General methods for evolutionary quantitative genetic inference from generalised mixed models

1 Methods for inference and interpretation of evolutionary quantitative genetic pa2 rameters, and for prediction of the response to selection, are best developed for traits 3 with normal distributions. Many traits of evolutionary interest, including many life 4 history and behavioural traits, have inherently non-normal distributions. The gen5 eralised linear mixed model (GLMM) framework has become a widely used tool for 6 estimating quantitative genetic parameters for non-normal traits. However, whereas 7 GLMMs provide inference on a statistically-convenient latent scale, it will often be 8 desirable to estimate quantitative genetic parameters on the scale upon which traits 9 are expressed. The parameters of a fitted GLMM, despite being on a latent scale, 10 fully determine all quantities of potential interest on the scale on which traits are 11 expressed. We provide expressions for deriving each of such quantities, including 12 population means, phenotypic (co)variances, variance components including additive 13 genetic (co)variances, and parameters such as heritability. The expressions require 14 integration of quantities determined by the link function, over distributions of latent 15 values. In general cases, the required integrals must be solved numerically, but ef16 ficient methods are available and we provide an implementation in an R package, 17 QGglmm. We show that known formulae for quantities such as heritability of traits 18 with Binomial and Poisson distributions are special cases of our expressions. Addi19 tionally, we show how a fitted GLMM can be incorporated into existing methods for 20 predicting evolutionary trajectories. We demonstrate the accuracy of the resulting 21 method for evolutionary prediction by simulation, and apply our approach to data 22 from a pedigreed vertebrate population. 23 . CC-BY 4.0 International license not peer-reviewed) is the author/funder. It is made available under a The copyright holder for this preprint (which was . http://dx.doi.org/10.1101/026377 doi: bioRxiv preprint first posted online Sep. 8, 2015;