The statistics test is:
\[\begin{align} F=\frac{\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}(\bar{x}_j - \bar{x})^2 / (j-1)}
{\sum^s_{j=1} \sum^{n_j}_{i=1} w_{ij}(\bar{x}_j - x_{ij})^2 / ((\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}-1) (j - 1))} \end{align}\]
where:
\(x_{ij}\) is the value of the \(i\)th of \(n_j\) observations in the \(s\) groups, where \(x_{ij}\) has been 'centered' such that \(\sum^s_{j=1} x_{ij} = 0\forall i \),
\(\bar{x}_j \) is the average in the \(j\)th group,
\( \bar{x} \) is the overall average,
\( w_{ij}\) is the calibrated weight,
\( p \approx \Pr(F_{(s-1)\epsilon,(\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}-1) (s - 1))\epsilon} \ge F )\), and
\( \epsilon\) is computed using the Greenhouse-Geisser method.[1]
See Also
ANOVA-Type Tests - Comparing Three or More Groups
How to do Planned ANOVA-Type Test
References
- Greenhouse SW, Geisser S (1959) On methods in the analysis of profile data. Psychometrika, 24, 95-112.