The Choice of Initial Configurations in Multidimensional Scaling: Local Minima, Fit, and Interpretability
Multidimensional scaling (MDS) algorithms can easily end up in local minima, depending on the starting configuration. This is particularly true for 2-dimensional ordinal MDS. A simulation study shows that there can be many local minima that all have an excellent model fit (i.e., small Stress) even if they do not recover a known latent configuration very well, and even if they differ substantially among each other. MDS programs give the user only one supposedly Stress-optimal solution. We here present a procedure for analyzing all MDS solutions resulting from using a variety of different starting configurations. The solutions are compared in terms of fit and configurational similarity. This allows the MDS user to identify different types of solutions with acceptable Stress, if they exist, and then pick the one that is best interpretable.
Bentler PM, Weeks DG (1978). Restricted multidimensional scaling models. Journal of Mathematical Psychology, 17, 138–151.
Borg I, Bardi A, Schwartz S (2016). Does the value circle exist within persons or only across persons? Journal of Personality. Forthcoming.
Borg I, Groenen PJF (2005). Modern Multidimensional Scaling. 2nd edition. Springer, New York.
Borg I, Groenen PJF, Mair P (2013). Applied Multidimensional Scaling. Springer, New York.
Borg I, Leutner D (1985). Measuring the similarity of MDS configurations. Multivariate Behavioral Research, 20, 325–334.
Borg I, Lingoes JC (1980). A model and algorithm for multidimensional scaling with external constraints on the distances. Psychometrika, 45, 25–38.
Cox TF, Cox MAA (2000). Multidimensional scaling. 2nd edition. Chapman & Hall, London.
Davison ML (1978). Multidimensional Scaling. Wiley, New York.
De Leeuw J, Heiser WJ (1980). Multidimensional scaling with restrictions on the configuration. In PR Krishnaiah (ed.), Multivariate Analysis, Volume V, pp. 501–522. North-Holland, Amsterdam.
De Leeuw J, Mair P (2009). Multidimensional Scaling Using Majorization: SMACOF in R. Journal of Statistical Software, 31(3), 1–30. URL http://www.jstatsoft.org/v31/i03/.
Ekman G (1954). Dimensions of color vision. Journal of Psychology, 38, 467–474.
Fellows I (2014). wordcloud: Word Clouds. R package version 2.5, URL https://CRAN.R-project.org/package=wordcloud.
Groenen PJF (1993). The Majorization Approach to Multidimensional Scaling: Some Problems and Extensions. Ph.D. thesis, University of Leiden.
Kruskal JB (1964). Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis. Psychometrika, 29, 1-27.
Kruskal JB, Wish M (1978). Multidimensional Scaling. Sage, Beverly Hills, CA.
Mair P, Borg I, Rusch T (2016). Goodness-of-fit Assessment in Multidimensional Scaling and Unfolding. Multivariate Behavioral Research. Forthcoming.
Mair P, De Leeuw J, Groenen PJF (2015). Multidimensional scaling in R: smacof. Technical report. URL https://cran.r-project.org/web/packages/smacof/smacof.pdf.
R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org.
Schwartz S, Bilsky W (1987). Toward a psychological structure of human values. Journal of Personality and Social Psychology, 53, 550–562.
Schwartz SH, Cieciuch J, Vecchione M, Davidov E, Fischer R, Beierlein C, Ramos A, Verkasalo, M, Lonquist JE, Demirutku K, Dirilen-Gumus O, Konty M (2014). Refining the theory of basic individual values. Journal of Personality and Social Psychology, 103, 663–688.
Shepard RN (1962). Analysis of Proximities: Multidimensional Scaling with an Unknown Distance Function. Psychometrika, 27, 125-140.
Spence I, Ogilvie JC (1973). A table of expected stress values for random rankings in nonmetric multidimensional scaling. Multivariate Behavioral Research, 8, 511–517.
Torgerson WS (1958). Theory and Methods of Scaling. John Wiley & Sons, New York.
Wish M (1971). Individual differences in perceptions and preferences among nations. In CW King, D Tigert (eds.), Attitude Research Reaches New Heights. American Marketing Association, Chicago.
Wish M, Deutsch M, Biener L (1972). Differences in perceived similarity of nations. In AK Romney, RN Shepard, SB Nerlove (eds.), Multidimensional scaling: Theory and Applications in the Behavioral Sciences, pp. 289–313. Academic Press, New York.
How to Cite
The Austrian Journal of Statistics publish open access articles under the terms of the Creative Commons Attribution (CC BY) License.
The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed.
Copyright on any research article published by the Austrian Journal of Statistics is retained by the author(s). Authors grant the Austrian Journal of Statistics a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
Manuscripts should be unpublished and not be under consideration for publication elsewhere. By submitting an article, the author(s) certify that the article is their original work, that they have the right to submit the article for publication, and that they can grant the above license.