The Polygenic Model

Abstract

The standard polygenic model of biomedical genetics can be motivated by considering a quantitative trait determined by a large number of loci acting independently and additively [9]. In a pedigree of m people, let X ^k_i be the contribution of locus k to person i. The trait value $X_i = \sum\nolimits_k {X_i^k }$ for person i forms part of a vector $X = (X_l , \ldots X_m )^t$ of trait values for the pedigree. If the effects of the various loci are comparable, then the central limit theorem implies that X follows an approximate multivariate normal distribution [13, 15]. Furthermore, independence of the various loci implies $Cov(X_i ,X_j ) = \sum\nolimits_k {Cov(X_i^k ,X_j^k )}$. From our covariance decomposition for two non-inbred relatives at a single locus, it follows that

$$ Cov(X_i ,X_j ) = 2\Phi _{ij} \sigma _a^2 + \Delta 7_{ij} \sigma _a^2 , $$

where σ ²_a and σ ²_d are the additive and dominance genetic variances summed over all participating loci. These covariances can be expressed collectively in matrix notation as $Var(X) = 2\sigma _a^2 \Phi + \sigma _d^2 \Delta _7$. Again it is convenient to assume that X has mean E(X) = 0. Although it is an article of faith that the assumptions necessary for the central limit theorem actually hold for any given trait, one can check multivariate normality empirically.