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Figure 5.8 shows the distributions of measurements of healthy people and ill individuals. An ideal clinical test would be able to totally separate healthy and diseased people—there would be no value at which the distributions of healthy and diseased population overlap. If there is overlap between distributions of test results, some diseased individuals will have normal test results. Below are several terms commonly used to denote the groups:A true positive(TP) is a positive test result obtained for a patient in whom disease is present. (The test result correctly classifies the patient as having the disease). A true negative(TN) is a negative test result obtained for a patient in whom disease is absent. (The test result correctly classifies the patient as not having the disease). A false positive(FP) is a positive test result obtained for a patient in whom disease is absent.
(The test result incorrectly classifies the patient as having the disease). A false negative(FN) is a negative test result obtained for a patient in whom disease is present. (The test result incorrectly classifies the patient as not having the disease). In Figure 5.8, setting different cutoff values between “normal” and “abnormal” across the continuous range of possible values changes the proportions of FPs and FNs for the two populations. As the cutoff increases, the number of FNs increases and FPs decreases. At a given cutoff point, we can construct a 2×2 contingency table, as shown in Table 5.2, to summarize the number of patients in each group. In anideal test, there would be no FN and FP results. However, in reality erroneous test results do occur and contingency tables are used to define the test performance and reflect these errors.Figure 5.8 Distribution of test results in healthy and diseased individuals.Owens & Sox (2006)Table 5.2. A 2×2 contingency table for test results. Test resultDisease presentDisease absentTotalpositiveTPFPTP+FPnegativeFNTNFN+TNTP+FNFP+TNSensitivity and specificity are often used to measure the performance of a classification method. Sensitivity(SE), also called true-positive rate, is the likelihood that a diseased patient has a positive test result. Specificity(SP), also called true-negative rate, is the likelihood that a non-diseased patient has a negative test result.SE=TPTP+FNSP=TNTN+FP
Figure 5.9 Two example ROC curves representing the performance of two classifiers. The Red line represents a better classifier, because the AROC of the Red line is larger than that of Black line. When a classifier is imperfect, we try to achieve a balance between sensitivity and specificity. To do this, we can plot true positive rate (equals to SE) versus false positive rate (equals to 1-SP), at various threshold settings (Swets, 1998); this is called a receiver operating characteristic (ROC) curve.