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Three-Class ROC Analysis—Toward a General Decision Theoretic Solution

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3 Author(s)
Xin He ; Sch. of Med., Dept. of Radiol., Johns Hopkins Univ., Baltimore, MD, USA ; Gallas, B.D. ; Frey, E.C.

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.

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Medical Imaging, IEEE Transactions on  (Volume:29 ,  Issue: 1 )