# Sample size calculation for a three independent arm RCT with a categorical outcome

I have three study arms on interventions A, B, and C, which are independent of each other. The primary outcome is categorical (positive or negative). The null hypothesis is A = B = C.

From prior studies, the likely proportion of subjects with positive outcomes in arms A, B, and C are 0.8, 0.7, and 0.6, respectively.

What is the sample size needed to prove that the null hypothesis is wrong, assuming a power of 0.8 and alpha of 0.05?

PS: I am able to calculate the sample size using GPower (chi-square tests) with effect size (w) and df (2). However, is there any way to calculate the sample size directly using the above proportions rather than the effect size (w)?

Statistical testing never proves that a hypothesis is wrong. In the frequentist (classical) statistics world you bring evidence against the supposition that the null hypothesis is right. Also, never use the observed results from a previous study in doing a power calculation. Power should be computed from the minimal effects you would be embarrassed to miss. And a power of 0.8 is suggesting that you care about noise 4x as much as you care about signal ($$\alpha=0.05, \beta=0.2$$).
A binary outcome has minimum information and will require very large sample sizes. A sample size necessary to estimate one proportion is 96 with a $$\pm 0.1$$ margin of error. To estimate a difference of proportions with this (somewhat large) margin of error requires $$4 \times$$ that sample size.