ABSTRACT
Computational formulae are provided for AUCs under various operating characteristics defined in the previous chapter. The important ones, namely AUCs under the AFROC and the weighted AFROC are detailed. Formulae for other AUCs (e.g., FROC and inferred ROC) are in the online appendix to this chapter. Small dataset examples, which permit hand-calculations, are used to illustrate the AFROC and wAFROC AUC, allowing the reader to appreciate that the AFROC gives undue importance to cases with more lesions, while the wAFROC corrects this deficiency. Two theorems are derived. The first shows the equivalence between the empirical AUC under the wAFROC and a quasi-Wilcoxon statistic. This theorem is the free-response equivalent of Bamber's theorem relating the empirical ROC AUC to a Wilcoxon statistic. The second theorem derives an expression for the area under the straight-line extension from the empirical end-point to (1,1). The contribution of this area increases as the abscissa of the end-point of the wAFROC decreases, i.e., as more non-diseased cases are not marked and as ordinate of the end-point increases, i.e., as more lesions, especially those with greater weights, are marked. Other online appendices have details of the proofs and numerical-integration based demonstrations of the AUC vs. quasi Wilcoxon statistic equivalences.
