Sample size calculation for expected percentage change in continuous score

I know how to conduct a sample size calculation comparing two independent means in a randomized controlled trial. However, I am confused by the following situation.

There are 2 groups: intervention (IG) and control (CG). The outcome of interest is a severity score (0-100, the higher the worse).

The baseline score in both groups is 20. Within a year the score increases by 5 in the general population. The intervention is supposed to cut the score by 50%. So, given a baseline of 20, the IG is expected to have a score of 22.5 after a year, while the CG is expected to have a score of 25. Let‘s assume the standard deviation of the score in both groups is 10. Aiming for an alpha of 5% (two-sided test) and a power of 80% what sample size is needed to identify the difference?

Using the standard formula 2*(Ζα∕2 + Ζ1-β)² devided by (mean1-mean2)/SD gives about 250 per group.

However, I was told the approach is not correct and I should instead calculate the sample size needed to identify that the score is 50% lower in the IG than in the CG. How can that be done? Isn’t that the same?

Best, Joey

• Since the score scale [0,100] is limited on both sides and have only 101 vales, the starting mean 20 and standard deviation of 10 don't adequately describe the population. I don't believe there can be any closed form solution that might give us the sample size, especially if you are going to use a non-parametric test, which is strongly preferred here. If you have a real-world distribution of the score in different age groups it might serve a basis for a Monte Carlo simulation to attempt to estimate the sample size for the target power of 80%.
– Alex
Commented Oct 1, 2022 at 10:45
• Thanks, Alex. The numbers were just an example. Disregarding that a mean of 20 and SD of 10 may not be plausible, what would be the correct formula to calculate the sample size for a percentage change (as in the example, i.e. „the IG experiences a 50% lower increase in the score than the CG“)?
– Joey
Commented Oct 1, 2022 at 12:37
• By there is no closed form solution I meant no formula :-)
– Alex
Commented Oct 1, 2022 at 14:23
• It seems you need to clarify with whomever is running the study what "50% reduction" means. Do they mean the score in control will be 50% of the treatment? Do they mean the change in scores will be 50% of what would have been otherwise? You need clarity on what the estimand should be. Commented Sep 18, 2023 at 19:44
• Percent change is an asymmetric measure so is not appropriate for use as a dependent variable. See this. Commented Sep 18, 2023 at 19:55