Drunkard's Walk - Leonard Mlodinow [60]
Finally, we ask, how many people are there in each of these classes? Suppose we consider an initial population of 10,000. We can estimate, employing statistics from the Centers for Disease Control and Prevention, that in 1989 about 1 in those 10,000 heterosexual non-IV-drug-abusing white male Americans who got tested were infected with HIV.6 Assuming that the false-negative rate is near 0, that means that about 1 person out of every 10,000 will test positive due to the presence of the infection. In addition, since the rate of false positives is, as my doctor had quoted, 1 in 1,000, there will be about 10 others who are not infected with HIV but will test positive anyway. The other 9,989 of the 10,000 men in the sample space will test negative.
Now let’s prune the sample space to include only those who tested positive. We end up with 10 people who are false positives and 1 true positive. In other words, only 1 in 11 people who test positive are really infected with HIV. My doctor told me that the probability that the test was wrong—and I was in fact healthy—was 1 in 1,000. He should have said, “Don’t worry, the chances are better than 10 out of 11 that you are not infected.” In my case the screening test was apparently fooled by certain markers that were present in my blood even though the virus this test was screening for was not present.
It is important to know the false positive rate when assessing any diagnostic test. For example, a test that identifies 99 percent of all malignant tumors sounds very impressive, but I can easily devise a test that identifies 100 percent of all tumors. All I have to do is report that everyone I examine has a tumor. The key statistic that differentiates my test from a useful one is that my test would produce a high rate of false positives. But the above incident illustrates that knowledge of the false positive rate is not sufficient to determine the usefulness of a test—you must also know how the false-positive rate compares with the true prevalence of the disease. If the disease is rare, even a low false-positive rate does not mean that a positive test implies you have the disease. If a disease is common, a positive result is much more likely to be meaningful. To see how the true prevalence affects the implications of a positive test, let’s suppose now that I had been homosexual and tested positive. Assume that in the male gay community the chance of infection among those being tested in 1989 was about 1 percent. That means that in the results of 10,000 tests, we would find not 1 (as before), but 100 true positives to go with the 10 false positives. So in this case the chances that a positive test meant I was infected would have been 10 out of 11. That’s why, when assessing test results, it is good to know whether you are in a high-risk group.
BAYES’S THEORY shows that the probability that A will occur if B occurs will generally differ from the probability that B will occur if A occurs.7 To not account for this is a common mistake in the medical profession. For instance, in studies in Germany and the United States, researchers asked physicians to estimate the probability that an asymptomatic woman between the ages of 40 and 50 who has a positive mammogram actually has breast cancer if 7 percent of mammograms show cancer when there is none.8 In addition, the doctors were told that the actual incidence was about 0.8 percent and that the false-negative rate about 10 percent. Putting that all together, one can use Bayes’s methods to determine that a positive mammogram is due to cancer in only about 9 percent of the cases. In the German group, however, one-third of the physicians concluded that the