Editor—Hutchon is correct that Bayes's original theorem concerned probabilities rather than odds. We should have stated more clearly that expressing this is an application of Bayes's theorem expressed as odds. Odds and probabilities are interchangeable quantities [P = 1/(1 - odds); and Odds = (1 - P)/(P)]. This can be expressed more precisely by using Bayes's original equations as follows, although these are cumbersome and not as intuitively useful as the version that we presented in the paper.
Posterior odds are P(D|data)/P(not D|data) (where P = probability and D = having the disease). By Bayes's rule, the numerator is P(D)P(data|D)/P(data) and the denominator is P(not D)P(data|not D)/P(data). Dividing these out gives P(D)/P(not D) times P(data|D)/P(data|notD) which is prior odds times the likelihood ratio. So by using Bayes's rule for probabilities the odds relation can be shown.
In response to the astute remarks of Chitty regarding the similarity between infectious disease diagnosis and detective work, we couldn't agree more. In fact, one reason why Gill loves his clinical work so much is the sense that we're on the hunt for a “villain”—who can often be captured and tamed, once identified.
We enjoyed McCrossin's hilarious encounter between doctor and patient. We cannot resist the temptation to extend his hypothetical dialogue.
Doctor: In fact, if I did this test on many patients with the same signs, in the long run I wouldn't go far wrong.
Johnny (a precocious, although short, 6 year old): But, Doc, what about my height? I'm so much shorter than all my friends.
Doctor: Yes, this is clearly true. But on the basis of standardised growth curves distributed by the Centers for Disease Control and Prevention in Atlanta, 5% of children will be at or below the 5th centile for growth. So I would have anticipated this.
Johnny: Doc, what illnesses might make me so short?
Doctor: It could be the case, in fact, it probably is the case, that you're not eating enough. Lack of adequate food intake is a major cause of malnutrition, and hence short stature.
Johnny: That's funny. I seem to be eating all the time. In fact, I eat six meals a day, and snack constantly, but somehow, I always feel hungry.
Doctor: That's clearly excessive eating. I want to refer you to our dietitian to educate you and your mother on the risks of overfeeding young children.
Johnny: But, Doc, if that's not the cause, what else might it be?
Doctor: Well, statistically speaking, the most likely cause of short stature is a condition called “congenital short stature.” This means you are short now because you were born to be short.
Johnny: But my dad isn't short. Actually, he's pretty tall. Mum, how tall is Dad?
Parent (mother): He's 6 feet 6 inches—and is a goalkeeper for Manchester United.
Doctor: But has he ever been tested for abnormal growth hormones? Perhaps he should come in for a test...
Stay tuned for the next episode of “Statistics and your health.”
Competing interests: None declared.