I took one applied math course in college, and in it I learned a bunch of things I have never since applied to anything, anywhere, ever. I really liked the professor, though; his name was Dr. Elderkin and he had these enormous hands that he would flap open and closed while he was lecturing, creating gale force winds that blew chalk dust around the room. Mrs. T.G. recently pointed out that I do this myself sometimes, so apparently the habit had quite an effect on me.
One of the useful things I learned from Dr. Elderkin is why screening for, e.g., diseases across populations is counter-productive and makes for bad social policy. Here's an example of the painfully stretched metaphor I tried to construct above.
- Say the test for HIV antibodies in the blood test correctly identifies the presence of the antibodies 99% of the time, and 99% of the time it will correctly tell an uninfected person that he or she is not infected. Let's further guess that one million people in the US are infected with HIV (I'm making all these numbers up, but they're reasonably close to the actual numbers).
- We test all adults--say, 100 million people, for HIV, and again we'll estimate that 1 million people are actually HIV positive.
- The test is 99% effective, so (.99 * 1,000,000 = ) 990,000 HIV positive people learn that they are HIV positive. But it also gives a false positive 1% of the time, so of the remaining population, (.01 * 99,000,000 = ) 990,000 are given false positive diagnoses. That's 1,980,000 positive results, half of which are wrong.
- Your HIV test, which is quite accurate for the individual, turns out to be only 50% accurate across an entire population.
Next: inference vis à vis implication!