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 > >__________________________________________________________________ > >Gesendet von Yahoo! Mail - http://mail.yahoo.de >Logos und Klingeltöne fürs 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: [log in to unmask] # ######################################################################