Sharp Bounds for Treatment Effect Generalization under Outcome Distribution Shift

Generalizing treatment effects from a randomized trial to a target population requires the assumption that potential outcome distributions are invariant across populations after conditioning on observed covariates. This assumption fails when unmeasured effect modifiers are distributed differently between trial participants and the target population. We develop a sensitivity analysis framework that bounds how much conclusions can change when this transportability assumption is violated. Our approach constrains the likelihood ratio between target and trial outcome densities by a scalar parameter $Λ\geq 1$, with $Λ= 1$ recovering standard transportability. For each $Λ$, we derive sharp bounds on the target average treatment effect – the tightest interval guaranteed to contain the true effect under all data-generating processes compatible with the observed data and the sensitivity model. We show that the optimal likelihood ratios have a simple threshold structure, leading to a closed-form greedy algorithm that requires only sorting trial outcomes and redistributing probability mass. The resulting estimator runs in $O(n \log n)$ time and is consistent under standard regularity conditions. Simulations demonstrate that our bounds achieve nominal coverage when the true outcome shift falls within the specified $Λ$, provide substantially tighter intervals than worst-case bounds, and remain informative across a range of realistic violations of transportability.

💡 Research Summary

Generalizing the causal effect estimated in a randomized clinical trial to a broader target population typically relies on the assumption of conditional outcome transportability: after adjusting for observed covariates X, the distribution of potential outcomes Y(a) is identical in the trial and the target. In practice, unmeasured effect modifiers (U) often have different distributions across the two populations, violating this assumption and potentially biasing naïve generalization methods.

The authors introduce an outcome‑shift sensitivity model that directly bounds the discrepancy between the conditional outcome densities in the trial (f_r) and the target (f_o). Specifically, for each treatment arm a and covariate value x, the likelihood ratio L(y;a,x)=f_o(y|a,x)/f_r(y|a,x) is constrained to lie in the interval