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Learning Robust Treatment Rules for Censored Data

Yifan Cui, Junyi Liu, Tao Shen, Zhengling Qi, Xi Chen

TL;DR

A sampling-based difference-of-convex algorithm is developed for learning optimal treatment rules with censored survival outcomes, and theoretical justifications for them are provided.

Abstract

There is a fast-growing literature on estimating optimal treatment rules directly by maximizing the expected outcome. In biomedical studies and operations applications, censored survival outcome is frequently observed, in which case the truncated mean survival time and survival probability are of great interest. In this paper, we propose two robust criteria for learning optimal treatment rules with censored survival outcomes; the former one targets an optimal treatment rule maximizing the truncated mean survival time, where the cutoff is specified by a given quantile such as median; the latter one targets an optimal treatment rule maximizing buffered survival probabilities, where the predetermined threshold is adjusted to account for the truncated mean survival time. We develop a sampling-based difference-of-convex algorithm for learning the proposed optimal treatment rules, and provide theoretical justifications for them. In simulation studies, our estimators show improved performance compared to existing methods. We also demonstrate the proposed method using AIDS clinical trial data.

Learning Robust Treatment Rules for Censored Data

TL;DR

A sampling-based difference-of-convex algorithm is developed for learning optimal treatment rules with censored survival outcomes, and theoretical justifications for them are provided.

Abstract

There is a fast-growing literature on estimating optimal treatment rules directly by maximizing the expected outcome. In biomedical studies and operations applications, censored survival outcome is frequently observed, in which case the truncated mean survival time and survival probability are of great interest. In this paper, we propose two robust criteria for learning optimal treatment rules with censored survival outcomes; the former one targets an optimal treatment rule maximizing the truncated mean survival time, where the cutoff is specified by a given quantile such as median; the latter one targets an optimal treatment rule maximizing buffered survival probabilities, where the predetermined threshold is adjusted to account for the truncated mean survival time. We develop a sampling-based difference-of-convex algorithm for learning the proposed optimal treatment rules, and provide theoretical justifications for them. In simulation studies, our estimators show improved performance compared to existing methods. We also demonstrate the proposed method using AIDS clinical trial data.
Paper Structure (14 sections, 9 theorems, 51 equations, 2 figures, 1 table)

This paper contains 14 sections, 9 theorems, 51 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Setup and original value function framework
  4. The CVaR criterion: maximizing truncated mean survival time at a given quantile
  5. The buffered criterion: maximizing buffered survival probabilities
  6. Accounting for censoring: identification and learning
  7. Sampling-based learning algorithm
  8. Empirical DC formulation
  9. Sampling-based DCA
  10. Extension to the buffered criterion
  11. Asymptotic properties
  12. Simulations
  13. Real data application
  14. Discussion

Key Result

Lemma 2.1

Let $V^2_\tau(d)$ be defined as in eq:VVVV with $\tau\in(\inf T(d),\mathbb{E}[T(d)]]$. Then Consequently, maximizing $V^2_\tau(d)$ over $d$ is equivalent to minimizing the following over $d$,

Figures (2)

  • Figure 1: An illustrative example of three criteria
  • Figure 2: Boxplots of empirical value functions.

Theorems & Definitions (11)

  • Remark 1: Relation to restricted mean survival time
  • Remark 2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • ...and 1 more