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 :) -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. To unsubscribe from this group and stop receiving emails from it, send an email to morphmet+unsubscr...@morphometrics.org. -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. To unsubscribe from this group and stop receiving emails from it, send an email to morphmet+unsubscr...@morphometrics.org. -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. To unsubscribe from this group and stop receiving emails from it, send an email to morphmet+unsubscr...@morphometrics.org. -- David C. Katz, Ph.D. Evolutionary Anthropology University of California, Davis Young Hall 204 ResearchGate profile <https://www.researchgate.net/profile/David_Katz29> Personal webpage <https://davidckatz.wordpress.com/> -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. To unsubscribe from this group and stop receiving emails from it, send an email to morphmet+unsubscr...@morphometrics.org.