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Design-based inference for generalized causal effects in randomized experiments

Xinyuan Chen, Fan Li

Abstract

Generalized causal effect estimands, including the Mann-Whitney parameter and causal net benefit, provide flexible summaries of treatment effects in randomized experiments with non-Gaussian or multivariate outcomes. We develop a unified design-based inference framework for regression adjustment and variance estimation of a broad class of generalized causal effect estimands defined through pairwise contrast functions. Leveraging the theory of U-statistics and finite-population asymptotics, we establish the consistency and asymptotic normality of regression estimators constructed from individual pairs and per-unit pair averages, even when the working models are misspecified. Consequently, these estimators are model-assisted rather than model-based. In contrast to classical average treatment effect estimands, we show that for nonlinear contrast functions, covariate adjustment preserves consistency but does not admit a universal efficiency guarantee. For inference, we demonstrate that standard heteroskedasticity-robust and cluster-robust variance estimators are generally inconsistent in this setting. As a remedy, we prove that a complete two-way cluster-robust variance estimator, which fully accounts for pairwise dependence and reverse comparisons, is consistent.

Design-based inference for generalized causal effects in randomized experiments

Abstract

Generalized causal effect estimands, including the Mann-Whitney parameter and causal net benefit, provide flexible summaries of treatment effects in randomized experiments with non-Gaussian or multivariate outcomes. We develop a unified design-based inference framework for regression adjustment and variance estimation of a broad class of generalized causal effect estimands defined through pairwise contrast functions. Leveraging the theory of U-statistics and finite-population asymptotics, we establish the consistency and asymptotic normality of regression estimators constructed from individual pairs and per-unit pair averages, even when the working models are misspecified. Consequently, these estimators are model-assisted rather than model-based. In contrast to classical average treatment effect estimands, we show that for nonlinear contrast functions, covariate adjustment preserves consistency but does not admit a universal efficiency guarantee. For inference, we demonstrate that standard heteroskedasticity-robust and cluster-robust variance estimators are generally inconsistent in this setting. As a remedy, we prove that a complete two-way cluster-robust variance estimator, which fully accounts for pairwise dependence and reverse comparisons, is consistent.
Paper Structure (21 sections, 13 theorems, 23 equations, 1 figure, 4 tables)

This paper contains 21 sections, 13 theorems, 23 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Regression estimators for generalized causal effect
  3. Notation and framework
  4. Regression adjustment using individual pairs
  5. Regression adjustment using per-unit pair averages
  6. Asymptotic properties of regression estimators
  7. Preliminaries
  8. Regression estimators using individual pairs
  9. Regression estimators using per-unit pair averages
  10. Estimating the causal net benefit
  11. Estimators from previous regression models
  12. Estimators from the probabilistic index models
  13. Variance estimation for regression estimators
  14. Estimators using individual pairs
  15. Estimators using per-unit pair averages
  16. ...and 6 more sections

Key Result

Theorem 1

Under complete randomization and Conditions cd:randomization-and-dimension and cd:outcome-covariates-order, $\widehat{\lambda}_\mathrm{I}(a,1-a)=\lambda(a,1-a)+o_\mathbb{P}(1)$; if further $V\{\overline\epsilon_i(a,1-a)\}\nrightarrow0$, then $N^{1/2}\{\widehat{\lambda}_\mathrm{I}(a,1-a)-\lambda(a,1-

Figures (1)

  • Figure 1: Bias, coverage percentages of 95% CIs, and empirical standard errors (SEs) for $\widehat{\lambda}(1,0)$ from simulation studies I - IV. The number of units $N=500$.

Theorems & Definitions (17)

  • Remark 1: Further examples of target estimand
  • Remark 2: An alternative estimand
  • Remark 3: Connection with rank regression
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Theorem 4
  • Theorem 5
  • Proposition 2
  • ...and 7 more