A cautionary note on two measures of explained variation in survival analysis

Abstract

In the paper we cast serious doubt on the usefulness of the index of concordance (C-index) and the coefficient of determination for survival models. The index of concordance is known for not being useful when selecting the best of several competing models because of its nonsensitivity. We show that it can even go down when a significant covariate is added to a correctly specified model, including cases when there is no censoring and no tied event times or covariate values, meaning that the usual suspects are not responsible. The coefficient of determination was originally suggested because its formula gives the usual R² when used in linear regression. But, the analogy with the linear model is gone when we use survival analysis models, since it crucially depends on the null model which changes with the change in the fitted model. We illustrate the expected behaviour of such measures and compare with an alternative Schemper–Henderson measure.