Semiparametric Estimation of the Relationship between ROC Operating Points and the Test-result Scale: Application to the Proper Binormal Model

Abstract

Semiparametric methods provide smooth and continuous receiver operating characteristic (ROC) curve fits to ordinal test results and require only that the data follow some unknown monotonic transformation of the model's assumed distributions. The quantitative relationship between cutoff settings or individual test-result values on the data scale and points on the estimated ROC curve is lost in this procedure, however. To recover that relationship in a principled way, we propose a new algorithm for "proper" ROC curves and illustrate it by use of the proper binormal model. Several authors have proposed the use of multinomial distributions to fit semiparametric ROC curves by maximum-likelihood estimation. The resulting approach requires nuisance parameters that specify interval probabilities associated with the data, which are used subsequently as a basis for estimating values of the curve parameters of primary interest. In the method described here, we employ those "nuisance" parameters to recover the relationship between any ordinal test-result scale and true-positive fraction, false-positive fraction, and likelihood ratio. Computer simulations based on the proper binormal model were used to evaluate our approach in estimating those relationships and to assess the coverage of its confidence intervals for realistically sized datasets. In our simulations, the method reliably estimated simple relationships between test-result values and the several ROC quantities. The proposed approach provides an effective and reliable semiparametric method with which to estimate the relationship between cutoff settings or individual test-result values and corresponding points on the ROC curve.