Feasible Dose-Response Curves for Continuous Treatments Under Positivity Violations

Positivity violations can complicate estimation and interpretation of causal dose-response curves (CDRCs) for continuous interventions. Weighting-based methods are designed to handle limited overlap, but the resulting weighted targets can be hard to interpret scientifically. Modified treatment policies can be less sensitive to support limitations, yet they typically target policy-defined effects that may not align with the original dose-response question. We develop an approach that addresses limited overlap while remaining close to the scientific target of the CDRC. Our work is motivated by the CHAPAS-3 trial of HIV-positive children in Zambia and Uganda, where clinically relevant efavirenz concentration levels are not uniformly supported across covariate strata. We introduce a diagnostic, the non-overlap ratio, which quantifies, as a function of the target intervention level, the proportion of the population for whom that level is not supported given observed covariates. We also define an individualized most feasible intervention: for each child and target concentration, we retain the target when it is supported, and otherwise map it to the nearest supported concentration. The resulting feasible dose-response curve answers: if we try to set everyone to a given concentration, but it is not realistically attainable for some individuals, what outcome would be expected after shifting those individuals to their nearest attainable concentration? We propose a plug-in g-computation estimator that combines outcome regression with flexible conditional density estimation to learn supported regions and evaluate the feasible estimand. Simulations show reduced bias under positivity violations and recovery of the standard CDRC when support is adequate. An application to CHAPAS-3 yields a stable and interpretable concentration-response summary under realistic support constraints.

💡 Research Summary

This paper tackles a fundamental challenge in causal inference for continuous treatments: practical violations of the positivity (overlap) assumption. When certain treatment levels are rarely or never observed within subpopulations defined by baseline covariates, standard methods—such as inverse‑probability weighting or trimming—either become unstable or change the target population, making scientific interpretation difficult. The authors motivate the problem with the CHAPAS‑3 trial, where plasma efavirenz concentrations in HIV‑positive children show strong heterogeneity and limited empirical support at the extremes of the clinically relevant range.

To address this, they introduce two novel concepts. First, the non‑overlap ratio, a population‑level diagnostic that quantifies, for each target dose a, the proportion of individuals whose covariate profile places a outside a high‑density region (HDR) of the conditional treatment distribution. By fixing a probability mass level α (e.g., 0.95), they define for each covariate vector ℓ a set Aα(ℓ) containing the treatment values that collectively account for α of the conditional probability mass. The non‑overlap ratio is simply the fraction of the population for which a∉Aα(ℓ). This provides a scale‑aware, interpretable measure of practical positivity violations for continuous exposures.

Second, they define the individualized most feasible intervention. For a desired dose a and covariate profile ℓ, if a lies inside Aα(ℓ) it is retained; otherwise it is mapped to the nearest value a∗(ℓ) within Aα(ℓ). This creates a covariate‑adaptive rule that preserves the original dose‑response relationship wherever data support exists, while substituting an attainable dose where support is lacking.

Using this rule they formulate the feasible dose‑response curve (FDRC):
 m_FDRC(a) = E