Balding–Nichols model

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

${\displaystyle B\left({\frac {1-F}{F}}p,{\frac {1-F}{F}}(1-p)\right)}$
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 ${\displaystyle {\frac {F-(1-F)p}{3F-1}}}$ ${\displaystyle Fp(1-p)\!}$ ${\displaystyle {\frac {2F(1-2p)}{(1+F){\sqrt {F(1-p)p}}}}}$ ${\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{{\frac {1-F}{F}}+r}}\right){\frac {t^{k}}{k!}}}$ ${\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!}$

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.

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. Archived from the original (PDF) on 2008-07-03. Retrieved 2009-02-19.