Balding–Nichols model

{{Short description|Model in population genetics}}

{{Probability distribution

| name = Balding-Nichols

| type = density

| pdf_image = 352px

| cdf_image = 352px

| parameters = (real)
(real)
For ease of notation, let
$\backslash alpha=\backslash tfrac\{1-F\}\{F\}p$, and
$\backslash beta=\backslash tfrac\{1-F\}\{F\}(1-p)$

| support = $x\; \backslash in\; (0;\; 1)\backslash !$

| pdf = $\backslash frac\{x^\{\backslash alpha-1\}(1-x)^\{\backslash beta-1\}\}\; \{\backslash mathrm\{B\}(\backslash alpha,\backslash beta)\}\backslash !$

| cdf = $I\_x(\backslash alpha,\backslash beta)\backslash !$

| mean = $p\backslash !$

| median = $I\_\{0.5\}^\{-1\}(\backslash alpha,\backslash beta)$ no closed form

| mode = $\backslash frac\{F-(1-F)p\}\{3F-1\}$

| variance = $Fp(1-p)\backslash !$

| skewness = $\backslash frac\{2F(1-2p)\}\{(1+F)\backslash sqrt\{F(1-p)p\}\}$

| kurtosis =

| entropy =

| mgf = $1\; +\backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash left(\; \backslash prod\_\{r=0\}^\{k-1\}\; \backslash frac\{\backslash alpha+r\}\{\backslash frac\{1-F\}\{F\}+r\}\backslash right)\; \backslash frac\{t^k\}\{k!\}$

| char = $\{\}\_1F\_1(\backslash alpha;\; \backslash alpha+\backslash beta;\; i\backslash ,t)\backslash !$

}}

In population genetics, the Balding–Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population.{{cite journal |last1=Balding |first1=DJ |last2=Nichols |first2=RA |year=1995 |title=A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity. |journal=Genetica |volume=96 |issue=1–2 |pages=3–12 |publisher=Springer |doi=10.1007/BF01441146 |pmid=7607457 |s2cid=30680826 }} 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

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

where B is the Beta distribution. This distribution has mean p and variance Fp(1 – p).{{cite journal |author1=Alkes L. Price |author2=Nick J. Patterson |author3=Robert M. Plenge |author4=Michael E. Weinblatt |author5=Nancy A. Shadick |author6=David Reich |year=2006 |title=Principal components analysis corrects for stratification in genome-wide association studies |journal=Nature Genetics |volume=38 |issue=8 |pages=904–909 |doi=10.1038/ng1847 |url=http://genepath.med.harvard.edu/~reich/Price%20et%20al.pdf |pmid=16862161 |s2cid=8127858 |access-date=2009-02-19 |archive-url=https://web.archive.org/web/20080703180251/http://genepath.med.harvard.edu/~reich/Price%20et%20al.pdf |archive-date=2008-07-03 |url-status=dead }}

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.