# F-Test (ANOVA)

The test statistics is:

${\displaystyle F={\frac {\sum _{j=1}^{s}\sum _{i=1}^{n_{j}}w_{ij}({\bar {x}}_{j}-{\bar {x}})^{2}/(j-1)}{\sum _{j=1}^{s}\sum _{i=1}^{n_{j}}w_{ij}({\bar {x}}_{j}-x_{ij})^{2}/(\sum _{j=1}^{s}\sum _{i=1}^{n_{j}}w_{ij}-j)}}}$

where:

${\displaystyle x_{ij}}$ is the value of the ${\displaystyle i}$th of ${\displaystyle n_{j}}$ observations in the ${\displaystyle j}$th of ${\displaystyle s}$ groups,
${\displaystyle {\bar {x}}_{j}}$ is the average in the ${\displaystyle j}$th group,
${\displaystyle {\bar {x}}}$ is the overall average
${\displaystyle w_{ij}}$ is the calibrated weight.
and ${\displaystyle F}$ is evaulated using the F-distribution with ${\displaystyle j-1}$ and ${\displaystyle \sum _{j=1}^{s}\sum _{i=1}^{n_{j}}w_{ij}-j}$ degrees of freedom.

(This is Type III Sum of Squares ANOVA.)