Re: PROC MDS with preference rankings

On Jul 10, 12:39 am, John Uebersax <jsueber...@xxxxxxxxx> wrote:
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).

That depends on which convention you follow for determining loadings
and scores in PCA. I follow the factor-analytic tradition in which the
loadings -- in this case, the locations of the objects -- are the unit-
length eigenvectors multiplied by the square roots of the eigenvalues,
with the scores having unit variances, but I realize that there are
those who take the loadings as being just the eigenvectors, with the
scores having variances equal to the eigenvalues.


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.

Yes. Here's how to get from D to C. Let m = the # of objects,
b(j,k) = d(j,k)^2, b(j) = SUM_k b(j,k)/m, and b() = SUM_j b(j)/m.
Then c(j,k) = (-1/2)*(b(j,k) - b(j) - b(k) + b()).


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.

I meant only that each subject's data would be unidimensional. All we
can say ahead of time about the dimensionality of the overall solution
is that it will be < m, because the m'th eigenvalue of C should be
exactly 0 (except for roundoff error).


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.

If SAS does follow the "other" convention, you should multiply the
loadings on each factor by the square root of the corresponding
eigenvalue.


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

One more comment. If you want the solution to give you numbers that
don't automatically get bigger (or smaller) as you increase (or
decrease) the number of subjects, you should divide C by the number of
subjects before doing the PCA. (This is equivalent to dividing D by
the square root of the number of subjects before doing the MDS.)
.

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