The length of the receiver operating characteristic curve and the two cutoff Youden index within a robust framework for discovery, evaluation, and cutoff estimation in biomarker studies involving improper receiver operating characteristic curves.

During the early stage of biomarker discovery, high throughput technologies allow for simultaneous input of thousands of biomarkers that attempt to discriminate between healthy and diseased subjects. In such cases, proper ranking of biomarkers is highly important. Common measures, such as the area under the receiver operating characteristic (ROC) curve (AUC), as well as affordable sensitivity and specificity levels, are often taken into consideration. Strictly speaking, such measures are appropriate under a stochastic ordering assumption, which implies, without loss of generality, that higher measurements are more indicative for the disease. Such an assumption is not always plausible and may lead to rejection of extremely useful biomarkers at this early discovery stage. We explore the length of a smooth ROC curve as a measure for biomarker ranking, which is not subject to directionality. We show that the length corresponds to a divergence, is identical to the corresponding length of the optimal (likelihood ratio) ROC curve, and is an appropriate measure for ranking biomarkers. We explore the relationship between the length measure and the AUC of the optimal ROC curve. We then provide a complete framework for the evaluation of a biomarker in terms of sensitivity and specificity through a proposed ROC analogue for use in improper settings. In the absence of any clinical insight regarding the appropriate cutoffs, we estimate the sensitivity and specificity under a two-cutoff extension of the Youden index and we further take into account the implied costs. We apply our approaches on two biomarker studies that relate to pancreatic and esophageal cancer.