Probstat/notes/anova

This is part of probstat.

Variance estimations

Estimator 1. The first estimator for $\sigma ^{2}$ is

${\frac {1}{m}}\sum _{i=1}^{m}\sum _{j=1}^{n}{\frac {(X_{ij}-{\bar {X}}_{i\centerdot })^{2}}{n-1}}.$

This estimator is always correct. (Even when $H_{0}$ is not true.)

Estimator 2. The second estimator is calculated from the sample means. It will be correct only when $H_{0}$ is true.

$n\sum _{i=1}^{m}{\frac {({\bar {X}}_{i\centerdot }-{\bar {X}}_{\centerdot \centerdot })^{2}}{m-1}}$

Hypothesis testing

Then compute

$SS_{w}=\sum _{i=1}^{m}\sum _{j=1}^{n}(X_{ij}-{\bar {X}}_{i\centerdot })^{2}$

and

$SS_{b}=n\sum _{i=1}^{m}({\bar {X}}_{i\centerdot }-{\bar {X}}_{\centerdot \centerdot })^{2}$

Recall that

${\frac {SS_{w}}{\sigma ^{2}}}\sim \chi _{m(n-1)}^{2}$,

and

${\frac {SS_{b}}{\sigma ^{2}}}\sim \chi _{m-1}^{2}$.

Also, since $S_{i}^{2}$ is independent of ${\bar {X}}_{i\centerdot }$, $SS_{w}$ and $SS_{b}$ are independent. Thus, the statistic

${\frac {SS_{b}/(m-1)}{SS_{w}/m(n-1)}}\sim F_{m-1,m(n-1)}.$

Therefore, we can use the table for the F-distribution to test for $H_{0}$.