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Bayesian Statistics: An Introduction, 4th Edition by Peter M. Lee

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5.6 Comparison of two proportions; the $2\times 2$ table

5.6.1 Methods based on the log-odds ratio

In this section, we are concerned with another two sample problem, but this time one arising from the binomial rather than the normal distribution. Suppose

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and that we are interested in the relationship between π and ρ. Another way of describing this situation is in terms of a 2 × 2 table (sometimes called a 2 × 2 contingency table)

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We shall suppose that the priors for π and ρ are such that  and  , independently of one another. It follows that the posteriors are also beta distributions, and more precisely if

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then

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We recall from Section 3.1 on the binomial distribution that if

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then  , so that

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