# Multidimensional scaling

Multidimensional scaling (MDS) is a set of related statistical techniques often used in data visualisation for exploring similarities or dissimilarities in data. MDS is a special case of ordination. An MDS algorithm starts with a matrix of item-item similarities, then assigns a location of each item in a low-dimensional space, suitable for graphing or 3D visualisation.

## Categorization of MDS

MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:

Classical multidimensional scaling
also known as Torgerson Scaling or Torgerson-Gower scaling – takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain.[1] (pp. 207-212).
Metric multidimensional scaling
A superset of classical MDS that generalizes the optimisation procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress which is often minimized using a procedure called Stress Majorization.
Generalized multidimensional scaling (GMDS)
A superset of metric MDS that allows for the target distances to be non-Euclidean.
Non-metric multidimensional scaling
In contrast to metric MDS, non-metric MDS both finds a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distance between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression.

## Applications

Applications include scientific visualisation and data mining in fields such as cognitive science, information science, psychophysics, psychometrics, marketing and ecology.

### Marketing

In marketing, MDS is a statistical technique for taking the preferences and perceptions of respondents and representing them on a visual grid, called perceptual maps.

Potential customers are asked to compare pairs of products and make judgements about their similarity. Whereas other techniques (such as factor analysis, discriminant analysis, and conjoint analysis) obtain underlying dimensions from responses to product attributes identified by the researcher, MDS obtains the underlying dimensions from respondents’ judgements about the similarity of products. This is an important advantage. It does not depend on researchers’ judgments. It does not require a list of attributes to be shown to the respondents. The underlying dimensions come from respondents’ judgements about pairs of products. Because of these advantages, MDS is the most common technique used in perceptual mapping.

#### Multidimensional scaling procedure

There are several steps in conducting MDS research:

1. Formulating the problem - What brands do you want to compare? How many brands do you want to compare? More than 20 is cumbersome. Less than 8 (4 pairs) will not give valid results. What purpose is the study to be used for?
2. Obtaining Input Data - Respondents are asked a series of questions. For each product pair they are asked to rate similarity (usually on a 7 point Likert scale from very similar to very dissimilar). The first question could be for Coke/Pepsi for example, the next for Coke/Hires rootbeer, the next for Pepsi/Dr Pepper, the next for Dr Pepper/Hires rootbeer, etc. The number of questions is a function of the number of brands and can be calculated as Q = N (N - 1) / 2 where Q is the number of questions and N is the number of brands. This approach is referred to as the “Perception data : direct approach”. There are two other approaches. There is the “Perception data : derived approach” in which products are decomposed into attributes which are rated on a semantic differential scale. The other is the “Preference data approach” in which respondents are asked their preference rather than similarity.
3. Running the MDS statistical program - Software for running the procedure is available in most of the better statistical applications programs. Often there is a choice between Metric MDS (which deals with interval or ratio level data), and Nonmetric MDS (which deals with ordinal data). The researchers must decide on the number of dimensions they want the computer to create. The more dimensions, the better the statistical fit, but the more difficult it is to interpret the results.
4. Mapping the results and defining the dimensions - The statistical program (or a related module) will map the results. The map will plot each product (usually in two dimensional space). The proximity of products to each other indicate either how similar they are or how preferred they are, depending on which approach was used. The dimensions must be labelled by the researcher. This requires subjective judgement and is often very challenging. The results must be interpreted ( see perceptual mapping).
5. Test the results for reliability and Validity - Compute R-squared to determine what proportion of variance of the scaled data can be accounted for by the MDS procedure. An R-square of .6 is considered the minimum acceptable level. Other possible tests are Kruskal’s Stress, split data tests, data stability tests (ie.: eliminating one brand), and test-retest reliability.

## Bibliography

1. Borg, I. and Groenen, P.: "Modern Multidimensional Scaling: theory and applications", Springer-Verlag New York, 1997
• Abdi, H. "[1] (2007). Metric multidimensional scaling. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
• Abdi, H., Valentin, D., O'Toole, A.J., Edelman, B. "[2]. (2005). DISTATIS: The analysis of multiple distance matrices. Proceedings of the IEEE Computer Society: International Conference on Computer Vision and Pattern Recognition. (San Diego, CA, USA). pp. 42-47.".
• Abdi, H., Valentin, D., Chollet, S., & Chrea, C. (in press, 2007).. "[3]. Analyzing assessors and products in sorting tasks: DISTATIS, theory and applications. Food Quality and Preference, 18, 627-640. .".
• Bronstein, A. M, Bronstein, M.M, and Kimmel, R. (2006), Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching, Proc. National Academy of Sciences (PNAS), Vol. 103/5, pp. 1168-1172.
• Cox, M.F., Cox, M.A.A., (2001), Multidimensional Scaling, Chapman and Hall.
• Coxon, Anthony P.M. (1982): "The User's Guide to Multidimensional Scaling. With special reference to the MDS(X) library of Computer Programs." London: Heinemann Educational Books.
• Green, P. (1975) Marketing applications of MDS: Assessment and outlook, Journal of Marketing, vol 39, January 1975, pp 24-31.
• Kruskal, J. B., and Wish, M. (1978), Multidimensional Scaling, Sage University Paper series on Quantitative Application in the Social Sciences, 07-011. Beverly Hills and London: Sage Publications.
• Torgerson, W. S. (1958). Theory & Methods of Scaling. New York: Wiley.