Re: PROC MDS with preference rankings

Ray,

The usual metric MDS analysis of that distance matrix will give the
same object space as the PCA

Absolutely right. I didn't quite understand what you were suggesting,
not being familiar with the Hair text. Metric MDS on Euclidean
distances and PCA of Pearson correlations have a simple algebraic
connection. They supply the same solution, except that MDS weights
the dimensions of the object space proportionally to the eigenvalues
of the PCA solution, whereas if you plot points in the PCA space all
dimensions are normalized to unit variance (I think).

This close relationship, in fact, leads to a nice trick: armed with
nothing but a Euclidean distance matrix and software for PCA, you can
perform metric MDS.

The individual-differences models are not meant for data in which each
subject's distance matrix is constructed from a set of ratings and is
therefore exactly unidimensional.

Do you mean the overall solution will be unidimensional? Certainly
each subject's data will be unidimensional.

would give you no more information than the PCA. The two object spaces
would be the same, and the subject-specific weights in the individual-
differences model would be proportional to the component scores in the
PCA.

That sounds about right.

So, in any case, Jennifer it seems like both Ray and I are suggesting
you forego the individual differences route. All you need to do is to
perform usual PCA on your data, and use the regular SAS options to
plot variables (brand 1, brand 2, ...) in the PC space.

To get a solution closer to MDS, you could also weight the principle
component loadings by the size of the corresponding eigenvalues.

Example:

Suppose you perform a PCA (across brands) and get exactly three
principle components. (For simplicity I'll call principle components
'factors').

Solution t = 0

PCA Solution

Eigenvalue Size Cumul. Pct.
1 1.5 50%
2 1.0 87%
3 .5 100%

You decide to examine only the first two factors. The factor loadings
are, e.g.:

Factor 1 Factor 2

brand 1 .8 .3
brand 2 .9 .5
brand 3 .4 .7
brand 4 .3 .8
brand 5 .2 .9

Now I believe you could just plot the above information, locating
brands in the 2-dimensional factor space -- a SAS option, in fact.

However you can also multiply the factor loadings times the factors'
corresonding eigenvalues, i.e., the loadings on Factor 1 times 1.5,
and loadings on Factor 2 times 1.0 -- and then plot the
results. That would be the same as the two-dimensional MDS solution.

It's been a while since I've done this, though, and I might not have
the details exactly correct.
Maybe Ray could offer an opinion.

HTH.

John Uebersax, PhD


.

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