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.

Keywords: G matrix; additive genetic variance; evolution; generalized linear model; quantitative genetics; statistics; theory.

Figure 1

Example of the relationships between the three scales of the GLMM using a Poisson distribution and a logarithm-link function. Deterministic relationships are denoted with shaded solid arrows, whereas stochastic relationships are denoted with shaded dashed arrows. Note that the latent scale is depicted as a simple Gaussian distribution for the sake of simplicity, whereas it is a mixture of Gaussian distributions in reality.

Figure 2

Simulated $R/S$ (evolutionary response over selection differential or the realized heritability) on the latent (top panels) or observed (bottom panels) data scales against the corresponding scale heritabilities. Each data point is the average over 10,000 replicates of 10,000 individuals for various latent heritabilities $h_{\mathrm{lat}}^2$ (0.1, 0.3, 0.5, 0.8), latent population mean (μ from 0 to 3, from left to right), and proportion of selected individuals (5%, 10%, 20%, 30%, 50%, 70%, 80%, 90%, and 95%, varying from black to blue). The 1:1 line is plotted in black. The breeder’s equation is predictive on the latent scale (top panels), but approximate on the observed data scale (bottom panels), because phenotypes and breeding values are not jointly multivariate normal on that scale.

Figure 3

Predicted $R_{\mathrm{obs}}$ (phenotypic evolutionary response on the observed scale, see Equation 34) against the simulated $R_{\mathrm{obs}}$, via evolutionary predictions applied on the latent scale. Each data point is the average over 10,000 replicates of 10,000 individuals for various latent heritabilities $h_{\mathrm{lat}}^2$ (0.1, 0.3, 0.5, 0.8), latent population mean (μ from 0 to 3), and proportion of selected individuals (5%, 10%, 20%, 30%, 50%, 70%, 80%, 90%, and 95%, varying from black to blue). The 1:1 line is plotted in black. The top panels (“Immediate”) show simulations for the response after a single generation, which include both a permanent and a transient response to selection arising from linkage disequilibrium. The bottom panels (“Permanent”) show simulation results modified to reflect only the permanent response to selection.