Power calculation for distinguishing linear regression slopes between two groups

Question

I would like to estimate the sample size required to statistically distinguish the linear relationship between two variables, x and y, between two groups, A and B, and specifically the slopes of the two linear relationships (but also the intercept if possible). How many samples from A and B are needed to achieve some specific power?

I know the expected slopes, beta1 and beta2, and I know the uncertainty in the slope estimate (via the standard error of the slope estimate, or alternatively via the standard deviation in x and y).

I have been unable to find guidance on how to perform this calculation. Any help would be very much appreciated.

Answer 1

This is the kind of question that you really need a simulation to answer. There are so many parameters that could change, you need to specify the values for them, generate data, and analyze it.

However, Abelson's 42% rule is useful here to at least get you started. If you have a positive slope for one group, and p = 0.05, what slope does a second group, of the same size, need to have for the difference in the slopes to be statistically significant at the 0.05 level? 42% of the magnitude, in the opposite direction.

So if your slope for one group is 0.5, and p = 0.05, when you add a second group of the same size, the slope needs to be 42% of 0.5, in the opposite direct (i.e. -0.21).

Finding moderator effects is hard, and requires large sample sizes.