In concept mapping, a multidimensional scaling analysis creates a map of points that represent the set of statements brainstormed, based on the similarity matrix that resulted from the sorting task in Step 4: Structure Ideas. The output from the two-dimensional multidimensional scaling is a set of x-y values that can be plotted, as well as some diagnostic statistical information. The plot is called the &quotpoint map" and consists of dots representing the statements, each of which is identified by a number. In studies where we have examined other than two-dimensional solutions, we have almost universally found the two-dimensional solution to be acceptable, especially when coupled with cluster analysis as Kruskal and Wish (1978) suggest. So,

Multidimensional Scaling

Nonmetric multidimensional scaling is a technique that takes any similarity (e.g., proximity) matrix and represents it in any number of dimensions as distances between the original items in the matrix.

A simple example of the principle that underlies multidimensional scaling can help clarify the concept. If you were given a geographic map of the United States and asked to construct a table of distances between three major cities, say New York, Chicago, and Los Angeles, you could accomplish this fairly easily. You might take a ruler and measure the distances between each pair of cities and enter them into a 3 x 3 table of ruler-scale relative distances. However, if you were given only a table of distances between the three cities and were asked to draw a map which located the three cities on it as points in a way that fairly represented the relative distances in the table, the task would be slightly more difficult. You might begin by arbitrarily placing two points on a page to represent two of the cities and then try to draw in a third point so that distances from the third city to the first two cities was proportionate to the distances given in the table. You would be able to accomplish this if the table consisted of three cities, but for more cities the task would become extremely complex. Multidimensional scaling is a multivariate analysis that accomplishes this task. It takes a table of similarities (or distances) and iteratively places points on a map so that the original table is as fairly represented as possible.

In a traditional multidimensional scaling analysis, the analyst specifies the number of dimensions into which the set of points will be fit. If a one-dimensional solution is requested, all of the points will be arrayed along a single line. A two-dimensional solution places the set of points into a bivariate distribution that is suitable for plotting on an X-Y graph. The analyst could compute any number of dimensions from 1 to N-1, where N is the number of statements, but, it is difficult to graph and interpret solutions that have more than three dimensions. (The literature on multidimensional scaling discusses this dimensionality issue extensively.) One view is that the analyst should fit a number of solutions (e.g., one- to five-dimensional solutions) and examine diagnostic statistics to see whether a particular dimensional solution is most compelling. Those of you familiar with factor analysis will recognize that this is analogous to examining J-plots of eigenvalues in order to decide on the number of factors.

Another view suggests that in certain contexts, a priori use of two-dimensional configurations might make sense. Kruskal and Wish (1978) state the following:

&quotSince it is generally easier to work with two-dimensional configurations than with those involving more dimensions, ease of use considerations are also important for decisions about dimensionality. For example, when an MDS configuration is desired primarily as the foundation on which to display clustering results, then a two-dimensional configuration is far more useful than one involving three or more dimensions" (p. 58).

Stress in Multidimensional Scaling

As noted, the output from the two-dimensional multidimensional scaling is a set of x-y values that can be plotted, as well as some diagnostic statistical information. The most important of the diagnostic statistics is the indicator of &quotStress.”

To get an idea of what Stress is, recall that the input to the multidimensional scaling analysis consists of the square matrix of similarities based on the sorting task. Ideally, the result is the two-dimensional picture of the statements that yields the closest approximation to this original input matrix. Now, imagine that you were to construct a new square matrix for the set of statements by measuring the straight-line distances between all statement pairs in a point map. We would consider the x-y point plot to be a good representation of the data if there is a strong relationship between the input matrix and the distances on the map. In fact, Stress measures the degree to which the distances on the map are discrepant from the values in the input similarity matrix. High Stress values imply that there is a greater discrepancy and that the map does not represent the input data as well; low Stress values imply a better fit. Some (mainly those who work with extremely well-behaved data like the perception of the similarities of colors or sounds) argue that it is desirable to have a Stress value of .10 or lower, but this will seldom be attained in concept mapping. However, it should be recognized that their low Stress-value expectations are based on experience with much better controlled psychometric testing environments--not usually the case in concept mapping.

In concept mapping, the facilitator should use the Stress indicator as a rough guideline of the degree to which the map represents the grouping data. High Stress values (e.g., .25 or greater) may imply that there is more complexity in the similarity matrix than can be represented well in two dimensions, that there was considerable variability or noise in the way people grouped the statements, or both. The idea of Stress is in many ways akin to the idea of reliability of measurement. In general, Stress values will be lower (i.e., the map will be a better fit) when there are more statements and more people rating the statements than otherwise. A high Stress value (i.e., greater than .35) may warn the facilitator that there may be some difficulty in interpreting the map sensibly.