I have $3$ groups of patients: baseline, test 1 and test 2. Their sample sizes are $N_0, N_1, N_2$, respectively. The number of observed positive patients are $n_0, n_1, n_2$, respectively. I want to construct the $(1-\alpha)\%$ confidence intervals (CI) for the pairwise changes: $\frac{n_1}{n_0} - 1, \frac{n_2}{n_0} - 1, \frac{n_1}{n_2} - 1$.
My questions are:
- Am I in the case of multiple comparison test? that is, for each change, I need to calculate $(1-\frac{\alpha}{3})\%$ CI, instead of $(1-\alpha)\%$ CI (suppose Bonferroni correction is used)
- How to calculate these intervals? I tried to model the number of positive patients by binominal distribution, then approximate them by normal distribution (assume the sample size is large enough), then I came up with the ratio of $2$ non-centered gaussians, which turns out to have a very complicated law.