Objective Statistic for Comparing Clusters I wondered if anybody knew of any objective measures for comparing clusters on ordinal data?
For example, suppose I was to run a standard clustering routine such as partitioning around medoids/k-medoids on a bunch of ordinal health data.
In particular, my health data might describe well-being on a Likert-like scale, and if I had 4 clusters; this might group those maximally dissimilar.
However, is there an objective measure out there which potentially expresses that one clusters is relatively better than another, given the ordinality of the data/ordinality of the clusters?
I appreciate this is potentially a scandalous question, but would be good to hear any feedback.

After cluster analysis, I always run ANOVA (Kruskal-Wallis) and chi-squared contingency table analysis on all the cluster input variables to see how their means and count proportions differ across clusters (groups).  This allows me to describe to a user/customer what the characteristics are for each cluster.  After all, without knowing if age, gender, race, well-being, SES, anxiety, depression, stress scores are significantly different across clusters, you won't know anything about subjects that were assigned (agglomerated into) to a cluster.
After running ANOVA/chi-squared analysis, you may learn e.g. that cluster 1 subjects have the lowest mean scores for well-being and have the highest age, whereas e.g. cluster 3 subjects are the youngest and have the greatest mean for well-being.
Once you tackle the issue of describing who the subjects are in each cluster based on the mean (count proportion) differences, you can begin to explore if one cluster is better than the other.  But without knowing the mean/proportion differences across clusters, I believe you won't be able to compare clusters to determine if "one is better than the other."
