Why not try a Bayesian route? For example, suppose that you want to estimate the posterior distribution of the difference or ratio of a proportion between two groups.
Step 1: Open R.
Step 2: Simulate the posterior distributions of the proportion in each group with two calls of the
rbeta() function, parameterized with the success and failure counts in the focal category (i.e., the "success") of the appropriate group. You could do 1,000,000 draws from each posterior in very little time. If you have at least one observation in each category, you can use an improper beta prior with shape and scale parameters equal to 0. If you don't, you can use a flat and still somewhat uninformative prior by adding a one to the counts in each category.
Step 3: Use the
- operator to subtract one empirical distribution from the other to obtain the empirical distribution of the difference. Or use the
/ operator if you are interested in a ratio.
Step 4: Use the
quantile() function to find the 2.5th, 50th, and 97.5th pecentile. Use the
plot(density()) functions to visualize the distribution.
Asymptotically, the answer you get will be the same as with a frequentist method, except this way is easier. You can do virtually the same thing with any other exponential family distribution by looking up the conjugate prior of the sampling distribution, and how to parameterize the posterior.
Disclaimer This isn't a critique of frequentist methods, just a proposal of a simpler way to solve this particular problem using Bayesian inference.