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Efficient and Debiased Learning of Average Hazard Under Non-Proportional Hazards

Xiang Meng, Lu Tian, Kenneth Kehl, Hajime Uno

Abstract

The hazard ratio from the Cox proportional hazards model is a ubiquitous summary of treatment effect. However, when hazards are non-proportional, the hazard ratio can lose a stable causal interpretation and become study-dependent because it effectively averages time-varying effects with weights determined by follow-up and censoring. We consider the average hazard (AH) as an alternative causal estimand: a population-level person-time event rate that remains well-defined and interpretable without assuming proportional hazards. Although AH can be estimated nonparametrically and regression-style adjustments have been proposed, existing approaches do not provide a general framework for flexible, high-dimensional nuisance estimation with valid sqrt{n} inference. We address this gap by developing a semiparametric, doubly robust framework for covariate-adjusted AH. We establish pathwise differentiability of AH in the nonparametric model, derive its efficient influence function, and construct cross-fitted, debiased estimators that leverage machine learning for nuisance estimation while retaining asymptotically normal, sqrt{n}-consistent inference under mild product-rate conditions. Simulations demonstrate that the proposed estimator achieves small bias and near-nominal confidence-interval coverage across proportional and non-proportional hazards settings, including crossing-hazards regimes where Cox-based summaries can be unstable. We illustrate practical utility in comparative effectiveness research by comparing immunotherapy regimens for advanced melanoma using SEER-Medicare linked data.

Efficient and Debiased Learning of Average Hazard Under Non-Proportional Hazards

Abstract

The hazard ratio from the Cox proportional hazards model is a ubiquitous summary of treatment effect. However, when hazards are non-proportional, the hazard ratio can lose a stable causal interpretation and become study-dependent because it effectively averages time-varying effects with weights determined by follow-up and censoring. We consider the average hazard (AH) as an alternative causal estimand: a population-level person-time event rate that remains well-defined and interpretable without assuming proportional hazards. Although AH can be estimated nonparametrically and regression-style adjustments have been proposed, existing approaches do not provide a general framework for flexible, high-dimensional nuisance estimation with valid sqrt{n} inference. We address this gap by developing a semiparametric, doubly robust framework for covariate-adjusted AH. We establish pathwise differentiability of AH in the nonparametric model, derive its efficient influence function, and construct cross-fitted, debiased estimators that leverage machine learning for nuisance estimation while retaining asymptotically normal, sqrt{n}-consistent inference under mild product-rate conditions. Simulations demonstrate that the proposed estimator achieves small bias and near-nominal confidence-interval coverage across proportional and non-proportional hazards settings, including crossing-hazards regimes where Cox-based summaries can be unstable. We illustrate practical utility in comparative effectiveness research by comparing immunotherapy regimens for advanced melanoma using SEER-Medicare linked data.
Paper Structure (53 sections, 7 theorems, 74 equations, 10 figures, 1 table)

This paper contains 53 sections, 7 theorems, 74 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Causal Quantity: Average Hazard
  3. Causal definition
  4. Identification of the Causal Parameter
  5. Related causal estimands for time-to-event outcomes.
  6. Debiased estimation
  7. Preliminaries: data structure and EIF for marginal survival
  8. Efficient influence function for log average hazard via functional delta method
  9. Cross-fitted plug-in estimator for log AH via efficient survival
  10. Large-Sample Theory for the AH Estimator
  11. Assumptions
  12. Main results
  13. Simulation study
  14. Simulation results
  15. Data Analysis
  16. ...and 38 more sections

Key Result

Theorem 2.2

Fix $a\in\{0,1\}$ and $\tau>0$. Under (A1)--(A3), the causal average hazard $\eta_a(\tau)=F_a(\tau)/R_a(\tau)$ defined in eq:AH is identified by where $S_0(t\mid a,w)=\Pr(T>t\mid A=a,W=w)$.

Figures (10)

  • Figure 1: Main simulation summary for the log average-hazard ratio at $\tau=12$ months. Columns correspond to non-PH (left) and PH with linear exponential censoring (right). Rows report percent bias, variance $\times 100$, MSE $\times 100$, and empirical coverage of nominal 95% confidence intervals.
  • Figure 2: Estimated average hazard (AH) ratios with 95% Wald confidence intervals for comparisons of nivolumab plus ipilimumab, nivolumab monotherapy, and ipilimumab monotherapy at 24- and 36-month horizons. The vertical dashed line indicates no difference in average hazard.
  • Figure 3: Non-PH DGP: Monte Carlo percent bias, variance (rescaled by 100) and mean squared error (rescaled by 100).
  • Figure 4: Non-PH DGP: empirical coverage of nominal 95% confidence intervals. The dashed horizontal line indicates the nominal 0.95 level.
  • Figure 5: Cross-A DGP: Monte Carlo percent bias, variance (rescaled by 100) and mean squared error (rescaled by 100).
  • ...and 5 more figures

Theorems & Definitions (16)

  • Definition 2.1: Average hazard
  • Remark 2.1: Interpretability under non-proportional hazards and study-design dependence of Cox HR
  • Remark 2.2: Consistency with stochastic ordering
  • Theorem 2.2: Causal identification of AH
  • Proposition 2.1: Observed-data identification under right censoring
  • Remark 2.3: Comparison to an alternative marginalization approach
  • Proposition 3.1: Efficient influence function of $S_a(t)$, westling2024inference
  • Theorem 3.1: Efficient influence function of log AH via survival
  • Remark 3.1: Structure of the log-AH EIF
  • Remark 3.2: Inheritance of double robustness
  • ...and 6 more