Analyzing rectangular tables by joint and constrained multidimensional scaling

Abstract

Abstract This paper selectively discusses Multidimensional Scaling methods, addressing both theory and applications. Most of the discussion is focussed upon the analysis of rectangular tables. The term joint MDS is introduced for an analysis which gives a meaningful joint representation of the set of row and the set of column objects of such a table so that the dissimilarities either within sets or between sets are optimally approximated. From this perspective, it is possible to give an unified account of the relationships between classical Multidimensional Scaling, Principal Components analysis and Correspondence Analysis. By imposing constraints the general model can be structurally changed in a number of ways. For example, it can be made discrete; that is, in principle it subsumes subset models and quadratic assignment as special cases. It can also be made more flexible in the sense that external information may be used in order to tune the analysis to specific hypotheses. Finally, an elegant joint MDS model called Multidimensional Unfolding is discussed. As a technique, Unfolding unfortunately suffers from being extremely sensitive to ill-conditioned data. A new way of coping with this problem is indicated and an application is given in which the Bootstrap technique succesfully establishes the stability of the results.