-------- 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
>
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