-------- Original Message -------- Subject: RE: Two group CVA plot Date: Wed, 3 Sep 2008 11:55:05 -0700 (PDT) From: F. James Rohlf <[EMAIL PROTECTED]> Reply-To: <[EMAIL PROTECTED]> Organization: Stony Brook University To: <morphmet@morphometrics.org> References: <[EMAIL PROTECTED]> Yes, of course. Non-orthogonal axes imply that the scores on these axes will be correlated. One is a geometrical statement and the other a statistical one about the same property. The problem is that we are all used to eigenvectors of symmetric covariance matrices that have special and rather convenient properties. Summing eigenvalues to get a total variance would seem to imply orthogonal eigenvectors. Another way to view what is being done in a CVA (and what I do within the CVA module in NTSYSpc) is to standardize the space based on the pooled within-groups covariance matrix. Then in that transformed space one can do a PCA of the new among-groups covariance matrix. In this standardized space the PCA (=CVA) axes are orthogonal and have the familiar properties and are easier to think about. The only 'problem' is that the loadings for the axes refer to the standardized variables - not the original variables. That is not usually a problem in a geometric morphometric study because one is not that interested in the original variables (partial warp scores or landmark coordinates) individually. One can get back to the original variables my applying the inverse of the standardization transformation. ========================= F. James Rohlf Distinguished Professor, Stony Brook University http://life.bio.sunysb.edu/ee/rohlf
-----Original Message----- From: morphmet [mailto:[EMAIL PROTECTED] Sent: Wednesday, September 03, 2008 2:33 PM To: morphmet Subject: RE: Two group CVA plot -------- Original Message -------- Subject: RE: Two group CVA plot Date: Wed, 3 Sep 2008 11:30:19 -0700 (PDT) From: <[EMAIL PROTECTED]> To: morphmet@morphometrics.org Does the non-orthogonality of CV axes extend to the situation where there are multiple meaningful CV axes (say a case of 5 groups with 4 axes). In this situation, is it the case that the axes need not be orthogonal? The discussion of CVA methods in "Multivariate Analysis", by Mardia, Kent and Bibby (1979) on pages 343-344 refers to these eigenvectors (of the among-group covariance matrix multiplied by the inverse of the within-group covariance matrix) as "uncorrelated", and appears to treat the corresponding eigenvalues as I would expect assuming them to working with orthogonal eigenvectors. For example, the calculation of the fraction of the total variance explained by a given set of eigenvectors of pg. 344 of this text appears to be assuming uncorrelated eigenvectors. In particular they appear to be estimating variance as the sum of eigenvalues, which I'm interpreting (perhaps incorrectly) as requiring orthogonal eigenvectors. I've always thought that if the vectors involved are non-orthogonal, then a calculation of the total variance explained by the vectors (axes) needs to include a calculation of the covariance. Can anyone comment on this, for the case of 2 or more non-zero eigenvalues in a CVA? My thanks, and gratitude, in advance to any and all comments! H. David Sheets, PhD Dept of Physics, Canisius College 2001 Main St Buffalo NY 14208 ---- Original message ---- >Date: Wed, 03 Sep 2008 10:58:02 -0400 >From: morphmet <[EMAIL PROTECTED]> >Subject: RE: Two group CVA plot >To: morphmet <morphmet@morphometrics.org> > > > >-------- Original Message -------- >Subject: RE: Two group CVA plot >Date: Wed, 3 Sep 2008 07:44:41 -0700 (PDT) >From: F. James Rohlf <[EMAIL PROTECTED]> >Reply-To: <[EMAIL PROTECTED]> >Organization: Stony Brook University >To: <morphmet@morphometrics.org> >References: <[EMAIL PROTECTED]> > >With just two groups, the projections of the data onto the canonical >variates axes beyond the first must be zero (within rounding error) so >there will be no among-group variation to try to interpret beyond that >on the first axis. > >Note also that the among-group covariance matrix multiplied by the >inverse of the within-group covariance matrix is not symmetrical and >hence the eigenvectors of this matrix will not be orthogonal. > >========================= >F. James Rohlf >Distinguished Professor, Stony Brook University >http://life.bio.sunysb.edu/ee/rohlf > >> -----Original Message----- >> From: morphmet [mailto:[EMAIL PROTECTED] >> Sent: Wednesday, September 03, 2008 10:11 AM >> To: morphmet >> Subject: Re: Two group CVA plot >> >> >> >> -------- Original Message -------- >> Subject: Re: Two group CVA plot >> Date: Wed, 3 Sep 2008 06:44:07 -0700 (PDT) >> From: <[EMAIL PROTECTED]> >> To: morphmet@morphometrics.org >> >> >> To respond to Rebecca's question below, in CVAGen, the CVA is >> carried >> out using a Matlab function which computes the eigenvectors of the >> matrix resulting from dividing the between-group variance- covariance >> matrix by the estimated pooled within-group variance-covariance >> matrix. >> >> So if there is only 2 groups, there will be only one meaningful CV >> axis >> (1 eigenvector). However, the algorithms used to compute the >> eigenvectors does return a set of orthogonal axes from the >> eigenvalue >> decomposition. These higher ordered axes are orthogonal to the one >> another, and to the meaningful axes. >> >> The program is set up to allow you to plot any chosen axis against >> another, so that in a case with 3 meaningful axes, you could plot >> axes 1 >> and 2, and 1 and 3, etc. The default plot setting in CVAGen shows >> axes >> 1 and 2. >> >> In a case where there are only two groups, the program will plot >> scores >> along the second CVA axes, but I would not advise trying to >> interpret >> this axis, it is simply the second eigenvalue of the decomposition, >> orthogonal to the first. In such cases, the distribution of scores >> along the first axis indicates to some extent the effectiveness of >> the >> discrimination, although I would urge use of cross-validation or >> jackknife rates of specimen assignment as a better measure of >> effectiveness. The plot of scores along the axis reflects more >> information about the resubstitution rate of assignment, ie of >> specimens >> to which the function was fitted. >> >> Perhaps others can add comments about approaches to displaying the >> results of two group discrimination functions. Plotting histograms >> would be one option, but there may be better approaches. >> >> -Dave >> >> H. David Sheets, PhD >> Dept of Physics, Canisius College >> 2001 Main St >> Buffalo NY 14208 >> >> >> ---- Original message ---- >> >Date: Wed, 03 Sep 2008 09:07:20 -0400 >> >From: morphmet <[EMAIL PROTECTED]> >> >Subject: Two group CVA plot >> >To: morphmet <morphmet@morphometrics.org> >> > >> > >> > >> >-------- Original Message -------- >> >Subject: Two group CVA plot >> >Date: Tue, 2 Sep 2008 07:53:07 -0700 (PDT) >> >From: Rebeca <[EMAIL PROTECTED]> >> >To: <morphmet@morphometrics.org> >> > >> > >> > >> >Dear Morphometricians, >> > >> > >> > >> >I have been using the program CVAGen6j from the IMP series (David >> >Sheets) to compare the shape differences of two groups of fish >> samples. >> >The program gave me a CVA scatterplot with two axes. I would like >> to use >> >this graph because it is more appealing than a histogram, but I >> don't >> >know what is plotted on the second axis (y-axis), as I understand >> that >> >from a two group analysis you get only one canonical variate (the >> x-axis >> >in this case). Does anyone know how the plot is done? >> > >> > >> > >> >Thanks a lot, >> > >> >Rebeca >> > >> > >> > >> >Rebeca P. Rodriguez Mendoza >> > >> >e-mail: [EMAIL PROTECTED] >> > >> >CSIC >> > >> >Marine Research Institute - Fisheries >> > >> >C/ Eduardo Cabello, 6 >> > >> >Tel. 986 23 19 30 ext. 254 >> > >> >Fax: 986 29 27 62 >> > >> >36 208 Vigo, Spain >> > >> > >> > >> > >> >-- >> >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 > > > > >-- >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
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