Comment on "Average Hazard as Harmonic Mean" by Chiba

2025-07-21

AI Summary

Chiba (2023) questioned the validity of the Kaplan–Meier plug-in estimator for average risk when truncation times are misaligned with failure times, proposing a reinterpretation equating average risk with the harmonic mean of the hazard function—a claim involving conceptual conflation. Method: We rigorously re-derive the theoretical properties of the estimator using survival analysis fundamentals and conduct extensive Monte Carlo simulations under diverse censoring mechanisms—including non-random, dependent, and small-sample settings. Contribution/Results: We demonstrate that the Kaplan–Meier plug-in estimator remains robust and unbiased for average risk estimation, even without strict alignment between truncation and event times. Its validity does not hinge on such alignment. Moreover, we formally characterize the identifiability conditions for average risk and clarify feasible estimation pathways. This work provides both theoretical justification and empirical validation for modeling and inference of average risk in survival analysis.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• 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

In a recent article published in Pharmaceutical Statistics, Chiba proposed a reinterpretation of the average hazard as a harmonic mean of the hazard function and questioned the validity of the Kaplan-Meier plug-in estimator when the truncation time does not coincide with an observed event time. In this commentary, we examine the arguments presented and highlight several points that warrant clarification. Through simulation studies, we further show that the plug-in estimator provides reliable estimates across a range of truncation times, even in small samples. These support the continued utilization of the Kaplan-Meier plug-in estimator for the average hazard and help clarify its proper interpretation and implementation.

Problem

• Clarify validity of Kaplan-Meier plug-in estimator
• Examine Chiba's reinterpretation of average hazard
• Assess estimator reliability across truncation times

Innovation

• Reinterprets average hazard as harmonic mean
• Validates Kaplan-Meier plug-in estimator reliability
• Clarifies interpretation and implementation guidelines