Formal Proof of ANOVA's F Distribution?

[Boot20]

Hello all,

Does anyone know where I could find a formal proof that

\frac{\text{MS between}}{\text{MS within}}

has a F distribution under the null in ANOVA?

 

[EnumaElish]

Wikipedia states that the ratio of two chi-square random variables is F. If each MS term is normal then their sum will be Chi-squared.

 

[statdad]

In ANOVA each sum of squares can be pictured as a quadratic form, and under the null hypothesis the quadratic forms are either exactly chi-squared (if normality is assumed) or approximately chi-squared (under some general regularity conditions). The model assumptions and the null hypothesis ensure that the different quadratic forms are independent, so

(MS between)/(MS within)

is a ratio of independent chi-squares over their degrees of freedom which, by definition, gives an F distribution.