Copula modeling of receiver operating characteristic and predictiveness curves

Copula modeling for the estimation of measures of marker classification and predictiveness performance with survival outcomes

The discriminative and predictive power of a continuous-valued marker for survival outcomes can be summarized using the receiver operating characteristic and predictiveness curves, respectively. In this paper, fully parametric and semi-parametric copula-based constructions of the joint model of the marker and the survival time are developed for characterizing, plotting, and analyzing both curves along with other underlying performance measures. The formulations require a copula function, a parametric specification for the margin of the marker, and either a parametric distribution or a non-parametric estimator for the margin of the time to event, to respectively characterize the fully parametric and semi-parametric joint models. Estimation is carried out using maximum likelihood and a two-stage procedure for the parametric and semi-parametric models, respectively. Resampling-based methods are used for computing standard errors and confidence bounds for the various parameters, curves, and associated measures. Graphical inspection of residuals from each conditional distribution is employed as a guide for choosing a copula from a set of candidates. The performance of the estimators of various classification and predictiveness measures is assessed in simulation studies, assuming different copula and censoring scenarios. The methods are illustrated with the analysis of two markers using the familiar primary biliary cirrhosis data set.