As part of the alpha-diversity analysis there is a Kruskal-Wallis pairwise test. I am a bit lost in terms of interpreting the results. More specifically, I do not know what the H value refers to and whether I should take it into account, and I am not sure of whether I should look at the adjusted q-value or at the p-value for significance.

I have read on the forum that the q-value is a p-value with a Benjamini & Hochberg correction. So is q-value same as the Benjamini-Hochberg critical value - (i/m)Q? I have read online ( http://www.biostathandbook.com/multiplecomparisons.html ) that you need to choose a false discovery rate for this type of correction, and that the largest p-value with p<(i/m)Q is significant, and so are all the other smaller p-values. Does that sound right?

As far as I know, H is the effect size, and should be reported along with the P value. This test uses the H value to determine whether the median rank distribution for at least one sample is statistically significant from another.

probably q-value, though wikipedia (the infallible, canonical source of all knowledge ) suggests using uncorrected p-values for post-hoc testing.

No, q is just the corrected p-value, not the critical Q value. The "Benjamini-Hochberg adjusted P value" is discussed part-way down the page you linked to.

Thanks for the explanation. That does make it a lot clearer!

Following up on that, a more basic question I had was what is the difference between the Kruskal Wallis for the alpha diversity and Anosim/Permanova for beta-diversity in terms of interpreting results? I know that alpha and beta diversity donâ€™t measure the exact thing, but in my head the two statistical tests show the same thing. For example, obtaining a significant value for Kruskal Wallis suggests that the species composition is different between the two categories. But the same conclusion would be drawn from a significant p-value for Permanova/Anosim, wouldnâ€™t it?

Alpha diversity is telling you about the diversity (e.g., richness) within each individual sample. How many species are present? How much of the phylogenetic tree is covered? When comparing groups of samples (e.g., with a Kruskal-Wallis test), you are asking whether these values differ between groups. Does group A have more unique species/phylotypes than group B? A significant P value will indicate that yes, group A is different from group B.

Beta diversity is telling you about how different samples are from one another. This may be phylogenetic (UniFrac) or non-phylogenetic (e.g., Bray-Curtis). This may be weighted by abundance or relate to the presence/absence of individual organisms. You really need to understand what the diversity metric measures (this is true for both alpha and beta diversity). A permanova test is going to tell you whether those differences partition based on some metadata value â€” is at least one group significantly different from at least one other group. A significant result is essentially telling you that samples in group A are more similar to each other than they are to samples in group B, and vice versa.

So these tests are telling you very different things, and the specific interpretation depends greatly on the metrics that you are using.