1}^N q_{j}^{(\tau_0)}$. Then, under $\cte=\tau_0$,

Figures (1)

  • Figure A.1: Boxplots of the ratios of the plug-in estimators of asymptotic variances of $\rsnu$ (left panel) and $\rsna$ (right panel) for $\nu = 1/3$ and various other values of $\nu$, in simulation Setting 4(a) (see \ref{['subsec:simulationsApp']} for details on the simulation setup).

Theorems & Definitions (109)

  • proposition 1: Asymptotic null distribution of $t_N$
  • remark 1: Choice of the tie-breaking method
  • definition 1: Breakdown point
  • remark 2: Implication of our notion of asymptotic breakdown point for confidence intervals
  • theorem 1: Asymptotic breakdown point of $\rsnu$
  • remark 3: On \ref{['ACjs']}
  • theorem 2: Local asymptotic normality of $t_N$
  • theorem 3: CLT for the estimator $\rsnu$
  • definition 2: Asymptotic relative efficiency
  • theorem 4: Efficiency lower bound
  • ...and 99 more