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Orthogonal Survival Learners for Estimating Heterogeneous Treatment Effects from Time-to-Event Data

Dennis Frauen, Maresa Schröder, Konstantin Hess, Stefan Feuerriegel

TL;DR

This paper introduces a general, model-agnostic toolbox of Neyman-orthogonal survival learners for estimating heterogeneous treatment effects from time-to-event data under censoring. By incorporating a customizable weighting function that targets different overlap violations (treatment, censoring, and survival) and deriving an EIF-based orthogonal loss, the method achieves robust, semi-parametrically efficient HTE estimates in survival settings. The framework encompasses existing Survival-DR and Survival-R learners and adds novel variants tailored to overlap challenges, with theoretical guarantees of orthogonality and consistency. Empirical results on synthetic and real-world data demonstrate improved accuracy (lower PEHE) and convergence in low-overlap regimes, highlighting practical impact for randomized and observational survival studies.

Abstract

Estimating heterogeneous treatment effects (HTEs) is crucial for personalized decision-making. However, this task is challenging in survival analysis, which includes time-to-event data with censored outcomes (e.g., due to study dropout). In this paper, we propose a toolbox of novel orthogonal survival learners to estimate HTEs from time-to-event data under censoring. Our learners have three main advantages: (i) we show that learners from our toolbox are guaranteed to be orthogonal and thus come with favorable theoretical properties; (ii) our toolbox allows for incorporating a custom weighting function, which can lead to robustness against different types of low overlap, and (iii) our learners are model-agnostic (i.e., they can be combined with arbitrary machine learning models). We instantiate the learners from our toolbox using several weighting functions and, as a result, propose various neural orthogonal survival learners. Some of these coincide with existing survival learners (including survival versions of the DR- and R-learner), while others are novel and further robust w.r.t. low overlap regimes specific to the survival setting (i.e., survival overlap and censoring overlap). We then empirically verify the effectiveness of our learners for HTE estimation in different low-overlap regimes through numerical experiments. In sum, we provide practitioners with a large toolbox of learners that can be used for randomized and observational studies with censored time-to-event data.

Orthogonal Survival Learners for Estimating Heterogeneous Treatment Effects from Time-to-Event Data

TL;DR

This paper introduces a general, model-agnostic toolbox of Neyman-orthogonal survival learners for estimating heterogeneous treatment effects from time-to-event data under censoring. By incorporating a customizable weighting function that targets different overlap violations (treatment, censoring, and survival) and deriving an EIF-based orthogonal loss, the method achieves robust, semi-parametrically efficient HTE estimates in survival settings. The framework encompasses existing Survival-DR and Survival-R learners and adds novel variants tailored to overlap challenges, with theoretical guarantees of orthogonality and consistency. Empirical results on synthetic and real-world data demonstrate improved accuracy (lower PEHE) and convergence in low-overlap regimes, highlighting practical impact for randomized and observational survival studies.

Abstract

Estimating heterogeneous treatment effects (HTEs) is crucial for personalized decision-making. However, this task is challenging in survival analysis, which includes time-to-event data with censored outcomes (e.g., due to study dropout). In this paper, we propose a toolbox of novel orthogonal survival learners to estimate HTEs from time-to-event data under censoring. Our learners have three main advantages: (i) we show that learners from our toolbox are guaranteed to be orthogonal and thus come with favorable theoretical properties; (ii) our toolbox allows for incorporating a custom weighting function, which can lead to robustness against different types of low overlap, and (iii) our learners are model-agnostic (i.e., they can be combined with arbitrary machine learning models). We instantiate the learners from our toolbox using several weighting functions and, as a result, propose various neural orthogonal survival learners. Some of these coincide with existing survival learners (including survival versions of the DR- and R-learner), while others are novel and further robust w.r.t. low overlap regimes specific to the survival setting (i.e., survival overlap and censoring overlap). We then empirically verify the effectiveness of our learners for HTE estimation in different low-overlap regimes through numerical experiments. In sum, we provide practitioners with a large toolbox of learners that can be used for randomized and observational studies with censored time-to-event data.
Paper Structure (24 sections, 5 theorems, 92 equations, 7 figures, 3 tables)

This paper contains 24 sections, 5 theorems, 92 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem setup
  4. Background on orthogonal learning
  5. A general toolbox for obtaining orthogonal survival learners
  6. Orthogonal survival learners
  7. Experiments
  8. Extended related work
  9. Background on influence functions and orthogonal learning
  10. Estimation of nuisance functions
  11. Extensions to marginalized effects
  12. Extension to the continuous-time setting
  13. Extensions to further causal estimands
  14. Treatment-specific survival probability
  15. (Restricted) mean survival times
  16. ...and 9 more sections

Key Result

Theorem 5.1

We define the (population) loss function where and with and where we used the convention $S_{-1}(x,a) = G_{-1}(x, a) = 1$. Then, $\mathcal{L}_{f}(g, \eta_t)$ is orthogonal with respect to the nuisance functions $\eta_t$.

Figures (7)

  • Figure 1: Causal graph for censored time-to-event data. Yellow variables are observed while blue variables are unobserved. Intuitively, our goal is to recover the red arrow from $A$ to $T$ based on the observed variables.
  • Figure 2: Overview of the three steps of our toolbox.
  • Figure 2: PEHE in Scenario 1: Mean and standard deviation of PEHE averaged over the first time steps across 10 runs. Targeted learners per setting (column) in gray background. $\Rightarrow$ Overall, targeted weighting improves performance.
  • Figure 3: Benefit of targeted weighting over time. Ratios of PEHE of the targeted learner wrt. the learner without the correct target (data scenario I across 10 runs). Blue: Low censoring overlap scenario. Green: Low survival overlap scenario. Red: Low treatment overlap scenario.
  • Figure 4: Twins: 10- and 30-day effects across 10 runs $\Rightarrow$ Estimated survival effects align with the literature. Variance decreases for weighted learners.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem D.1
  • proof
  • Lemma H.1
  • proof
  • Lemma H.2
  • proof
  • ...and 2 more