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Receiver Operating Characteristic Curves and Their Use in Radiology¹

¹ From the Department of Biostatistics and Epidemiology/Wb4, Cleveland Clinic Foundation, 9500 Euclid Ave, Cleveland, OH 44195. Received May 8, 2001; revision requested June 11; revision received August 1; accepted August 2. Address correspondence to the author (e-mail: nobuchow@bio.ri.ccf.org).

View larger version (21K): [in a new window]   Figure 1. Graph of the empirical and fitted ROC curves for the mammography study. The points on the empirical curve are marked with open circles and are estimated in Table 3. The points labeled 1 and 2 on the curve correspond to the first and second cut points, respectively, that are defined in the note to Table 1.

View larger version (20K): [in a new window]   Figure 2. Graph shows the binormal distribution that best fits the mammography study data. By convention, the distribution of unobserved variables for the patients without cancer is centered at zero (ie, µ1 = 0) with variance equal to 1. For these data, the center of the distribution of the unobserved variables for the patients with cancer is estimated to be 1.59 (ie, µ2 = 1.59) with variance estimated to be 1.54. The binormal distribution can be described by its two parameters (4), a and b, as a = (µ1 - µ2)/2 and b = 1/2. The four cut points z1, z2, z3, and z4 define the five categories of test results. That is, a variable with a value below the point defined by z1 indicates a normal result; a variable with a value between z1 and z2, a benign result; a variable with a value between z2 and z3, a probably benign result; a variable with a value between z3 and z4, a suspicious result; and a variable with a value above the point defined by z4, a malignant result. Note that the binormal variables exist only in the mind of the reader (ie, they are unobserved). When the reader applies the cut points z1, z2, z3, and z4 to the unobserved variables, we obtain the observed five categories of test results.

View larger version (24K): [in a new window]   Figure 3. Graph shows comparison of three ROC curves. A perfect test has an area under the ROC curve of 1.0. The chance diagonal has an ROC area of 0.5. Tests with some discriminating ability have ROC areas between these two extremes.

View larger version (19K): [in a new window]   Figure 4. Graph shows two crossing ROC curves. The ROC areas of the two tests are the same at 0.80; however, for the clinically important range (ie, an FPR of less than 0.20), test A is preferable to test B.

View larger version (38K): [in a new window]   Figure A1. Fictitious data set and example of how to calculate the area under the empirical ROC curve.