5 Analysis of variance

Consider the linear model $\boldsymbol{Y} = \mathbf{1}_n\alpha + \mathbf{Z}\boldsymbol{\beta} + \boldsymbol{\varepsilon}$ where $\mathbf{X}=(\mathbf{1}_n^\top, \mathbf{Z}^\top)^\top$ is a full rank $n \times p$ design matrix. Let as usual $\mathrm{TSS} = \boldsymbol{y}^\top\mathbf{M}_{\mathbf{1}_n}\boldsymbol{y}$, the total sum of square, and $\mathrm{RSS}= \boldsymbol{y}^\top\mathbf{M}_{\mathbf{X}}\boldsymbol{y}$, the sum of squared residuals. Under the assumptions of the Gaussian linear model (or asymptotically), the $F$-test statistic for testing the null hypothesis $\mathrm{H}_0: \boldsymbol{\beta}=\mathbf{0}_{p-1}$ against the alternative $\mathrm{H}_a: \boldsymbol{\beta} \in \mathbb{R}^{p-1}$ assuming the larger model is correctly specified is $$F = \frac{(\mathrm{TSS}-\mathrm{RSS})/(p-1)}{\mathrm{RSS}/(n-p)}.$$ Under the null hypothesis, $F \sim \mathcal{F}(p-1, n-p)$.

An ANOVA table (anova) arranges the information about the sum of squares decomposition, the degree of freedom and the value of the $F$ test statistic in the following manner.

Sum of squares	degrees of freedom	scaled sum of squares	test statistic	$P$-value
ESS	$p-1$	$\mathrm{ESS}/(p-1)$	$F$	$1-\texttt{pf}(F, p-1, n-p)$
RSS	$n-p$	$\mathrm{RSS}/(n-p)$