On comparing 2 correlated C indices with censored survival data

As new biomarkers and risk prediction procedures are in rapid development, it is of great interest to develop valid methods for comparing predictive power of 2 biomarkers or risk score systems. Harrell C statistic has been routinely used as a global adequacy assessment of a risk score system, and the difference of 2 Harrell C statistics as a test statistic has been suggested in recent literature for comparison of predictive power of 2 biomarkers for censored outcome. In this study, we showed that such a test can have severely inflated type I errors as the difference between the 2 Harrell C statistics does not converge to zero under the null hypothesis of equal predictive power measured by concordance probabilities, as illustrated by 2 counterexamples and corresponding numerical simulations. We further investigate a necessary and sufficient condition under which the difference of 2 Harrell C statistics converges to zero under the null hypothesis.