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Conformal Prediction for Dose-Response Models with Continuous Treatments

Jarne Verhaeghe, Jef Jonkers, Sofie Van Hoecke

TL;DR

This work addresses uncertainty quantification for dose–response models with continuous treatments by reframing the problem as covariate shift and introducing propensity-weighted conformal prediction (PWCP). The method combines generalized propensity scores, interventional distributions, and kernel-based local weighting to construct prediction intervals that maintain finite-sample coverage across all doses $t\in [t_L,t_U]$, including both global and local calibration strategies. It provides theoretical finite-sample guarantees under weighted exchangeability and demonstrates the approach on synthetic and semi-synthetic data, highlighting the impact of covariate shift and overlap on interval width and reliability. The framework is model-agnostic and extendable to conformal predictive systems, with practical implications for personalized dosing and other interventions where reliable uncertainty estimates are essential.

Abstract

Understanding the dose-response relation between a continuous treatment and the outcome for an individual can greatly drive decision-making, particularly in areas like personalized drug dosing and personalized healthcare interventions. Point estimates are often insufficient in these high-risk environments, highlighting the need for uncertainty quantification to support informed decisions. Conformal prediction, a distribution-free and model-agnostic method for uncertainty quantification, has seen limited application in continuous treatments or dose-response models. To address this gap, we propose a novel methodology that frames the causal dose-response problem as a covariate shift, leveraging weighted conformal prediction. By incorporating propensity estimation, conformal predictive systems, and likelihood ratios, we present a practical solution for generating prediction intervals for dose-response models. Additionally, our method approximates local coverage for every treatment value by applying kernel functions as weights in weighted conformal prediction. Finally, we use a new synthetic benchmark dataset to demonstrate the significance of covariate shift assumptions in achieving robust prediction intervals for dose-response models.

Conformal Prediction for Dose-Response Models with Continuous Treatments

TL;DR

This work addresses uncertainty quantification for dose–response models with continuous treatments by reframing the problem as covariate shift and introducing propensity-weighted conformal prediction (PWCP). The method combines generalized propensity scores, interventional distributions, and kernel-based local weighting to construct prediction intervals that maintain finite-sample coverage across all doses , including both global and local calibration strategies. It provides theoretical finite-sample guarantees under weighted exchangeability and demonstrates the approach on synthetic and semi-synthetic data, highlighting the impact of covariate shift and overlap on interval width and reliability. The framework is model-agnostic and extendable to conformal predictive systems, with practical implications for personalized dosing and other interventions where reliable uncertainty estimates are essential.

Abstract

Understanding the dose-response relation between a continuous treatment and the outcome for an individual can greatly drive decision-making, particularly in areas like personalized drug dosing and personalized healthcare interventions. Point estimates are often insufficient in these high-risk environments, highlighting the need for uncertainty quantification to support informed decisions. Conformal prediction, a distribution-free and model-agnostic method for uncertainty quantification, has seen limited application in continuous treatments or dose-response models. To address this gap, we propose a novel methodology that frames the causal dose-response problem as a covariate shift, leveraging weighted conformal prediction. By incorporating propensity estimation, conformal predictive systems, and likelihood ratios, we present a practical solution for generating prediction intervals for dose-response models. Additionally, our method approximates local coverage for every treatment value by applying kernel functions as weights in weighted conformal prediction. Finally, we use a new synthetic benchmark dataset to demonstrate the significance of covariate shift assumptions in achieving robust prediction intervals for dose-response models.
Paper Structure (28 sections, 1 theorem, 35 equations, 30 figures, 6 tables, 3 algorithms)

This paper contains 28 sections, 1 theorem, 35 equations, 30 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Related Work
  4. Method
  5. Introduction to Conformal Prediction
  6. Inductive Conformal Prediction
  7. Weighted Conformal Prediction
  8. Conformal Predictive Systems
  9. Proposed methodology: Propensity Weighted Conformal Prediction
  10. Experiments
  11. Synthetic Data
  12. Setup 1
  13. Setup 2
  14. Setup 3
  15. Implementation
  16. ...and 13 more sections

Key Result

Proposition 1

Assume $(X_i, T_i, Y_i) \overset{i.i.d.}{\sim} P_X \times P_{T|X} \times P_{Y|T,X}$, $i=1,...,n$; the likelihood ratio $w(X,T)\propto\frac{d\Tilde{P}_{T|X}}{dP_{T|X}}$; and the estimated likelihood ratio $\hat{w}(X,T)$. Using WCP to construct $\hat{C}(X,T)$, the following finite-sample bounds apply:

Figures (30)

  • Figure 1: Barplot of the mean coverage calculated over 40 treatment values in 50 experiments for setup 3 scenario 1. The black dotted line is the ideal coverage.
  • Figure 2: Barplot of the mean coverage calculated over 45 treatment values in 100 experiments for the AMICAS semi-synthetic evaluation. The black dotted line is the ideal coverage.
  • Figure 3: CADRF UQ Example on Setup 3 Scenario 1 using estimated propensity
  • Figure 4: A Ceteris Paribus curve generated with Local Propensity WCP.
  • Figure 5: Barplot of the mean coverage calculated over 40 treatment values in 50 experiments for setup 3 scenario 2. Black dotted line is the ideal coverage.
  • ...and 25 more figures

Theorems & Definitions (8)

  • Proposition 1: following tibshirani_conformal_2019lei_conformal_2021
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6