Elahe and Ari,

If your dependent observations have more than a few dimensions, such as is
typical with landmark data or even a collections of linear measurements,
then I think the common covariance matrix recommendation from Dr. Rohlf is
the more standard approach. A typical strategy for removing group structure
is to perform your analysis on the pooled within group covariance matrix.
Chs.11 and 12 of Zelditch et al's *Geometric Morphometrics for Biologists*
provides an overview and citations, as well as the equation for
partitioning overall covariance into within and between group components.

However, Dr. Belk is correct that, in principle, a mixed model suits your
needs. The mixed model provides a way to partition random deviances
attributable to population history and structure from common (fixed)
effects of developmental stage and sex. You will have to think about how
you want to model conditionality of sexual dimorphism with respect to
stage. For example, if dimorphism does not arise until later stages of
development in your subject species, it does not make much sense to model a
single dimorphism coefficient across the entire sample.

To make the model go, you will need data with which to estimate pairwise
evolutionary relationships among the groups in your sample. The computation
of random effects depends upon it. Stone et al (http://rstb.
royalsocietypublishing.org/content/366/1569/1410) describe some of the
challenges of estimating relatedness where there is gene flow among the
sample clusters (i.e., where your groups are populations rather than
non-reticulating species). A paper from me+colleagues, linked to below,
provides an overly simple but (we think) reasonable solution to this.

This leaves one more potential problem: dimensionality. If the dependent
observation is univariate (centroid size, the first shape principal
component, etc.) or very low dimensional, the MCMCglmm package (see
http://onlinelibrary.wiley.com/doi/10.1111/j.1420-9101.2009.01915.x/full)
could suit your mixed model needs. However, if your data is truly
multivariate (GM or anything more than a handful of linear measurements), I
would recommend looking at the BSFG package. We used it to estimate sex and
climate effects on the human cranium, using a subset of populations from
the Howells linear craniometric collection. You can find that paper on my
ResearchGate page (ResearchGate profile
<https://www.researchgate.net/profile/David_Katz29>). You will also find a
poster where we extend the application to cranial and mandibular 3D
landmark data. In the next few months, we should have a paper out that
combines high-dimensional mixed model analysis with 3DGM in a more
satisfying way than was managed in the poster.

Unfortunately, unless the developers have finished updating the BSFG
package, it is not quite plug and play. It takes some time to figure out.
You need familiarity with Bayesian analysis, and with Matlab. However, the
outputs are worth the effort. For example, with GM data, your fixed effect
posterior will estimate shape contrasts for sex and developmental stage
effects for the whole shape configuration, rather than on synthetic,
orthogonal subsets (PCs).

Conclusion: if your data is multidimensional and you don't think you can
get going with BSFG, for the time being, the common covariance matrix
approach is probably your best option.

David

On Thu, Jan 19, 2017 at 5:24 PM, Mark Belk <mark_b...@byu.edu> wrote:

What you describe – samples from multiple populations – is best considered
as a random effect in a typical generalized linear model format.  You have
randomly sampled some populations from all of those that might be
available.  If I understand your data correctly, to evaluate allometry, use
a mixed model approach where some trait measurement is the response
variable and some measure of body size would be the predictor variable,
then population would be included as a random effect in the model.  This
structure has the advantage of accounting for and adjusting for covariation
among populations before the fixed effect is evaluated.  Appropriately
crafted mixed models can rigorously account for a range of complicated
covariance structures within the context of one model.  Several examples of
the use of mixed models in ecology and evolution can be found in the
literature.

Hope that helps,



Mark



Mark C. Belk, Professor of Biology

Brigham Young University

Editor, *Western North American Naturalist*

801-422-4154 <(801)%20422-4154>



*From:* Ariadne Schulz [mailto:ariadne.sch...@gmail.com]
*Sent:* Wednesday, January 18, 2017 1:27 PM
*To:* Elahe
*Cc:* MORPHMET
*Subject:* Re: [MORPHMET] eliminating the effect of population differences



I would like to hear any responses to this as well. I did something similar
and I wasn't sure how to approach this question. In future studies I would
like to address precisely this issue. My inclination would be that first
you would want to determine how much morphological variation you're getting
between sites. You could then look at sexual dimorphism within each site
and/or you could look at variation of only females and only males over all
sites. But this is all rather clunky and does not eliminate any
interpopulation variation. If anyone has already proposed or can propose a
better methodology I'd be interested in it as well.

Best,

Ari



On Wed, Jan 18, 2017 at 5:29 PM, Elahe <ellie.parv...@gmail.com> wrote:

Dear all,



I have pooled samples from 7 different populations of one species in order
to study the allometric growth and sexual dimorphism in that species. As
different populations may have subtle differences in terms of body
dimensions with each other, I want to remove their effects.

Can anyone suggest a way to eliminate population effects and maybe finding
some residuals that are homogeneous and can be used for further analyses?



I would appreciate any helps :)

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Evolutionary Anthropology
University of California, Davis
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ResearchGate profile <https://www.researchgate.net/profile/David_Katz29>
Personal webpage <https://davidckatz.wordpress.com/>

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