if got the a question concerning sample size calculation:

When doing a sample size calculation for a randomized clinical trial for comparing to group means I know the following formula:

\( n= \frac{r+1}{r} \frac{\sigma^2 (Z_{\beta}+Z_{\alpha/2})^2}{difference^2} \)

Now my question: How does the sample size change when I have a non-randomized trial instead of a randomized one? Which formula to use?

Thanks in advance for your help.

Best regards

sternchen