Steve, Even PCA should require multivariate normality, because the method is based on using either the covariance or the correlation matrix. Both of these require multivariate normality to some extent to be meaningful. The variance is only a good estimate of the "true" variance if the distribution is normal or transformable to normality, and so, normality is required. Correlations, to be meaningful, also require normality, as the statistic program is not using a covariance matrix based on ranks (Spearman). Finally, if the new components (reduced number of variables) are being used in a new analysis (hypothesis testing) - for them to make sense, their distributions should also fit the assumptions of the test (if ANOVA, normality of the residuals and equality of the variances, for example). If the original variables were far from normal, then the reduced number of variables based on the correlation or covariance matrix are problematic as well. And of course, we all know that the normality assumption is fairly robust, more so than the equality of variance assumption.
Cheers, Jim Steve Brewer wrote: > Please allow me to clarify one comment I made regarding multivariate > normality. When I was talking about the multivariate normality > requirement, it was in relation to doing discriminant analysis and > MANOVA, not PCA. I believe that multivariate normality is required for > testing significance using these techniques. If I am wrong, then > several multivariate textbooks are wrong also. > > Indeed, multivariate normality is not required for PCA. PCA does not > involve hypothesis testing. Having said that, several have shown using > simulations that, when certain aspects of multivariate normality do > not hold (e.g., when there are lots of zero values), other exploratory > techniques (e.g., non-metric multidimensional scaling) perform better. > I have seen some use Principal Coordinates Analysis (using distance > measures other than correlation) to examine morphometric differences > among taxa. Presumably, this performs better than PCA under certain > circumstances. > > One problem I have seen is that some investigators become attached to > a particular technique. When I ask them why, many respond that it is > the most commonly used analysis in their particular field of study. > Hopefully, we can all agree that *that* is not an adequate > justification for using a particular technique. Personally, I prefer > to analyze multivariate data using several different techniques > (including PCA). When they provide different results, I become > suspicious and am encouraged to find out why. > > Steve Brewer > > > At 6:05 AM -0400 8/24/06, Highland Statistics Ltd. wrote: >> > >>> >>> Hope this helps some. Let me know if you want information about >>> SuperAnova or PC-Ord. >>> >>> Steve >>> >>> >>> >>> >>> >>> At 7:31 PM -0500 8/21/06, Chris Taylor wrote: >>>> Hey Steve. What do you run those nested discriminant analyses with? >>>> Hope all is well! >>>> >>>> Chris >>>> >>>> At 11:18 AM 8/21/2006, you wrote: >>>>> Matthew, >>>>> >>>>> You may also want to do a nested discriminant analysis to determine >>>>> whether the mean morphology differs among populations, while >>>>> controlling for species. The nesting of populations within species >>>>> should "correct for phylogeny", unless there is something I'm missing >>>>> here (e.g., phylogenetic relationships among populations within >>>>> species). Don't really see the need for PICs. Make sure the >>>>> assumptions of multivariate normality are met. >>>>> >>>>> Steve >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >> >> Matthew, >> >>>>> At 10:30 AM -0400 8/18/06, Matthew Gifford wrote: >>>>>> I am looking for advice regarding principal components analysis. My >>>>>> situation is as follows: I have a >>>>>> data set of morphological measurements for 6 "taxa" (4 populations >>>>>> of one species and 2 >>>>>> populations of another). I read somewhere that in order to do a PCA >>>>>> appropriately, one needs to >>>>>> have more "taxa" (i.e., rows) than measurement variables (i.e., >>>>>> columns). >> >> This is to avoid negative eigenvalues. But if you only focus on the >> first >> few eigenvalues, this should be no problem. >> >> If I use mean values for >>>>>> each "taxon" then I viiolate this assumption. To circumvent this, >>>>>> is it valid to do a PCA on all data >>>>>> and use mean PC scores? >> >> No need to do this. And if you do, it doesn't solve the engative >> eigenvalue problem. >> >> >> No need for multivariate normality neither. >> >> >> I will be using this information in >>>>>> phylogenetically independent contrasts >>>>>> analysis looking at ecomorphological relationships. >> >> >> The real problem with morphometric data is that the first axes become >> size >> and shape axes. See: >> >> Jolliffe IT (2002) Principal Component Analysis. Springer: New York >> >> and: >> >> Claude, J., Jolliffe, I.T., Zuur, A.F., Ieno, E.N. and Smith, G.M. >> Multivariate analyses of morphometric turtle data size and shape. >> Chapter 30 in Zuur, AF., Ieno, EN, Smith. GM. (Expected publication >> date: >> March 2007). Springer >> >> >> Kind regards, >> >> Alain Zuur >> www.highstat.com >> >> >> >> >> >> >> Any >>>>>> thoughts/opinions are most appreciated. >>>>>> >> >>>>Best, >>>>>> >>>>>> Matthew E. Gifford >>>>>> Ph.D. Candidate >>>>>> Washington University, St. Louis, MO >>>>>> http://www.biology.wustl.edu/larsonlab/people/Gifford/Matt's_webpage.ht >>>>>> >> ml >>>>> >>>>> >>>>> -- >>>>> Department of Biology >>>>> PO Box 1848 >>>>> University of Mississippi >>>>> University, Mississippi 38677-1848 >>>>> >>>>> Brewer web page - http://home.olemiss.edu/~jbrewer/ >>>>> >>>>> FAX - 662-915-5144 >>>>> Phone - 662-915-1077 >>>> >>>> *************************************************************** >>>> Christopher M. Taylor >>>> Associate Professor of Biological Sciences >>>> Dept. of Biological Sciences >>>> Mississippi State University >>>> Mississippi State, MS 39762 >>>> Phone: 662-325-8591 >>>> Fax: 662-325-7939 >>>> Email: [EMAIL PROTECTED] >>>> http://www2.msstate.edu/~ctaylor/ctaylor.htm >>> >>> >>> -- >>> Department of Biology >>> PO Box 1848 >>> University of Mississippi >>> University, Mississippi 38677-1848 >>> >>> Brewer web page - http://home.olemiss.edu/~jbrewer/ >>> >>> FAX - 662-915-5144 >>> Phone - 662-915-1077 >>> ========================================================================= >>> > > -- ------------------------------------- James J. Roper, Ph.D. Universidade Federal do Paraná Depto. de Zoologia Caixa Postal 19020 81531-990 Curitiba, Paraná, Brasil ===================================== E-mail: [EMAIL PROTECTED] Phone/Fone/Teléfono: 55 41 33611764 celular: 55 41 99870543 ===================================== Zoologia na UFPR http://zoo.bio.ufpr.br/zoologia/ Ecologia e Conservação na UFPR http://www.bio.ufpr.br/ecologia/ ------------------------------------- http://jjroper.sites.uol.com.br Currículo Lattes http://lattes.cnpq.br/2553295738925812
