Summary: Given a non-Euclidean distance matrix in which some entries represent replicate observations of an entity, is it (statistically) valid to calculate the arithmetic mean of pairwise distances across replicates?
Details: From a matrix of sample by species incidence, I have used a non-Euclidean distance metric (an index of the difference of ecological communities) to calculate a pairwise sample distance matrix. The ~70 samples represent only ~65 sites; that is, ~5 sites have been sampled multiple times, and are each represented by ~2-3 replicate observations.
I wish to sensibly collapse each set of ~2-3 replicate samples into a single observation for each site. Since the species data are binary presence/absence, taking an average at this level does not seem appropriate. I have computed the PCoA centroids for use in some analyses, but I wish to use the distance matrices themselves in other non-ordination analyses.
For this non-Euclidean distance matrix, is it valid to take the arithmetic mean (or some other average) of pairwise distances between replicates and other samples? For example, given a complete pairwise distance matrix for replicates 1A and 1B and unique samples 2 to 70, could I simply replace the two sets of rows and columns for 1A and 1B with a single row and column containing their average?
One issue I foresee is that the self-distance of replicates condensed in this way will be >0. Would it then be valid to set self-distance to 0, or would this somehow render other pairwise distances nonsensical?