# Relationship between Fisher Test and Student Test

When performing linear regression with one variable, one can compute Fisher's test with value $$F$$, and derive Student's test $$T=\sqrt{F}$$.

When there is more than one variable, the relationship $$T=\sqrt{F}$$ no longer holds. Is there a relationship between $$F$$ and the different Student tests $$T_i$$ (one for each variable)? Perhaps a system of equations?

EDIT :

I am referring to the F test in Excel's linear regression report:

Note : crossposted here with no answer.

• What "Fisher's test" are you referring to? Oct 11, 2022 at 13:00
• @utobi I am reffering to the F test which is run by Excel when performing linear regression. Oct 11, 2022 at 15:27
• and by "more than one variable" do you mean more than one predictor? Oct 11, 2022 at 17:07
• Yes, by variable I mean explanatory variable / predictor. Oct 12, 2022 at 7:57
• the relation between $t$ and $F$ is always true no matter what is the number of predictors, as long as you test a single predictor. If you want to test jointly more than one predictor, you have to switch to an $F$ or a Wald test. Oct 12, 2022 at 8:22

I'm not sure that this is what you mean, but the $$T^2$$ Hotelling test is a generalization of the $$t$$-test for multivariate normal means.
Suppose that $$X \sim N(\mu, \Sigma)$$, where the number of dimensions is $$p$$. The number of observations is $$n$$. The test statistic is $$T = n (\bar{X} - \mu)' \hat{\Sigma}^{-1} (\bar{X} - \mu)$$, which using SVD decomposition for $$\hat{\Sigma}$$ is, suppose that $$S$$ is a diagonal matrix of the eigen-values and $$U$$ is the eigenvectors of $$\hat{\Sigma}$$. You obtain
$$T= n (U(\bar{X} -\mu))' S^{-1} U(\bar{X} - \mu),$$ which can be viewed as sum of squared $$t$$-statistics.
The statistic is related to $$F$$ distribution by $$\frac{n-p}{p(n-1)}T \sim F_{p, n-p}$$.