Extensions of Grier's computational formulas for A′ and B″ to below-chance performance.

Abstract

Oner's (1971) computing expressions for nonparametric indices of sensitivity and bias are briefly reviewed, and shown to be incorrect when performance is below chance levels. Modifications of Grier's expressions, for cases when false alarms exceed hits, are presented, Pollack and Norman (1964) suggested general procedures for developing a nonparametric index of sensitivity, A', for recognition and detection experiments in which only a single data point is obtained in the unit square probability space representing hits and false alarms. Later Hodos (1970) suggested a nonparametric index of response bias, B", also based on the geometry of the square probability space. Finally, Grier (1971) derived explicit functional expressions for computing both of these indices. Grier's formulas have been widely used during the past 15 years, and have proved to be useful when the normal-distribution assumptions underlying traditional signal detection procedures fail, or when the assumptions are not testable.The application of Grier's (1971) formulas to detection data from individual subjects, some of whom perform below chance, can yield bizarre numbers. Some values of A', which should represent the area under an "averaged" ROC curve, can be negative rather than positive, and they can also be greater than 1.0 in absolute value. Further, values of B" for points to the left of the equal-bias diagonal can be negative rather than positive as they should be. The magnitudes of such erroneous A' and B" values can be large even when performance is only slightly below the chance diagonal. Discussions with colleagues suggested that some were unaware of the below-chance problem with Grier's formulas, and had reported values of A' and B* averaged over groups, without examining the individual subjects' data. If only a few subjects in a subset of conditions perform even slightly below chance, the averages would be distorted, but in ways that the researcher might not notice. Thus, we felt that it would be useful to provide a brief note on modifications of Grier's formulas that are appropriate for below-chance performance. Grier's FormulasSensitivity index. First, let us briefly review Grier's (1971) computing expressions. In Figure 1 -left, let y denote the proba-

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