I'm a bit of a newbie with stats and I'm still trying to fully grasp things. I'm a bit confused on what exactly the test statistics are telling me for PERMANOVA (pseudo-F), ADONIS (R2) and ANOSIM (R), and what exactly I should be using for my question, that is: are there significant differences between locations?

From what I understand of permanova, pseudo-F tells me the ratio of between-group variation and within-group variation. A larger pseudo-F value means that the between-group variation is greater than the within-group variation (so like, the variation between locations is greater than the variation between samples within one location?) But I'm not sure what I'm comparing this number against. Are values 1-3 really low? Also, I noticed that this value is the same as the F.Model value in the ADONIS output. What is the relationship between PERMANOVA and ADONIS?

For ANOSIM, the R-value tells me if groups of samples are significantly different. In my case 0.44...is probably not statistically different? What does it mean that the R-value is 0.7 between two locations but the total R-value is much lower than that? Doesn't that explain some significant difference?

And for ADONIS, the R2-value tells me how much can be explained by the variable. So in my case, only 19% of variation can be explained by location differences.

Problem is, 0.44 doesn't seem suuuuper low to me...so I feel like there is a difference between locations, but why is the R2 value so low? Am I just being biased and misinterpreting?

See here for more description of this test and its interpretation. See also the original PERMANOVA paper, the vegan-R adonis documentation, and other links on that webpage.

They are different implementations of the same test. The adonis action in QIIME 2 supports multi-factor PERMANOVA tests. The beta-group-significance action only supports one factor but generates useful plots.

No, the P value provides that information. Read more about this test and interpretation here.

Yes all of these tests appear significant (look at the P values) but only a portion of the variation is explained by location. This is normal and your R2 values actually look quite good; R2=0.19 may not sound like a lot, but many many factors shape microbiomes at different sites and so R2 values are often < 0.5, simply because the microbiome is not very simple.

In terms of R intrepetation, can I link you to a recent Nature paper where they had an Adonis R^{2} less than 0.06. (Check figure 1). Im actually amazed youâ€™re getting an R^{2} in the double digits, your explanatory factors must be large, or your sample homogenous and/or relatively small.

Thanks both @Nicholas_Bokulich and @jwdebelius for your replies! I will check these papers out. I donâ€™t know how when I google these cites donâ€™t come up, or I just miss them, so thank you for taking the time to link them to me. I was looking up other factors to explain variation, such as species and plant part type (leaf or stem) the R2 values were about 4-8% for that, so good to know that this is normal.

And yes @jwdebelius, unfortunatley my sample size is quite small (n < 50)

And for species, for example, the R2 value of species on itâ€™s own was 0.08934, but with the other factors taken into account, itâ€™s now 0.068488. Interestingly, the R2 value for Location stayed the same. Is this what it means by â€śmulti-way ADONIS testsâ€ť? Is it basically like, some factors may have been influencing others and the test is now taking that into account? And with what I did before, the test just looked at one factor discretely and it didnâ€™t take into account the influence of other factors? (Sorry for the messy sentences)

Yes, yes, and yes! You can even make more complicated formulae, e.g., see if there is an interaction between factors by using the * operator instead of + (but beware, interactions can get very complicated to interpret! especially if you have > 2 levels of interaction)