Permutational analysis of variance

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:

${\displaystyle SS_{T}={\frac {1}{N}}\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}d_{ij}^{2}}$

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:

${\displaystyle SS_{W}={\frac {1}{n}}\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}d_{ij}^{2}\delta _{ij}}$

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 (${\displaystyle SS_{A}}$) can be calculated as the difference between the overall and the within groups sum-of-squares:

${\displaystyle SS_{A}=SS_{T}-SS_{W}}$

Finally, a pseudo F-statistic is calculated:

${\displaystyle F={\frac {\left({\dfrac {SS_{A}}{p-1}}\right)}{\left({\dfrac {SS_{W}}{N-p}}\right)}}}$

where p is the number of groups.

Finally, the PERMANOVA procedure draws significance for the actual F statistic by performing multiple permutations of the data. In each permutation ${\displaystyle \pi }$ the items are shuffled between groups, and the F-ratio is calculated for it, ${\displaystyle F^{\pi }}$. The P-value is then calculated by:

${\displaystyle P={\frac {{\text{number of permutations with }}F^{\pi }\geq F}{\text{total number of permutations}}}}$

PERMANOVA is widely used in the field of ecology and is implemented in several software packages including PERMANOVA[2] software, PRIMER and R (programming language) Vegan and lmPerm[3] packages.