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I'm studying about analysis of variance and in my study material it was stated the following:

For the F-test in ANOVA, calculate the degrees of freedom as follows:

$$F(\alpha, \beta)$$

$\alpha$ is the degrees of freedom for variance between groups.

$\beta$ is the degrees of freedom for variance within groups.

$\alpha$ = number of groups $-1$

$\beta$ = total number of observations $-$ number of groups.

The definitions for $\alpha$ and $\beta$ were just defined as such with no further explanation. So my question is:

How do we arrive in the given definitions?

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    The simple answer is the degrees of freedom (DF) correspond to dimensionalities of subspaces. When you write SS(Between) + SS(Within) = SS(Total), that is a multi-demensional Pythagorean Theorem. SS ("sum of squares") is the squared length of a vector. The data vector is a vector in $gr$-dimensional space ($g$ groups, $r$ replications within groups). One dimension is 'used' to estimate $\mu$ of the model $Y_{ij} = \mu + \alpha_i + e_{ij}.$. Then $g-1$ dim (DF) for Between and $g(r-1)$ DF for Within. ANOVA table ignores est of $\mu.$ DF(Btw) + DF(W'in) = $(g-1)+g(r-1) = DF(Tot) = gr-1$ – 2017-02-20
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    Thank you for your help! not yet easy to grasp the full idea, but this will help :) – 2017-02-21
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    Insightful question. Full grasp will come if you use an approach to ANOVA and regression based on linear algebra, in which the model is specified by a 'design matrix'. Of course my 'simple answer' is way short of the whole story. One step at a time. – 2017-02-21

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