A general area theorem for the <Emphasis Type="Italic">samedifferent </Emphasis> paradigm

Green & Swets, 1966), the area under the receiver operating characteristic (ROC) in the psychophysical yes–no (YN) paradigm is equal to the proportion of correct responses in the two-interval two-alternative forced choice (2I2AFC) paradigm. This “area theorem” has both theoretical and practical value. It establishes a link between two psychophysical paradigms, and it allows experimenters to convert hit and false alarm rates measured using the YN paradigm into an equivalent proportion of correct responses in the less bias prone 2I2AFC paradigm. In an insightful text on the decision rules for various psychophysical paradigms, Noreen (1981) proposed an extension of Green’s original area theorem to another popular psychophysical paradigm: the two-interval same–different (also known as AX or, more precisely, 2IAX) paradigm. Noreen predicted that the area under the ROC in that paradigm should equal the proportion of correct responses in the four-interval same–different (4IAX) paradigm, PC4IAX. In the latter paradigm, also known as the dual-pair paradigm (Creelman & Macmillan, 1979; Macmillan, Kaplan, & Creelman, 1977; Rousseau & Ennis, 2001), the observer is presented on each trial with two pairs of stimuli. One pair contains two stimuli from the same distribution (the same stimuli), while the other pair contains stimuli from different distributions (the different stimuli). The task is to indicate which of the two pairs contains the different stimuli. Noreen remarked that each stimulus within a pair in the 2IAX paradigm could be likened to the single stimulus presented on a trial of the YN paradigm, and that each pair of stimuli in the 4IAX paradigm could be likened to one of the two stimuli presented on a trial of the 2I2AFC paradigm. On the basis of these analogies, Noreen suggested that it followed from Green’s area theorem that the area under the ROC for the 2IAX paradigm should equal PC4IAX. In this note, we describe a mathematical proof of Noreen’s (1981) conjecture. The proof is general, in the sense that it does not require that the sensory observations have a specific (e.g., Gaussian) distribution, or that they be statistically independent. This is important, because the degree of statistical dependence between the sensory observations has been shown to influence the decision rule of an optimal observer in both the 2IAX and the 4IAX paradigms (Dai, Versfeld, & Green, 1996; Micheyl & Messing, 2006). It has been shown that when (Gaussian) observations are highly correlated, the “differencing” strategy, which involves comparing the two stimuli in a pair, is optimal in the likelihood ratio sense. However, when the observations are uncorrelated, a more elaborate independent observations decision rule becomes available, and the “differencing” strategy is no longer optimal. Dependent observations are likely to arise in experiments in which the stimuli are randomized (or roved ) across trials, so that the two (or four) stimuli presented on a trial are more likely to fall closer together in the parameter space than stimuli from different trials.1 The mathematical proof described below applies equally well to the results of such experiments as to those of experiments in which no roving was used, and regardless of which decision rule is assumed.