Review of the Development of Multidimensional Scaling Methods

Multidimensional scaling methods are now a common statistical tool in psychophysics and sensory analysis. The development of these methods is charted, from the original research of Torgerson (metric scaling), Shepard and Kruskal (non-metric scaling) through individual differences scaling and the maximum likelihood methods proposed by Ramsay. 1 Introduction-what is multidimensional scaling? The set of procedures referred to as multidimensional scaling methods are concerned with constructing a configuration of n points, usually in Euclidean space, from information about the pairwise 'distances' among a set of n objects or individuals. The initial research into multidimensional scaling methods was carried out by scientists working in psychophysics and sensory analysis, in which the objects are stimuli, and the distances between them are assessments of either their similarity or dissimilarity as made either by an individual or by a panel of judges. Most multidimensional scaling methods have been developed to analyse dissimilarities, and so, since similarities can be expressed as dissimilarities subtracted from a constant, I shall generally refer only to dissimilarities in the remainder of this review. The dissimilarities are usually presented in an (n x n) lower triangular matrix, the entry in the ith row andjth column being the dissimilarity, dij, between object i and objectj. In some cases the assessments of dissimilarity are not made directly, but are obtained from observations on p variables for each of the n objects. Appropriate methods for calculating such measures of similarity and dissimilarity are discussed by Chatfield & Collins (1980), section 10.2. Jardine & Sibson (1971) defined one class of dissimilarity functions, which they called 'dissimilarity coefficients', which are required to satisfy the following conditions: