Suppose we have three different treatments A, B, and C and we are modeling response(0,1). I want to know how to do sample size determination for each group(A,B,C) given say normal priors on each group.
I need to show how we can (and how much we can reduce) the sample size and get similar testing of differences in main effects if we have prior info.
the most important thing is being able to show how we can reduce the sample size if we incorporate the prior info.
I'm in agreement with Adam. A very clear analytic power calculation likely isn't going to suit you, and simulation is probably the way to go.
I'd then run a series of simulated studies, drawing different sample sizes each time. For example, 10,000 studies each with 100 subjects, 10,000 studies with 150 subjects...etc. etc.
For each, you then run both a normal logistic regression and your bayesian analysis with the specified priors, and measure what % of the studies correctly identify that there is an effect, and low and behold, you have your power. You should then be able to graph two curves, showing the rise in power as sample size goes up for both the bayesian and frequentist analysis, and show (hopefully) that the curve for the bayesian analysis arrives a the desired value for beta at a smaller sample size.
This is most probably going to be achieved using simulation. Using normal priors for the log odds ratios with "almost 0" precision, estimating the logistic regression model, and computing the parameters' posterior mode gives you approximate frequentist inference. Increasing this precision can be used to represent the strength of prior information. The power of your Bayesian test will, of course, depend on both the precision and the location of the priors so be certain to state explicitly how this information is used.
