title: General Methods for Evolutionary Quantitative Genetic Inference from Generalized Mixed Models.
abstract: 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 behavioral traits, have inherently nonnormal distributions. The generalized linear mixed model (GLMM) framework has become a widely used tool for estimating quantitative genetic parameters for nonnormal 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 formulas 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.
authors: Laboratoire d'Écologie Alpine, Centre National de la Recherche Scientifique Unité Mixte de Recherche 5553, Université Joseph Fourier, 38041 Grenoble Cedex 9, France.
authors: Pierre
authors: 0000-0002-8791-6104
authors: P
authors: de Villemereuil
authors: Department of Evolutionary Biology, Bielefeld University, 33615 Bielefeld, Germany.|Institute of Ecology, Friedrich Schiller University, 07743 Jena, Germany.
authors: Holger
authors: 0000-0002-9124-2261
authors: H
authors: Schielzeth
authors: Evolution and Ecology Research Centre, University of New South Wales, Sydney, New South Wales 2052, Australia.
authors: Shinichi
authors: 0000-0002-7765-5182
authors: S
authors: Nakagawa
authors: School of Evolutionary Biology, University of St. Andrews, St. Andrews KY16 9TH, United Kingdom michael.morrissey@st-andrews.ac.uk.
authors: Michael
authors: 0000-0001-6209-0177
authors: M
authors: Morrissey
mesh_terms: D019143:Evolution, Molecular; D005828:Genetics, Population / Q000379:methods; D008957:Models, Genetic; D019655:Quantitative Trait, Heritable; D012984:Software
keywords: G matrix; additive genetic variance; evolution; generalized linear model; quantitative genetics; statistics; theory