Measuring the Similarity of MDS Configurations

The similarity of multidimensional scaling configurations can be assessed by either correlating over corresponding point coordinates, after Procrustean similarity transformations, or by directly assessing the similarity of corresponding distances. The product-moment correlation is, however, an inadmissible measure in the latter case, and hence the statistical norms reported by Poor and Wherry (1976) are worthless. What can be used is the congruence coefficient. Norms are derived for this coefficient under a wide range of parameters. We thus have two measures of configurational similarity, but even though they are both admissible, it is shown that they are not related in a simple way. Moreover, their statistical evaluation with respect to the question whether two configurations are "significantly similar" is likely to lead to different conclusions. Alternative ways to assess configurational similarity are discussed.