Abstract
Various mistakes that users tend to make when using MDS are discussed, from conceptual fuzziness, over using MDS for the wrong type of data, or using MDS programs with suboptimal specifications, to misinterpreting MDS solutions.
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Notes
- 1.
The correlations of test items from two of the subgroups {NP,...,NR}, {SLP} and {CCR, ILR}, respectively, are always smaller than the correlations of test items from the same group.
- 2.
Unfortunately, few statistics packages offer Procrustean transformations and if they do (such as Systat, for example), then the proper modules can be difficult to find and use. If the users have a program for matrix algebra (e.g., MatLab or R), they can quite easily compute Procrustean fittings themselves. Let \(\mathbf X \) be the target configuration and \(\mathbf Y \) the configuration to be fitted. Then, compute \(\mathbf C = \mathbf X ^{\prime }\mathbf ZY \), and find for it the singular value decomposition \(\mathbf C = \mathbf P \mathbf \Phi \mathbf Q ^{\prime }\). (In the centering matrix \(\mathbf Z =\mathbf I -n^{-1}\mathbf 1 \mathbf 1 ^{\prime }\), \(\mathbf I \) is the identity matrix and \(\mathbf 1 \) is a vector of ones.) The optimal rotation/reflection for \(\mathbf Y \) is done by \(\mathbf T = \mathbf QP ^{\prime }\); the optimal central dilation factor is \(s = \text{ trace}(\mathbf X ^{\prime }\mathbf ZYT )/ \text{ trace}(\mathbf Y ^{\prime }\mathbf ZY )\); the optimal translation vector is \(\mathbf t = n^{-1}(\mathbf X - s\mathbf YT )^{\prime }\mathbf 1 \). Hence, the solution is \(\hat{\mathbf Y } = s\mathbf YT +\mathbf 1t ^{\prime }\) (Borg and Groenen 2005).
- 3.
The optimal weights are the non-standardized or raw weights of the multiple regression solution (betas).
- 4.
That this scale cannot be easily fitted into the MDS solution can be seen, for example, from the closeness of the points representing Israel and the USA, two countries with vastly different numbers of inhabitants.
- 5.
To generate Fig. 7.8, we used hierarchical single-linkage cluster analysis. Choosing the “average” criterion leads to a solution where the Congo does not remain a singleton, but it is included into one cluster together with Egypt and India.
- 6.
Cluster analysis identifies cluster structures in the total space of the data: if there are clear clusters in this space, then the major clusters also tend to appear in the plane spanned by the first two principal axes of this space or in an MDS plane.
- 7.
Guttman (1977) comments on this: “To say that one ‘wants to construct’ a scale... towards something ... is almost analogous to saying that one ‘wants’ the world to be flat... To throw away items that do not ‘fit’ unidimensionally is like throwing away evidence that the world is round” (p. 105).
- 8.
Flattened weights are computed differently. Alscal in Spss first rescales the dimension weights of each person such that they add to 1.00; it then drops dimension 2 and subtracts the mean from each weight for dimension 1. This yields the flattened weights.
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Borg, I., Groenen, P.J.F., Mair, P. (2013). Typical Mistakes in MDS. In: Applied Multidimensional Scaling. SpringerBriefs in Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31848-1_7
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