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I've found no answer in the papers I have read regarding PERMANOVAs or on the web so I'm really hoping someone could help me answer this question. An example would be testing PERMANOVA on dissimilarity matrix Y by effects C, B, and A, wherein C is nested in B, and B is nested in A. I'm using the adonis function in the R vegan package.

Thank you

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This answer is based on the adonis function in the vegan package for R. Nesting can enter a permanova in two ways: your model and your permutation structure (the strata parameter).

In your model formula, nesting is simply interaction as specified in R with the forward slash or colon (A/B/C or A:B:C); you are specifying that the effect of C should differ based on the value of B and A. This is like a fixed effect. Also note that forward slash differs from colon in that it includes terms for all layers in the nesting hierarchy: A+A:B+A:B:C.

With strata=A:B, you limit your permutations to be between only those samples with the same levels of A and B. This implicitly treats data from each A:B interaction level as coming from the separate distributions/having disparate error structure. It is therefore somewhat similar to a random effect.

Depending on what your question is, you might include your nesting in either place or in both (see link in this answer).

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