-------- Original Message --------
Subject: Re: Analyzing ontogenetic trajectory angles
Date: Thu, 21 Jul 2011 14:32:35 -0400
From: Dean Adams <[email protected]>
To: [email protected]
Folks,
This is a very cogent point that Philipp makes. One must retain the
'biology' of the traits in question when discussing similarities in
ontogenetic trajectories, and not perform multivariate comparisons of
trajectories without keeping this information in mind. To rephrase
Philipp's point:
Similarity at young ages due largely to one portion of anatomy, and
similarity at older ages due largely to another portion of anatomy, does
not necessarily imply parallel ontogenetic trajectories.
One must always be cognizant to compare apples to apples: both in the
analytics, and in relation to the data analysis and the biological
hypothesis under investigation.
Dean
--
Dr. Dean C. Adams
Associate Professor
Department of Ecology, Evolution, and Organismal Biology
Department of Statistics
Iowa State University
Ames, Iowa
50011
www.public.iastate.edu/~dcadams/
phone: 515-294-3834
Similarity between in early stages due to
On 7/21/2011 12:32 PM, morphmet wrote:
-------- Original Message --------
Subject: Re: Analyzing ontogenetic trajectory angles
Date: Thu, 21 Jul 2011 11:47:04 -0400
From: Philipp Mitteröcker<[email protected]>
To: [email protected]
CC: Simon Huttegger<[email protected]>
I would like to make a few comments on the analysis of ontogenetic
trajectories in addition to the excellent summaries posted so far.
1.) The discussion of the geometry of two linear trajectories is biased
by our two-dimensional intuition. For example, two multidimensional
trajectories can be oblique, that is, they neither converge, diverge,
nor are they parallel. The distance between the two starting points of
the trajectories might be the same as the distance between the two end
points, but the trajectories need not be parallel. (In the first two
principal components, however, two oblique trajectories tend to appear
parallel; see Mitteroecker et al. 2004, 2005).
2.) This implies that the direction in shape space along which the two
starting points differ need not be the same direction along which the
end points differ. In such a situation one should better not speak about
convergence or divergence because phenotypic distances in different
directions usually are not comparable. I discussed this in a recent
paper with Simon Huttegger (Huttegger S, Mitteroecker P. in press.
Invariance and Meaningfulness in Phenotype spaces. Evolutionary
Biology). Consider, for example, two organisms (or species) that differ
in facial morphology at age 1, and in neurocranial morphology at age 2.
The two amounts of difference depend, among other things, on the number
of measurements (e.g., landmarks) in the face and the neurocranium. By
changing the number of landmarks it is possible to change diverging
trajectories into converging ones.
3.) Hence, I think, one should just talk about convergence and
divergence if the two trajectories approximately lie in a plane, i.e.,
the trajectories differ in the same direction in shape space at all age
stages. Only in this case, angles between trajectories can be
biologically meaningful. In the paper mentioned above, we argued that
angles are not in general meaningful in phenotype spaces (including
shape spaces), but for three trajectories A, B, and C that all lie in a
single plane one can say that the angle between A and B is smaller than
the angle between A and C if B is in between A and C (but you may not be
able to say whether the angle AB is larger than BC).
4.) Of course all of that assumes that the trajectories are
approximately linear.
5.) Testing the statistical significance of a geometric statement that
is unlikely to correspond to any biological property is of little help.
Best regards,
Philipp
___________________________________
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: [email protected]
homepage: http://theoretical.univie.ac.at/people/mitteroecker
Am 20.07.2011 um 21:34 schrieb morphmet:
-------- Original Message --------
Subject: Re: Analyzing ontogenetic trajectory angles
Date: Wed, 20 Jul 2011 14:31:32 -0400
From: Dean Adams<[email protected]>
To: [email protected]<[email protected]>
David,
Over the years there have been quite a few approaches for comparing
ontogenetic (or other types of phenotypic) trajectories. Note first that
we are discussing the angle between two multivariate regressions of
shape~logCS (not simply PC1 vs. logCS). The regression line is simply a
vector, and so the angle between two vectors can be obtained as the
inner product of the vectors. Here I will summarize a few methods from
the literature, as they relate to your questions.
1: Klingenberg and Leamy 2001 (Evol.) compared angles between estimated
G and P matrices from morphometric data. Here they obtained the angle
between vectors, and compared them to a distribution of angles obtained
from sets of random vectors. One could adapt this approach to compare
the angles between ontogenetic trajectories.
2: Zelditch et al. 2000 (Evol.) performed multivariate regressions of
shape~logCS for each species. They then obtained the angles between
these vectors, and used a bootstrap of individuals to obtain CI to
evaluate these angles for biological importance. Quite a few variants
on this approach have been proposed over the years.
Both of these methods allow one to evaluate whether or not ontogenetic
trajectories are oriented similarly in size-shape morphospace. However,
ontogenetic trajectories can still vary in other distinct ways (see
Mitteroecker et al. 2005: Evol. Devel). Thus, there may be some
additional interesting patterns that one may wish to identify.
3: Mike Colllyer and I developed an approach for comparing multivariate
phenotypic trajectories: Collyer and Adams 2007 (Ecol), Adams and
Collyer 2007 (Evol), Adams and Collyer 2009 (Evol). In short, one
obtains the trajectories of interest, and then compares their
magnitudes, their directions, and their shapes (if applicable) to
determine how they may vary.
For ontogenetic trajectories, one fits species-specific regressions of
shape~logCS. Angles between these are obtained, and assessed
statistically using residual randomization of individuals (see our
papers for why this resampling scheme is appropriate for such designs).
The lengths of the ontogenetic trajectories could also be compared by
adapting the magnitude comparisons, such that they assess magnitudes as
defined by the smallest and largest individual within each species. This
might be useful for identifying heterochronic changes that affect the
'length' of ontogenetic trajectories, as might be observed under
progenesis or hypermorphosis.
4: As Paolo mentioned in his post, approach #3 was further extended to
test yet another hypothesis. Here ontogenetic convergence, divergence,
and parallelism can be assessed by comparing the distance between
ontogenetic trajectories at their 'starting points' vs. the distances
between trajectories at their 'ending points'. (Piras et al. 2010: Evol.
Devel.; also Adams and Nistri 2010: BMC Evol. Biol.). Convergence is
found when larger individuals are more similar in shape (e.g., Adams and
Nistri 2010).
5: Mitteroecker et al. 2005 (Evol. Devel.) used slightly different test
statistics, but followed the same general logic that extends through
methods 2-4. Here they fit species-specific regressions of shape~logCS.
Then then test for 'identical regressions' by obtaining SSresid for all
trajectories, permuting species assignments, and repeating. Next, they
alter their test statistic somewhat for evaluating overlapping
trajectories (see their). The general idea here is to go through several
such tests sequentially to determine whether ontogenetic trajectories
differ in their direction, their length, or other attributes.
Again, this is NOT an exhaustive list of approaches. Rather it is
intended to provide a flavor of what has been done, and the
commonalities of the approaches (both in implementation and in
hypothesis testing).
Hope this is helpful.
Best,
Dean
--
Dr. Dean C. Adams
Associate Professor
Department of Ecology, Evolution, and Organismal Biology
Department of Statistics
Iowa State University
Ames, Iowa
50011
www.public.iastate.edu/~dcadams/
phone: 515-294-3834
On 7/19/2011 12:28 PM, morphmet wrote:
-------- Original Message --------
Subject: Analyzing ontogenetic trajectory angles
Date: Mon, 18 Jul 2011 14:46:19 -0400
From: David Katz<[email protected]>
To: [email protected]
Hello,
I have read several morphometrics papers which test for significant
differences in ontogenetic trajectory between two groups (species,
subspecies, etc) by calculating the "angle" between their growth
trajectories. However, parts of (or even lots of) the method remain
unclear to me.
First, it seems that calculation of the angle requires calculation of
two simple regression lines, one for each group, with the angle being
the arc subtended by the two lines. One axis for these regression
plots/calculations is the distribution of specimens along a PC which is
significantly correlated with size or age (usually the first PC). But
it is not clear to me what the second axis is. Log centroid size?
Second, the significance of the angle is tested by randomly permuting
specimens between the two groups 1000 or more times, then calculating a
new angle for each permutation. Significance is then tested against the
distribution of permutation-generated angles. But what is the
permutation procedure? Do we only permute a single specimen each time?
Third, if I'm understanding the method correctly, then absent an
additional scaling step, the two lines from which the angle is
calculated are not likely to be of the same length. How is this
accounted for, if at all?
Any help, or a reference to a good, explicit journal or book
explanation, would be very much appreciated.
Thanks.
David Katz
Doctoral Candidate
Department of Anthropology--Evolutionary Wing
University of California, Davis
Young Hall 204
/--Trying to focus on one distraction at a time/--
