Probability of worthwhile effect of monotone-response treatments

2026-07-15

AI Summary

This study addresses the challenge of identifying the distribution of treatment effects when only the marginal distributions of the treated and control groups are observed, and their joint distribution is unidentifiable. Under the assumption of monotone treatment response—where individual treatment effects are non-negative—the authors explicitly construct extremal dependence structures that satisfy this constraint. They reformulate the problem of bounding the probability that the treatment effect exceeds any given threshold as a risk aggregation problem with order constraints and an optimal transport problem under a specific cost function. Leveraging theories of risk aggregation under dependence uncertainty and monotonicity-constrained optimal transport, the paper derives sharp analytical bounds for this probability and provides explicit coupling constructions that attain these bounds, thereby offering a computable and practically applicable framework for distributional identification in causal inference.

Abstract

Experiments may, by design, prevent one from observing on a single subject both the response to a treatment and to its absence. Because of this, marginal distributions for both cases may be observable but not their joint distribution, thus obscuring the distribution of the treatment effect. We examine the case where we impose that the treatment effect is nonnegative, also called monotone treatment response, a common assumption relevant to many practical applications. We solve the problems of best- and worst-case probabilities that the treatment effect exceeds a given value, using an explicit construction for the dependence scheme in each case. Such problems can equivalently be described, in different contexts, as risk aggregation under dependence uncertainty and an order constraint, and as optimal transport with a particular cost function.

Problem

• monotone treatment response
• treatment effect
• dependence uncertainty
• marginal distributions
• probability bounds

Innovation

• monotone treatment response
• dependence uncertainty
• optimal transport
• risk aggregation
• causal inference