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 risk varies over time, e.g.\ in case of crossing hazard rates. We show that, in the limit, concordance is maximized if and only if the risk score is concordant with the hazard rate, in the sense that for a comparable pair where the first event time is observed, the risk score is concordant with the hazard rate at this first event time. Thus, we suggest using the hazard rate as the 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., machine learning models.

中文翻译：

---

关于具有时变风险的模型的一致性指数

Harrel 的一致性指数是生存模型常用的判别指标，特别是对于个体风险的相对排序与时间无关的模型，例如比例风险模型。关于如何将其扩展到风险随时间变化的模型（例如，在交叉危险率的情况下），有一些建议，但没有达成共识。我们表明，在极限情况下，当且仅当风险评分与危险率一致时，一致性最大化，即对于观察到第一个事件时间的可比较对，风险评分与危险率一致在第一次活动时间。因此，我们建议在计算一致性时使用危险率作为风险评分。通过模拟，