I am currently dealing with an issue regarding cluster validation.
I have a pairwise distance matrix (based on the hamming distance between a set of core genes of different strains of a bacterium). I performed k-medoids clustering on the pairwise distances, attempting to validate my initial suspicion that there are 3 clusters. Surprisingly, the average silhouette index is around 0.3 for k = 2 and around 0.2 for k = 3, suggesting that the clustering is "better" for the k = 2 case. I understand that the silhouette index is only one of many cluster validation methods and that the silhouette index cannot be wrong per se.
I have the suspicion that the silhouette index supports k = 2 over k = 3 because my putative clusters are of different inter-cluster distance as well as of different variability, as indicated by an nMDS plot as well as boxplots of (real) pairwise distances between clusters. Specifically, I appear to have two tight clusters that are close to each other (cluster 1 and 2), while the third cluster is further away and less well defined (cluster 3).
I believe that the average silhouette index is smaller for k = 2 as the neighbouring cluster in that case is far away (for both clusters), where in the k = 3 case the average index is smaller since the two well defined clusters' neighbouring cluster are much closer (the other well defined cluster, respectively).
I am aware that the nMDS representation is not necessarily representative of the true pairwise distances.
I would like to know how to find out whether it would be ok to dismiss k = 2 in favor of k = 3 and, if so, how to make that point in the most elegant fashion.