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.
Value-Monotonicity Matters: A Concordance Loss for Deep Survival Prediction
Department of Radiation Oncology, University of Colorado School of Medicine（科罗拉多大学医学院放射肿瘤学系）；Department of Radiation Oncology, University of Texas Southwestern Medical Center（德克萨斯大学西南医学中心放射肿瘤学系）
Deep Learning for Patient-Specific Kidney Graft Survival Analysis
University Paris Descartes（巴黎笛卡尔大学）；Institut Hypercube（超立方体研究所）；Montreal Institute for Learning Algorithms（蒙特利尔学习算法研究所）；École Polytechnique de Montréal（蒙特利尔理工学院）
Concordance in basal cell carcinoma diagnosis. Building a proper ground truth to train Artificial Intelligence tools
Hospital Universitario Virgen Macarena Seville, Spain（塞维利亚大学医院维珍玛卡伦娜医院）；Dpt. Teoría de la Señal y Comunicaciones Escuela Técnica Superior de Ingeniería Universidad de Sevilla, Seville, Spain（塞维利亚大学电信与通信系）；Hospitales Quironsalud Infanta Luisa y Sagrado Corazón Seville, Spain（quironsalud医院）
