Multidimensional Scaling Introductory Overview - Computational Approach

Multidimensional Scaling is not so much an exact procedure as rather a way to "rearrange" objects in an efficient manner, so as to arrive at a configuration that best approximates the observed distances. The program actually moves objects around in the space defined by the requested number of dimensions, and checks how well the distances between objects can be reproduced by the new configuration. In more technical terms, the program uses a function minimization algorithm that evaluates different configurations with the goal of maximizing the goodness-of-fit (or minimizing "lack of fit").

Measures of goodness-of-fit: Stress
The most common measure that is used to evaluate how well (or poorly) a particular configuration reproduces the observed distance matrix is the stress measure. The raw stress value Phi of a configuration is defined by:

Phi = S[dij - f (dij)]2

In this formula, dij stands for the reproduced distances, given the respective number of dimensions, and dij (Deltaij) stands for the input data (i.e., observed distances). The expression f(dij ) indicates a nonmetric, monotone transformation of the observed input data (distances). Thus, the program will attempt to reproduce the general rank-ordering of distances between the objects in the analysis.

There are several similar related measures that are commonly used; however, most of them amount to the computation of the sum of squared deviations of observed distances (or some monotone transformation of those distances) from the reproduced distances. Thus, the smaller the stress value, the better is the fit of the reproduced distance matrix to the observed distance matrix.

Shepard diagram
You can plot the reproduced distances for a particular number of dimensions against the observed input data (distances). This scatterplot is referred to as a Shepard diagram. This plot shows the reproduced distances plotted on the vertical (y) axis versus the original similarities plotted on the horizontal (x) axis (hence, the generally negative slope). This plot also shows a step-function. This line represents the so-called D-hat values, that is, the result of the monotone transformation f(dij ) of the input data. If all reproduced distances fall onto the step-line, then the rank-ordering of distances (or similarities) would be perfectly reproduced by the respective solution (dimensional model). Deviations from the step-line indicate lack of fit.