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 (Pc) 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 Pc 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.