Permutational analysis of variance
Permutational multivariate analysis of variance (PERMANOVA),
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:
where is the total number of objects, and is the squared distance between objects i and j.
Similarly, the within groups sum-of-squares is determined as:
where is 1 if the observations i and j belong to the same group, and 0 otherwise. Then, the between groups sum-of-squares () can be calculated as the difference between the overall and the within groups sum-of-squares:
Finally, a pseudo F-statistic is calculated:
where p is the number of groups.
Drawing significance
Finally, the PERMANOVA procedure draws significance for the actual F statistic by performing multiple permutations of the data. In each permutation the items are shuffled between groups, and the F-ratio is calculated for it, . The P-value is then calculated by:
Implementation and use
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.
References
- .
- ^ Anderson, Marti J. (2005). "Permutational Analysis of Variance" (PDF).
- ^ Wheeler, Bob; Torchiano, Marco (2016). "lmPerm: Permutation Tests for Linear Models". Retrieved 2019-02-08.