## 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 where σ

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

_{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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## References

- [1]Boerwinkle E, Chakraborty R, Sing CF (1986) The use of measured genotype information in the analysis of quantitative phenotypes in man. I. Models and analytical methods.
*Ann Hum Genet*50:181–194zbMATHCrossRefGoogle Scholar - [2]Cannings C, Thompson EA, Skolnick MH (1978) Probability functions on complex pedigrees.
*Adv Appl Prob*10:26–61zbMATHCrossRefMathSciNetGoogle Scholar - [3]Daiger SP, Miller M, Chakraborty R (1984) Heritability of quantitative variation at the group-specific component (Gc) locus.
*Amer J Hum Genet*36:663–676Google Scholar - [4]Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood estimation with incomplete data via the EM algorithm (with discussion).
*J Roy Stat Soc*539:1–38MathSciNetGoogle Scholar - [5]Elston RC, Stewart J (1971) A general model for the genetic analysis of pedigree data.
*Hum Hered*21:523–542CrossRefGoogle Scholar - [6]Falconer DS (1965) The inheritance of liability to certain diseases, estimated from the incidences among relatives.
*Ann Hum Genet*29:51–79CrossRefGoogle Scholar - [7]Falconer DS (1967) The inheritance of liability to diseases with variable age of onset, with particular reference to diabetes mellitus.
*Ann Hum Genet*31:1–20Google Scholar - [8]Fernando RL, Stricker C, Elston RC (1994) The finite polygenic mixed model: An alternative formulation for the mixed model of inheritance.
*Theor Appl Genet*88:573–580CrossRefGoogle Scholar - [9]Fisher RA (1918) The correlation between relatives on the supposition of Mendelian inheritance.
*Trans Roy Soc Edinburgh*52:399–433CrossRefGoogle Scholar - [10]Holt SB (1954) Genetics of dermal ridges: Bilateral asymmetry in finger ridge-counts.
*Ann Eugenics*18:211–231Google Scholar - [11]Hopper JL, Mathews JD (1982) Extensions to multivariate normal models for pedigree analysis.
*Ann Hum Genet*46:373–383.zbMATHCrossRefGoogle Scholar - [12]Jennrich RJ, Sampson PF (1976) Newton-Raphson and related algorithms for maximum likelihood variance component estimation.
*Technometrics*18:11 – 17zbMATHCrossRefMathSciNetGoogle Scholar - [13]Lange K (1978) Central limit theorems for pedigrees.
*J Math Biol*6:59–66zbMATHCrossRefMathSciNetGoogle Scholar - [14]Lange K (1997) An approximate model of polygenic inheritance.Google Scholar
- [15]Lange K, Boehnke M (1983) Extensions to pedigree analysis. IV. Covariance component models for multivariate traits.
*Amer J Med Genet*14:513–524CrossRefGoogle Scholar - [16]Lange K, Boehnke M, Weeks D (1987) Programs for pedigree analysis: MENDEL, FISHER, and dGENE.
*Genet Epidemiology*5:473–476Google Scholar - [17]Lange K, Westlake J, Spence MA (1976) Extensions to pedigree analysis. III. Variance components by the scoring method.
*Ann Hum Genet*39:484–491CrossRefGoogle Scholar - [18]Morton NE, MacLean CJ (1974) Analysis of family resemblance. III. Complex segregation analysis of quantitative traits.
*Amer J Hum Genet*26:489–503Google Scholar - [19]Ott J (1979) Maximum likelihood estimation by counting methods under polygenic and mixed models in human pedigrees.
*Amer J Hum Genet*31:161–175Google Scholar - [20]Peressini AL, Sullivan FE, Uhl JJ Jr (1988)
*The Mathematics of Nonlinear Programming.*Springer-Verlag, New YorkzbMATHCrossRefGoogle Scholar - [21]Rao CR (1973)
*Linear Statistical Inference and its Applications*, 2nd ed. Wiley, New YorkzbMATHCrossRefGoogle Scholar - [22]Strieker C, Fernando RL, Elston RC (1995) Linkage analysis with an alternative formulation for the mixed model of inheritance: The finite polygenic mixed model.
*Genetics*141:1651–1656Google Scholar

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