The F Distribution

Description

Density, distribution function, quantile function and random generation for the F distribution with df1 and df2 degrees of freedom (and optional non-centrality parameter ncp).

Usage

df(x, df1, df2, ncp, log = FALSE)
pf(q, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
qf(p, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
rf(n, df1, df2, ncp)

Arguments

Details

The F distribution with df1 = \nu_1 and df2 = \nu_2 degrees of freedom has density

f(x) = \frac{\Gamma(\nu_1/2 + \nu_2/2)}{\Gamma(\nu_1/2)\Gamma(\nu_2/2)} \left(\frac{\nu_1}{\nu_2}\right)^{\nu_1/2} x^{\nu_1/2 -1} \left(1 + \frac{\nu_1 x}{\nu_2}\right)^{-(\nu_1 + \nu_2) / 2}%

for x > 0.

The F distribution's cumulative distribution function (cdf), F_{\nu_1,\nu_2} fulfills (Abramowitz & Stegun 26.6.2, p.946) F_{\nu_1,\nu_2}(qF) = 1 - I_x(\nu_2/2, \nu_1/2) = I_{1-x}(\nu_1/2, \nu_2/2), where x := \frac{\nu_2}{\nu_2 + \nu_1*qF}, and I_x(a,b) is the incomplete beta function; in R, = pbeta(x, a,b).

It is the distribution of the ratio of the mean squares of \nu_1 and \nu_2 independent standard normals, and hence of the ratio of two independent chi-squared variates each divided by its degrees of freedom. Since the ratio of a normal and the root mean-square of m independent normals has a Student's t_m distribution, the square of a t_m variate has a F distribution on 1 and m degrees of freedom.

The non-central F distribution is again the ratio of mean squares of independent normals of unit variance, but those in the numerator are allowed to have non-zero means and ncp is the sum of squares of the means. See Chisquare for further details on non-central distributions.

Value

df gives the density, pf gives the distribution function qf gives the quantile function, and rf generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is determined by n for rf, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Note

Supplying ncp = 0 uses the algorithm for the non-central distribution, which is not the same algorithm used if ncp is omitted. This is to give consistent behaviour in extreme cases with values of ncp very near zero.

The code for non-zero ncp is principally intended to be used for moderate values of ncp: it will not be highly accurate, especially in the tails, for large values.

Source

For the central case of df, computed via a binomial probability, code contributed by Catherine Loader (see dbinom); for the non-central case computed via dbeta, code contributed by Peter Ruckdeschel.

For pf, via pbeta (or for large df2, via pchisq).

For qf, via qchisq for large df2, else via qbeta.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapters 27 and 30. Wiley, New York.

See Also

Distributions for other standard distributions, including dchisq for chi-squared and dt for Student's t distributions.

Examples

## Equivalence of pt(.,nu) with pf(.^2, 1,nu):
x <- seq(0.001, 5, length.out = 100)
nu <- 4
stopifnot(all.equal(2*pt(x,nu) - 1, pf(x^2, 1,nu)),
          ## upper tails:
 	  all.equal(2*pt(x,     nu, lower.tail=FALSE),
		      pf(x^2, 1,nu, lower.tail=FALSE)))

## the density of the square of a t_m is 2*dt(x, m)/(2*x)
# check this is the same as the density of F_{1,m}
all.equal(df(x^2, 1, 5), dt(x, 5)/x)

## Identity (F <-> t):  qf(2*p - 1, 1, df) == qt(p, df)^2  for  p >= 1/2
p <- seq(1/2, .99, length.out = 50); df <- 10
rel.err <- function(x, y) ifelse(x == y, 0, abs(x-y)/mean(abs(c(x,y))))
stopifnot(all.equal(qf(2*p - 1, df1 = 1, df2 = df),
                    qt(p, df)^2))

## Identity (F <-> Beta <-> incompl.beta):
n1 <- 7 ; n2 <- 12; qF <- c((0:4)/4, 1.5, 2:16)
x <- n2/(n2 + n1*qF)
stopifnot(all.equal(pf(qF, n1, n2, lower.tail=FALSE),
                    pbeta(x, n2/2, n1/2)))