An extension of Green’s area theorem.

In signal detection theory (SDT), the well-known Green’s area theorem establishes an equivalence between the area under the receiver operating characteristic (ROC) curve with the hit probability ( H ) and the false-alarm probability ( F ) in the yes–no paradigm and the proportion of correct responses ( P c ) of an unbiased observer in the two-alternative forced-choice (2AFC) paradigm. This theoretical note extends Green’s area theorem from the 2AFC to the specified ‘ M + N ’, which is a generalization of many sensory discrimination forced-choice methods including the 2AFC as its special case. It is demonstrated that the area under the ROC curve with the hit probability h ( H ,N) and the false-alarm probability f ( F ,M) equals the P c of an unbiased observer in the specified ‘ M + N ’. An analytical expression of a general psychometric function for the specified ‘ M + N ’ is derived. A logistic variant of the general psychometric function is also provided. [ABSTRACT FROM AUTHOR]