F-test = t-test squared - how to show? (beginner question)
Zootal
to show that an F-statistic is equal to a t-statistic squared, or IOW F =
t^2. I am working with linear regression and extra sum of squares tests. How
would one go about showing this? Or, can some kind soul nudge me in the
right direction?
Jack Tomsky
The easiest way is to work with the canonical forms.
F(m,n) = [X2(m)/m]/[X2(n)/n],
where F(m,n) is an F with m and n degrees of freedom, X2(m) is a chi-square with m degrees of freedom, X2(n) is a chi-square with n degrees of freedom, and X2(m) and X2(n) are independent.
In particular, for m = 1,
F(1,n) = X2(1)/[X2(n)/n] = = z^2/[X2(n)/n] =
[z/sqrt(X2(n)/n)]^2 = [t(n)]^2,
where z ~ N(0,1) and t(n) is Student's t with n degrees of freedom.
Thus an F with 1 and n degrees of freedom is the square of a t with n degrees of freedom.
Adole
The reason this is not a great explanation is because you don't
explain (or show) how X2 and Z2 are approximated to be the same.
Therefore, this equations carries assumptions which from the student's
perspective aren't demonstrated to be valid.