TDist
The Student t Distribution
Density, distribution function, quantile function and random
generation for the t distribution with df degrees of freedom
(and optional non-centrality parameter ncp).
- Keywords
- distribution
Usage
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)Arguments
- x, q
vector of quantiles.
- p
vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.- df
degrees of freedom (\(> 0\), maybe non-integer).
df = Infis allowed.- ncp
non-centrality parameter \(\delta\); currently except for
rt(), only forabs(ncp) <= 37.62. If omitted, use the central t distribution.- log, log.p
logical; if TRUE, probabilities p are given as log(p).
- lower.tail
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
Details
The \(t\) distribution with df \(= \nu\) degrees of
freedom has density
$$
f(x) = \frac{\Gamma ((\nu+1)/2)}{\sqrt{\pi \nu} \Gamma (\nu/2)}
(1 + x^2/\nu)^{-(\nu+1)/2}%
$$
for all real \(x\).
It has mean \(0\) (for \(\nu > 1\)) and
variance \(\frac{\nu}{\nu-2}\) (for \(\nu > 2\)).
The general non-central \(t\)
with parameters \((\nu, \delta)\) = (df, ncp)
is defined as the distribution of
\(T_{\nu}(\delta) := (U + \delta)/\sqrt{V/\nu}\)
where \(U\) and \(V\) are independent random
variables, \(U \sim {\cal N}(0,1)\) and
\(V \sim \chi^2_\nu\) (see Chisquare).
The most used applications are power calculations for \(t\)-tests:
Let \(T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}}\)
where
\(\bar{X}\) is the mean and \(S\) the sample standard
deviation (sd) of \(X_1, X_2, \dots, X_n\) which are
i.i.d. \({\cal N}(\mu, \sigma^2)\)
Then \(T\) is distributed as non-central \(t\) with
df\({} = n-1\)
degrees of freedom and non-centrality parameter
ncp\({} = (\mu - \mu_0) \sqrt{n}/\sigma\).
Value
dt gives the density,
pt gives the distribution function,
qt gives the quantile function, and
rt generates random deviates.
Invalid arguments will result in return value NaN, with a warning.
The length of the result is determined by n for
rt, 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.
References
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.
See Also
Distributions for other standard distributions, including
df for the F distribution.
Examples
library(stats)
# NOT RUN {
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")
# }