Optimal cutoff selection for U-shaped prognostic variable in survival data with competing risk

Selecting clinically meaningful cutoff values for continuous prognostic variables is challenging when the association with risk is U-shaped and competing risks are present.We propose a Competing Risk C-index based Cutoff (CRCC) method that identifies two cutoff values by pairing points with equal log-hazard heights and selecting the pair that maximizes the Inverse probability of censoring weight (IPCW) C-index.Our approach first fits a smoothing spline to the log relative hazard from the Fine-Gray (FG) and cause specific hazard (CSH) model to confirm the U-shape relationship.This model then generates candidate cutoff pairs at equal heights on either side of the nadir, partitioning patients into the central low-risk group and the high-risk tail groups.From these candidates, we select the optimal pair that maximizes the IPCW C-index.For comparison, we also evaluated Gönen-Heller's concordance probability estimate (CPE), the minimum p-value (Min-P), and the percentile (Q1,Q3) methods.Monte Carlo simulations spanning symmetric, moderately asymmetric, and severely asymmetric U-shapes with 20% and 50% censoring show that CRCC using FG model consistently achieves the lowest Bayesian information criterion (BIC) and smallest standard errors (SEs), and CRCC using CSH model is the second best.The percentile method is highly stable but modestly inferior by BIC.Min-P tends to select more variable, wider cutoff values, and CPE often yields extreme or unstable cutoff values.Overall, CRCC based cutoff value selection (particularly with the FG model) offers a stable, discriminative, and well-fitting strategy for risk stratification when the continuous prognostic variable exhibits U-shaped relationships under survival data with competing risks.The proposed method is further illustrated with a real kidney-transplant data.