Concerning Cardini's question about procrustes distance as an averall variation measure/ Several years ago I also tried to play this game, as the idea of PRD is a strightfoward one. However the results appeared to be discouraging and I left the idea. At that time, I tried to clear the point in discussion with the late Les Marcus. Below are a part of our correspondence on that matter (my letter marked by >, and Les' respond), someone may find it relevant.
At 05:03 PM 9/5/00 +0400, you wrote: Procrustes distances as the formulas are given, do not take into account the number of landmarks as you know. I would generally expect larger ones for more landmarks - so your results most interesting. In permutation tests - which Fred likes so much - you can compare differences between mean shapes for subsets of your sample (sexes, localities etc), and the test does not depend on the within group variability interestingly. Though it doesn't hurt things if you do include it - is because TSS = ASS + WSS (Total, Among, Within). Am curious in your data set if the same points are included in the outline as in the landmarks - that is do you augment the landmarks with more points? I think there is a seed of something interesting here, but don't know where to go next. >Dear Les, I communicated with Fred Bookstein about use of Procrustes >distance (PD) in analyses of within-sample variation. His >answer appeared less than promising: number of landmarks matters. >I calculated PDs for my vole molar datasets which are of two kinds, one is >based on 12 landmarks and >another on 50 outline points (semi-landmarks). The average PD values >appeared to be the following: > >LANDMARK DATA OUTLINE POINT DATA >0.174 0,221 >0.184 0.180 >0.170 0.177 >0.207 0.271 >0.183 0.261 >0.168 0.277 > >As you see, PDs for less numerous landmarks are generally lower than those >for more numerous outline points. However, in >two cases they are nearly the same (in one of them the ratio is even >opposite). So, one may conclude empirically that - see the above >Bookstein's comment. However, the relation is not linear and there are >some other factors impacting on it. > >Sure, it would be great to find some general (math) resolution of the >task. However, as long as it is not resolved yet, how do you >think wouldn't it be possible to overcome the problem some empirical >"partial" way - for instance, by using the same number of landmarks? >You understand that I mean those vole molars which variation seem to be >quite attractive to study. I guess I wouldn't expect it to be linear - but what to expect I don't know. I would think using the same number of landmarks would at least control for that variability. Since one can partition Total Procrustes distance into among groups and wihtin; I would think that you could use some idea like dividing through by the number of landmarks or their square root - to remove the number of landmarks. I do that to get an idea of what is going on, but not sure if it has been published. >Bookstein mentioned your results of analyses of mammal skull shape >variation. I saw them published in Hystrix. However, I think that >the position of cetaceans is just one due to "Pinoccio effect": in >setaceans the nostrils are displaced from the tip of rostrum (as in most of >the mammals) onto the crown. And to me it would be more surprising if the >method wouldn' indicate that. I agree with you here. I will look at Fred's answer to you again, and see if I can think of anything else. If you divided through by number of landmarks - before taking square root, you would expect that the ouline would increase by 2 ~ sqrt(50/12) but that doesn't happen at all. Remind me if you are looking at exactly the same part of the tooth. ALso what happens to centroid size in this case? I'll try to think some more - but so far empty handed. ----- Original Message ----- From: <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, August 11, 2003 7:56 PM Subject: multivariate measure of shape variation > > Dear morphometricians, > I'd like to have your precious advice about possible simple measures of > shape variation for a sample in a multivariate space. I discussed about > this with Prof. Rohlf about one year ago, and I know that it's hard to > find a 'good analogue' of SD in univariate analyses. > However, I liked the way Dr. Polly coped with this problem in his paper > PALEOPHYLOGEOGRAPHY: THE TEMPO OF GEOGRAPHIC DIFFERENTIATION IN MARMOTS > (MARMOTA) (J. Mammal., Volume 84, Number 2, 2003). > > " Variation within samples was calculated as the mean of pairwise > distances between individuals in a sample: > ..... > where dii was the multidimensional Euclidean distance between specimen i > and i + 1 in sample x. > This distance is analogous, but not equal to the sample standard > deviation; dx is larger than the standard deviation." > > I wish to do something similar with average Procrustes distance (aPRD) > in samples of marmot crania of different species (same landmark > configuration in all samples, of course). I was wondering whether a > ratio between aPRD in a species and aPRD in the pooled sample of all > species could be used as an index of intraspecific variation compared to > interspecific variation. Also, the same could be done to compare > digitizing error (say, 10 digitizations of the same specimen) with > biological variation in the sample under study (a marmot species or the > pooled sample of all marmot species). > Does it make any sense? If yes, are there any references on this topic? > And, what are the caveats if aPRD is used as an approximate measure of > variation in a sample? > > I thank you in advance all of you who may reply for your kind help. > Please, let me know if you do not want that I redirect your answers to > colleagues that may ask me to read them. > > All the best > > Andrea > > > Andrea Cardini > Istituto di Medicina Legale e delle Assicurazioni > Universit degli Studi di Milano > via Mangiagalli 37 > Milano 20133, Italy > tel. 0039 02 50315678/9 > [EMAIL PROTECTED] > > and > > Division of Mammals, Department of Vertebrate Zoology > NHB 390, MRC 108 > National Museum of Natural History, Smithsonian Institution > P.O. BOX 37012 > Washington, DC 20013-7012, USA > tel. 202 3571476 > [EMAIL PROTECTED] > > > ----------------------------------------------------- > > Salve, il messaggio che hai ricevuto > stato inviato per mezzo del sistema > di web mail interfree. Se anche tu vuoi > una casella di posta free visita il > sito http://club.interfree.it > Ti aspettiamo! > > ----------------------------------------------------- > == > Replies will be sent to list. > For more information see http://life.bio.sunysb.edu/morph/morphmet.html. > == Replies will be sent to list. For more information see http://life.bio.sunysb.edu/morph/morphmet.html.
