| TDist {base} | R Documentation |
The Student t Distribution
Description
These functions provide information about the t distribution with
df degrees of freedom (and optional noncentrality parameter
ncp). dt gives the density, pt
gives the distribution function, qt gives the quantile function
and rt generates random deviates.
Usage
dt(x, df)
pt(q, df, ncp=0)
qt(p, df)
rt(n, df)
Arguments
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations to generate. |
df |
degrees of freedom ( |
ncp |
non-centrality parameter |
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 a the distribution of
T_{\nu}(\delta) := \frac{U + \delta}{\chi_{\nu}/\sqrt{\nu}}
where U and \chi_{\nu} are independent random
variables, U \sim {\cal N}(0,1), and
\chi^2_\nu
is chi-squared, see pchisq.
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.
N(\mu,\sigma^2).
Then T is distributed as non-centrally t with
df= n-1
degrees of freedom and non-centrality parameter
ncp= \mu - \mu_0.
References
Lenth, R. V. (1989). Algorithm AS 243 –
Cumulative distribution function of the non-central t distribution,
Appl.\ Statist. 38, 185–189.
See Also
df for the F distribution.
Examples
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))
image(tt,ncp,ptn, zlim=c(0,1),main=t.tit <- "Non-central t - Probabilities")
persp(tt,ncp,ptn, zlim=0:1, r=2, phi=20, theta=200, main=t.tit)