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.


Gamete Contribution Ridge Count Polygenic Model Observe Information Matrix Additive Genetic Correlation 
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© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Kenneth Lange
    • 1
  1. 1.Department of Biostatistics and MathematicsUniversity of MichiganAnn ArborUSA

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