Multidimensional Scaling

Conceptual Overview and Discussion

The researcher starts with a set of objects or stimuli that he or she is interested in scaling, such as a set of cars. The research questions of interest are about the underlying dimensions that describe this set of objects and how the objects themselves are organized relative to each other. The research participant is presented with pairs of these stimuli and asked for his or her ratings, typically in terms of the objects' similarity (or dissimilarity), although the ratings can be on other continua as well. All pairs are presented, and care should be taken in presenting each pair in both orders, AB and BA if possible, because it has been found that people's ratings sometimes differ depending on how the stimuli are presented.

The raw data for the analysis, then, is a potentially nonsymmetric matrix of all pair-wise ratings of similarity (or dissimilarity/distance/disparity) between the objects in the set. The data matrix is analyzed with MDS software such as the one available in SPSS or specialized programs available from several Web sites. (A useful feature of the SPSS program is that it provides an option of using correlation coefficients computed from multivariate data as the input matrix, with the correlations being interpreted as similarities.) The MDS analysis provides the researcher with a geometric model of the objects' representation with two key features: (1) the number of dimensions that are required to describe the set of objects under study, and (2) the placements, in the form of geometric coordinates, of these objects in this space. In the classic approach, the coordinates of the objects, x1, x2, …, xm, where m is the number of dimensions, are estimated from the similarities, sij, from the model equation

The researcher makes two key decisions for fitting the model to the data. One is about the [Page 576]number of dimensions, m, and the second is the type of metric to be used, represented by r in the model equation above. (A metric is a mathematical function with specific properties that assigns numbers to distances; several different types of metrics have been proposed, and in MDS two typical metrics come from what is called the Minkowski family of metrics.) Any positive value of r is possible, but two of the most commonly used and most easily interpretable metrics are the “city-block” (where r = 1) and Euclidean (r = 2). The city-block metric measures the distance by simply adding the distances the two objects are apart along each dimension, in a grid-like fashion as one would walk in a city. The Euclidean metric measures the distance “as the crow flies.”

...