This is in response to the postings by Jiri, Dennis and Miquel.
For Jiri's problem, I think the first question is: are there landmarks
or just outlines?
If there are corresponding landmarks, then the method for object
symmetry according to Mardia et al. (2000) should work. Some additional
information and illustration is in the following paper:
Klingenberg, C. P., M. Barluenga, and A. Meyer. 2002. Shape analysis of
symmetric structures: quantifying variation among individuals and
asymmetry. Evolution 56:19091920.
This should work in combination with semilandmarks, if you have them in
equal numbers and corresponding spacing between the landmarks so that
you can define correspondences across the two possible axes of symmetry.
The analyses will end up averaging over the four quadrants for the
symmetric component of shape variation. The remaining information is in
the two asymmetry components. The two-factor model could be expanded
simply by having one effect for each of the two axes of symmetry -- for
instance, left and right sides and anterior and posterior ends. However,
if there is no way to distinguish 'left' and 'right' sides and
'anterior' and 'posterior' then there is a bit of a problem for how to
define the respective main effects (and directional asymmetry). This
requires some careful thought, because it is qualitatively different
from the standard situation with one symmetry axis and unambiguous left
and right sides.
If there are just outlines, it's probably easier with the method of
'circulant symmetry' described in the following paper: Kent, J. T., I.
L. Dryden, and C. R. Anderson. 2000. Using circulant symmetry to model
featureless objects. Biometrika 87:527544. As far as I know, this
approach has not yet made it into the biological morphometrics
community. This method is exactly for the kind of symmetry that exists
in diatoms and other sorts of algae, where there may be two or more axes
of symmetry, and it should also work for Scenedesmus. The problem here
is that it may not be straightforward to adapt this approach to the
two-factor ANOVA model devised for asymmetry studies by Leamy and Palmer
& Strobeck. Because you never can be sure from which side you're looking
at an organism with multiple symmetry axes, the ANOVA approach may not
work in the usual way. But nobody has tried it yet...
***
On Miquel's question:
This concerns matching symmetry with paired structures that are separate
from each other on the two body sides (see Mardia et al. 2000;
Klingenberg et al. 2002).
Personally, I would rather not code dummy variables for the different
effects in the ANOVA model because I know I would make mistakes at some
point and screw up the analysis... Miquel, if you are more reliable than
I am, go ahead (I think you have too many dummy variables, as their
number should match the degrees of freedom for the respective effect,
all dummies 0 is coding for a level).
There is an alternative. The idea is to export the data after the
Procrustes superimposition (of all configurations, with those from one
body side reflected to their mirror images) and do the rest of the
analysis in a standard statistics package. You can export the data
either as the coordinates of the superimposed configurations or as the
partial warp scores AND uniform components.
There are two alternatives for the analysis:
a) Procrustes ANOVA -- this implicitly assumes isotropic shape variation
(see the recent discussion on Morphmet). Run a two-factor ANOVA a la
Palmer and Strobeck for each of the variables, sum up the resulting sums
of squares across variables. You'll have to adjust the degrees of
freedom by multiplying with the shape dimension (twice the number of
landmarks - 4). Tests can be done as F tests (assuming isotropy and
normality) or by permutation (still with the isotropy assumption).
b) MANOVA with all the shape variables, which does not make the
assumption of isotropic variation. With the partial warp scores and
uniform components, this should be straightforward, and with the
coordinates of superimposed landmarks, you just have to make sure that
your statistics package can handle singular covariance matrices (by
using a generalized inverse). Test can be done with parametric methods
(the standard multivariate tests included in program packages) or by
permutation.
The permutation test needs to be performed separately for each effect in
the model -- each is associated with a different null hypothesis and
therefore needs to be simulated differently with the permutation
approach. Sides effect: exchange sides randomly within individuals (but
not between individuals). Individual effect: randomly reassign a left
side and a right side to each individual, but keep the left and right
sides separate. Interaction effect: first eliminate the main effects by
subtracting the deviation of each individual from the grand mean mean
and by subtracting the average deviation of each side (directional
asymmetry), and then you permute all the rows of the modified data set
freely.
Personally, I would stick with the ANOVA approach and call FA just the
side*individual interaction. If DA is nonsignificant because of limited
power (small sample size), then calling all asymmetry FA would be
misleading -- just collect more specimens and the results might look
differently. If the estimate of DA is negligible (the mean square for
sides is very close to zero) then this is a moot point because almost
all asymmetry will be in the side*individuals interaction anyway.
I think the strength of the two-factor ANOVA approach is that
it does not treat FA and DA in an either-or manner, but that they can
exist side by side and can be estimated separately. For estimating
individual-level FA scores, you can subtract the mean asymmetry from
each individual asymmetry, and then calculate the lengths of the
mean-corrected asymmetry vectors as Procrustes distances.
I hope this helps.
Best wishes,
Chris
******************************************************
Christian Peter Klingenberg
School of Biological Sciences
University of Manchester
3.614 Stopford Building
Oxford Road
Manchester M13 9PT
United Kingdom
Telephone: +44 161 2753899
Fax: +44 161 2753938
E-mail: [EMAIL PROTECTED]
Web: http://www.sbs.man.ac.uk/chrisk
******************************************************
==
Replies will be sent to list.
For more information see http://life.bio.sunysb.edu/morph/morphmet.html.
