MULTIDIMENSIONAL SCALING (MDS)

Multidimensional scaling (MDS) refers to a family of models in which the structure in a set of data is represented graphically by the relationships between a set of points in a space. MDS can be used on a wide variety of data, using different models and allowing different assumptions about the level of measurement.

In the simplest case, a data matrix giving information about the similarity (or dissimilarity) between a set of objects is represented by the proximity (or distance) between corresponding points in a low-dimensional space. Given a set of data, interpreted as “distances,” it finds the map locations that generated them.

For example, in a study of how people categorize drugs, a list of drugs was elicited from a sample of users and nonusers; 28 drugs were retained for a free-sorting experiment, and the co-occurrence frequency was used as the measure of similarity. The [Page 670]data were scaled in two dimensions, producing Figure 1.

Figure 1

In the case of perfect data, the correspondence between the dissimilarity data and the distances of the solution will be total, but with imperfect data, the degree of fit is given by the size of the normalized stress (residual sum of squares) value, which is a measure of badness of fit. In this case, the fit is excellent (stress1 = 0.097), and it is three times smaller than random expectation. The solution configuration yields three highly distinct clusters (confirmed by independent hierarchical clustering) of mild stimulant drugs (core prototypes: tobacco, caffeine), hard recreational (core: cocaine, heroin), and household prescription (core: aspirin, penicillin).

MDS can be either exploratory (simply providing a useful and easily assimilable visualization of a data set) or explanatory (giving a geometric representation of the structure in a data set, in which the assumptions of the model are taken to represent the way in which the data were produced). Compared to other multivariate methods, MDS models are usually distribution free, make conservative demands on the structure of the data, are unaffected by nonsystematic missing data, and can be used with a wide variety of measures; the solutions are also usually readily interpretable. The chief weaknesses are relative ignorance of the sampling properties of stress, proneness to local minima solutions, and inability to represent the asymmetry of causal models.