General methods for evolutionary quantitative genetic inference from generalized mixed models
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Methods for inference and interpretation of evolutionary quantitative genetic parameters, and for prediction of the response to selection, are best developed for traits with normal distributions. Many traits of evolutionary interest, including many life history and behavioural traits, have inherently non-normal distributions. The generalised linear mixed model (GLMM) framework has become a widely used tool for estimating quantitative genetic parameters for non-normal traits. However, whereas GLMMs provide inference on a statistically-convenient latent scale, it is often desirable to express quantitative genetic parameters on the scale upon which traits are measured. The parameters of fitted GLMMs, despite being on a latent scale, fully determine all quantities of potential interest on the scale on which traits are expressed. We provide expressions for deriving each of such quantities, including population means, phenotypic (co)variances, variance components including additive genetic (co)variances, and parameters such as heritability. We demonstrate that fixed effects have a strong impact on those parameters and show how to deal with this by averaging or integrating over fixed effects. The expressions require integration of quantities determined by the link function, over distributions of latent values. In general cases, the required integrals must be solved numerically, but efficient methods are available and we provide an implementation in an R package, QGGLMM. We show that known formulae for quantities such as heritability of traits with Binomial and Poisson distributions are special cases of our expressions. Additionally, we show how fitted GLMM can be incorporated into existing methods for predicting evolutionary trajectories. We demonstrate the accuracy of the resulting method for evolutionary prediction by simulation, and apply our approach to data from a wild pedigreed vertebrate population.
de Villemereuil , P , Schielzeth , H , Nakagawa , S & Morrissey , M 2016 , ' General methods for evolutionary quantitative genetic inference from generalized mixed models ' , Genetics , vol. 204 , no. 3 , pp. 1281-1294 . https://doi.org/10.1534/genetics.115.186536
Copyright © 2016 de Villemereuil et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
DescriptionP.d.V. was supported by a doctoral studentship from the French Ministère de la Recherche et de l’Enseignement Supérieur. H.S. was supported by an Emmy Noether fellowship from the German Research Foundation (SCHI 1188/1-1). S.N. is supported by a Future Fellowship, Australia (FT130100268). M.M. is supported by a University Research Fellowship from the Royal Society (London). The collection of the Soay sheep data is supported by the National Trust for Scotland and QinetQ, with funding from the Natural Environment Research Council, the Royal Society, and the Leverhulme Trust.
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