Equivalence of binormal likelihood-ratio and bi-chi-squared ROC curve models.

A basic assumption for a meaningful diagnostic decision variable is that there is a monotone relationship between it and its likelihood ratio. This relationship, however, generally does not hold for a decision variable that results in a binormal receiver operating characteristic (ROC) curve. As a result, ROC curve estimation based on the assumption of a binormal ROC-curve model produces improper ROC curves, which have 'hooks', are not concave over the entire domain and cross the chance line. Although in practice this 'improperness' is usually not noticeable, sometimes it is evident and problematic. To avoid this problem, Metz and Pan proposed basing ROC-curve estimation on the assumption of a binormal likelihood-ratio (binormal-LR) model, which states that the decision variable is an increasing transformation of the likelihood-ratio function of a random variable having normal conditional diseased and nondiseased distributions. However, their development is not easy to follow. I show that the binormal-LR model is equivalent to a bi-chi-squared model in the sense that the families of corresponding ROC curves are the same. The bi-chi-squared formulation provides an easier-to-follow development of the binormal-LR ROC curve and its properties in terms of well-known distributions.