Paired samples collected on the same day - best way to analyze them?

Yep, please see some discussion here:

Since BAL is one of the states you are comparing, it should not also be used as the group column.

The general goal with this test is that you may have a set of paired samples in two different treatment groups, and this test will determine if one group experiences greater divergence between states (e.g., pre/post treatment) than the other. This works whether the pairs are of patients' samples collected at two different times, or in your case two different collection sites. So this would apply to your case if you are collecting BAL and brush samples from two different groups — e.g., one group is receiving some treatment.

Another option is to run PCoA on this sample set, and then input the PCoA data to pairwise-differences. This will determine whether you see a directional effect on beta diversity.

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