﻿ Multidimensional Scaling

Graziano & Raulin
Research Methods (9th edition)

## Multidimensional Scaling

Multidimensional scaling techniques refer to a group of statistical procedures that attempt to map variables in an N-dimensional space on the basis of their similarity or distance from the other variables (Carroll & Arabie, 1998). This abstract definition is difficult to comprehend, but a simple example might illustrate the logic of these procedures. Imagine that you have 20 towns in the state of Kansas and you measure the distance between each pair of towns. There are 190 distances in this data set, which you could organize in a 20 X 20 grid. You would list all of the towns across the top of the grid and the same list along the side. Each cell shows the distance between the towns represented by that row and column. A road atlas often lists distance information in just this format. But such an arrangement of distances between cities simply organizes the data; it does not analyze the data.

Multidimensional scaling techniques analyze such data by seeing if a small number of dimensions could account for the observed distances between variables. In our example, the variables are towns and the distances are actual distances in miles. But the variables could be anything, and the distances are usually measures of similarity. For example, the variables could be 20 wines, where people rate the similarity of each pair of wine. The greater the similarity, the shorter the distance. Through a complex mathematical process, the distance or similarity ratings are used to plot the variables in first a one-dimensional space (a straight line), then a two-dimensional space, a three-dimensional space, and so on. At each step, the program computes how accurately the distances can be represented. We would expect that two dimensions would reproduce the distances between Kansas towns reasonably accurately because the towns are essentially on a relatively flat two-dimensional surface (assuming that there are few mountains or lakes that force roads to deviate significantly from straight lines between towns). If our example were distances between stars, we would expect that a three-dimensional space would reproduce those distances accurately because the stars are distributed in a three-dimensional space. With the towns and stars examples, the dimensions are physical space. But what about wines? How many dimensions might we find there, and what would those dimensions represent? In this example, we are not talking about a physical space, but rather a conceptual space. We might find that a single dimension represented the similarity ratings of wines well. Perhaps that dimension would represent the wine's sweetness. Perhaps it would take two dimensions, with one dimension representing sweetness and another representing tartness. This would be an interesting study to run, and it would likely give us clues about human taste perception.

Multidimensional scaling techniques seek to identify underlying factors that account for complex patterns of scores. It is presumed that such underlying factors correspond to real variables in life. For example, the sweetness dimension that we might uncover with our wine example would correspond to the level of sugar in the wine. If we knew nothing about how sugar contributed to taste, but we identified a dimension of sweetness and could reliably order wines on this dimension, we could then look systematically for factors to account for this observed ordering. Presumably such a strategy would quickly focus us on sugar as the underlying factor that accounted for the sweetness dimension.

It was once believed that the mathematical wizardry of multidimensional scaling procedures would help us to make sense of a vast and complex world (Kruskal & Wish, 1978). That proved to be a unrealistic expectation. Nevertheless, these procedures have led to a much greater understanding of our psychological world (e.g., Katsikitis, 1997; Samson, Zatorre, & Ramsey, 1997), and furthermore, have stimulated even newer and more promising techniques such as path analysis and taxometric search procedures.

## References

Carroll, J. D., & Arabie, P. (1998). Multidimension scaling. In M. H. Birnbaum (Ed.), Measurement, judgment, and decision making: Handbook of perception and cognition (2nd ed.) (pp. 179-250). San Diego: Academic Press.

Katsikitis, M. (1997). The classification of facial expressions of emotion: A multidimensional scaling approach. Perception, 26, 613-626.

Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling. Beverly Hills, CA: Sage.

Samson, S., Zatorre, R. J., & Ramsey, J. O. (1997). Multidimensional scaling of synthetic musical timbre: Perception of spectral and temporal characteristics. Canadian Journal of Experimental Psychology, 51, 307-315.