title: Three-class ROC analysis--toward a general decision theoretic solution.
abstract: Multiclass receiver operating characteristic (ROC) analysis has remained an open theoretical problem since the introduction of binary ROC analysis in the 1950s. Previously, we have developed a paradigm for three-class ROC analysis that extends and unifies decision theoretic, linear discriminant analysis, and probabilistic foundations of binary ROC analysis in a three-class paradigm. One critical element in this paradigm is the equal error utility (EEU) assumption. This assumption allows us to reduce the intrinsic space of the three-class ROC analysis (5-D hypersurface in 6-D hyperspace) to a 2-D surface in the 3-D space of true positive fractions (sensitivity space). In this work, we show that this 2-D ROC surface fully and uniquely provides a complete descriptor for the optimal performance of a system for a three-class classification task, i.e., the triplet of likelihood ratio distributions, assuming such a triplet exists. To be specific, we consider two classifiers that utilize likelihood ratios, and we assumed each classifier has a continuous and differentiable 2-D sensitivity-space ROC surface. Under these conditions, we proved that the classifiers have the same triplet of likelihood ratio distributions if and only if they have the same 2-D sensitivity-space ROC surfaces. As a result, the 2-D sensitivity surface contains complete information on the optimal three-class task performance for the corresponding likelihood ratio classifier.
authors.affiliation: Department of Radiology, Johns Hopkins School of Medicine, Baltimore, MD 21287, USA. xinhe@jhmi.edu
authors.forename: Xin
authors.initials: X
authors.lastname: He
authors.forename: Brandon D
authors.initials: BD
authors.lastname: Gallas
authors.forename: Eric C
authors.initials: EC
authors.lastname: Frey
mesh_terms: D000465:Algorithms; D003661:Decision Support Techniques; D003952:Diagnostic Imaging / Q000379:methods*; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D012372:ROC Curve