## Sample size estimation for parallel design with more than 2 treatments [Power / Sample Size]

Hello all!

I have a question regarding sample size estimation for parallel designs with more than 2 treatment arms (with more than 2 groups). Package PowerTOST in R covers sample size estimation for parallel design for just 2 parallel groups (known.designs()) and not for more.
Lets say we are conducting a (pilot) parallel study with 3 test formulations and 1 reference. How do we estimate sample size in this case?

I would say that we just double the necessary sample size estimated (in R with sampleN.TOST) for 2 parallel groups (to get 4 independent groups for 3 test formulations and 1 reference). For example for CV=0.2 (parallel design and theta0=0.95) for 2 parallel groups we get sample size of 36 subjects (total sample size for 2 groups), so if we double it we get 72 subjects (for 4 groups?). Is this assumption and sample size estimation a correct one?

On the other hand I have read this article regarding multiple treatments comparison in parallel design and I dont exactly understand why should we just control Type I Error and change alpha when estimating sample size with multiple treatments? So in our case (when using Bon­ferroni adjustment) we would get the total sample size (for 4 groups) of 50 subjects (parallel design, CV=0.2, theta0=0.95, alpha=0.0167)? Is this correct? What is more, 50 is not even dividable by 4. I am confused and I probably didnt get this explanation in the article right. Nevertheless I wouldnt mind this being the right way as the needed sample size is smaller

Regarding alpha adjustment in the first case (for doubling sample size from 36 to 72) I would say that it is not necessary as it is a pilot study. On the other hand, if this was a pivotal study and we would want to show equivalence for just 1 test formulation (among 3) then we would have to adjust alpha (to 0.0167) and then I would say we would double the estimated sample size for 2 groups the same way as described above (so 50x2=100 subjects)?

So which sample size estimation is a correct one?

BEQool