How to determine the number of clusters when using correlation as the distance? How does using 1 - correlation as the distance influence the determination of the number of clusters when doing kmeans?
Is it still valid to use the classical indices (Dunn, Davies-Bouldin...)?
 A: First. It is odd to use $1-r$ distance with K-means clustering, which internally operates with euclidean distance. You could easily turn r into true euclidean d by the formula derived from cosine theorem: $\sqrt{2(1-r)}$.
Second. I wonder how you manage to input distance matrix into K-means clustering procedure. Does R allow it? (I don't use R, and the K-means programs I know require casewise data as input.) Note: it is possible to create raw casewise data out of euclidean distance matrix.
Third. There is a great number of internal "clustering criterions" (over 100 I believe) helpful to decide what cluster solution is "better". They differ in assumptions. Some (like cophenetic correlation or Silhouette Statistic) are very general and can be used with any distance or similarity measure. Some (like Calinski-Harabasz or Davies-Bouldin) imply euclidean distance (or at least metric one) in order to have geometrically sensible meaning. I haven't heard of Dunn's index you mention.
P.S. Reading Wikipedia page on Dunn index suggests that this index is of general type.
A: Note that k-means is designed for Euclidean distance. The mean may or may not be an appropriate estimator for the cluster center with other distances. So be careful when using other distance functions with k-means. Consider using a more modern clustering algorithm!
