I have a conceptual issue that I am wrestling with. I have a scenario with three treatment arms, C, T1, and T2, where C is the control. The outcome is binary, and I am interested in Odds Ratios. In particular I am interested in the OR_21 of T2 relative to T1, and I am interested in the P(OR_21 > 1). I want a prior that encapsulates P(OR_21 < 0.8) < 0.025 and P(OR_21 > 1.2) < 0.025. If I eliminated treatment arm C, and just compared T2 with T1 with this model: logit(Y) = a + b*(rx =T2), then b is the OR, and I could state prior on b on the log scale as N(0, 0.3537), since exp(b) = OR_21.
However, if I have C in the mix, I have logit(Y) = a + b1*(rx=T1) + b2*(rx=T2), and now the odds ratio OR_21 = exp(b2 - b1). In this case, how do I specify the priors of b1 and b2 so that the prior for OR_21 meets the requirement that P(OR_21 < 0.8) < 0.025 and P(OR_21 > 1.2) < 0.025?
Am I thinking of this completely wrong?