Abstract
The different purposes of MDS are explained: MDS for visualizing proximity data; MDS for uncovering latent dimensions; MDS as a psychological theory about judgments of similarity; MDS for testing structural hypotheses; unfolding as a psychological theory about judgments of preference.
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Notes
- 1.
Typing data() gives you a listing of all the data sets available in the R packages loaded previously by library(); data(wish) loads the data set wish; help(wish) provides information about wish.
- 2.
Most MDS programs are set, by default, to deliver a two-dimensional solution for data assumed to have an ordinal scale level.
- 3.
This is easy to see from an example: If point i has the coordinates (0, 0) and j the coordinates (3, 2), we get the intra-dimensional distances \(|0-3|=3\) and \(|0-2|=2\), respectively. The overall distance \(d_{ij}\), with \(p=1\), is thus equal to \(2+3=5.00\). For \(p=2\), the overall distance is 3.61. For \(p=10\), it is equal to 3.01.
- 4.
The solution was computed with Systat. Neither Proxscal nor smacof offer city-block distances. In Systat, the city-block metric is invoked by setting the “R-metric:” option in the GUI in Fig. 1.5 equal to “1”.
- 5.
The data set is contained in smacof. It is loaded automatically when calling smacof. You can check it by typing attributes(PVQ40) or head(PVQ40), for example. There is no need to explicitly load the data by typing data(PVQ40), but it would give you an error message if a file with this name does not exist.
- 6.
Note that we use the first two characters of the variables’ codes (i.e., “se”, “co”, etc.) in ind=codes to group the points.
- 7.
We show here how this is done, but the result is also directly available in smacof in the file PVQ40agg.
- 8.
Expressed more formally, in external unfolding either the person points or the object points are fixed and the other points are then optimally fitted into this fixed point configuration.
- 9.
We here also first center each person’s ratings, i.e., subtract the mean of his/her ratings from his/her rating scores to generate “relative value priorities.” This leads to a simpler model by reducing the dimensionality of the solution. See Sect. 8.1.
- 10.
This model is ratio unfolding. This is the default of the unfolding() function.
- 11.
You can generate a nice plot by simply typing plot(result). We here show some ways to customize such plots. You can also use R graphics or special graphics packages for customized plotting. However, smacof offers some easy-to-use arguments for plotting convex hulls (see Fig. 2.12) or confidence regions of the points (see Fig. 3.8) that are particularly useful in the MDS context.
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Borg, I., Groenen, P.J.F., Mair, P. (2018). The Purpose of MDS and Unfolding. In: Applied Multidimensional Scaling and Unfolding. SpringerBriefs in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-73471-2_2
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