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Learning Optimal Distributionally Robust Individualized Treatment Rules Integrating Multi-Source Data

Wenhai Cui, Wen Su, Xingqiu Zhao

TL;DR

This work proposes a prior information-based distributionally robust ITR (PDRO-ITR) that maximizes the worst-case policy value over a covariate-dependent distributional uncertainty set, ensuring robust performance under posterior shift.

Abstract

Integrative analysis of multiple datasets for estimating optimal individualized treatment rules (ITRs) can enhance decision efficiency. A central challenge is posterior shift, wherein the conditional distribution of potential outcomes given covariates differs between source and target populations. We propose a prior information-based distributionally robust ITR (PDRO-ITR) that maximizes the worst-case policy value over a covariate-dependent distributional uncertainty set, ensuring robust performance under posterior shift. The uncertainty set is constructed as an individualized combination of source distributions, with weights combining prior source-membership probabilities and deviation terms constrained to the probability simplex to accommodate posterior shift. We derive a closed-form solution for the PDRO-ITR and develop an adaptive procedure to tune the uncertainty level. We establish risk bounds for the PDRO-ITR estimator, which guarantees robust performance under the worst case. Extensive simulations and two real-data applications demonstrate that the proposed method achieves superior performance compared to existing approaches.

Learning Optimal Distributionally Robust Individualized Treatment Rules Integrating Multi-Source Data

TL;DR

This work proposes a prior information-based distributionally robust ITR (PDRO-ITR) that maximizes the worst-case policy value over a covariate-dependent distributional uncertainty set, ensuring robust performance under posterior shift.

Abstract

Integrative analysis of multiple datasets for estimating optimal individualized treatment rules (ITRs) can enhance decision efficiency. A central challenge is posterior shift, wherein the conditional distribution of potential outcomes given covariates differs between source and target populations. We propose a prior information-based distributionally robust ITR (PDRO-ITR) that maximizes the worst-case policy value over a covariate-dependent distributional uncertainty set, ensuring robust performance under posterior shift. The uncertainty set is constructed as an individualized combination of source distributions, with weights combining prior source-membership probabilities and deviation terms constrained to the probability simplex to accommodate posterior shift. We derive a closed-form solution for the PDRO-ITR and develop an adaptive procedure to tune the uncertainty level. We establish risk bounds for the PDRO-ITR estimator, which guarantees robust performance under the worst case. Extensive simulations and two real-data applications demonstrate that the proposed method achieves superior performance compared to existing approaches.
Paper Structure (16 sections, 4 theorems, 38 equations, 4 figures, 6 tables)

This paper contains 16 sections, 4 theorems, 38 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Methodology
  4. Distributionally robust ITRs
  5. Distributionally robust ITR with prior information
  6. Motivating Example
  7. Prior information-based distributionally robust ITR
  8. Estimation procedure for PDRO-ITR
  9. Theoretical Results
  10. Simulation Studies
  11. Simulation Designs
  12. Simulation Results
  13. Applications
  14. The AIDS Clinical Trials Group Study
  15. Oregon Health Insurance Experiment
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{P}=\{ \boldsymbol{\rho}=(\rho_1, \ldots,\rho_{|\mathcal{S}|}) \quad|\quad \sum_{s=1}^{|\mathcal{S}|}\rho_s=1, \rho_s \geq0\}$. The distributionally robust ITR is $d^{*}_{dro}= \mathbb{I}(\sum_{s=1}^{|\mathcal{S}|} \rho_{s}^{*} C\{\boldsymbol{X}; \mathbb{P}_{Y(1), Y(0)|\boldsymbol{X}}^{

Figures (4)

  • Figure 1: Geometric illustration of the simplex surface $\mathcal{P}$ and the prior information-based surfaces $\widetilde{\mathcal{P}}(\boldsymbol{x},\delta)$ evaluated at $x = 0, ({1}/{3}), ({2}/{3}), 1$ with $\delta = 0.8$; The weight functions in the $\widetilde{\mathcal{P}}(\boldsymbol{x},\delta)$ are defined as $\omega^{(s)}_{0}( {x})= {\exp(\beta_s x)}/\{\sum_{s \in \mathcal{S}} \exp(\beta_s x)\}$ for $s=1, 2, 3$, where $(\beta_1, \beta_2, \beta_3) = (1, -1, 0.5)$; The true curve is defined as $\left[\mathbb{P}(S^{(t)}=1|{X}^{(t)}={x}), \mathbb{P}(S^{(t)}=2|{X}^{(t)}={x}), \mathbb{P}(S^{(t)}=3|{X}^{(t)}={x})\right],$ and we assume that $\mathbb{P}(S^{(t)}=s|{X}^{(t)}={x}) = \omega^{(s)}_{0}( {x})$, i.e., the source and target domains share the same subgroup structure.
  • Figure 2: Plot of the smoothed surrogate function $\Phi_h(u)$ under varying $h$.
  • Figure 3: Policy value plots under the worst-case for Scenarios 1 and 2, with varying $\delta$, $n=2000$, and $200$ repetitions.
  • Figure 4: Policy value plots under the worst-case for Scenarios 3 and 4, with varying $\delta$, $n=2000$, and $200$ repetitions.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Theorem 3