Density, distribution function, quantile function and random generation for the t distribution with
df degrees of freedom (and optional non-centrality parameter
dt(x, df, ncp, log = FALSE) pt(q, df, ncp, lower.tail = TRUE, log.p = FALSE) qt(p, df, ncp, lower.tail = TRUE, log.p = FALSE) rt(n, df, ncp)
- x, q
- vector of quantiles.
- vector of probabilities.
- number of observations. If
length(n) > 1, the length is taken to be the number required.
- degrees of freedom (> 0, maybe non-integer).
df = Infis allowed.
- non-centrality parameter delta; currently except for
rt(), only for
abs(ncp) <= 37.62. If omitted, use the central t distribution.
- log, log.p
- logical; if TRUE, probabilities p are given as log(p).
- logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].
The t distribution with
df = n degrees of freedom has density f(x) = Γ((n+1)/2) / (√(n π) Γ(n/2)) (1 + x^2/n)^-((n+1)/2) for all real x. It has mean 0 (for n > 1) and variance n/(n-2) (for n > 2).
The general non-central t with parameters (df, Del)
= (df, ncp) is defined as the distribution of T(df, Del) := (U + Del) / √(V/df) where U and V are independent random variables, U ~ N(0,1) and V ~ χ^2(df) (see Chisquare).
The most used applications are power calculations for t-tests:
Let T= (mX - m0) / (S/sqrt(n)) where mX is the
mean and S the sample standard deviation (
sd) of X_1, X_2, ..., X_n which are i.i.d. N(μ, σ^2) Then T is distributed as non-central t with
df= n - 1 degrees of freedom and non-centrality parameter
ncp = (μ - m0) * sqrt(n)/σ.
Invalid arguments will result in return value
NaN, with a warning.
The length of the result is determined by
rt, and is the maximum of the lengths of the numerical parameters for the other functions. The numerical parameters other than
n are recycled to the length of the result. Only the first elements of the logical parameters are used.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (Except non-central versions.)
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapters 28 and 31. Wiley, New York.
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.
Distributions for other standard distributions, including
df for the F distribution.
require(graphics) 1 - pt(1:5, df = 1) qt(.975, df = c(1:10,20,50,100,1000)) tt <- seq(0, 10, len = 21) ncp <- seq(0, 6, len = 31) ptn <- outer(tt, ncp, function(t, d) pt(t, df = 3, ncp = d)) t.tit <- "Non-central t - Probabilities" image(tt, ncp, ptn, zlim = c(0,1), main = t.tit) persp(tt, ncp, ptn, zlim = 0:1, r = 2, phi = 20, theta = 200, main = t.tit, xlab = "t", ylab = "non-centrality parameter", zlab = "Pr(T <= t)") plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32), main = "Non-central t - Density", yaxs = "i")
Documentation reproduced from R 3.0.2. License: GPL-2.