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I am a long time user of the forum but first time poster. My question is unrelated to a specific dataset but on the internal workings of a PERMANOVA in R (adonis2 function, vegan package) vs a nested linear model in R (linear mixed model, package lmer or nlme). I will explain step by step what my concern is.

Lately I have been working on ecological experiments with complex designs, meaning that we have several sampling times, two spatially nesting variables and two treatments with two levels each. However, the number of replicates per treatment combination was of 4. This means that the number of degrees of freedom (or, in other words, independent data points) that we have in order to perform tests were extremely limited.

When performing a linear model on one of our response variables with all the aforementioned explanatory variables and covariates, including two spatial nests, the test could not be performed due to lack of degrees of freedom (this was identical for the lmer and the nlme packages).

However, reading the statistical analyses of other similar experiments in the literature, I came across the PERMANOVA test. The experiments had the same number of replicates and comparatively complicated designs, and therefore I decided to test them. I am aware that the PERMANOVA test is multivariate in opposition of the univariate nature of the linear model, but it can be used with a single response variable. This unique response variable is transformed into a distance matrix and the analysis is performed.

To my surprise, the test worked correctly, with absolutely no errors in the output. However, I am not sure if I can confidently use this test until I understand why the number of degrees of freedom was not enough for the linear model whereas it was enough for the PERMANOVA. My only educated guess is that is related to the permutational nature of the test, but I could not find an answer neither in the original literature explaining the PERMANOVA test nor in papers or online forums.

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PERMANOVA uses a permutation test for the tests of covariates. The key assumption is that the observations (or residuals) are exchangeable under the NULL hypothesis. It doesn't sound like your data are exchangeable under the NULL model implied by the default permutation scheme, which is simple randomization.

You may be able to specify the constraints implied by your design in the permutation test; vegan uses the permute package for permutation schemes so look at the help for permute to see if your design fits within the possibilities.

The other way to deal with these issues is to include the experimental design in the model by including extra terms.

You don't give enough information on the design etc to speculate further.

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  • $\begingroup$ Thank you very much for your answer. I understand that one of the assumptions in PERMANOVA is the excheangability of the permutational units. In order for the units to be exchangeable, I added the nesting design to the PERMANOVA to avoid the violation of this assumption. I we assume that I added all the correct covariates to my PERMANOVA model (including the nesting design), I am still unsure why it is possible to run it whereas in the case of the linear mixed model this is impossible. Therefore, my question is still partially unanswered. $\endgroup$ Sep 8 '20 at 16:29

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