Permanova outputs with or without random factor My question is similar to this previous post but remained unanswered. I am having trouble understanding the output of an adonis models with and without the argument “strata” in the formula.
I want to compare microbial communities associated with healthy and diseased reef algae. 
Healthy and diseased algae have been collected in different islands. I am more interested in the difference between the Healthy and Diseased group, I am not really interested in comparing the islands but still would like to take the islands into account. From what I read it looks like a way to do it is to include Island as a random factor in my model.
Variable: Bray-Curtis dissimilarity matrix (dist)
Fixed factor: Disease (Dis)
Random factor: Island
adonis1<- adonis(dist ~ Dis, strata = Island, data = meta)
adonis2<- adonis(dist ~ Dis, data = meta)
However, I get the exact same output for both models:
]1
Could someone please explain to me why outputs are identical?
Thanks a lot.

Thanks for your response. 
In order to provide you with a reproducible example, I was preparing a subset of my data and ran the two models again and got the same results as your example with the dune  data set. That is: exactly the same output besides a different p-value. 
Do you think you could explain what is it that adonis makes differently when running with or without strata? I am sorry if that sounds trivial but I don’t really understand why R2, SumsOfSqs and MeanSqs are exactly identical with or without the strata argument. I’d like to understand better what’s behind it. 
 A: I am afraid I cannot reproduce your similar results.. see below for a reproducible example, maybe you can provide the example dataset so that we can explore it further:
library(vegan)
data(dune)
data(dune.env)

table(dune.env$Management)

BF HF NM SF 
 3  5  6  6 

set.seed(111)
adonis(dune ~ A1, data = dune.env,strata=dune.env$Management) 

Blocks:  strata 
Permutation: free
Number of permutations: 999

Terms added sequentially (first to last)

          Df SumsOfSqs MeanSqs F.Model      R2 Pr(>F)  
A1         1    0.7230 0.72295  3.6389 0.16817  0.031 *
Residuals 18    3.5761 0.19867         0.83183         
Total     19    4.2990                 1.00000     

adonis(dune ~ A1, data = dune.env)

Permutation: free
Number of permutations: 999

Terms added sequentially (first to last)

          Df SumsOfSqs MeanSqs F.Model      R2 Pr(>F)   
A1         1    0.7230 0.72295  3.6389 0.16817  0.007 **
Residuals 18    3.5761 0.19867         0.83183          
Total     19    4.2990                 1.00000   

R2, SumsOfSqs and MeanSqs are calculated based on your observed data, and it depends on your model, so for both codes (with or without strata) this is unchanged.
To understand why the p-values might be different, adonis "uses a permutation test with pseudo-F ratios." (see vignette). This would entail permutating the rows of the dissimilarity matrix (or whatever is on the LHS), and re performing the test under the condition where association between the response variable (distance in this case) and your variable of interest is lost. It collects this permutated F values and calculates how many of these are larger than your actual F value (in my dune example it is 3.6389), giving you a p-value. Maybe check out this for more explanation
If there is a structure to your data, you might want to restrict the way this permutation goes, for example in your case, you want to preserve the effect of island, and randomize the disease effect, then you specify the strata to be island. In my example, I specify it to be Management.
In your situation, you might want to check whether there is any difference in the dissimilarity matrix between island. If not, then most likely the strata is unlikely to have an effect.      
