An application of Harrell's C‐index to PH frailty models

Frailty models are encountered in many medical applications, yet little research has been devoted to develop measures that quantify the predictive ability of these models. In this paper, we elaborate on the concept of the concordance probability to clustered data, resulting in an 'Overall Conditional C-index' or bfC(O, C) and an 'Overall Marginal C-index' or C(O, M) . Both Overall C-indices can be split up into a 'Between Conditional' or C(B, C) and a 'Between Marginal C-index' or C(B, M) and into a 'Within Conditional' or C(W, C) and a 'Within Marginal C-index' or C(W, M) . For PH frailty models of the power variance family, C(W, C) and C(W, M) are equivalent resulting in one 'Within C-index' C(W) . We propose an application of Harrell's C-index to estimate the proposed indices within a likelihood and a Bayesian context and the performances of their point estimates and confidence/credible intervals are compared in an extensive simulation study. This simulation study shows that the point estimates of C(W) and C(B, M) perform good within both a likelihood and Bayesian context but that the point estimates of C(B, C) show less bias for the Bayesian approach than for the likelihood approach. The 95 per cent confidence/credible intervals also possess good coverage properties, given that the point estimates perform good. The performance of the C-indices is evaluated on a real data set.