Receiver Operating Characteristic (ROC) Curve

In practice, the outcomes of diagnostic tests are mostly interpreted and operationalized as binary—that is, as positive or negative—for the presence of a target condition. However, the actual outcome of a test is rarely a binary one. For example, the results of laboratory tests are typically measured on continuous scales, and the same applies to measures summarizing scans with modern imaging modalities, such as the standardized uptake value in positron emission tomography. When test results are measured on an explicitly defined and observed scale as in these examples, a binary outcome is defined on the basis of an explicit threshold for test positivity. When tests involve interpretation by a human observer, a similar model with a threshold for test positivity has been used widely. However, in this case the test result and the threshold are conceptualized as occurring on a latent scale, measuring the [Page 954]interpreter's degree of suspicion about the presence of the target condition.

Because binary test outcomes are obtained through thresholds on an observed or a latent scale, these thresholds affect all the usual measures of diagnostic and predictive performance, including sensitivity, specificity, positive predictive value, negative predictive value, and likelihood ratios. The dependence of measures of test performance on the threshold for test positivity is a fundamental tenet of diagnostic test evaluation. In particular, this dependence induces the well-known trade-off between test sensitivity and specificity as the threshold for positivity is moved across its possible values.

Figure 1 shows hypothetical distributions of test results for individuals with and without the target condition and a threshold for test positivity. If the likelihood of having the target condition increases with the test score, the sensitivity of the test is measured by the area under the “condition-present” curve, to the right of the threshold. Similarly, the specificity is measured by the area under the “condition-absent” curve, to the left of the threshold. In the formulation of Figure 1, the sensitivity of the test is a decreasing function of the threshold value, and the specificity of the test is an increasing function of the threshold value.

The receiver operating characteristic (ROC) curve of a test is the graph of all possible pairs of (1 – specificity, sensitivity) obtained by varying the positivity threshold across its entire range of possible values. As can be seen from Figure 1, when the threshold moves to the left end of its range, sensitivity becomes 1 and specificity becomes 0. The converse occurs when the threshold moves to the right end of its range. Figure 2 shows a typical ROC curve.