Just a pre-announcement because I have a nice graph.
I am looking into tests for binomial proportions, especially equivalence (TOST) and non-inferiority tests.
Power and significance levels in testing for proportions have a saw tooth pattern because the observed proportions are discrete, see for example this SAS page
Unfortunately for my unit testing, I have not found any equivalence tests for proportions in R. Currently, I'm just trying to match some examples that I found on the internet.
And here is the plot for my power function. It shows the power as a function of the sample size, for either the normal approximation or the binomial distribution, of the test for equivalence, TOST two one-sided tests. The TOST test itself is based on the normal approximation.