ABSTRACT
The binormal model, widely used since the 1960s, to fit ROC data in various, mostly non-medical imaging contexts, predicts an inappropriate chance line crossing and a hook that is a patently false prediction. While excuses have been made as to why this discrepancy can be ignored, researchers have been working on methods of fitting proper ROC curves, defined as those that do not show such inappropriate behavior. Related issues are data degeneracy and the ability to fit datasets with no interior ROC operating points. Expert radiologists frequently produce such datasets, and indeed the RSM predicts this behavior for experts. The RSM-based proper ROC curve fitting method was described in Chapter 19. This chapter describes three classical methods for fitting proper ROC curves. The likelihood ratio is defined and is shown that an observer using a likelihood ratio based decision variable is ideal in the Neyman-Pearson sense. The proper ROC, implemented in PROPROC software, uses a complex transformation from the binormal model Z-sample scale to a likelihood ratio scale. PROPROC cannot fit degenerate datasets. The contaminated binormal model (CBM) is simpler, has a resemblance to the RSM, and can fit practically any dataset, including degenerate ones. Finally, for historical completeness, the bigamma model is reviewed. Its basic assumption of a gamma distributed decision variable is inconsistent with the central limit theorem of statistics. Online appendices detail how to view ROC curves and their slopes, and how to plot PROPROC, CBM, and bigamma model predicted ROC curves.
