Feasible Dose-Response Curves for Continuous Treatments Under Positivity Violations

TL;DR

This paper tackles the challenge of estimating causal dose--response curves for continuous treatments when positivity is limited. It introduces an HDR-based diagnostic to quantify practical support via the non-overlap ratio and defines a covariate-adaptive feasible intervention to construct a Feasible Dose–Response Curve (FDRC) that remains close to the target CDRC while avoiding extrapolation. The authors contrast FDRC with the weighted CDRC and a trimmed variant, show through simulations that FDRC offers reduced bias under positivity violations, and demonstrate its applicability with CHAPAS-3 efavirenz data, yielding stable, interpretable concentration--response summaries under realistic support constraints. The work is implemented in the CICI R package, enabling researchers to diagnose overlap and estimate the FDRC in practical settings, thereby enhancing causal inference for continuous exposures in public health and pharmacology. Overall, the approach preserves scientific interpretability where support is adequate and provides robust inference when positivity is limited, offering a valuable tool for causal analysis of continuous interventions.

Abstract

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.

Figures (6)

• Figure 1: Illustration of the highest density regions (HDRs) used to define covariate-specific intervention support. For fixed $\vb l$ and $\alpha=0.95$, well-supported intervention values are those on the $x$-axis in $\mathcal{A}_\alpha(\vb l)$. The green shaded region indicates probability mass of $A\mid \vb L=\vb l$ inside $\mathcal{A}_\alpha(\vb l)$; the grey region indicates mass outside.
• Figure 2: Illustration of the most feasible intervention defined in Equations (\ref{['eqn:data_adaptive']}) and (\ref{['eqn:most_feasible']}). Panel (a) shows the conditional density of treatment values given covariates, with HDRs (green) representing well-supported regions and grey areas denoting poorly supported regions. Target interventions (red arrows) are either retained if they fall inside HDRs or reassigned to the nearest feasible value within HDRs (green arrows). Panel (b) extends this principle across confounder strata, where targets (red circles) are reassigned to the closest supported value (green circles) whenever they fall outside HDRs, while feasible targets remain unchanged. Together, the panels illustrate the principle of the most feasible intervention: retain the target when supported, and otherwise make the smallest necessary adjustment to ensure feasibility.
• Figure 3: Results of Simulation 1
• Figure 4: Results of Simulation 2
• Figure 5: Results of Simulation 3

Theorems & Definitions (1)

• Remark 4.1: Grid-based versus closed-form HDR construction