As we mentioned in Example 3.14, diagnostic accuracy alone is not enough to determin...

As we mentioned in Example 3.14, diagnostic accuracy alone is not enough to determine whether a test is good. In this problem, we will introduce the concept of receiver operating characteristic (ROC) curves with a simple example. Consider the situation described in Problem 3.26; suppose the mean and variance of the test value for the normal subjects are and with arbitrary units. The mean and variance of the test value for the diseased subjects are with arbitrary units.

(a) For threshold value t₀ in the interval plot the curve of Sensitivity (vertical axis) against 1 − specificity (horizontal axis). This is the ROC curve.

(b) What is the ROC curve for a perfect diagnostic test?

(c) For a given ROC curve, if we define the optimal test to be the one that is closest to the perfect test, determine the optimal test (threshold) for the above case.