A general area theorem for the same-different paradigm

According to a well-known theorem in psychophysics (Green & Swets, 1966), the area under the receiver operating characteristic (ROC) for the yes-no paradigm equals the proportion of correct responses of an unbiased observer in the two-interval, two-alternative, forced choice paradigm (2I2AFC). Here, we demonstrate a similar relationship between the ROC area in the two-interval same-different (AX or 2IAX) paradigm, and the proportion correct in the four-interval same-different (4IAX, also known as dual-pair comparison) paradigm. The theorem demonstrated here is general, in the sense that it does not require that the sensory observations have a specific distribution (e.g., Gaussian), or that they be statistically independent.