The Polygenic Model

Abstract

The standard polygenic model of biometrical genetics can be motivated by considering a quantitative trait determined by a large number of loci acting independently and additively [12]. In a pedigree of m people, let X ^suk_ini be the contribution of locus k to person i. The trait value X _i = Σ _kX ^suk_ini for person i forms part of a vector X = (X ₁,...,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. (See the references [19, 21] and Appendix B.) Furthermore, independence of the various loci implies Cov(X ₁, X _j) = Σ _k Cov(X ^suk_ini , X ^suk_ink . From our covariance decomposition for two non-inbred relatives at a single locus, it follows that Cov(X ₁, X _j) = 2Φ_ijσ ^su2_inα + Δ7ijσ ^su2_ind , where σ ^su2_inα and σ ^su2_ind are the additive and dominance genetic variances summed over all participating loci. These covariances can be expressed collectively in matrix notation as Var(X) = 2σ ^su2_inα Φ + σ ^su2_ind Δ7. 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.