ABSTRACT
The equal variance binormal model was described in Chapter 02. In Chapter 04 it was shown, for a single clinical dataset, that the unequal variance binormal model fit the data better. This is a universal finding in many ROC studies dating to the 1960s, not just limited to medical imaging. The main aim of this chapter is to demystify statistical curve fitting. It starts with a description of the binormal model, and how it accommodates data binning. The invariance of the binormal model to arbitrary monotone transformations of the ratings is demonstrated with an
R
example – this gives the model unusual robustness with respect to deviations from strict normality. Expressions for sensitivity and specificity are derived. Two notations used to characterize the binormal model are explained. Expressions for the pdfs of the binormal model are derived. A simple linear fitting method is illustrated – this used to be the only recourse a researcher had before Dorfman and Alf's seminal 1969 publication. The maximum likelihood method for estimating parameters of the binormal model is detailed and an R-implementation is compared to a website implementation of ROCFIT, a commonly used software for fitting the binormal model to ROC data. Validation of the fitting method is described, i.e., how can one be confident that the fitting method, which makes normality and other assumptions, is valid for a dataset arising from an unknown distribution. An Appendix has a detailed derivation, originally published in a terse paper on the partial area under the ROC curve, a special case of which yields the total area under the binormal model fitted curve. An online appendix describes methods for debugging R code and displaying plots. Another appendix describes calculation of variance of binormal fitted AUC.
