-------- Original Message --------
Subject: RE: MorphoJ problem: erroneous canonical coefficients?
Date: Tue, 6 Mar 2012 14:56:43 -0500
From: F. James Rohlf <ro...@life.bio.sunysb.edu>
Reply-To: <ro...@life.bio.sunysb.edu>
Organization: Stony Brook University
To: <morphmet@morphometrics.org>
The counterintuitiveness is somewhat like what one finds when one tries
to interpret partial regression coefficients in a multiple regression
analysis. One is tempted to interpret the coefficients at face value and
assume that variables with large coefficients (either positive or
negative) are "important" and those with very small coefficients are
"unimportant". That is true as far as constructing the optimal
prediction of a dependent variable. However, if the independent
variables are correlated then one will find the paradoxical result that
of two very highly correlated variables one of them may have a very
large partial regression coefficient and the other one could have a
very small coefficient. It does not seem reasonable to conclude that
one of a pair of highly correlated variables has little or no
relationship to the dependent variable that it is also correlated with.
One see a similar problem in canonical correlation analysis. However,
2-block partial least-squares analysis gives a result that can be
interpreted in a more straight-forward way (and is the reason why I
recommend it over canonical correlation analysis and why many prefer
partial least-square regression or other alternatives over multiple
regression).
Perhaps the main reason I do not suggest the direct interpretation of
the canonical variates coefficient vectors is that they often do not
give sensible results just as in a naive interpretation of partial
regression coefficients in a multiple regression analysis.
A little math:
In a PCA one computes projections as P = Y*E, and thus one can construct
a PCA biplot based on the relationship Y = P*E'. Biplots are very useful
for the interpretation of the relationships between the variation of
observations and variables.
In a CVA the scores are computed as P = Y*C. To make a biplot one would
use the relationship Y = P*inv(C) but C is not an orthonormal matrix so
one cannot just transpose it or invert it. One solution is to compute a
least-squares generalized inverse. That is equivalent to the regression
method mentioned in earlier emails.
I hope the above is clear.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
F. James Rohlf, John S. Toll Professor, Stony Brook University
The much revised 4th editions of Biometry and Statistical Tables are now
available:
http://www.whfreeman.com/Catalog/product/biometry-fourthedition-sokal
http://www.whfreeman.com/Catalog/product/statisticaltables-fourthedition-rohlf
-----Original Message-----
From: morphmet [mailto:morphmet_modera...@morphometrics.org]
Sent: Tuesday, March 06, 2012 5:30 AM
To: morphmet
Subject: Re: MorphoJ problem: erroneous canonical coefficients?
-------- Original Message --------
Subject: Re: MorphoJ problem: erroneous canonical coefficients?
Date: Mon, 5 Mar 2012 10:45:16 -0500
From: Louis Boell <lbo...@evolbio.mpg.de>
To: morphmet@morphometrics.org
Hi Chris, hi Phillip,
thanks to both of you for the interesting and helpful comments. I can confirm
what Philipp says concerning multiple groups. Just for fun I digitized the
lollipop graphs resulting from CVA in MorphoJ for 11 groups and the resulting
"coefficients" matched very closely the loadings from a PCA on the group
means.
I find this somewhat counterintuitive in so far as CVA is supposed to quantify
distinctness, not just variation between groups in general. Personally I find
that the mismatch between displayed shape changes and coefficients proper
can be confusing/misleading, because both features result from the same
analysis which is supposed to address distinctness (analogous to phenotypic
FSTs). What, if not this distinctness, do the shape changes in MorphoJ CVA
reflect?
Best regards,
Louis
----- Ursprüngliche Mail -----
Von: "morphmet" <morphmet_modera...@morphometrics.org>
An: "morphmet" <morphmet@morphometrics.org>
Gesendet: Montag, 5. März 2012 16:14:07
Betreff: Re: MorphoJ problem: erroneous canonical coefficients?
-------- Original Message --------
Subject: Re: MorphoJ problem: erroneous canonical coefficients?
Date: Mon, 5 Mar 2012 06:01:43 -0500
From: Philipp Mitteröcker <mitte...@univie.ac.at>
To: morphmet@morphometrics.org
This is an interesting issue raised by Chris. In a recent paper, we explained
why shape changes corresponding to CVs should be visualized based on the
CV coefficients, just as for PCA. They are vectors in tangent space. For two
groups, the regression approach advocated by Chris leads to a visualization of
the group mean difference vector, not the CV.
I am aware that this is a controversial topic and I am happy to discuss it!
Mitteroecker P, Bookstein FL (2011) Classification, linear discrimination, and
the visualization of selection gradients in modern morphometrics.
Evolutionary Biology 38, 100-114
Best wishes,
Philipp
Am 04.03.2012 um 15:56 schrieb morphmet:
>
>
> -------- Original Message --------
> Subject: Re: MorphoJ problem: erroneous canonical coefficients?
> Date: Wed, 29 Feb 2012 06:38:43 -0500
> From: Chris Klingenberg <c...@manchester.ac.uk>
> Reply-To: c...@manchester.ac.uk
> Organization: University of Manchester
> To: morphmet@morphometrics.org
>
> Dear Louis
>
> There is nothing erroneous with the canonical coefficients provided by
> MorphoJ, as far as I know.
>
> The way you phrase your query suggests that you expect that the shape
> changes associated with canonical variates (CVs) are just a scaled
> version of the CV coefficients, as it holds for principal components.
> This expectation is mistaken, because it misses the crucial difference
> that CVs are not computed in shape (tangent) space itself, but in a
> transformed space.
>
> This problem has been extensively discussed in geometric morphometrics.
> The computations of shape changes associated with CVs that are used in
> MorphoJ are based on the solution proposed by Rohlf et al. (1996):
> Rohlf, F. J., A. Loy, and M. Corti. 1996. Morphometric analysis of Old
> World Talpidae (Mammalia, Insectivora) unsing partial-warp scores. Syst.
> Biol. 45:344–362.
>
> More discussion on transformed spaces and discriminant/canonical
> variate analysis can be found here:
> Klingenberg, C. P., and L. R. Monteiro. 2005. Distances and directions
> in multidimensional shape spaces: implications for morphometric
> applications. Syst. Biol. 54:678–688.
>
> I hope this is useful.
>
> Best wishes,
> Chris
>
>
>
> On 2/27/2012 5:23 PM, morphmet wrote:
>>
>>
>> -------- Original Message --------
>> Subject: MorphoJ problem: erroneous canonical coefficients?
>> Date: Thu, 23 Feb 2012 08:18:03 -0500
>> From: Louis Boell <lbo...@evolbio.mpg.de>
>> To: morphmet@morphometrics.org
>>
>> Dear colleagues,
>>
>> I am encountering a peculiar problem in MorphoJ: after performing
>> CVA, the Canonical Coefficients given in the results do not
>> correspond at all to the vector lengths in the lollipop shape change
>> graphs. Either the graphs or the Coefficients appear to be erroneous,
>> because they contradict each other. This only happens with CVA, not
>> with PCA, for which the results are nicely congruent. Did anyone else
encounter this?
>> Any explanation?
>> Thanks for any help
>> Best regards,
>>
>> Louis
>>
>>
>
> --
>
**********************************************************
*****
> Christian Peter Klingenberg
> Faculty of Life Sciences
> The University of Manchester
> Michael Smith Building
> Oxford Road
> Manchester M13 9PT
> United Kingdom
>
> Telephone: +44 161 275 3899
> Fax: +44 161 275 5082
> E-mail: c...@manchester.ac.uk
> Web: http://www.flywings.org.uk
> Skype: chris_klingenberg
>
**********************************************************
*****
>
>
___________________________________
Dr. Philipp Mitteroecker
Department of Theoretical Biology
University of Vienna
Althanstrasse 14
A-1090 Vienna, Austria
Tel: +43 1 4277 56705
Fax: +43 1 4277 9544
email: philipp.mitteroec...@univie.ac.at
homepage: http://theoretical.univie.ac.at/people/mitteroecker
