Using a constrained formulation based on probability summation to fit receiver operating characteristic (ROC) curves

A constrained ROC formulation from probability summation is proposed for measuring observer performance in detecting abnormal findings on medical images. This assumes the observer's detection or rating decision on each image is determined by a latent variable that characterizes the specific finding (type and location) considered most likely to be a target abnormality. For positive cases, this 'maximum- suspicion' variable is assumed to be either the value for the actual target or for the most suspicious non-target finding, whichever is the greater (more suspicious). Unlike the usual ROC formulation, this constrained formulation guarantees a 'well-behaved' ROC curve that always equals or exceeds chance- level decisions and cannot exhibit an upward 'hook.' Its estimated parameters specify the accuracy for separating positive from negative cases, and they also predict accuracy in locating or identifying the actual abnormal findings. The present maximum-likelihood procedure (runs on PC with Windows 95 or NT) fits this constrained formulation to rating-ROC data using normal distributions with two free parameters. Fits of the conventional and constrained ROC formulations are compared for continuous and discrete-scale ratings of chest films in a variety of detection problems, both for localized lesions (nodules, rib fractures) and for diffuse abnormalities (interstitial disease, infiltrates or pnumothorax). The two fitted ROC curves are nearly identical unless the conventional ROC has an ill behaved 'hook,' below the constrained ROC.