There is a method called common PCA which seems to overcome the problem of non-multinormality of overall sample that includes several subsamples all with different central momenta. The source to read is:
Flury B. 1988. Common principal components and relat�d multivariate models. NY: Wiley. 258 p. Cheers, Igor ----- Original Message ----- From: <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Tuesday, May 18, 2004 10:09 PM Subject: Re: size correction & discriminant functions analyses > Dear Brett and Marta, > > I think the problem you are encountering may not be the size-versus-shape issue, but a Normal distribution issue. PCA Analysis assumes multivariate normality. I know for human beings the distribution of men and women combined is often not Multivariate Normal. It is bi-modal and the male and female variance-covariance structure is different. This dramatically affects the correlation and covariance matrices and provides misleading components. I would assume this could be true for catsharks as well, and suspect that is why you found such a large amount of variation seemingly explained by your first component. We have found that for humans the lack of Normality is big enough that it requires doing separate PCA analyses for men and women, and in some cases separate analyses by ethnicity as well. In addition, it sounds to me that you have additional modes or non-normalities due to age. (I generally only work with adults.) Have you checked to see if your data is Normally d! > istributed? If it isn't you could consider separating your samples into subgroups (gender and age groups) that are normally distributed, prior to PCA analysis. In other words, you would do a PCA analysis for each group, rather than just one PCA for all of them combined. I don't know how difficult this may be, not knowing your data. Or you might check into classification methods that do not depend upon the normality assumption. > > Most discriminant analyses also assume that the attributes of the entities within each group are Multivariate Normal, and that the variance-covariance structures of the entity attributes are equal across groups. You might be OK with the within-group normality assumption, but if there are important shape differences due to age or gender as you say then you may not be OK with the assumption of equal variance-covariance across groups. For example, there may be a strong correlation (covariance) between two attributes in younger growing catsharks that disappears when they reach adulthood. This would cause a difference in the covariance structure. You could break your data into groups and look at the differences/similarities in the variance/covariance matrices. This will tell you a lot about the similarities and differences between your groups as well. > > Hope this is helpful, > > Kathleen M. Robinette, Ph.D. > Principal Research Anthropologist > Air Force Research Laboratory > AFRL/HEPA > 2800 Q Street > Wright-Patterson AFB, OH 45433-7947 > (937) 255-8810 > DSN 785-8810 > FAX (937) 255-8752 > e-mail:[EMAIL PROTECTED] > > > > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of [EMAIL PROTECTED] > Sent: Tuesday, May 18, 2004 9:50 AM > To: [EMAIL PROTECTED] > Subject: Re: size correction & discriminant functions analyses > > > Dear Brett, > > I have the same problem. I found several approaches in the literature, bbut non > efficient or clear review... well there were some, but too mathematic for me as > a simple biologist. > By what I know, it is complicated to work with ratios (which have difficult > statistical properties). On the other way, you also have the problem of > colinearity between variables (I imagine). > I found some approaches to solve this, but none was universal or definitive. > There is an article by Leonart et al. > that proposes a simple formula, but it has been much discussed, and a > statistical lecturer told me that it is not recent. > On the other way, in Ade4 lab, I saw in the other day that they standartise the > columns with the mean. I tred this, and it was very good... gave much clearer > results. > My supervisor said to use PCA, as it is and simply consider that the first > component is 'size'... however this did not gave clear images of the data... > thus I am as traped in the beggining. I suppose in the end all this hypothesis > are possible and correct, and most will give very similar answers. > > I am also puzzled by the range of multivariate techniques, that give similar > answers... particularly because in many cases different authors (and > statistical packages) call the same techniques with different names, which > really messes the things. I started to do a summary of it (which I can send > you), of information I found in several books... as well, in the end, as I saw > it now, things are much simpler, and mainly consist in a couple of method with > variations, which arises different names. On the other way, people from the R > list have discussed a lot stepwise analysis, and some do not recommend it at > all... so some care should be taken in this point as well. Anyway, I can adive you of a free online manual from the VEGAN package (from > www.R-project.org) which for me was very good and compares many methods using > the same data: http://cc.oulu.fi/%7Ejarioksa/opetus/metodi/index.html > > hope this helps somehow, or at least shows solidarity with your question ;-) > > Please let me know if if you finally find 'a' answer :-) > > Best wishes, > > Marta > > > > Quoting [EMAIL PROTECTED]: > > > Dear morphometrician, > > > > I have recently reviewed 3 genera of catsharks that display a great > > deal of morphological conservation within the genera, however, there > > is also prominent sexual dimorphism present (profoundly so in some > > species). There is quite a bit of shape variation between juveniles > > and adults, in one genus in particular, but I think that the shape > > variation is being obscured by the size component. > > > > I have a sizeable morphometric data set (# measures >> # taxa & > > specimens) and have used principal components analysis on the raw data > > to explore shape variation within each of the genera (not between). > > The first component was always a general component and accounted for > > more than 85-90% of the variation in most instances, therefore the > > bipolar components only contributed relatively little to the overall > > shape variation resulting in crowded PCA plots. > > > > The main reference I have used for the analyses to date has been > > 'Pimental. 1979. Morphometrics. The multivariate analysis of > > biological data' however, it doesn't deal with size correction. Can > > anyone suggest a review that deals with size correction, or can I > > convert my data to ratios and then log transform the data? > > > > I am also looking for reviews of canonical discriminant functions > > analysis and stepwise discriminant function analysis in an attempt to > > quantitate differences between species within a genus. > > > > Thanks for your help. > > > > Brett > > > > ************************************ > > Brett Human > > Shark Researcher > > 27 Southern Ave > > West Beach SA 5024 > > Australia > > 61 8 8356 6891 > > [EMAIL PROTECTED] > > ************************************ > > > > > > > > == > > Replies will be sent to list. > > For more information see > > http://life.bio.sunysb.edu/morph/morphmet.html. > > > > > > > ------------------------------------------------- > This mail sent through IMP: http://horde.org/imp/ > > > > == > 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. == Replies will be sent to list. For more information see http://life.bio.sunysb.edu/morph/morphmet.html.
