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 1963-65, 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
__________________________________________________________________
Gesendet von Yahoo! Mail - http://mail.yahoo.de Logos und Klingelt�ne f�rs Handy bei http://sms.yahoo.de
###################################################################### # 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. # # E-mail: [EMAIL PROTECTED] # ######################################################################
