# Balding–Nichols model

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Parameters Probability density function Cumulative distribution function ${\displaystyle 0(real) ${\displaystyle 0 (real) For ease of notation, let ${\displaystyle \alpha ={\tfrac {1-F}{F}}p}$, and ${\displaystyle \beta ={\tfrac {1-F}{F}}(1-p)}$ ${\displaystyle x\in (0;1)\!}$ ${\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!}$ ${\displaystyle I_{x}(\alpha ,\beta )\!}$ ${\displaystyle p\!}$ ${\displaystyle I_{0.5}^{-1}(\alpha ,\beta )}$ no closed form ${\displaystyle {\frac {F-(1-F)p}{3F-1}}}$ ${\displaystyle Fp(1-p)\!}$ ${\displaystyle {\frac {2F(1-2p)}{(1+F){\sqrt {F(1-p)p}}}}}$ ${\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{{\frac {1-F}{F}}+r}}\right){\frac {t^{k}}{k!}}}$ ${\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!}$

In population genetics, the Balding–Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population.[1] With background allele frequency p the allele frequencies, in sub-populations separated by Wright's FST F, are distributed according to independent draws from

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

where B is the Beta distribution. This distribution has mean p and variance Fp(1 – p).[2]

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.

${\displaystyle \left\{F(x-1)xf'(x)+f(x)(F(-p)+3Fx-F+p-x)=0\right\}}$

## References

1. ^ Balding, DJ; Nichols, RA (1995). "A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity.". Genetica. Springer. 96: 3–12. doi:10.1007/BF01441146. PMID 7607457.
2. ^ 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.