Is the area measure a historical anomaly?

Green's well-known area theorem establishes an equivalence between the area under the yes-no ROC curve and the percent correct of an unbiased observer in a two-alternative forced-choice (2AFC) task with equivalent stimuli. In this article, we show that this conversion from yes-no detection data to hypothetical performance in a 2AFC task is unnecessary: The same yes-no detection data that are used to compute the area statistic can always be used to compute the percent correct of an unbiased observer in the yes-no detection task itself. We also show that the ROC curve may not be the ideal graphical device for many investigators. A more natural representation of the difficulty of a discrimination task is obtained by plotting the distribution of the posterior betting odds under equal base rates, which can be estimated from their distributions under unequal base rates. Finally, unlike the area measure and other traditional detection theory statistics, both the yes-no percent correct measure and the odds distributions generalize in an obvious and direct way to classification paradigms with more than two responses (e.g., identification).