Efficient Transported Distributional and Quantile Treatment Effects with Surrogate-Assisted Missing Primary Outcomes

TL;DR

This paper develops efficient methods for estimating distributional and quantile treatment effects in target populations using surrogate outcomes and partial primary outcome data, with theoretical guarantees and practical estimators.

Contribution

It introduces a novel framework that leverages surrogates to improve efficiency in transportability of treatment effects without relying on traditional surrogacy assumptions.

Findings

Derived the nonparametric efficient influence function for the estimands.

Established asymptotic linearity and Bahadur representations for quantile estimators.

Provided conditions under which efficiency gains are achieved from surrogate data.

Abstract

We study target-population distributional and quantile treatment effects when a source study observes treatment and post-treatment surrogates for all source units but observes a long-run primary outcome only for a validation subset, while the target population contributes only baseline covariates. The target estimands are transported counterfactual distribution functions $\psi_a(y)=P(Y^a\le y\mid R=0)$, their quantiles $q_a(\tau)$, and the quantile treatment effect $\Delta(\tau)=q_1(\tau)-q_0(\tau)$. The surrogate is not treated as a replacement endpoint and no Prentice-type surrogacy condition is imposed. Instead, the surrogate is used only to improve efficiency under missing-at-random primary-outcome sampling. We derive the nonparametric efficient influence function, which has three orthogonal components corresponding to target covariate sampling, the source surrogate process, and missing primary outcomes. This yields a closed-form cross-fitted one-step estimator after nuisance estimation. We establish identification, the canonical gradient, exact drift identities, ratio-level robustness, pointwise and uniform asymptotic linearity for transported CDFs, Bahadur representations for quantiles under explicit local inverse-map conditions, high-level multiplier-bootstrap simultaneous bands under explicit estimated-process and density conditions, and quantile-specific efficiency gains from observing surrogates. We also give lower-level nuisance-rate verification for a deliberately restricted class of analyzable bounded finite-dimensional or finite-rank implementations based on sieve ridge regression, ridge logistic regression, calibrated density-ratio estimation, finite-rank kernel ridge regression, and isotonic projection under explicit grid, eigenvalue, source, and entropy conditions.