The way PERMANOVA handling continuous explanatory variable

Question

I have read some tutorials which explain the mechanism behind PERMANOVA, e.g., enter link description here. Based on the tutorials, my understanding is PERMANOVA can only handle categorical explanatory variables as grouping variables. However, adonis2 can handle continuous explanatory variables.

Here is an example I tried in R:

beta <- betadisper(s_distance, sample_data(human)$weight)
plot(beta)

The graph: Based on the graph, it seems adonis2 treats continuous explanatory variables as categorical variables. Also in the example shown in the adonis document enter link description here, A1 in adonis2(dune ~ Management*A1, data = dune.env) is a continuous variable. My questions are:

1. Does adonis2 really treats continuous variables as categorical variables? If not, how?
2. Could the multivariate multiple regression be a more reasonable choice when both categorical and continuous explanatory variables exist? If not, why?

Thanks in advance for your help!

Answer 1

In my opinion, adonis2 does not treat continuous variables as categorical variables.

If it does, df (degree of freedom) of a continuous variable would be the number of distinct values of the varilable minus one. But, df of continuous varialbes in adonis2 is always 1.

According to a little bit similar question (posted on researchgate) that has an answer with an excerpt from an email written by J. Oksanen (a main author of package vegan), it is likely that adonis(2) performs simple linear regression when explanatory (grouping) variables are continuous.

https://www.researchgate.net/post/How_ADONIS_is_calculated_for_continues_variable "Good question. I should probably change that definition. Variation explained is directly analogous to that of general linear models. With a continuous variable, it acts like simple linear regression, where each point is associated with its own "centroid" which is the best fit linear approximation."

And, I think that this is the reason why df of continuous varialbes is always 1. According to a comment by J. Oksanen on a question posted at stackoverflow, "continuous variable in any (linear) model uses single df".

https://stackoverflow.com/questions/53932311/why-do-i-always-get-1-for-df-when-running-adonis-function-permanova

I don't have any ideas on your 2nd question.