Hi,
I have paired data. Let's say pre-treatment and post-treatment. I would like to know whether composition changed after treatment. In my simplest idea, I would calculate the difference of PCoA of this two condition applying similar idea of one sample t-test. In such condition, I can simply calculate one sample t-test on PC1 which may serve as an approximate. However, I wonder if there is some way I can calculate directly on paired multidimensional data?

timanix,
Thank you for the reply. I worked on pairwise distances. I am not quite sure but I think it works on distances (therefore always value ≥ 0) instead of vector (should have positive and negative values). I noticed that pairwise-differences can work with beta diversity values as stated in the tutorial (no example provided). Here, I have some trouble with the --p-metric that I don't know what to give for that parameter when applying a PCoAResults artifact as the metadata file.

I am not sure if you can just use PCOA_results.qza artifact as metadata input.
However, if you extract/export/open as archive this file, you should be able to find a txt file "ordination.txt":

If you skip all rows above and include "Site", you will get a table with samples as index and values columns, X is a first column and Y is the second. Hope that these values are what you are looking for. You can add them to your metadata and run some tests in qiime or outside it. I indicated technical possibility to run it like this but I am afraid that I can't say anything about how valid this approach is.

Thank you. I think the columns are likely to be PC1, PC2, ......
For the qiime longitudinal pairwise-differences, I think I can only select one parameter at a time which dose not resolve the multidimensional nature of PCs. I also did some search and found that the Hotelling's T-squared distribution which is a generalization of t-test may help and there is also available python module (hotelling · PyPI). However, this test seems not to encompass the weighted nature of PCs.
I am not expertise in this field indeed. At present, the best way I can think of is to present statistical analyses on the first three PCs which may be representatives of the PCs.