I agree with much of what Fred says but I disagree with the implication that one should never use non-metric MDS.
When these analyses differ appreciably it is because non-metric MDS relaxes PCA/PCOORD's requirement that distances in the reduced 2 or 3- dimensional ordination space should be the result of a simple rotation and projection. Non-metric MDS relaxes this and just requires that distances in the ordination tend to have a monotone relationship with the distances in the original high-dimensional space. This means that a unit distance in different parts of the MDS ordination plot need not mean the same amount of difference in the original data. On the other hand, if ones purpose is to look for patterns, clusters, and trends in the ordination plot then the fact that MDS often seems to pack more information about the relative proximity of points into fewer dimensions can be a real benefit. Note also that because a PCA is determining axes that account for the maximal variance, PCA ordinations best preserve distances among the most distant pairs of points and often represents the relative distances among close points very poorly. This means that one should only try to interpret the overall gross patterns in a PCA ordination and try to ignore patterns among points that are relatively close together. Often those largest distances are not very interesting. MDS gives equal weight to distances at all scales. The best way to appreciate this is to compare Sheppard diagrams for non-metric MDS and PCOORD/PCA. I discussed some of these points in: Rohlf, F. J. (1972) Empirical comparison of three ordination techniques in numerical taxonomy. Systematic Zool., 21:271-280. In that paper I point out that if you want to look at an ordination plot to find interesting patterns in data then a PCA maximizes the wrong criterion. One does not usually care how much of the variance is accounted for by 2 or 3 dimensions but how well the distances one sees in the ordination accurately reflect the distances in the multivariate space. That is what non-metric MDS is designed to optimize. If a monotone relation is too relaxed for your purposes then there is also a linear MDS method that can be used. Jim ----------------------- F. James Rohlf, Distinguished Professor & Graduate Program Director State University of New York, Stony Brook, NY 11794-5245 www: http://life.bio.sunysb.edu/ee/rohlf > -----Original Message----- > From: morphmet [mailto:[EMAIL PROTECTED] > Sent: Friday, September 30, 2005 6:50 PM > To: morphmet > Subject: Re: PWS and MDS? > > > September 30, 2005 > > This is a reply to Luke Finley's question of earlier > today about using nonmetric techniques with shape coordinates. > I'll refer to the questioner in the second person, as "you". > Your "rationale for using MDS" is actually the list of > reasons for using PCA, which in this specific context does > PRECISELY what you are asking MDS to do! > Remember that the Procrustes shape coordinates arise > all at once as an essentially unique representation of one > single distance function, namely, Procrustes distance. The > shape coords should not be thought of as separate variables, > but only as one joint set. (The same is true of the partial > warp scores, which are just a rotation of the natural basis > [assuming you've preserved the zeroth PW, which is to say, > the uniform component in the right metric].) It makes no > sense to do a "nonparametric" analysis of coordinates that > have the metric built right in down to the level of the > fundamental definition the way these do. > In particular, there is just no point to a technique > like MDS in this connection, as there is no choice about the > projection metric you MUST use: it MUST BE the principal > coordinates of the shape coordinate data (the best > representation of the original distance metric that's > already been enforced), and for the first pair this is > identical with the ordinary RW1/RW2 plot, by definition. > I'm glad you observed that in your example, because you need > to be using the RW's, not any other output from MDS: this > plane is, by definition, the unique best representation of > the original Procrustes distances -- you just needn't bother > with MDS. > If your results aren't IDENTICAL between MDS and PCA of the > PW scores, it can only be that you have done something wrong > with one computation or the other. > Likewise there is just > no point to using "similarity in the RW1/RW2 plane" or > anywhere else. The Procrustes technique has already locked > you into a measure of shape distance, and you gain nothing > whatever by using only the part in the first two dimensions > -- that's just a limitation of your retina, having nothing > to do with krill OR statistics. > So your rationale for using MDS instead of PCA is > actually the argument for PCA, or, rather, the argument for > PCOORD (which is identical with PCA but is reported using > different words), and you have to use PCOORD because you > already assumed that Procrustes distance was worth paying > attention to when you used the Procrustes shape > representation in the first place, and it doesn't matter if > you're using the PW scores vs. the shape coordinates because > they have exactly the same metric. > > But the analysis of shape against geography doesn't > go very well in terms of these distances. In other words, > your last concern, "whether samples that are far apart are > more different than samples close together," isn't actually > a very good way to ask a question, as it presumes two > distance measures when you don't actually have to presume even one. > Scientifically, it's better to do a Partial Least Squares > analysis of shape coordinates against geographical > coordinates (or ecological, or whatever). For the > latitude/longitude example, see Frost et al., Cranial > allometry, phylogeography, and systematics of large-bodied > papionins (Primates/Cercopithecinae) inferred from geometric > morphometric analysis of landmark data. > {\sl Anatomical Record} 275A:1048--1072 (with cover), 2003, > where we do it for baboon skulls over a map of Africa. > What you get are prediction gradients that have (finally) > lifted the requirement of symmetry that underlies the > Procrustes method, and you get to learn what is predicted by > latitude and by longitude separately. Only if these are the > same does it make any sense to proceed in terms of distance > apart, which was the question as you posted it (so, to > repeat, I think that's not a good way to phrase hte question > -- unless you know for certain that you're looking at a > shape diffusion process rather than a selective process! -- > such knowledge is rarely granted to ordinary mortals, even > though Sewall Wright published some astonishingly good > examples in the '40's). > > Thus I see no role for either ANOSIM or MDS in a > good competent analysis of the data you've described for us. > Every biometric data set has problems; yours is about > average in this regard. > > > Fred Bookstein > [EMAIL PROTECTED] > > -- > Replies will be sent to the list. > For more information visit http://www.morphometrics.org > -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
