Feasible Dose-Response Curves for Continuous Treatments Under Positivity Violations

1. Problem Statement

In causal inference, estimating the Causal Dose–Response Curve (CDRC) for continuous interventions (e.g., drug concentrations) is complicated by positivity violations.

• The Core Issue: The standard CDRC, $m(a) = E[Y^a]$, requires that for every covariate stratum $L=l$, the conditional density of the treatment $f(A|L=l)$ is strictly positive for all target levels $a$.
• Practical Violations: In real-world data (especially pharmacokinetics), certain treatment levels are theoretically possible but empirically unsupported for specific subpopulations (e.g., a "fast metabolizer" cannot achieve high drug concentrations even with high doses).
• Limitations of Existing Methods:
  • Trimming: Excludes unsupported strata, changing the target population and losing data.
  • Weighted CDRC: Redefines the estimand using density-based weights to handle low support, but the resulting parameter often lacks a clear scientific interpretation (it becomes a mix of causal and associational effects).
  • Modified Treatment Policies (MTPs): Shift observed treatments to nearby values but typically target a policy-defined effect rather than the original scientific question of "what happens if we set everyone to dose $a$?"

The authors argue that existing methods either sacrifice the scientific target (trimming/weights) or do not directly address the specific question of "what is the outcome if we try to set everyone to dose $a$?"

2. Methodology

The authors propose a framework that preserves the original scientific target as much as possible while ensuring the intervention is feasible (supported by the data) for every individual.

A. Diagnostic: The Non-Overlap Ratio

To quantify the severity of positivity violations, the authors introduce the Non-Overlap Ratio, $\tau_\alpha(a)$.

1. Highest Density Regions (HDRs): For each individual with covariates $L=l$, they define a support set $A_\alpha(l)$ containing the treatment values with the highest conditional density, covering a probability mass of $\alpha$ (e.g., 95%).
2. Definition: $\tau_\alpha(a) = P(a \notin A_\alpha(L))$.
   • This represents the proportion of the population for whom the target dose $a$ is unsupported (falls outside the HDR) given their covariates.
   • $\tau_\alpha(a) \approx 0$ implies good support; $\tau_\alpha(a) \approx 1$ implies severe positivity violation.

B. The Feasible Dose–Response Curve (FDRC)

The core contribution is the definition of a new estimand, the Feasible Dose–Response Curve (FDRC), denoted $m_{feasible}(a)$.

1. Most Feasible Intervention Rule ($d_{0,\alpha}$):
   For a target dose $a$ and individual covariates $l$, the rule maps $a$ to a feasible value:
   $$d_{0,\alpha}(a, l) = \begin{cases} a & \text{if } a \in A_\alpha(l) \quad (\text{Target is supported}) \\ \Pi_{A_\alpha(l)}(a) & \text{if } a \notin A_\alpha(l) \quad (\text{Target is unsupported}) \end{cases}$$
   where $\Pi_{A_\alpha(l)}(a)$ is the projection of $a$ onto the nearest value within the supported set $A_\alpha(l)$.

2. The Estimand:
   $$m_{feasible}(a) = E[Y^{d_{0,\alpha}(a, \cdot)}]$$
   Interpretation: "If we attempt to set everyone to dose $a$, but for some individuals this is not realistically attainable, what is the expected outcome after shifting those individuals to their nearest attainable concentration?"

C. Estimation Strategy

The authors propose a Plug-in g-computation estimator:

1. Density Estimation: Estimate the conditional treatment density $\hat{f}(a|l)$ using flexible methods (e.g., parametric, kernel, or hazard-binning).
2. Construct HDRs: Calculate the HDR cutoffs $\hat{f}_{\alpha}(l)$ and supported sets $\hat{A}_{\alpha}(l)$ for each individual.
3. Feasible Mapping: For each target $a$ and individual $j$, compute the feasible dose $\hat{d}_{n,\alpha}(a, L_j)$.
4. Outcome Regression: Fit a model $\hat{\mu}(a, l) \approx E[Y|A=a, L=l]$.
5. Prediction: Average the predicted outcomes using the feasible doses: $\hat{m}_{feasible}(a) = \frac{1}{n} \sum \hat{\mu}(\hat{d}_{n,\alpha}(a, L_j), L_j)$.

3. Key Contributions

1. Novel Diagnostic: Introduction of the Non-Overlap Ratio based on covariate-specific Highest Density Regions, providing a scale-aware measure of positivity violations for continuous treatments.
2. New Estimand (FDRC): A scientifically coherent estimand that answers the original dose-response question while respecting empirical constraints. It avoids the interpretability issues of weighted CDRCs and the population restriction of trimming.
3. Individualized Intervention Rule: A deterministic mapping that reassigns infeasible targets to the nearest feasible value, ensuring identifiability without relying on extrapolation into unsupported regions.
4. Software Implementation: The method is implemented in the R package CICI, utilizing hazard-binning for robust conditional density estimation.

4. Results

Simulation Studies

The authors conducted three simulation scenarios:

• Scenario 1 (Model Misspecification): Compared standard CDRC, FDRC, and Trimming under linear, logarithmic, and sinusoidal outcome functions.
  • Result: Under model misspecification (non-linear outcomes) and positivity violations, the FDRC exhibited significantly lower absolute bias than the standard CDRC and the trimming estimand. The standard CDRC failed due to unreliable extrapolation; trimming was unstable due to small sample sizes in retained strata.
• Scenario 2 (Overlap Severity): Compared strong vs. weak overlap.
  • Result: FDRC maintained low bias even when positivity was violated, whereas the standard CDRC bias increased sharply in low-overlap regions.
• Scenario 3 (Complex Real-World Data): Mimicked the CHAPAS-3 trial structure.
  • Result: FDRC provided stable estimates in regions where the standard curve diverged or became unidentifiable (e.g., near zero concentration).

Real-Data Application (CHAPAS-3 Trial)

• Context: Analyzed the relationship between efavirenz (EFV) concentration and viral failure in HIV-positive children.
• Findings:
  • Diagnostics: The non-overlap ratio revealed that extreme concentrations (<1 mg/L and >3.5 mg/L) were unsupported for a significant portion of children due to metabolic and clinical constraints.
  • Curves: In the well-supported mid-range (1–3.5 mg/L), the FDRC and standard CDRC were nearly identical.
  • Divergence: At extreme concentrations, the standard CDRC showed high variability and relied on extrapolation. The FDRC remained stable, reflecting the realistic outcome if children were shifted to their nearest attainable concentration.
  • Conclusion: The FDRC offers a stable, interpretable summary of the concentration-response relationship under realistic clinical constraints.

5. Significance and Implications

• Scientific Coherence: Unlike weighted methods that blend causal and associational parameters, the FDRC retains a clear causal interpretation: it describes the outcome of a "best-effort" intervention policy.
• Robustness: By explicitly mapping unsupported targets to supported ones, the method reduces reliance on extrapolation, which is the primary source of bias in continuous treatment settings with limited overlap.
• Practical Utility: The approach is particularly valuable in pharmacology and public health where biological or clinical constraints prevent certain treatment levels from being achievable for specific subgroups. It allows researchers to answer "what if" questions without ignoring the reality of the data.
• Limitations: The method relies on the stability of the estimated conditional density and the choice of the support level $\alpha$. It does not provide double robustness (a trade-off for interpretability) and requires careful handling of the dependency between the data and the estimand (addressed via bootstrapping or sample splitting).

In summary, this paper provides a rigorous framework for causal inference with continuous treatments that respects the limits of empirical support, offering a practical solution to the pervasive problem of positivity violations.