Balding–Nichols model

Model in population genetics

Balding-Nichols
Probability density function
Cumulative distribution function
Parameters	$0<F<1$ {\displaystyle 0<F<1}(real) $0<p<1$ {\displaystyle 0<p<1} (real) For ease of notation, let $\alpha ={\tfrac {1-F}{F}}p$ {\displaystyle \alpha ={\tfrac {1-F}{F}}p}, and $\beta ={\tfrac {1-F}{F}}(1-p)$ {\displaystyle \beta ={\tfrac {1-F}{F}}(1-p)}
Support	$x\in (0;1)\!$ {\displaystyle x\in (0;1)\!}
PDF	${\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!$ {\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!}
CDF	$I_{x}(\alpha ,\beta )\!$ {\displaystyle I_{x}(\alpha ,\beta )\!}
Mean	$p\!$ {\displaystyle p\!}
Median	$I_{0.5}^{-1}(\alpha ,\beta )$ {\displaystyle I_{0.5}^{-1}(\alpha ,\beta )} no closed form
Mode	${\frac {F-(1-F)p}{3F-1}}$ {\displaystyle {\frac {F-(1-F)p}{3F-1}}}
Variance	$Fp(1-p)\!$ {\displaystyle Fp(1-p)\!}
Skewness	${\frac {2F(1-2p)}{(1+F){\sqrt {F(1-p)p}}}}$ {\displaystyle {\frac {2F(1-2p)}{(1+F){\sqrt {F(1-p)p}}}}}
MGF	$1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{{\frac {1-F}{F}}+r}}\right){\frac {t^{k}}{k!}}$ {\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!}}}
CF	${}_{1}F_{1}(\alpha ;\alpha +\beta ;i,円t)\!$ {\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i,円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 F_ST F, are distributed according to independent draws from

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

{\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.

References

[edit ]

^ 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. 96 (1–2). Springer: 3–12. doi:10.1007/BF01441146. PMID 7607457. S2CID 30680826.
^ 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. S2CID 8127858. Archived from the original (PDF) on 2008年07月03日. Retrieved 2009年02月19日.

v t e Population genetics
Key concepts	Hardy–Weinberg principle Genetic linkage Identity by descent Linkage disequilibrium Fisher's fundamental theorem Neutral theory Shifting balance theory Price equation Coefficient of inbreeding Coefficient of relationship Selection coefficient Fitness Heritability Population structure Constructive neutral evolution
Selection	Natural Artificial Sexual Ecological
Effects of selection on genomic variation	Genetic hitchhiking Background selection
Genetic drift	Small population size Population bottleneck Founder effect Coalescence Balding–Nichols model
Founders	R. A. Fisher J. B. S. Haldane Sewall Wright
Related topics	Biogeography Evolution Evolutionary game theory Fitness landscape Genetic genealogy Landscape genetics and genomics Microevolution Population genomics Phylogeography Quantitative genetics
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Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda