ROC analysis for multiple markers with tree-based classification

Multiple biomarkers are frequently observed or collected for detecting or understanding a disease. The research interest of this article is to extend tools of receiver operating characteristic (ROC) analysis from univariate marker setting to multivariate marker setting for evaluating predictive accuracy of biomarkers using a tree-based classification rule. Using an arbitrarily combined and-or classifier, an ROC function together with a weighted ROC function (WROC) and their conjugate counterparts are introduced for examining the performance of multivariate markers. Specific features of the 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 the ROC and WROC functions, and area under curve and concordance probability. With emphasis on population average performance of markers, the proposed procedures and inferential results are useful for evaluating marker predictability based on multivariate marker measurements with different choices of markers, and for evaluating different and-or combinations in classifiers.