# 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 . 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.

## 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 50:181–194
2. 
Cannings C, Thompson EA, Skolnick MH (1978) Probability functions on complex pedigrees. Adv Appl Prob 10:26–61
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 539:1–38
5. 
Elston RC, Stewart J (1971) A general model for the genetic analysis of pedigree data. Hum Hered 21:523–542
6. 
Falconer DS (1965) The inheritance of liability to certain diseases, estimated from the incidences among relatives. Ann Hum Genet 29:51–79
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 88:573–580
9. 
Fisher RA (1918) The correlation between relatives on the supposition of Mendelian inheritance. Trans Roy Soc Edinburgh 52:399–433
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.
12. 
Jennrich RJ, Sampson PF (1976) Newton-Raphson and related algorithms for maximum likelihood variance component estimation. Technometrics 18:11 – 17
13. 
Lange K (1978) Central limit theorems for pedigrees. J Math Biol 6:59–66
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 14:513–524
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 39:484–491
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 York
21. 
Rao CR (1973) Linear Statistical Inference and its Applications, 2nd ed. Wiley, New York
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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