Abstract
Multidimensional scaling (MDS) is a set of related statistical techniques to explore and visualize relative positions among members in a group in respect of some feature(s). It starts with a distance matrix giving pair-wise differences (in scores or ranks or some other indicators), uses some least-squares principle, and eventually yields a point for each individual on a low-dimensional plane. We have metric vs non-metric MDS as also one-matrix, replicated (unweighted), and weighted MDS with different types of data and of distance models. The optimization exercise involved is quite complicated in some types of MDS. MDS bears some analogy with factor analysis, though the two serve different purposes and proceed on different lines. MDS finds applications in many fields, especially in market research. Steps involved in developing the distance matrix, applying the “stress”-minimizing algorithm, locating the objects to be compared as points in a two-dimensional plane, and interpreting the output have been demonstrated in terms of an example.
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References and Suggested Readings
Carroll, J. D., & Chang, J. J. (1970). Psychometrika, 35, 238–319. (A key paper: Provides the first workable WMDS algorithm and one that is still in very wide use. Generalizes singular value (Eckart-Young) decomposition to N-way tables).
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Kruskal, J. B. (1964). Psychometrika, 29, 1–27; 115–129. (Completes the second major MDS breakthrough started by Shepard by placing Shepard’s work on a firm numerical analysis foundation. Perhaps the most important paper in the MDS literature).
Kruskal, J. B., & Wish, M. (1977). Multidimensional Scalling. Beverly Hills, CA: Sage Publications. (Very readable and accurate brief introduction to MDS that should be read by everyone wanting to know more).
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Young, F. W. (1981). Psychometrika, 46, 357–388. (A readable overview of nonmetric issues in the context of the general linear model and components and factor analysis).
Young, F. W. (1984). In H. G. Law, C. W. Snyder, J. Hattie, & R. P. MacDonald (Eds.). Research methods for multimode data analysis in the behavioral sciences. (An advanced treatment of the most general models in MDS. Geometrically oriented. Interesting political science example of a wide range of MDS models applied to one set of data).
Young, F. W., & Hamer, R. M. (1994). Theory and applications of multidimensional scaling. Hillsdale, NJ: Eribaum Associates. (The most complete theoretical treatment of MDS and the most wide ranging collection of).
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Mukherjee, S.P., Sinha, B.K., Chattopadhyay , A. (2018). Multidimensional Scaling. In: Statistical Methods in Social Science Research. Springer, Singapore. https://doi.org/10.1007/978-981-13-2146-7_11
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DOI: https://doi.org/10.1007/978-981-13-2146-7_11
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