# How to calculate sample size (or power) for the experiment with multiple treatments and a binary outcome?

I have an experiment in which there are 13 treatment groups and the relevant outcome is binary. I'm trying to apply the test of proportion to determine the required sample size to be able to detect the minimum detectable effect (MDE). Here are the issues;

1. I don't expect the MDE to be uniform across treatment groups. For example the null hypothesis for Treatment 1 is $p_1 = 0.5$ against the alternate $p_1>0.5$, where $p_1$ is the proportion of sample who will have the relevant characteristic in treatment 1. For treatment 2 null is going to be $p_1=p_2$ against the alternate $p_2>p_1$ and so on until the comparison between $p_3$ and $p_4$. Apriori I expect $p_1=0.6$, $p_2=0.57$, $p_3\approx 0.54$, and $p_4\approx0.52$
2. I will also need to compare proportion from treatment 2 to 4 with proportion in treatment 1. Specifically at the end of experiment I will be testing hypothesis such as $p_1=p_2$, $p_1=p_3$ and $p_1=p_4$.
3. Then treatments 5 to 13 help estimate a parameter in my structural model. They basically vary wage rate to see how the outcome change. I want to make sure that sample size in these treatments is sufficient to identify the parameter of interest.

Now my heuristic approach towards this is to just calculate required sample size for treatment 1 and treatment 2 (using 10% MDE) and assume that every other treatment group (3-13) should have the same size. I feel that this is not a right way of doing this, so I'd appreciate any leads on how to go about this. I understand how the power calculation is done for comparing proportions in two samples but not beyond that.

P.S. I'm using Stata power twoproportions command to do the calculation for comparing two proportions.