Density, distribution function, quantile function and random
generation for the t distribution with df degrees of freedom
(and optional non-centrality parameter ncp).
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)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
= Inf is allowed.
non-centrality parameter \(\delta\);
currently except for rt(), only for abs(ncp) <= 37.62.
If omitted, use the central t distribution.
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
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.
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\({} = (\mu - \mu_0) \sqrt{n}/\sigma\).
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.
Distributions for other standard distributions, including
df for the F distribution.
# 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")
# }
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