```To Jim Rohlf and whomever else it may concern,

If you are concerned about the fit via a monotonic instead of a merely linear
function, what you can do is to fit the configuration (say in 2 dimensions) via
(presumably the classical) METRIC MDS procedure and then compute the STRESS for
THAT configuration.  One way to do this is to use this as the starting
configuration
for KYST or another nonmetric (two-way) MDS program, in which the initial
STRESS
(called, in KYST, KYST2 or KYST2-A, and other earlier versions of Kruskal's
nonmetric
MDS program (i.e., some of the later versions of MDSCAL) the STRESS for the
"zeroth iteration" if I recall this correctly.  (I believe there is also an
option
in the various versions of KYST to simply compute that "zeroth iteration"
STRESS
value-- using either STRESSFORM1 or STRESSFORM2-- but I assume the first of
these,
which is generally most appropriate for analyzing standard symmetric two-way
matrices of proximity data).  I believe if you do this, you would find that the
STRESS(Formula 1, which I will henceforth presume is used) computed this
way for
the METRIC solution will be almost the same (perhaps slightly larger, but
probably
not substantially so, and not enough to make up for the effective extra
parameters
being fit in the nonmetric 2-dimensional analysis) as that for the NONMETRIC
solution in the same dimensionality.  Assuming you have obtained the global
minimum in the nonmetric analysis,  that solution will necessarily yield at
least as low, and almost certainly a LOWER, STRESS computed this way for the
solution obtained via the metric 2-dimensional analysis-- but there is no
theoretically rigorous way to determine the number of effective additional
parameters the nonmetric analysis provides-- but it clearly DOES add what
amounts
to extra parameters to the overall model being fit, so that the STRESS will
necessarily be equal to or less than that of the metric solution used as the
"zeroth iteration" configuration in this way.  One way to verify this
empirically
is to note that the various versions of KYST actually use what amounts to a
form
of metric analysis-- but one I believe to be inferior in various ways to the
full classical metric analysis a la Torgerson et al-- and the final
solution obtained
always has a lower STRESS value.  A number of people have experimented with
using
the full classical two-way metric MDS solution as a starting point for KYST
analysis,
and always find a similar result-- which is empirical proof, I would argue,
that the
resulting nonmetric model with all parameters fit simultaneously
necessarily yields
lower STRESS than when the model is fit sequentially by first doing the
metric analysis
and then using the MFIT algorithm to fit the best OLS monotone function,
CONDITIONAL on
the distances being computed based on that metrically fitted
configuration.  These two
facts together imply that there are effectively at least SOME additional
parameters
implicitly being allowed via the nonmetric analysis-- which can only be
attributed, I
believe, to the addition of the monotone function as an integral part of
the nonmetric
model thus being fitted.  (If an unconstrained nonlinear function were
being fitted
to approximate the distances as a function of the proximity data, and this
function itself
had p parameters, then you could definitively state that p extra
parameters/degrees of
freedom are being added by allowing this nonlinear function.  When,
however, the function
being fit, while generally nonlinear, is defined simply by a constraint to
(weak) monotonicity,
NOONE (to my knowledge) has yet been able to come up with a definitive
theoretical OR
empirical way to assess the number of effectively added parameters.  Noone
questions the
fact that this DOES add effective additional parameters to the model being
fitted, but
I know of noone who has ever claimed to have a justifiable answer to the
question of how
many extra parameters this involves!)

While I've assumed above that one of the versions of KYST or another of the
Kruskal
(et al) MDS programs is used for the nonmetric analysis, largely because
I'm not
familiar with the specific software for MDS Jim mentioned in his earliest
e-mail
on this topic (the "NTSYSpc  software"), which he indicates, among many
other options,
allows some form of nonmetric or metric MDS.  Since I don't know exactly
what algorithm
Jim uses for MDS in this software, I cannot comment on using IT, or other
software, such
as Willem Heiser's PROXSCAL, recently incorporated into the professional
version of
SPSS (but not in the student version, nor, I assume in any of the earlier
versions of
any kind-- which, I believe, allow only ALSCAL, which does NOT optimize
STRESS, but a
different loss function called "SSTRESS", which many people-- myself
included-- feel is
less desirable than an algorithm for two-way metric or nonmetric MDS
optimizing STRESS;
the three-way case becomes even more complicated vis a vis comparison of
different
algorithms, but suffice it to say that I personally feel ALSCAL has some
especially serious
problems when conducting a three-way MDS analysis).  IF Jim Rohlf's
software DOES optimize
Kruskal's STRESS (in the two-way case-- there are MANY different ways to
generalize the
STRESS measure to the three-way case, so, strictly speaking, I cannot speak
to it in that
case-- but the analysis we are now discussing is strictly limited to the
simplest and most
straightforward case of two-way MDS, so it is quite easy to determine
whether or not it is
an algorithm optimizing Kruskal's STRESS in this case), then it is possible
that the same
approach (to computing STRESS for the configuration obtained as the
solution to the two-way
metric MDS program, whether based on the classical method associated with
Torgerson or a
different type of two-way metric MDS analysis).  In practice, it may come
down to very
detailed program features, such as whether there is a provision to use a
user provided
configuration as the starting configuration, and whether in that case the
best fitting OLS monotonic function relating data to distances is computed
(and provided as output) for that
starting configuration-- and then STRESS computed for this "zeroth
iteration" solution (or,
if this is not ordinarily done, whether there is an option to LIMIT the
total number of iterations to zero, which really means simply computing the
monotonic function and then the
STRESS for that input configuration ONLY, and not iterating the gradient
based algorithm at
all beyond that point-- effectively simply computing the STRESS value for
the user provided
configuration and nothing more.  IF the NTSYSpc software BOTH uses an
algorithm essentially
equivalent to Kruskal's in that it optimizes (minimizes) STRESS, not some
other loss function,
AND if it allows one or both of these options, then what I suggest COULD be
done using this
software Jim Rohlf provides as well as by using some version of KYST
(say).  It would also be
fairly easy to write a standalone program, if one has Kruskal's MFIT
algorithm available as a
subroutine-- or is willing and able to reproduce it-- that would go through
these steps and
thus calculate the STRESS for ANY configuration provided as input.

In short-- T\there are many ways to implement what I suggest (although not
all would work for
a matrix as large as 400x400-- none of Kruskal's algorithms used in the way
described earlier
could manage matrices that large for example-- but I'm sure this goal could
be achieved one
way or another easily enough.  If it were desired to do this numerous times
(or even to devise
a new approach consisting of analyzing the data via a metric MDS method--
say the classical
two-way metric MDS method-- and then computing the STRESS via use of a
procedure, equivalent
to the standalone program mentioned earlier, designed only to carry out
that specific operation!

(More than) enuf for now,

Doug Carroll.

At 07:58 PM 11/12/2002 -0500, F. James Rohlf wrote:
>I agree with Doug's observations but the problem is that I don't know how to
>tell whether the nonmetric-MDS will give a much better solution unless I try
>it and see what the stress value is. Linear-MDS takes less computing and
>avoids some of the degeneracy problems that Doug alludes to. I think they
>are usually worth a try as 2D nonmetric-MDS can sometimes give a better fit
>than a 3D metric-MDS. PCA, PCoord (classical scaling) can be much worse for
>relationships among close points.
>
>Jim Rohlf
>
> > -----Original Message-----
> > From: Classification, clustering, and phylogeny estimation
> > Sent: Tuesday, November 12, 2002 9:40 PM
> > Subject: Re: MDS with a 400X400 Matrix
> >
> >
> > To whomever it may concern (especially Jim Rohlf and David Dubin),
> >
> > I sent an e-mail response to this same broadcast e-mail sent out via the
> > CLASS-L system, but addressed it directly to Jim Rohlf, since, for some
> > reason I misinterpreted this as being an inquiry initiated by him, not
> > one from someone else (Dirk Meurer, I now see) simply passed on by Jim
> > as administrator of CLASS-L.  I had thought at the time that perhaps
> > David and Mark Rorvig, prior to Rorvig's tragic and untimely death last
> > year, had put together such a program.  While I gather from this message
> > from David that, instead, Michael Trosset (who was also involved
> > in the same
> > DIMACS Workshop on MDS Algorithms I mentioned in that earlier
> > to Jim) has done so.  I would therefore endorse David's
> > recommendation that
> > this software developed by Trosset be used for this purpose.  I personally
> > think that, in practice, it seldom makes a great difference
> > whether one uses
> > an MDS procedure that is purely "metric" (as long as you're
> > reasonably careful
> > to make sure the data are transformed, if necessary, to a form as
> > consistent
> > as possible with the assumptions that the procedure is based on-- in
> > particular,
> > most likely that they be DISsimilarities-- which may simply
> > entail reversing
> > the scale, if data are similarities, by, say, subtracting all
> > values from the
> > largest similarity value in the matrix, and possibly, if the
> > particular method
> > assumes RATIO SCALE instead of merely INTERVAL SCALE dissimilarities, that
> > these
> > dissimilarity values are transformed so that it is reasonable to
> > assume that a
> > dissimlarity of zero (0) corresponds to a DISTANCE of zero (0) in the
> > recovered
> > multidimensional representation-- or "nonmetric" (in which the
> > similarities or
> > dissimilarities are assumed only to be monotonic with the proximities--
> > monotone
> > non-increasing or monotone non-decreasing respectively for the two types--
> > sim.'s
> > or dissim.'s-- of proximity data).  The metric MDS procedures are
> > sufficiently
> > robust under even  fairly severe nonlinearities in the monotonic "distance
> > function"
> > transforming proximities into distances in the underlying representation
> > that the
> > solutions are usually almost totally indistinguishable (as long
> > as the same
> > dimensionality is assumed in two analyses, of course), although,
> > as is well
> > known,
> > an orthogonal rotation of a two-way MDS solution (based, as almost all MDS
> > algorithms are, on assumption of Euclidean metric in the
> > underlying space), no
> > matter HOW obtained, is necessary to bring even what are really identical
> > solutions
> > into exact congruence (since Euclidean spaces are defined only up to
> > similarity
> > transforms-- including an orthogonal rotation, as well as translation of
> > the origin
> > and, in some cases a possible uniform dilation resulting in multiplication
> > of all
> > distances by a positive constant-- but the latter two types of
> > transformations are
> > typically resolved by normalization conventions, so generally do
> > not need to be
> > considered).  In fact, not only are metric solutions (in the same
> > dimensionality)
> > usually as good as nonmetric ones, there is considerable empirical and
> > theoretical
> > evidence that, for certain types of data, they are in fact
> > BETTER-- being less
> > susceptible to various degeneracies and quasi-degeneracies that can affect
> > many
> > nonmetric MDS analyses, if the data exhibit certain characteristics that
> > are not at
> > all unusual in the case of realistic proximity data from various domains.
> >
> > Best,
> >
> > Doug Carroll.
> >
> > At 03:52 PM 11/12/2002 -0600, David Dubin wrote:
> > >Michael Trosset wrote a program for classical metric MDS that Mark Rorvig
> > >and I used with success on matrices much larger than 400 by 400. But it
> > >doesn't do nonmetric MDS. The program is written in Fortran and
> > requires the
> > >ARPACK libraries. I was able to compile with g77 on Linux  and
> > Solaris with
> > >little trouble.
> > >
> > >Dave Dubin
> > >
> > >
> > > > Dear Listmembers,
> > > > I would like to do a MDS with a 400X400 square, symetric matrix of
> > > > (dis)similaritys. Most MDS-Software is limited to a much
> > smaller amount
> > > > of variables though. I have been told, that SAS might be able
> > to process
> > > > the analyis I need, but this is quite inconvenient for me in terms of
> > > > access to the software, hardware-requirements etc. Could anybody tell
> > > > me, if there is a stand-alone program that can do MDS (preferably
> > > > nonmetric) with a matrix of that size?
> > > >
> > > > Thanks a lot for your help,
> > > > dirk
> > > >
> > > > P.S: My Matrix is not suitable for factor analysis and clustering does
> > > > not produce the results I need.
> > > >
> >
> >
> >
> >    ######################################################################
> >    # J. Douglas Carroll, Board of Governors Professor of Management and #
> >    #Psychology, Rutgers University, Graduate School of Management,      #
> >    #Marketing Dept., MEC125, 111 Washington Street, Newark, New Jersey  #
> >    #07102-3027.  Tel.: (973) 353-5814, Fax: (973) 353-5376.             #
> >    # Home: 14 Forest Drive, Warren, New Jersey 07059-5802.              #
> >    # Home Phone: (908) 753-6441 or 753-1620, Home Fax: (908) 757-1086.  #
> >    ######################################################################
> >

######################################################################
# J. Douglas Carroll, Board of Governors Professor of Management and #
#Psychology, Rutgers University, Graduate School of Management,      #
#Marketing Dept., MEC125, 111 Washington Street, Newark, New Jersey  #
#07102-3027.  Tel.: (973) 353-5814, Fax: (973) 353-5376.             #
# Home: 14 Forest Drive, Warren, New Jersey 07059-5802.              #
# Home Phone: (908) 753-6441 or 753-1620, Home Fax: (908) 757-1086.  #