Permutational analysis of variance

Calculation of the statistic

In the simple case of a single factor with p groups and n objects in each group, the total sum-of-squares is determined as:

${\displaystyle S{S}_T=\frac{1}{N}\sum _{i=1}^{N-1}\sum _{j=i+1}^N{d}_{ij}^2}$

where ${\displaystyle N=pn}$ is the total number of objects, and ${\displaystyle d_{ij}^2}$ is the squared distance between objects i and j.

Similarly, the within groups sum-of-squares is determined as:

${\displaystyle S{S}_W=\frac{1}{n}\sum _{i=1}^{N-1}\sum _{j=i+1}^N{d}_{ij}^2{\delta }_{ij}}$

where ${\displaystyle {\delta }_{ij}}$ is 1 if the observations i and j belong to the same group, and 0 otherwise. Then, the between groups sum-of-squares (${\displaystyle S{S}_A}$) can be calculated as the difference between the overall and the within groups sum-of-squares:

${\displaystyle S{S}_A=S{S}_T-S{S}_W}$

Finally, a pseudo F-statistic is calculated:

${\displaystyle F=\frac{\left({\displaystyle \frac{S{S}_A}{p-1}}\right)}{\left({\displaystyle \frac{S{S}_W}{N-p}}\right)}}$

where p is the number of groups.