Analysing the ANOVA table

Let’s go back to lecture 6 where we met the ANOVA.

The F statistic value: MS$_\mathrm{model}$/MS$_\mathrm{residuals}$ provides a test statistic that allows us to test whether there is any evidence that at least one of the model parameters is not zero.

$\newline$ The null hypothesis is

$\newline$ H$_0: \beta_1, \ldots, \beta_k = 0$.

$\newline$ which will be tested against the alternative that at least one of the parameters is not zero. If the null hypothesis is true, the statistic has an F(Df$_\mathrm{model}$, Df$_\mathrm{residuals}$) distribution. This implies that

$$F = { MS_\mathrm{model} \over MS_\mathrm{residuals}} \sim F(Df_\mathrm{model}, Df_\mathrm{residuals}).$$

If H$_0$ is false, we would expect MS$_\mathrm{residuals}$ to be smaller than MS$_\mathrm{model}$ and so large values of F should lead us to reject H$_0$. i.e. for large values of F the p-value will be small.