Why clinicians are natural bayesians

Editor—Clinicians are natural bayesians when it comes to diagnosis.¹ They have to be. The alternative approach might be to use the methods of classical hypothesis testing, but probably only once.

The Neyman-Pearson diagnosis of coeliac disease

(Assume that the sensitivity and specificity of transglutaminase IgA are both 95%.)

Parent: Well doctor, have you got the result of the test yet?

Doctor: Yes I have. When you brought little Johnny in with weight loss, short stature, and diarrhoea I thought it was worth checking for coeliac disease, and the test has come back positive.

Parent: Does that mean he has coeliac disease?

Doctor: I can't be certain, but it is likely.

Parent: Well, how likely?

Doctor: I can't actually tell you that, but given that he does not have coeliac disease, there was a 95% probability that the test would have been negative, and it in fact was positive.

Parent: Well, does that mean he has a 95% chance of having coeliac disease?

Doctor: No. I can't say that.

Parent: Well what can you say?

Doctor: Given that he does not have coeliac disease, there was a 95% probability that the test would have been negative, and only a 5% chance of obtaining this result.

Parent: We're going around in circles. What about the fact that he has the diarrhoea and weight loss?

Doctor: Well, that's why I did the test.

Parent: Well, does it make coeliac disease more likely?

Doctor: I can't say that.

Parent: Well, what do we do?

Doctor: We could do the test a few more times, and if it keeps coming up positive, it makes the diagnosis more likely.

Parent: How much more likely?

Doctor: I can't say, but more likely. In fact if I did this test on many patients with the same signs, in the long run I wouldn't go far wrong.

Parent: That won't help Johnny. Are you a complete idiot or what?