8 February 2008 by John.

My previous two posts have been about false research conclusions and false positives in medical tests. The two are closely related.

With medical testing, the prevalence of the disease in the population at large matters greatly when deciding how much credibility to give a positive test result. Clinical studies are similar. The proportion of potential genuine improvements in the class of treatments being tested is an important factor in deciding how credible a conclusion is.

In medical tests and clinical studies,  we’re often given the opposite of what we want to know. We’re given the probability of the evidence given the conclusion, but we want to know the probability of the conclusion given the evidence. These two probabilities may be similar, or they may be very different.

The analogy between false positives in medical testing and false positives in clinical studies is helpful, because the former is easier to understand that the latter. But the problem of false conclusions in clinical studies is more complicated. For one thing, there is no publication bias in medical tests: patients get the results, whether positive or negative. In research, negative results are usually not published.

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8 February 2008 by John.

The most commonly given example of Bayes theorem is testing for rare diseases. The results are not intuitive. If a disease is rare, then your probability of having the disease given a positive test result remains low. For example, suppose a disease effects 0.1% of the population and a test for the disease is 95% accurate. Then your probability of having the disease given that you test positive is only about 2%.

Textbooks typically rush through the medical testing example, though I believe it takes a more details and numeric examples for it to sink in. I know I didn’t really get it the first couple times I saw it presented.

I just posted an article that goes over the medical testing example slowly and in detail: Canonical example of Bayes’ theorem in detail. I take what may be rushed through in half a page of a textbook and expand it to six pages, and I use more numbers and graphs than equations. It’s worth going over this example slowly because once you understand it, you’re well on your way to understanding Bayes’ theorem.

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