Clustering quality: normalized mean square error or absolute error?

Question

I have divided my data (a matrix of proximities expressed by cosines between 94 objects) into clusters with Ward hierarchical method, and I am very happy with the results from a visual point of view. However, I'd like to check whether the quality of the clusters is good.

I've been told that for Ward method, statistics that can be good to check how good are the clusters are:

• cophenetic correlation coefficient,
• normalized mean square error,
• normalized mean absolute error.

However, I don't know how to read them. I have a very low correlation coeff. (0.56) which sounds bad, since good correlations are averagely above .80. Then I have a 351.14 NMSE and a 17.24 NMAE. Is that bad? Is that good? How do I tell?

Answer 1

Several points:

• Ward method is geometrically correct to use only with a matrix of squared Euclidean distances. You may not apply it to cosines unless they are internally converted by the clustering program into those distances. So, you should read help docs of the program you use to know how it treats cosines when Ward is used. Anyway, Euclidean $d^2=2(1-cos)$, for your information.
• Since Ward attempts to minimize within cluster sum of squared deviations it naturally follows that one should generally prefer an isomorphic clustering validation criterion - mean squared error. A number of popular clustering criterions are based on MSE, including famous Calinski-Harabasz criterion and Davies-Bouldin criterion. There exist many programs which compute them.
• Clustering criterions should mostly be taken as relative measures only. That is, one compares alternative clusterings with their help. Absolute magnitude of a criterion is of little use. See further discussion of the topic here.