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

2026-02-13

AI Summary

This study addresses the limitation of the conventional Cox model’s hazard ratio, which lacks a stable causal interpretation under non-proportional hazards. To overcome this, the authors propose the average hazard (AH) as an alternative effect measure and, for the first time, establish its pathwise differentiability under a nonparametric model, deriving the corresponding efficient influence function. Building on this, they develop a semiparametric inference framework that achieves both double robustness and √n-consistency. By integrating cross-fitting with machine learning algorithms, the method flexibly accommodates high-dimensional covariates, requiring only mild product-rate convergence conditions on nuisance parameters. Simulations demonstrate low bias and near-nominal confidence interval coverage across proportional, non-proportional, and crossing hazards scenarios. The approach is successfully applied to comparative effectiveness analysis of immunotherapies in advanced melanoma.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• non-proportional hazards
• average hazard
• causal estimand
• hazard ratio
• survival analysis

Innovation

• average hazard
• non-proportional hazards
• doubly robust
• semiparametric inference
• machine learning