Multiple-Event Forced-Choice Tasks in the Theory of Signal Detectability

Receiver operating characteristic (ROC) analysis is generalized to unidimensional forced-choice tasks involving three or more events. It is shown that the performance of an observer in a unidimensional identification task with n independent events can be represented in n! ROC spaces of dimension n. Each ROC space is associated with a unique pairing of the events and decisions. A hypersurface can be generated in each ROC space by manipulating the observer's decision criteria. Using information theory, a new measure of discriminability based on the hypervolumes under the hypersurfaces is defined. This measure, denoted D, is nonparametric and independent of the criteria. The value of D is shown to increase monotonically with n and to be equal to the channel capacity of an observer in an n-interval forced-choice task. Procedures to compute ROC hypersurfaces from ratings and from ROC curves are given.