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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 postthis 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?

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?

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?

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

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?