Hej @salias and @rahel_park,
Hopefully its okay if I QIIME in! (
)
Everything @salias has said is true! I just wanted to comment on one thing. I think about rarefaction like buttons on men's suits:
Beta: Sometimes
Alpha: Always
DA: Never
You need to rarefy before you calcualte things involving richness or unweighted metrics becuase youre looking at presence and absence and the depth is associated with the presence/absence. Rarefaction introduces zeros you need to help deal with differences in depths. I've had discussions with colleagues over drinks (
) about whether its the "right" solution for differences, but at least according to that recent Pat Schloss paper it seems expeditious. (SRS is an interesting alternative, if thats your direction).
Bray Curtis distance doesn't care as much about rarefaction, becuase Bray Curtis looks at abundant organsisms, which are less sensitive to depth. I still tend to recommend it (although there are other schools) but its not going to suffer if you don't rarefy. Jaccard or UniFrac distance, which look at the presence/absence of organisms will amplify differences in depth, though!
For compositional metrics and tests like ANCOM-BC, zeros are actually a big 
problem! For Aitchison (and I think ANCOM), we add a psuedocount as a constant, so all the zeros become 1s and all the 1s become 2s... etc because you can't take a log or a ratio of 0. The problem is that on the log scale, 1 + 1 = 2 is the same as 32 + 32 = 64. So, we get a heavier weighting toward rare features, which is how you end up being sensitive to depth.
(FWIW and unpublished, but I find DEICODE to be a nice alternative to aitchison distance that's less sensitive to depth).
Rarefaction is introducing more zeros to the data, and so you end up having more issues with your Aitchison distance!
I tend to think about Aitchison more with filtering (how can I get rid of rare organisms to decrease my zeros), but you may have different thoughts/constraints.
Hope this helps and adds on the awesome job @salias is doing! Thank you
Best,
Justine