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• #### Committee and Staff Biographies 408-414

be the point on the ROC curve where its slope (or tangent) is [(1 − 0.1) × 0.05]/[0.1 × 0.5] or 0.90, a point near the middle of most ROC curves, with modest true-positive and false-positive rates. If we use the distributions displayed in Figure 9.2 and the ROC curve displayed in Figure 9.3, the optimal criterion of positivity would correspond to a true-positive rate (sensitivity) of 80% and a false positive rate of 17% (a specificity of 83%). However, if one were considering screening a population in which the prevalence of disease is [subjunctive case is were] only 1%, then the best criterion for positivity would be the point on the ROC curve where its slope (or tangent) is [(1 − 0.01) × 0.05]/[0.01 × 0.5] or 9.9, a point nearer to the origin for most ROC curves, with both true-positive and false-positive rates low. Again, if we use the distributions displayed in Figure 9.2 and the ROC curve displayed in Figure 9.3, the optimal criterion of positivity would correspond to a true-positive rate (sensitivity) of 42% and a false-positive rate of 1% (a specificity of 99%).

In the special case when both probability density distributions (patients with and without disease) are normal or Gaussian in shape, the slope of the corresponding ROC at any point (the ratio of the heights of the corresponding density distributions) can be solved algebraically, although the equation is fairly complex. Because the normal distribution is

where x is the test result, is the mean, and σ is the standard deviation, the optimal cutoff criterion will be the value of x where

The value of x at which the equality holds can be found by successive approximations or using the “goal seek” function in a spreadsheet program.

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