Why do we make a F-Test rather than a Beta-Test in ANOVAs?

Question

When one performs an ANOVA, (s)he always end up calculating the observed F-ratio and comparing it to the appropriate F-distribution. From this post, I discovered that the coefficient of correlation $r^2$ follows a beta distribution under the null hypothesis and I am wondering why we are not using the $r^2$ rather than the F-ratio to calculate the P.value.

Could we use the calculated $r^2$ and compare it to the beta distribution (make a Beta-Test) rather than comparing the F ratio to the F-distribution? If yes, do we use a F-test rather than a Beta-Test just for historical reasons or is there any other reason behind this choice? Would we systematically get the same P.value?

Answer 1

The F and the beta are related by a simple transformation.

If $X \sim \text{Beta}(\nu_1/2,\nu_2/2)$ then $\frac{\nu_2 X}{\nu_1(1-X)} \sim \text{F}(\nu_1,\nu_2)$.

See Wikipedia on F-distribution

Indeed the result that one can derive a simple monotonic transform of $R^2$ as an $F$ is a standard one, covered in many texts; see What is the relationship between R-squared and p-value in a regression? for example

So there's really no difference; one is as simple as the other. (One might as readily ask "Why use the beta distribution when we could use the beta-prime distribution?")