The main idea...
As noted by Anderson (2001), ecological data sets rarely conform to the assumptions of MANOVAlike procedures (see MANOVA). For example, rare species inflate the data set with zeros while species with low abundances are unlikely to be normally distributed (the "bellshaped" curve will be 'cut' at zero, resembling a Poisson distribution with λ ~ 1). Nonetheless, the power of MANOVAlike procedures, especially in partitioning variation between multiple factors and their interactions, is in much demand.
She thus proposed a nonparametric multivariate analysis of variance (NPMANOVA or PERMANOVA) method that addresses the limits of these assumptions, allows the use of any dissimilarity measure between objects (rather than only Euclidean distances), and can partition variation between the various terms included in the NPMANOVA model (i.e. support analysis of multifactorial designs). Further, NPMANOVA is tolerant towards nonindependent variables. In a data set with sites × OTUs table, for example, it is unlikely that the presence or abundance of OTUs at a given site is independent. NPMANOVA is analogous to to a distancebased redundancy analysis (dbRDA) wherein the 'grouping variables' may be represented as dummy variables in the explanatory variable matrix. However, NPMANOVA, in addition to being simpler, has been found to have a more reliable Type I error rate (McArdle and Anderson, 2001).
The statistic
The test statistic used is a pseudo Fratio, similar to the Fratio in ANOVA. It compares the total sum of squared dissimilarities (or ranked dissimilarities) among objects belonging to different groups to that of objects belonging to the same group (Equation 1). Larger Fratios indicate more pronounced group separation, however, the significance of this ratio is usually of more interest than its magnitude.
Significance
In a oneway test (where the interest is on whether a statistic is either less than or greater than what can be expected by chance), the Pvalue calculated reports the proportion of permuted pseudo Fstatistics which are greater than or equal to the observed statistic, i.e. what proportion of the permuted data sets yield a better resolution of groups relative to the actual data set following an NPMANOVA. It is generally accepted that any separation between groups is not significant if more than ~ 5% of the permuted Fstatistics have values greater than that of the observed statistic (i.e. a Pvalue > 0.05).
It is vital that the correct permutational scheme is defined and only exchangeable units are permuted. In nested studies, this would mean restricting permutations to an appropriate subgroup of the data set. At times, exact permutation tests either cannot be done, or are restricted to so few objects, that they are not useful. See Anderson (2001, 2005) for examples of permutational schemes involving complex experimental or sampling designs.
Postanalysis: a posteriori testing
As in ANOVA, a significant result indicates that there is a significant difference between the groups defined; however, there is no way of knowing which groups are significantly separated. A posteriori testing, using NPMANOVA, of each pair of groups can be performed after a significant result to determine this. As these are pairwise comparisons, the test statistic involved is the nonparametric, multivariate analogue of the tstatistic, with significance determined by permutation, as above.
As this involves multiple testing, an appropriate correction should be applied.
Key assumptions
Warnings
Implementations
References

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