Balding–Nichols model

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Balding-Nichols
parameters:	$0 < F < 1$ (real) $0 < p < 1$ (real) For ease of notation, let $\alpha=\tfrac{1-F}{F}p$ , and $\beta=\tfrac{1-F}{F}(1-p)$
support:	$x \in (0; 1)\!$
pdf:	$\frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\!$
cdf:	$I_x(\alpha,\beta)\!$
mean:	$p\!$
median:	$I_{0.5}^{-1}(\alpha,\beta)$ no closed form
mode:	$\frac{F-(1-F)p}{3F-1}$
variance:	$Fp(1-p)\!$
skewness:	$\frac{2F(1-2p)}{(1+F)\sqrt{F(1-p)p}}$
mgf:	$1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\frac{1-F}{F}+r}\right) \frac{t^k}{k!}$
cf:	${}_1F_1(\alpha; \alpha+\beta; i\,t)\!$

In population genetics, the Balding–Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population. With background allele frequency p the allele frequencies, in sub-populations separated by Wright's F_ST F, are distributed according to independent draws from

$B\left(\frac{1-F}{F}p,\frac{1-F}{F}(1-p)\right)$

where B is the Beta distribution. This distribution has mean p and variance Fp(1 – p).^[1]

The model is due to David Balding and Richard Nichols and is widely used in the forensic analysis of DNA profiles and in population models for genetic epidemiology.

References

^ Alkes L. Price, Nick J. Patterson, Robert M. Plenge, Michael E. Weinblatt, Nancy A. Shadick & David Reich (2006). "Principal components analysis corrects for stratification in genome-wide association studies" (PDF). Nature Genetics 38 (8): 904–909. doi:10.1038/ng1847. PMID 16862161. http://genepath.med.harvard.edu/~reich/Price%20et%20al.pdf.

v · d · eTopics in population genetics

Key concepts	Hardy-Weinberg law · Genetic linkage · Linkage disequilibrium · Fisher's fundamental theorem · Neutral theory · Price equation

Selection	Natural · Sexual · Artificial · Ecological

Effects of selection on genomic variation	Genetic hitchhiking · Background selection

Genetic drift	Small population size · Population bottleneck · Founder effect · Coalescence · Balding–Nichols model

Founders	R. A. Fisher · J. B. S. Haldane · Sewall Wright

Related topics	Evolution · Microevolution · Evolutionary game theory · Fitness landscape · Genetic genealogy

List of evolutionary biology topics

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Balding–Nichols model

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