# Balding–Nichols model

(Redirected from Balding–Nichols distribution)
Parameters Probability density function Cumulative distribution function $0 < F < 1$(real) $0< p < 1$ (real) For ease of notation, let $\alpha=\tfrac{1-F}{F}p$, and $\beta=\tfrac{1-F}{F}(1-p)$ $x \in (0; 1)\!$ $\frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\!$ $I_x(\alpha,\beta)\!$ $p\!$ $I_{0.5}^{-1}(\alpha,\beta)$ no closed form $\frac{F-(1-F)p}{3F-1}$ $Fp(1-p)\!$ $\frac{2F(1-2p)}{(1+F)\sqrt{F(1-p)p}}$ $1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\frac{1-F}{F}+r}\right) \frac{t^k}{k!}$ ${}_1F_1(\alpha; \alpha+\beta; i\,t)\!$

In population genetics, the Balding–Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population.[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

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

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.

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

## 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.