Concordance probability as a meaningful contrast across disparate survival times

The performance of time-to-event models is frequently assessed in part by estimating the concordance probability, which evaluates the probabilistic pairwise ordering of the model-based risk scores and survival times. The standard definition of this probability conditions on any survival time pair ordering, irrespective of whether the times are meaningfully separated. Inclusion of survival times that would be deemed clinically similar attenuates the concordance and moves the estimate away from the contrast-of-interest: comparing the risk scores between individuals with disparate survival times. In this manuscript, we propose a concordance definition and corresponding method to estimate the probability conditional on survival times being separated by at least a minimum difference. The proposed estimate requires direct input from the analyst to identify a separable survival region and, in doing so, is analogous to the clinically defined subgroups used for binary outcome area under the curve estimates. The method is illustrated in two cancer examples: a prognostic score in clear cell renal cell carcinoma and two biomarkers in metastatic prostate cancer.