"Proper" Binormal ROC Curves: Theory and Maximum-Likelihood Estimation

J Math Psychol. 1999 Mar;43(1):1-33. doi: 10.1006/jmps.1998.1218.


The conventional binormal model, which assumes that a pair of latent normal decision-variable distributions underlies ROC data, has been used successfully for many years to fit smooth ROC curves. However, if the conventional binormal model is used for small data sets or ordinal-category data with poorly allocated category boundaries, a "hook" in the fitted ROC may be evident near the upper-right or lower-left corner of the unit square. To overcome this curve-fitting artifact, we developed a "proper" binormal model and a new algorithm for maximum-likelihood (ML) estimation of the corresponding ROC curves. Extensive simulation studies have shown the algorithm to be highly reliable. ML estimates of the proper and conventional binormal ROC curves are virtually identical when the conventional binormal ROC shows no "hook," but the proper binormal curves have monotonic slope for all data sets, including those for which the conventional model produces degenerate fits. Copyright 1999 Academic Press.