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 likelihood-ratio criterion.