# General methods for evolutionary quantitative genetic inference from generalised mixed models

## TL;DR

This paper introduces a general method to estimate evolutionary genetic parameters for non-normal traits using GLMMs, enabling accurate predictions of evolutionary responses.

## Contribution

A general framework for calculating quantitative genetic parameters on the observed scale for arbitrary GLMMs is introduced.

## Key findings

- Existing formulae for Binomial and Poisson distributions are shown as special cases within the new framework.
- Simulations confirm the method's accuracy in predicting evolutionary responses to selection.
- The approach is applied successfully to a natural study population with pedigree data.

## 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 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 will often be desirable to estimate quantitative genetic parameters on the scale upon which traits are expressed. The parameters of a fitted GLMM, 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. 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 a 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 pedigreed vertebrate population.