Nested Sensitivity Envelopes for Transported Quantile Treatment Effects

TL;DR

This paper develops a method to bound and estimate transported quantile treatment effects in observational studies with unmeasured confounding and population shifts, using sensitivity envelopes and semiparametric inference.

Contribution

It introduces a novel nested sensitivity envelope framework that improves bounds on transported quantile effects and provides sharp, attainable bounds with valid inference procedures.

Findings

Derived closed-form sharp target counterfactual CDF envelopes.

Developed semiparametric estimators with uniform Gaussian approximation.

Constructed honest confidence sets for quantile and QTE bounds.

Abstract

We study target-population quantile treatment effects when a source study may have unmeasured treatment confounding and may not transport to a target population after conditioning on observed covariates. The observed data consist of a source sample with treatment, outcome and covariates, and a target sample with covariates only. We impose two marginal sensitivity restrictions: an odds-ratio bound \(\Gam\) for source treatment assignment and a conditional likelihood-ratio bound \(\Lam\) for source-to-target potential-outcome distribution shift. For each treatment arm and threshold \(y\), we derive a closed-form sharp target counterfactual CDF envelope. The envelope nests a source marginal-sensitivity map inside a target outcome-shift map, preserving two normalizations and generally improving on a single product likelihood-ratio relaxation. We prove process-level sharpness, so the envelopes are attainable as entire CDFs and can be inverted to obtain sharp target quantile bounds and sharp interval-hull QTE bounds. We then develop semiparametric theory for these nonsmooth bound processes. On regular index sets, we give the canonical gradient, including the source propensity contribution required in observational studies, and construct cross-fitted Neyman-orthogonal one-step estimators with uniform Gaussian approximation. On full index sets with active-set ties or mass points, we use Hadamard directional differentiability and subsampling-valid inference, with a primitive finite-support route for the required weak convergence. Finally, we invert simultaneous monotone CDF bands to obtain honest confidence sets for quantile and QTE interval-hull processes, and formulate the two-dimensional \((\Gam,\Lam)\) breakdown frontier as level-set inference for interval-hull non-refutation.