All I can say is that you're using obsolete methodology that absolutely
NOONE I know
uses any more these days.  MDS users usually just ask for pairwise similarity
judgments on a simple rating scale (ranging from, say 10, for absolutely
identical to
0 for as dissimilar as possible, if the judgments are similarities--
reversing the
scale if dissimilarities are judged-- with, say,  0 meaning "no perceived
difference at all"
and 10, say, meaning "as dissimilar as possible").  I think dissimilarity
judgments are
really better than similarity judgments for this purpose, since they are
essentially
what could be called "psychological distances", and it is, after all,
distances that
you ultimately want to estimate (after estimating an additive constant to
transform
so-called "comparative distances"-- i.e., dissimilarities-- to "absolute
distances"
to use the terminology in Torgerson's book and 1952 paper.  Then you use the
equations involving doubly centering the matrix whose entries are
-1/2(dij**2) [in
words-- you doubly center the matrix of minus one half the SQUARED
Euclidean distances
to transform the "absolute distances" to approximate scalar products, and
then simply
apply an eigendecomposition (basically equivalent to a principal components
analysis
of the scalar product matrix, which in many respects is analogous to a
covariance
matrix) to get the R dimensional reduced representation (in principal axis
orientation--
then you usually have to rotate the configuration to get an interpretable
solution).

All that stuff you're talking about hasn't been used by anyone I'm aware of
as a method
for collecting the similarity or dissimilarity data (proximities, to use
the term originally
introduced by Coombs and later used by Shepard) in years.  I think Paul
Green and Frank Carmone may have used this as one of a whole variety of
data collection methods in their original book
on MDS Applied to Marketing-- they were trying all methodology then
available, but quickly
concluded that it was totally unnecessary and simply a great waste of
time-- to actually
do those "triadic judgments" that Torgerson talks about.  The number of
triples of
stimuli or other objects, of course, goes up as the cube of the number of
stimuli or objects,
whereas the pairwise judgments go up only roughly as the square, and
there's no indication
that the data you get are really better, at least as far as the solutions
you get are
concerned, and using one of these triadic judgment methods makes it
essentially impossible
to use more than about 10 or so stimuli in a study!  You're limited enough
when doing
pairwise judgments, and many people are now looking for ways to reduce the
data collection
demands even moreso for MDS, so as to allow analysis of much larger data
sets-- but I simply
cannot understand why you'd even want to use that now obsolete method,
unless you're doing
it for purely historical purposes, or something of that sort.  (You might
get slightly more
reliable data with triadic judgments, but the extra reliability doesn't
come anywhere close
to compensating for the other limitations use of such a method puts on data
collection and
the size of the data set you can deal with!)

Best regards,

Doug Carroll.


At 04:59 PM 8/21/2003 +0200, simmerl augustiner wrote:
>Dear Listmembers,
>
>I´m working with metric multidimensional scaling and
>trying to implement the Torgerson algorithm in
>SAS/IML. The terms I use are all shown in "Theory and
>methods of scaling", Torgerson (1958), p.263-268.
>Torgerson´s P-matrices are first transformed into
>x-matrices (based on Thurstone´s law of comparative
>judgment) which contain both estimated differences
>between stimuli and missing values.
>Computing the matrix of comparative distances in the
>next step I meet a problem:
>In the formula to solve for the comparative distances
>a few averages summed over different indices are
>required.
>Maybe the problem sounds trivial, but I don´t know if
>the missing values have to be regarded computing the
>means, or do I simply have to calculate these averages
>neglecting all missing values.
>
>Thanks a lot for your help,
>
>Simon Gollick
>
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>
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