title: A contaminated binormal model for ROC data: Part II. A formal model.
abstract: OBJECTIVE
  A contaminated binormal receiver operating characteristic (ROC) model is proposed to account for ROC data that have very few false-positive reports even though many healthy subjects are sampled. The model assumes that no signal information is captured for a proportion of abnormalities, and that these abnormalities have the same distribution as noise along the latent decision axis.
  
  
  METHODS
  The authors developed a formal psychophysical model, presented here in detail. They have specified the psychophysical assumptions of the theory, and have provided proofs that include all essential details, from assumptions to implications. With the technical details that are provided, this theory can be implemented with computer programs to fit data.
  
  
  RESULTS
  The new model can fit ROC data in which some or all of the ROC points have false-positive fractions of 0 and true-positive fractions of less than 1, without implying that performance is perfect. The resulting ROC curves are always proper, never exhibiting inappropriate chance line crossings. The model predicts that, under certain conditions, a bimodal categorical rating histogram will be observed for the signal distribution. The model predicts a relationship between the mean and standard deviation of the signal distribution and holds that, for expert decision makers, there are situations in which the prevalence and utility matrix preclude operating points in some ROC regions. The model has a straightforward extension to the joint detection and localization ROC curve.
  
  
  CONCLUSIONS
  The contaminated binormal model accounts for ROC data with few or no false-positive reports.
authors.affiliation: Department of Radiology, University of Iowa, Iowa City 52242, USA.
authors.forename: D D
authors.initials: DD
authors.lastname: Dorfman
authors.forename: K S
authors.initials: KS
authors.lastname: Berbaum
mesh_terms: D003662:Decision Theory; D003952:Diagnostic Imaging / Q000706:statistics & numerical data*; D006801:Humans; D015233:Models, Statistical; D011601:Psychophysics; D012372:ROC Curve