APPROXIMATE BAYES FACTORS AND ORTHOGONAL PARAMETERS, WITH APPLICATION TO TESTING EQUALITY OF 2 BINOMIAL PROPORTIONS

We use asymptotic expansions to approximate Bayes factors, improving on a method used by Jeffreys. Suppose that the hypothesis H0: psi = psi-0 is to be tested against H(A): psi not-equal psi-0 in the presence of a nuisance parameter-beta, and initially priors-pi-0(beta) under H-0 and pi(beta, psi) under H(A) are used. We consider the problem of assessing sensitivity of the Bayes factor to small changes in pi-0 and pi. We show that for local alternatives (which, for moderate sample sizes, are consistent with small or moderately large values of the Bayes factor in favour of the alternative), if beta and psi are what we call 'null orthogonal' parameters, then alterations in pi-0 have no effect on the Bayes factor up to order O(n-1). Under similar conditions we also derive an order O(n-1) approximation to the minimum Bayes factor overall priors-pi under H(A) such that the marginal prior on psi is normal with mean psi-0. We then go on to consider sensitivity to specific changes in the marginal prior on psi and show how asymptotics may be used for this, applying a second-order approximation due to Tierney and Kadane. We illustrate the results with a test of equality of two binomial proportions and briefly investigate the accuracy of the approximations is this context.