Abstract
Receiver operating characteristic (ROC) curves have application in analysis of the performance of diagnostic indicators used in the assessment of disease risk in clinical and veterinary medicine and in crop protection. For a binary indicator, an ROC curve summarizes the two distributions of risk scores obtained by retrospectively categorizing subjects as cases or controls using a gold standard. An ROC curve may be symmetric about the negative diagonal of the graphical plot, or skewed towards the left-hand axis or the upper axis of the plot. ROC curves with different symmetry properties may have the same area under the curve. Here, we characterize the symmetry properties of bi-Normal and bi-gamma ROC curves in terms of the Kullback-Leibler divergences (KLDs) between the case and control distributions of risk scores. The KLDs describe the known symmetry properties of bi-Normal ROC curves, and newly characterize the symmetry properties of constant-shape and constant-scale bi-gamma ROC curves. It is also of interest to note an application of KLDs where their asymmetry—often an inconvenience—has a useful interpretation.
Highlights
Receiver operating characteristic (ROC) curve analysis provides a basis for describing the performance of a diagnostic indicator when deployed in a binary diagnostic test
We refer to this kind of skew as true positive proportion (TPP)-asymmetry, and to the kind of skew where the curve clings to the top edge of the ROC space longer than it does to the left as true negative proportion (TNP)-asymmetry [17]
Notwithstanding, it is sometimes rather difficult to tell from a graphical plot whether an empirical ROC curve is symmetrical or only approximately so (e.g., Figure 2 in [24])
Summary
Receiver operating characteristic (ROC) curve analysis provides a basis for describing the performance of a diagnostic indicator when deployed in a binary diagnostic test. We can present the results graphically as frequency distributions of risk scores plotted separately for cases and controls. We are concerned with the properties of ROC curves based on continuous parametric models for the distributions of risk scores (e.g., [5,7]). The idea is that diagnostic indicators with ROC curves which pass close to the top left-hand corner of the graphical plot of TPP against FPP (high AUC) provide tests for which TPP and TNP are both high, offering good discrimination between cases and controls. Diagnostic indicators with ROC curves close to the main diagonal of the plot of TPP against FPP (low AUC) have little to offer in terms of discrimination between cases and controls.
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