PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / NON CENTRAL F

Non Central F distribution

What it represents

The noncentral F is a ratio of a noncentral and a central chi-square, each divided by its degrees of freedom. lambda represents an effect under an alternative.

Historical clue

It was developed for F statistics when the null is false. Its central role is therefore power analysis, not fitting any positive ratio.

Relationships that clarify its use

It links to noncentral chi-square and noncentral t: the square of a noncentral t with one numerator degree of freedom gives a noncentral F under the corresponding parameterization.

Data examples

  • ANOVA and regression-test power
  • signal detection and energy comparison with a present signal

Modelling warning

Do not read lambda as location or replace noncentral F by central F in power calculations.

The F distribution when the effect is not zero

Under a null hypothesis, the numerator of an F statistic leads to central Chi Square. Under an alternative with systematic signal, it becomes noncentral and the ratio follows Noncentral F. The parameter lambda combines effect with sample size, design, and error scale.

This makes the family the natural tool for power in ANOVA and regression. There is no universal lambda for a given substantive effect because design determines how it enters the statistic. At lambda=0, central F returns. Estimating noncentrality from one observed statistic can yield highly asymmetric intervals; for planning, deriving it from a proposed effect and design is more transparent.

Decision guide

A good candidate when: a ratio of mean squares includes an alternative with nonzero effect, especially in power calculations.

Compare it with: central F under the null and simulation when the design violates independence or homoscedasticity. Noncentrality depends on design, not only observed data.

References

Non Central F Distribution: equations and calculator

Distribution defintion

X∼NonCentralF(λ,n1,n2)X\sim\mathrm{NonCentralF}\left(\lambda,n_{1},n_{2}\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

λ∈R+,n1∈R+,n2∈R+\lambda\in\mathbb{R}^{+},n_{1}\in\mathbb{R}^{+},n_{2}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=∑j=0∞((12λ)jj!e−λ/2)In1x/(n2+n1x)(n12+j,n22)F_{X}\left(x\right)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\lambda/2}\right)I_{n_1x/(n_2+n_1x)}\left(\frac{n_1}{2}+j,\frac{n_2}{2}\right)

Probability density function

fX(x)=∑k=0∞e−λ/2(λ/2)kBeta(n22,n12+k)k!(n1n2)n12+k(n2n2+n1x)n1+n22+kxn1/2−1+kf_{X}\left(x\right)=\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ \text{Beta}\left(\frac{n_2}{2},\frac{n_1}{2}+k\right) k!}\left(\frac{n_1}{n_2}\right)^{\frac{n_1}{2}+k}\left(\frac{n_2}{n_2+n_1x}\right)^{\frac{n_1+n_2}{2}+k}x^{n_1/2-1+k}

Percent point function/Sample

SampleX=(∑i=1n1(λn1+Φ−1(ui))2)/n1(2P−1(n22,u))/n2\text{Sample}_{X}=\frac{\left(\sum_{i=1}^{n_1}\left(\sqrt{\frac{\lambda}{n_1}}+\Phi^{-1}\left(u_{i}\right)\right)^{2}\right)/n_1}{\left(2\text{P}^{-1}\left(\frac{n_2}{2},u\right)\right)/n_2}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=e−λ/2(n1n2)kΓ(n1/2−k)Γ(n1/2)∑r=0∞(1r!)(λ2)rΓ(n12+r+k)Γ(n12+r)\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=e^{-\lambda/2}\left(\frac{n1}{n2}\right)^{k}\frac{\Gamma\left(n_1/2-k\right)}{\Gamma\left(n_1/2\right)}\sum_{r=0}^{\infty }\left(\frac{1}{r!}\right)\left(\frac{\lambda}{2}\right)^{r}\frac{\Gamma\left(\frac{n_1}{2}+r+k\right)}{\Gamma\left(\frac{n_1}{2}+r\right)}

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=FX−1(12)\mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right)

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
ui:Uniform[0,1] random varibleu_{i}:\text{Uniform[0,1] random varible}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}