Roger Shepard has written a paper on this I recall one in an early issue
of Journal of Mathematical Psychology. Also, Jean Paul Benzecri, the French
statistician associated primarily with correspondence analysis sent Roger some
theorems he proved establishing that, under very general conditions, ordinal
proximity data leads asymptotically to a metric solution as n (number of
stimuli or other objects) grows large but I'm not at all sure that this
material was ever published (although Roger may refer to this in the paper
of his I recall in JMP). This was all done circa 196365, but I don't have
precise reference for Shepard's JMP paper. Maybe someone else on this
mailing list does. You might try searching on Roger Shepard and Journal of
Math Psych I don't think he had too many papers on the subject of nonmetric
MDS in that journal, so you should be able to find it fairly easily. I seem
to recall it's title was something like "Metric structure from nonmetric data."
Doug Carroll.
At 04:40 PM 11/20/2003 +0100, Simon Gollick wrote:
>Dear Listmembers,
>
>in my final paper I´m dealing with both metric MDS
>(Torgersons´s Triadic combinations from his book 1958)
>and nonmetric MDS (Kruskal´s algorithm) as "milestones
>of MDS". The most important advantage of nonmetric MDS
>are the low requirements to data, i.e. the algorithm
>requires only ordinal data to reconstruct a
>configuration of objects. Nevertheless the procedure
>yields metric information within the configuration. My
>problem is how to give a reasonable justification for
>that fact. I suppose it`s because of the constraints
>simultaneously concerning all objects.
>What is the minimum number of objects to have no
>noticeable difference between an ordinal and an metric
>configuration?
>Could anyone perhaps give me an statement on this
>subject or at least a reference for a quotation?
>
>Many thanks an best regards,
>
>Simon Gollick
>
>
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