# Sample size determination for analysis with several multiple regressions

In a proposed analysis that involves several multiple regressions, how can we determine the required sample size?

If, for example, we can tolerate a type I error rate of 5% ($$\alpha$$ = 0.05) and a type II error rate of 20% (power = 0.8). Also assume that our desired effect size (f2) is 0.01.

Where $$f2 = \frac{R^2}{1-R^2}$$ (where $$R^2$$ is the squared multiple correlation)

Then suppose that we intend to investigate the associations with three dependent variables. For the purpose of this question, let's assume that there are 7 predictors that are common to each regression and one unique.

What would our 'u' be in sample size estimation for these linear regressions. Is it additive, because we have greater risk of type II error as we introduce more regressions?

Using the pwr package in R, would this be as follows (u = 7):

pwr.f2.test(u=7, f2 = 0.01, power = 0.8, sig.level = 0.05)


or something like (u = 24):

pwr.f2.test(u=24, f2 = 0.01, power = 0.8, sig.level = 0.05)

• multiple regression or multivariate regression? – user158565 Dec 1 '18 at 2:55
• @user158565 revised – Simon Dec 1 '18 at 5:53