Event

The ROC curve is a popular graphical method used to study the diagnostic capacity of biomarkers. In its simplest form it plots true-positive rates against false-positive rates. Both practical and theoretical aspects of the properties of ROC curves have been extensively studied. Conventionally, it is assumed that the considered marker has a monotone relationship with the studied characteristic, that is, the upper (lower) values of the biomarker are associated with a higher (lower) probability of a positive result. In many real situations, however, both the lower AND the upper values of the marker are associated with higher probability of a positive result. We propose an ROC curve generalization which is useful in this context. All pairs of possible cut-off points, one for the lower and another one for the upper biomarker value, are considered and the best one is selected. Moreover, the empirical estimator for the curve is considered and its uniform consistence and asymptotic distribution derived. The theoretical framework we provide allow us to derive ROC curve generalizations for multivariate biomarkers.