Slopes of a Receiver Operating Characteristic Curve and Likelihood Ratios for a Diagnostic Test

This paper clarifies two important concepts in clinical epidemiology: the slope of a receiver operating characteristic (ROC) curve and the likelihood ratio. It points out that there are three types of slopes in an ROC curve--the tangent at a point on the curve, the slope between the origin and a point on the curve, and the slope between two points on the curve. It also points out that there are three types of likelihood ratios that can be defined for a diagnostic test that produces results on a continuous scale--the likelihood ratio for a particular single test value, the likelihood ratio for a positive test result, and the likelihood ratio for a test result in a particular level or category. It further illustrates mathematically and empirically the following three relations between these various definitions of slopes and likelihood ratios: 1) the tangent at a point on the ROC curve corresponds to the likelihood ratio for a single test value represented by that point; 2) the slope between the origin and a point on the curve corresponds to the positive likelihood ratio using the point as a criterion for positivity; and 3) the slope between two points on the curve corresponds to the likelihood ratio for a test result in a defined level bounded by the two points. The likelihood ratio for a single test value is considered an important parameter for evaluating diagnostic tests, but it is not easily estimable directly from laboratory data because of limited sample size. However, by using ROC analysis, the likelihood ratio for a single test value can be easily measured from the tangent. It is suggested that existing ROC analysis software be revised to provide estimates for tangents at various points on the ROC curve.