Beyond the Average: Distributional Causal Inference under Imperfect Compliance

TL;DR

This work tackles distributional causal effects in randomized trials with covariate-adaptive randomization and imperfect compliance by identifying and estimating the local distributional treatment effect (LDTE) for compliers using the assignment as an instrumental variable. It introduces a regression-adjusted distribution regression estimator with Neyman-orthogonal moment conditions and cross-fitting, applicable to various outcome types and proven to be semiparametrically efficient. The authors derive the estimator’s asymptotic distribution, establish the efficiency bound, and demonstrate finite-sample gains through simulations and an Oregon Health Insurance Experiment application. The approach yields richer, distributional insights into treatment effects and offers practically relevant tools for precise inference in settings with noncompliance and CAR.

Abstract

We study the estimation of distributional treatment effects in randomized experiments with imperfect compliance. When participants do not adhere to their assigned treatments, we leverage treatment assignment as an instrumental variable to identify the local distributional treatment effect-the difference in outcome distributions between treatment and control groups for the subpopulation of compliers. We propose a regression-adjusted estimator based on a distribution regression framework with Neyman-orthogonal moment conditions, enabling robustness and flexibility with high-dimensional covariates. Our approach accommodates continuous, discrete, and mixed discrete-continuous outcomes, and applies under a broad class of covariate-adaptive randomization schemes, including stratified block designs and simple random sampling. We derive the estimator's asymptotic distribution and show that it achieves the semiparametric efficiency bound. Simulation results demonstrate favorable finite-sample performance, and we demonstrate the method's practical relevance in an application to the Oregon Health Insurance Experiment.

Figures (4)

• Figure 1: The relationship between the variables. Solid arrows ($\longrightarrow$) represent direct causal pathways, while dashed arrows ($\dashrightarrow$) denote conditioning or derivation relationships rather than direct causality.
• Figure 2: Simulation results for LDTE estimators under a nonlinear, high-dimensional design ($n=1000$). RMSE, average 95% CI length, and empirical coverage are shown across quantiles $\{0.1, \dots, 0.9\}$ based on 1000 replications. Three estimators are compared: unadjusted, linearly adjusted, and ML-adjusted (gradient boosting with 2-fold cross-fitting). Both adjusted estimators improve RMSE and CI length; the unadjusted estimator attains near-nominal coverage, while ML adjustment slightly over-covers, reflecting conservative inference.
• Figure 3: RMSE reduction (%) of adjusted estimators relative to the unadjusted estimator across quantiles and sample sizes. Linear adjustment yields modest efficiency gains (1–10%), while ML adjustment achieves up to 50% reduction, with improvements becoming more pronounced as sample size increases.
• Figure 4: Oregon Health Insurance Experiment: Local Distributional Treatment Effect (LDTE) and Local Probability Treatment Effect (LPTE) of insurance coverage on number of emergency department (ED) visits. The left panels depict the empirical probability estimates, while the right panels present regression-adjusted estimates obtained using gradient boosting with 5-fold cross-fitting. Shaded regions and error bars represent 95% confidence intervals. Sample size: $n=17{,}021$.

Theorems & Definitions (9)

• Lemma 3.2: Local distributional treatment effect
• Theorem 5.2: Asymptotic Distribution
• Theorem 5.3: Semiparametric Efficiency Bound
• Definition B.1: Covering numbers
• Definition B.2: Envelope function
• Definition B.3: VC-type class
• proof
• proof
• proof