Nonparametric Relationships between Single-Interval and Two-Interval Forced-Choice Tasks in the Theory of Signal Detectability

Abstract

Green's relationship, A SI= P(C) 2I, which equates the area, A SI, under the receiver operating characteristic (ROC) curve in the single-interval forced-choice (SIFC) task with the proportion correct, P(C) 2I, in the two-interval forced-choice (2IFC) task, is rederived using the cross-correlation functions of the SIFC evidence distributions. The relationship is generalized to include discrete random variables, unidimensional decision axes that do not need to be monotonic with likelihood ratio, and arbitrary prior and guessing probabilities. A 2IFC difference decision rule is assumed. Further nonparametric relationships, including an equality between an entropy transform of A SI and the 2IFC channel capacity, nonparametric bounds on the area under the 2IFC ROC curve in terms of A SI, and methods for estimating 2IFC ROC curves based on information from the SIFC task, are developed. These relationships are investigated experimentally. Experiment I is a frequency-discrimination task where the evidence is known to be distributed as a discrete random variable. Experiment II is an amplitude-discrimination task where the theoretical evidence distributions are continuously distributed. The problem of observer inconsistency is addressed by repeating the experiments multiple times, using the same stimuli, then using group operating characteristic (GOC) analysis to remove unique noise. Results from Experiment I show excellent support for all the theoretical relationships, and results from Experiment II show partial support for the theoretical relationships.