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Applied Multidimensional Scaling and Unfolding pp 29–41Cite as

The Fit of MDS and Unfolding Solutions

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Abstract

Ways to assess the goodness of an MDS solution are discussed. The Stress measure is defined as an index that aggregates representation errors. Criteria for evaluating Stress are presented. Stress per Point (SPP) is defined as a way to assess the fit of single points.

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Notes

  1. 1.

    Transformations that convert similarity data into dissimilarities can be done in smacof by the function.sim2diss(). For example, if the data is given as the correlation matrix \(\mathbf {R}\), diss<-sim2diss(R, method = "corr") will generate a dissimilarity matrix that can be used in out <-mds(diss) to compute an MDS representation.

  2. 2.

    So, for example, in ratio MDS \(f(\delta _{ij}) = b \cdot \delta _{ij}\), where b (\(\ne 0\)) is a global scaling factor. In interval MDS, \(f(\delta _{ij}) = a + b \cdot \delta _{ij}\), where the additive constant a and the multiplicative constant b are picked so that the Stress is minimized. In ordinal MDS, f() is required to be monotonic.

  3. 3.

    Note that the sum \(\sum _{i<j}\) runs over the lower triangular part of the dissimilarities only, because it is assumed that the data are symmetric as, for example, in case of Tables 1.1 and 3.1. Asymmetric data require special models. See Sect. 5.4.

  4. 4.

    The square root has no deeper meaning here; its purpose is to make the resulting values less condensed by introducing more scatter.

  5. 5.

    You can display this result graphically. For example, by ex <-permtest(my.results); hist(ex$stressvec), xlim=c(ex$stress.obs-.05, max(ex$stressvec)+.05)); abline(v=ex$stress.obs, col="red"); points(ex$stress.obs, 0, cex = 2, pch = 16, col = "red"). The plot shows the distribution of the Stress values for the permuted data, with a red vertical line at the point of the Stress value for the observed data.

  6. 6.

    Simulations using pure random data show that in case of 12 points the SPP values scatter about 8.33 with an sd of 3.10; 96% of the SPPs are in the range from 14.53 to 2.13.

  7. 7.

    This heat map is generated by library(gplots); diss<-sim2diss(wish, method=7); res<-mds(diss, type="ordinal"); RepErr<-as.matrix((res$dhat - res$confdist) \({}^\wedge \) 2); yr<-colorRampPalette(c("lightyellow", "red"), space = "rgb")(100); heatmap.2( RepErr, cellnote=round(RepErr,2), Rowv = NA, Colv = "Rowv", lhei=c(0.05, 0.15), margins = c(8, 8), key=FALSE, notecol = "black", trace = "none", col = yr, symm = TRUE, dendrogram = "none") , where RepErr is the matrix of representation errors.

  8. 8.

    Note that we first compute an MDS solution and then use the solution as the first argument when calling boot. The second argument is the data file; the third is the type of proximity measure for the variables; and the fourth is the number of bootstrapping samples you want the function to draw. It is set here to 500.

References

  • Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). New York: Springer.

    MATH  Google Scholar 

  • De Leeuw, J., & Meulman, J. (1986). A special jackknife for multidimensional scaling. Journal of Classification, 3, 97–112.

    Article  Google Scholar 

  • Jacoby, W. G., & Armstrong, D. A. (2014). Bootstrap confidence regions for multidimensional scaling solutions. American Journal of Political Science, 58, 264–278.

    Article  Google Scholar 

  • Mair, P., Borg, I., & Rusch, T. (2016). Goodness-of-fit assessment in multidimensional scaling and unfolding. Multivariate Behavioral Research, 51, 772–789.

    Google Scholar 

  • Spence, I., & Graef, J. (1974). The determination of the underlying dimensionality of an empirically obtained matrix of proximities. Multivariate Behavioral Research, 9, 331–341.

    Article  Google Scholar 

  • Spence, I., & Ogilvie, J. C. (1973). A table of expected stress values for random rankings in nonmetric multidimensional scaling. Multivariate Behavioral Research, 8, 511–517.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Westfälische Wilhelms-Universität, Muenster, Germany

    Ingwer Borg

  2. Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands

    Patrick J. F. Groenen

  3. Department of Psychology, Harvard University, Cambridge, MA, USA

    Patrick Mair

Authors
  1. Ingwer Borg
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  2. Patrick J. F. Groenen
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  3. Patrick Mair
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Correspondence to Ingwer Borg .

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Borg, I., Groenen, P.J.F., Mair, P. (2018). The Fit of MDS and Unfolding Solutions. In: Applied Multidimensional Scaling and Unfolding. SpringerBriefs in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-73471-2_3

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