Bivariate marker measurements and ROC analysis.

This article considers receiver operating characteristic (ROC) analysis for bivariate marker measurements. The research interest is to extend tools and rules from univariate marker to bivariate marker setting for evaluating predictive accuracy of markers using a tree-based classification rule. Using an and-or classifier, an ROC function together with a weighted ROC function (WROC) and their conjugate counterparts are proposed for examining the performance of bivariate markers. The proposed functions evaluate the performance of and-or classifiers among all possible combinations of marker values, and are ideal measures for understanding the predictability of biomarkers in target population. Specific features of ROC and WROC functions and other related statistics are discussed in comparison with those familiar properties for univariate marker. Nonparametric methods are developed for estimating ROC-related functions (partial) area under curve and concordance probability. With emphasis on average performance of markers, the proposed procedures and inferential results are useful for evaluating marker predictability based on a single or bivariate marker (or test) measurements with different choices of markers, and for evaluating different and-or combinations in classifiers. The inferential results developed in this article also extend to multivariate markers with a sequence of arbitrarily combined and-or classifier.