Betadisper with distance matrix or principle coordinates for ecological data? When reading through the vegan::betadisper() function help document, I noticed that it is suggested that:
"However, better measures of distance than the Euclidean distance are available for ecological data. These can be accommodated by reducing the distances produced using any dissimilarity coefficient to principal coordinates, which embeds them within a Euclidean space. The analysis then proceeds by calculating the Euclidean distances between group members and the group centroid on the basis of the principal coordinate axes rather than the original distances."
I used to run betadisper directly on my distance matrix created using measurement such as bray-curtis or unifrac.
Given this notice, I wonder if it is better for me to use the pcoa coordinates generated by cmdscale, generate a Euclidian distance matrix based on the coordinates, and then run the betadisper based on the new distance matrix. If so, how many PCs should I use? Or probably it won't make a big difference?
Thanks!

Function vegan::betadisper is designed so so that you do not need to use principal coordinate analysis, but you can use original dissimilarities (or distances). Actually the math is the same, but in vegan::betadisper latter axes are not thrown out, but all are used – including axes associated with negative eigenvalues.
I can see some theoretical value of bypassing this work in vegan::betadisper and using just some first axes of PCoA, but if you do so, be prepared to give your arguments to the reviewers of your work. The argument would be that only the first axes have real information and the latter are noise. That is actually what you asked with "how many axes to use". Unfortunately, there won't be a firm answer to this (although various approximations can be suggested), and there is an added problem of handling (or ignoring) negative eigenvalues in PCoA with semimetric dissimilarities. I think it is easier to use vegan::betadisper like it was written.
