Multidimensional Scaling

Basic Principles

An easy way to explain the basic principles of MDS is to consider a simple example involving 20 cities (objects). The data consist of distances (as the crow flies) in kilometers between each pair of cities. These distances can be organized in a [Page 1106]20 × 20 symmetric matrix. With this distance matrix as the input, MDS produces a geographic map by computing the coordinates (basically, longitude and latitude) for each city. In MDS terminology, such a representation is called a configuration.

Applications in the social and behavioral sciences are typically not based on geographic input distances. Instead of distances, proximities (i.e., similarities or dissimilarities) are used as the input, which can be computed easily from an ordinary person × variables matrix. Popular proximity measures are correlation coefficients, a Euclidean distance measure, a Jaccard coefficient, and so on. Depending on whether the rows or the columns of the data matrix are subject to scaling, the proximity measure of choice is applied to either the rows or the columns. From this point in this entry, all explanations are limited to dissimilarities because most MDS software packages require dissimilarities as input. Note that similarities can be easily converted into dissimilarities (and vice versa).

Formally, MDS takes a symmetric input dissimilarity matrix Δ of dimension n × n and computes a configuration X in a space of dimension p. In the cities example provided, it was quite natural to choose two dimensions (map), even though three dimensions would have made sense as well (globe). Similar to principal components analysis, p needs to be fixed a priori. In MDS, researchers typically aim for a small p (e.g., p = 2 or p = 3) so that the configuration can be easily plotted. In order to compute X on the base of Δ, a target function needs to be formulated and solved (i.e., minimized). The most popular target function in MDS is Kruskal’s stress, which, in its simplest form, can be expressed as

$$\text{stress}={\displaystyle \sum _{i<j}{\left({\widehat{d}}_{ij}-d_{ij}(X)\right)}^2}.$$

Minimizing this function implies that the distances among the points in the MDS space should be as close as possible to the input dissimilarities δij, or, to be more precise, a transformed version of them: ${\widehat{d}}_{ij}=f({\delta }_{ij})$. The resulting outcomes are called disparities or, simply, d-hats. The most popular transformation functions (optimal scaling) are a monotone step function (which leads to Kruskal’s nonmetric or ordinal MDS) or a linear regression function of the form ${\widehat{d}}_{ij}=a+b{\delta }_{ij}$ (called interval MDS, or, if a = 0, ratio MDS). The choice of the transformation function gives the user some modeling flexibility (e.g., considering the input dissimilarities to be on an ordinal or on a metric scale level).

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