# The Polygenic Model

• Kenneth Lange
Chapter
Part of the Statistics for Biology and Health book series (SBH)

## 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 i k 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 a 2 and σ d 2 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.

## Keywords

Gamete Contribution Ridge Count Polygenic Model Observe Information Matrix Additive Genetic Correlation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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