Derivation of a detectability index for correlated responses in multiple-alternative forced-choice experiments

In a recent paper [J. Opt. Soc. Am. A 17, 206 (2000)], a modified detectability index, d'r, was proposed to accommodate correlations between internal responses in multiple-alternative forced-choice (MAFC) experiments. The derivation given in that work pertained only to two-alternative forced choice, although it was shown empirically that the result held for general MAFC tasks when the correlation between responses is constant. Here we present a rigorous derivation that shows that the d'r result generalizes to MAFC tasks in this case.