Mathematical derivation of the odds form of Bayes theorem

I had included some rather technical details on my web page about likelihood ratios, but I thought it would be best to move it to a separate page. Define the following terms:

With this notation, we can describe all the classic measures for diagnostic tests:

Bayes Theorem, a classic result of probability theory, tells us that

Notice that the denominators are the same for the above two fractions. So when you compute the ratio of these two probabilities, you get

When the test is negative, we get a similar result:

Note that Se/(1-Sp) is the formula for the likelihood ratio of a positive result and that (1-Se)/Sp is the formula for the likelihood ratio for a negative result.

Earlier versions are here and here.