Additive genetic variability and the Bayesian alphabet

Abstract

The use of all available molecular markers in statistical models for prediction of quantitative traits has led to what could be termed a genomic-assisted selection paradigm in animal and plant breeding. This article provides a critical review of some theoretical and statistical concepts in the context of genomic-assisted genetic evaluation of animals and crops. First, relationships between the (Bayesian) variance of marker effects in some regression models and additive genetic variance are examined under standard assumptions. Second, the connection between marker genotypes and resemblance between relatives is explored, and linkages between a marker-based model and the infinitesimal model are reviewed. Third, issues associated with the use of Bayesian models for marker-assisted selection, with a focus on the role of the priors, are examined from a theoretical angle. The sensitivity of a Bayesian specification that has been proposed (called "Bayes A") with respect to priors is illustrated with a simulation. Methods that can solve potential shortcomings of some of these Bayesian regression procedures are discussed briefly.

Figure 1.—

Ratio between coefficients of variation of the conditional posterior and prior distributions of the variance of the marker effect, as a function of the degrees of freedom ν of the prior.

Figure 2.—

Prior densities of the marker-specific variance (solid circles, ν = 4, S = 1; solid curve, ν = 10, S = 1; crosses, ν = 100, S = 1) and values of integrand in the Kullback–Leibler distance, for each of the three priors, shown as solid lines. The integrands are essentially indistinguishable from each other for all values of . Values of the integrand are drastically different (open circles) when 10 markers are assigned the same variance, so that .

Figure 3.—

Effect of scale parameter on the conditional posterior distribution of the variance of the marker effect. Open boxes, prior distribution; solid circles, conditional posterior distribution for c = 2 (standardized marker effect). The other three conditional distributions (solid lines) are barely distinguishable from the prior.

Figure 4.—

Positions (chromosome and marker number) and effects of markers (there were 280 markers, with 270 having no effect).