1. Joe: Doctor X! I have Ickitis, I need a Pill! Quick!
Dr. X: Hold on a minute Take a deep breath — how do you know? How sure are you?
Joe: (breathes) I read there was an epidemic with about 1000 cases in town so I bought the test and it’s 99% accurate, so okay I guess there’s a 1% chance that I don’t really have it and I don’t have any symptoms yet but they say people often don’t get symptoms until really late and I’ve got it and I need a cure.
Dr. X: You do know that there are a million people in this city, right? And we estimate that 1000 of them have Ickitis.
Joe: Yes, and I’m one of them — just my luck.
Dr. X: So before you took the test, if you didn’t know anything else, the odds of your having Ickitis were 1 yes to 999 no, okay? That’s an odds ratio, 1 to 999, right?
Joe: Yeah — actually I was really surprised.
Dr. X: And the odds of the test being right, if you didn’t know anything else, were 99 yes to 1 no, okay? That’s an odds ratio too.
Joe: Umm…yeah, I guess.
Dr. X: So when you get new evidence, you don’t forget the old evidence, you combine them.
Joe: Uhhh… What?
Dr. X: Actually you multiply them, like fractions: 1 / 999 * 99 / 1 = 99 / 999 = 11 / 111 and that’s 11 yes to 111 no, or you can call it 11 yes out of 122 total and that’s about 9% of the total. So you probably don’t have Ickitis, but you might and we’ll do some more tests.
2. Joe: Doctor, I’m really not a numbers guy and I don’t understand. Can you make that a little bit simpler?
Dr. X: Probably. Imagine we give everybody this test — that’s one million tests, right?
Joe: Well, sure.
Dr. X: And we know that about ten thousand of them will be wrong. That’s one percent.
Joe: It sounds pretty bad when you say it like that.
Dr. X: Not really, it’s better than most tests. Very few tests are conclusive all by themselves, but they help and a 99% – accurate test helps a lot. Anyway there are ten thousand wrong tests in this situation, some positive and some negative. And there are one thousand who really are infected, with ten of them having negative tests.
Joe: Okay, and you’re saying there are nine-hundred ninety-nine thousand who are not infected, but one percent of them will have positive tests?
Dr. X: Yes, and that’s nine thousand, nine hundred and ninety with positive tests that are wrong. That’s four groups of people — Okay so far?
Joe: I think so… there’s a very big bunch of people who are not infected and they know it; that’s almost a million. There’s a big bunch of people who are not infected but think they are, that’s almost ten thousand. There’s a small bunch of people who are infected and know it, that’s almost a thousand, and there’s a very small bunch — ten — who are infected but think they aren’t. Right?
Dr. X: Right. And what you know right now is that you have a positive test, which means that EITHER you’re one of the small bunch — 990 — who are infected and “know” it, OR you’re one of the big bunch — 9,990 — who aren’t infected but think they are. So your odds ratio for actually being infected is 990 yes to 9990 no, which is the same 11 yes to 111 no that we had before: it’s your chance of being in the 990 true positives out of a total of, umm, let me write this down, a total of 10,980 positives true and false, and it’s just about nine percent. See?
3. Joe: Yeah, so I guess it was pretty silly to take the test….
Dr. X: No — not at all. You were worried about a one-in-a-thousand chance, and that’s not silly. Now you have a one-in-eleven risk, and that’s definitely reason for more thorough testing. See the vampire down the hall, I’m writing a prescription for two blood tests and I’ll see you — probably very briefly — tomorrow, same time.
Daily Bayes 