A Proper Concordance Index for Time-Varying Risk

TL;DR

This paper introduces a proper concordance index for time-varying risk models in survival analysis, demonstrating its advantages over existing indices through theoretical proof and simulations, especially in models with crossing hazard rates.

Contribution

It proposes a new proper concordance index based on hazard rates for time-varying risks, ensuring correct model selection and evaluation.

Findings

Existing indices are not proper and can favor incorrect models.

Using hazard rate as risk score yields a proper concordance index.

Simulations show the new index improves model selection accuracy.

Abstract

Harrel's concordance index is a commonly used discrimination metric for survival models, particularly for models where the relative ordering of the risk of individuals is time-independent, such as the proportional hazards model. There are several suggestions, but no consensus, on how it could be extended to models where relative risk can vary over time, e.g.\ in case of crossing hazard rates. We show that these concordance indices are not proper, in the sense that they are maximised in the limit by the true data generating model. Furthermore, we show that a concordance index is proper if and only if the risk score used is concordant with the hazard rate at the first event time for each comparable pair of events. Thus, we suggest using the hazard rate as the time-varying risk score when calculating concordance. Through simulations, we demonstrate situations in which other concordance indices can lead to incorrect models being selected over a true model, justifying the use of our suggested risk prediction in both model selection and in loss functions in, e.g., deep learning models.