Multidimensional Scaling

Abstract

Multidimensional scaling is perhaps in concept the simplest and most intuitive of the various approaches to multivariate analysis, and this can rightly be regarded as a major advantage. It is difficult to misunderstand the principles upon which it is based. A multidimensional scaling takes as its starting point a matrix of similarity (or dissimilarity) scores between cases like the one in Table 22.9. The analysis consists of an iterative, trial-and-error process of creating a configuration of points, each representing one of the cases in the dataset. These points representing the cases are placed in space in such a way that the rank order of the distances between the pairs of points corresponds as well as possible to the rank order of the similarity coefficients in space. That is to say, the aim of the configuration is to place the two points representing the two most similar cases closer to each other than any other pair of points in the configuration. The two points representing the second highest similarity score should be the second-closest pair of points, and so on. Finally, the two cases with the lowest similarity score should be the two points farthest apart in the configuration. In this very simple way, multidimensional scaling attempts to draw a picture of the relationships between cases that are encapsulated in the matrix of similarity coefficients. Since only a rank order correlation is sought between similarity scores and distances between pairs of points, multidimensional scaling is sometimes referred to as nonmetric multidimensional scaling. The conceptual simplicity of multidimensional scaling masks the fiercely complex challenge of writing a program to produce such a configuration of points. A multidimensional scaling program must set up an initial configuration by placing points representing all the cases in space, and then tinker with that configuration, moving some points to new locations to see whether that improves the rank order correlation between distances between pairs of points and similarity coefficients between pairs of cases. This is done over and over until no improvement can be found. As multidimensional scaling was developed, it was not unusual to get different results from different programs, but the algorithms for this iterative procedure are now honed enough that all the programs currently in use are pretty much equivalent. Some, but not all, large statpacks will perform multidimensional scaling.