# What is the role of MDS in modern statistics?

I recently came across multidimensional scaling. I am trying to understand this tool better and its role in modern statistics. So here are a few guiding questions:

• Which questions does it answer?
• Which researchers are often interested in using it?
• Are there other statistical techniques which perform similar functions?
• What theory is developed around it?
• How does "MDS" relate to "SSA"?

I apologize in advance for asking such a mixed/unorganized question, but so is the nature of my current stage in this field.

In case you will accept a concise answer...

What questions does it answer? Visual mapping of pairwise dissimilarities in euclidean (mostly) space of low dimensionality.

Which researchers are often interested in using it? Everyone who aims either to display clusters of points or to get some insight of possible latent dimensions along which points differentiate. Or who just wants to turn a proximity matrix into points X variables data.

Are there other statistical techniques which perform similar functions? PCA (linear, nonlinear), Correspondence analysis, Multidimensional unfolding (a version of MDS for rectangular matrices). They are related in different ways to MDS but are rarely seen as substitutes of it. (Linear PCA and CA are closely related linear algebra space-reducing operations on square and rectangular matrices, respectively. MDS and MDU are similar iterative generally nonlinear space-fitting algorithms on square and rectangular matrices, respectively.)

What theory is developed around it? Matrix of observed dissimilarities $S$ is transformed into disparities $T$ in such a way as to minimize error $E$ of mapping the disparities by means of euclidean distances $D$ in $m$-dimensional space: $S \rightarrow T =^m D+E$. The transformation could be requested linear (metric MDS) or monotonic (non-metric MDS). $E$ could be absolute error or squared error or other stress function. You can obtain a map for a single matrix $S$ (classic or simple MDS) or a map for many matrices at once with additional map of weights (individual differences or weighted MDS). There are as well other forms like repeated MDS and generalized MDS. So, MDS is a diverse technique.

Update for the last point. This technote from SPSS leaves impression that SSA is a case of Multidimensional unfolding (PREFSCAL procedure in SPSS). The latter, as I've noted above, is MDS algo applied to rectangular (rather than square symmetric) matrices.

• (+1) Nice summary! Forrest Young (already cited by @whuber in his comment), Yoshio Takane and Jan de Leeuw have some very good papers on MDS. – chl Jun 28 '12 at 16:00
• @chl, thanks: useful articles on Takane's page in galore – ttnphns Jun 28 '12 at 16:07

@ttnphns has provided a good overview. I just want to add a couple of small things. Greenacre has done a good deal of work with Correspondence Analysis and how it is related to other statistical techniques (such as MDS, but also PCA and others), you might want to take a look at his stuff (for example, this presentation may be helpful). In addition, MDS is typically used to make a plot (although it is possible to just extract some numerical information), and he has written a book this general type of plot and put it on the web for free here (albeit only one chapter is about MDS plots per se). Lastly, in terms of a typical use, it is used very commonly in market research and product positioning, where researchers use it descriptively to understand how consumers think about the similarities between different competing products; you don't want your product to be poorly differentiated from the rest.

• (+1) Understanding Biplots, by Gower and coll. is also a great book (it comes with a binary R package, windows-only), with about 50 pp. on MDS and nonlinear biplots. – chl Jun 29 '12 at 6:41
• @chl, thanks for the tip, that book looks interesting. – gung - Reinstate Monica Jun 29 '12 at 15:25

One additional strength is that you can use MDS to analyze data for which you don't know the important variables or dimensions. The standard procedure for this would be: 1) have participants rank, sort, or directly identify similarity between objects; 2) convert the responses into dissimilarity matrix; 3) apply MDS and, ideally, find a 2 or 3D model; 4) develop hypotheses about the dimensions structuring the map.

My personal opinion is that there are other dimension reduction tools that are usually better suited for that goal, but that what MDS provides is the opportunity to develop theories about the dimensions that are being used to organize judgments. It's important to also keep in mind the degree of stress (distortion that results from the dimension reduction) and incorporate that into your thinking.

I think one of the best books out on MDS is "Applied Multidimensional Scaling" by Borg, Groenen, & Mair (2013).