Balding–Nichols model

Balding-Nichols
Probability density function

Cumulative distribution function

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.[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.


Differential equation

\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.