A note on ROC analysis and non-parametric estimate of sensitivity

In the signal detection paradigm, the non-parametric index of sensitivity A', as first introduced by Pollack and Norman (1964), is a popular alternative to the more traditional d' measure of sensitivity. Smith (1995) clarified a confusion about the interpretation of A' in relation to the area beneath proper receiver operating characteristic (ROC) curves, and provided a formula (which he called A) for this commonly held interpretation. However, he made an error in his calculations. Here, we rectify this error by providing the correct formula (which we call A) and compare the discrepancy that would have resulted. The corresponding measure for bias b is also provided. Since all such calculations apply to proper ROC curves with non-decreasing slopes, we also prove, as a separate result, the slope-monotonicity of ROC curves generated by the likelihood-ratio criterion.