# What does the F-test in linear regression measure when the null hypothesis is false?

For testing whether a group of multiple coefficients ($$\beta_{q+1}...\beta_p$$) in a linear regression of the form $$Y = \beta_0 + \beta_1 X_1\dots+\beta_{q}X_{q}+\beta_{q+1}X_{q+1}\dots+\beta_{p}X_{p}+\epsilon$$ (with $$\epsilon$$ ~ $$N(0,\sigma^2)$$) are all simultaneously zero using the F test, we would test $$H_0: {\beta_{q+1}...\beta_p = 0}$$ vs the alternative where at least one of $${\beta_{q+1}...\beta_p \neq 0}$$.

This is usually done with the following $$F$$ statistic: $$\frac{(\hat\sigma_{null}^2-\hat\sigma_{full}^2)/(p-q)}{\hat\sigma_{full}^2/(n-p-1)}$$ ~ $$F_{p-q,n-p-1}$$ where $$\hat\sigma_{null}^2$$ is the in sample mean sum of square residuals when fitting under the null hypothesis and $$\hat\sigma_{full}^2$$ is the same for the alternative.

If the null hypothesis is actually false, when trying to fit the null model (which ignores $$X_{q+1}\dots X_p$$), we would basically be lumping $$\beta_{q+1}X_{q+1}\dots+\beta_{p}X_{p}$$ with the noise and the form of this combination is not guaranteed to be Gaussian as assumed of the noise in the null model. If so, I believe that $$\frac{n(\hat\sigma_{null}^2-\hat\sigma_{full}^2)}{\sigma^2}$$ would no longer be $$\chi_{p-q}^2$$ as assumed in the $$F$$ statistic? If so, does this number mean anything, or does it simply serve the purpose of giving us a high number to reject the null hypothesis?

• F-tests in linear regression simply test the possibility that at least one of the parameters in a multiple regression are significant.
– user234562
May 2, 2020 at 16:30
• @user332577 Please refer to my liked post to see why the F-test doesn't say if any of the individual t-tests will give statistically significant results: stats.stackexchange.com/a/491239/247274.
– Dave
Oct 21, 2020 at 22:14