Slopes of a receiver operating characteristic curve and likelihood ratios for a diagnostic lest

This paper clarifies two important concepts in clinical epidemiology: the s lope of a receiver operating characteristic (ROC) curve and the likelihood ratio. II: 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 poi nt on the curve, and the slope between two points on the curve. It also poi nts 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 like lihood ratio for a particular single test value, the likelihood ratio for a positive test result, and the likelihood ratio far a test result in a part icular level or category. It further illustrates mathematically and empiric ally the following three relations between these various definitions of slo pes and likelihood ratios: I)the tangent at a point on the ROC curve corres ponds to the likelihood ratio for a single test value represented by that p oint; 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 positiv ity; and 3) the slope between two points on the curve corresponds to the li kelihood ratio for a test result in a defined level bounded by the two poin ts. 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 u sing ROC analysis, the likelihood ratio for a single test value can be easi ly measured from the tangent. It is suggested that existing ROC analysis so ftware be revised to provide estimates for tangents at various points on th e ROC curve.