Comparing Two Correlated C Indices with Right-censored Survival Outcome: a One-shot Nonparametric Approach

The area under the receiver operating characteristic curve is often used as a summary index of the diagnostic ability in evaluating biomarkers when the clinical outcome (truth) is binary. When the clinical outcome is right-censored survival time, the C index, motivated as an extension of area under the receiver operating characteristic curve, has been proposed by Harrell as a measure of concordance between a predictive biomarker and the right-censored survival outcome. In this work, we investigate methods for statistical comparison of two diagnostic or predictive systems, of which they could either be two biomarkers or two fixed algorithms, in terms of their C indices. We adopt a U-statistics-based C estimator that is asymptotically normal and develop a nonparametric analytical approach to estimate the variance of the C estimator and the covariance of two C estimators. A z-score test is then constructed to compare the two C indices. We validate our one-shot nonparametric method via simulation studies in terms of the type I error rate and power. We also compare our one-shot method with resampling methods including the jackknife and the bootstrap. Simulation results show that the proposed one-shot method provides almost unbiased variance estimations and has satisfactory type I error control and power. Finally, we illustrate the use of the proposed method with an example from the Framingham Heart Study.