Rank-Based Estimation of U-Shaped Biomarker Risk Curves and Critical Points for Time-to-Event Outcomes

2026-07-21

AI Summary

This study addresses the challenge of modeling U-shaped relationships between biomarkers and risk in survival analysis, particularly the difficulty in estimating their threshold points. The authors propose a semiparametric transformation model that explicitly parameterizes the turning point of the hazard curve. By integrating rank-based maximum C-index estimation with a smoothed Kaplan–Meier approach, the method enables joint estimation of subgroup-specific U-shaped risk curves and their corresponding thresholds. This work represents the first effort to directly model, estimate, and perform statistical inference on U-shaped risk thresholds within time-to-event analysis. Theoretical results establish the consistency and asymptotic normality of the proposed estimators, while simulation studies demonstrate favorable finite-sample performance. The approach is successfully applied to UK Biobank data, revealing a U-shaped association between BMI and all-cause mortality along with its critical threshold.

Abstract

U-shaped relationships between prognostic biomarker levels and adverse event risk are commonly observed across diseases, where both low and high biomarker values are associated with elevated risk, with a well-defined minimum -- the critical point -- marking the biomarker value of the lowest risk. The U-shaped risk curve, especially the location of the critical point, informs the identification of high- and low-risk subgroups. However, existing methods are limited: U-shaped risk models rarely accommodate survival outcomes, and existing survival analysis methods do not enable estimation of or formal inference for the critical point. To fill this gap, we propose a semiparametric transformation model that explicitly parameterizes the critical point, a rank-based maximum C-index estimator for the parametric component, and a smoothed Kaplan-Meier estimation approach for the nonparametric component. The resulting framework estimates both subgroup-specific U-shaped risk curves and their critical points within a single survival model. We establish consistency and asymptotic normality of the proposed estimators and demonstrate their finite-sample performance through numerical studies. We apply the proposed methods to UK Biobank data to characterize subgroup-specific U-shaped associations between body mass index and all-cause mortality and to identify the corresponding critical points.

Problem

• U-shaped risk
• critical point
• survival analysis
• biomarker
• time-to-event

Innovation

• U-shaped risk
• critical point
• semiparametric transformation model
• rank-based estimation
• survival analysis