Doctors, Odds Ratios, and Bayes’ Theorem
Today’s lecture largely focused on Bayes’ theorem, a powerful tool for updating beliefs based on evidence—say, whether or not you have a given disease, if you find out that you tested positive for it. If undergraduates in a math-related course at Cornell learn the math behind that, surely most practicing doctors know it…right?
You’d be surprised. News articles like the one linked pop up every few years, showing that doctors get these questions right only about 20% of the time. That wouldn’t be so bad if you were asking the doctors advanced calculus questions, but for something with so many day-to-day applications in their line of work, it’s rather concerning.
Fortunately, Bayes’ theorem has a very intuitive formulation, not in terms of probabilities but in terms of odds ratios. The odds ratio for a probability p is p / (1 – p); if p = 9/10, then it has odds ratio 9/10 / 1/10 = 9. (Typically, you’d say that the event has “9-to-1 odds”, written as 9 : 1; since the ratio is all that matters, you could equally well write this as 18 : 12 or 63 : 7.) Intuitively, this tells you that the even with probability 9/10 occurs 9 times as often as its negation. To go back from an odds ratio to a probability, you divide it by one plus itself; in our example, 9 / (9 + 1) = 9/10 gives us our probability back.
We worked through a toy example in class today of an event (in this case having a rare disease) with a prior probability of 1/100, and a test that’s accurate 90% of the time; in this case, a bunch of arithmetic showed us that the probability of having the disease given a positive test result was 1/12.
Arithmetic is fine, but not-arithmetic is even better. With a probability of 1/100 of having the disease, the odds ratio is 1 : 99. The test lets us multiply by an odds ratio of 9 : 1, in the following sense: we can think about the 1 and the 99 as two different populations, the 1 person having the disease and the 99 people not. We apply the disease screening test and 9/10 of the first group remains, compared with 1/10 of the second, for a new odds ratio of 9/10 : 99/10. Clearing the denominators reveals that we’ve just multiplied by an odds ratio of 9, which is the ratio of the two probabilities of getting a positive test result (9/10 in one case, 1/10 in the other).
Reducing the odds ratio 9 : 99 to 1 : 11 means that, for every one person with the disease (and a positive result), 11 people are without; converting back into probability land means a probability of 1/12 of having the disease, just like we calculated before.
I don’t know how much more time the 2850 staff plan to spend on the current mathematical digression of Bayes’ theorem. It’s not given that much more space in the textbook, and while this is the sort of thing I love to geek out on, this is a class with broad appeal and I am dimly aware that other people have different priorities. But once you’ve been through the probability arguments, I think the odds ratios are a great way to think about these sorts of problems. It’s how I form my intuition for them, it’s easy to describe, and the math is simple enough that, with some practice, you can do it all in your head. 1/100 -> 1 : 99 -> 9 : 99 -> 1 : 11 -> 1/12, no muss and no fuss.
I don’t like being a doctor, and I doubt most doctors like being mathematicians. Somehow, doctors as a collective profession manage to continue telling me, and the rest of the world, what to do to not die, and they do so in language that all us non-doctors can understand. It’s an admirable way that the often highly technical results of a profession get transmitted to a population that needs those results, but not the jargon behind them.
The odds ratio form of Bayes’ rule is one way mathematicians can give back to doctors. It can be explained over dinner on one side of a napkin. It’s intuitive enough that I think the idea would stick for longer than that. As a baseline, we should expect doctors to be able to interpret medical test results, and odds ratios are just the tool they need to do it.