The Student t Distribution
Description
Density, distribution function, quantile function and random
generation for the t distribution with df
degrees of freedom
(and optional noncentrality parameter ncp
).
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 noninteger). df
= Inf is allowed.

ncp 
noncentrality 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).

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 $\frac{\nu}{\nu2}$
(for $\nu > 2$
).
The general noncentral $t$
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 noncentral $t$
with
df
${} = n1$
degrees of freedom and noncentrality parameter
ncp
${} = (\mu  \mu_0) \sqrt{n}/\sigma$
.
The $t$
distribution's cumulative distribution function (cdf),
$F_{\nu}$
fulfills
$F_{\nu}(t) = \frac 1 2 I_x(\frac{\nu}{2}, \frac 1 2),$
for $t \le 0$
, and
$F_{\nu}(t) = 1 \frac 1 2 I_x(\frac{\nu}{2}, \frac 1 2),$
for $t \ge 0$
,
where
$x := \nu/(\nu + t^2)$
, and $I_x(a,b)$
is the
incomplete beta function, in R this is pbeta(x, a,b)
.
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 noncentral
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 nonzero 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
The central dt
is computed via an accurate formula
provided by Catherine Loader (see the reference in dbinom
).
For the noncentral case of dt
, C code contributed by
Claus Ekstrøm based on the relationship (for
$x \neq 0$
) to the cumulative distribution.
For the central case of pt
, a normal approximation in the
tails, otherwise via pbeta
.
For the noncentral case of pt
based on a C translation of
Lenth, R. V. (1989). Algorithm AS 243 —
Cumulative distribution function of the noncentral $t$
distribution,
Applied Statistics 38, 185–189.
This computes the lower tail only, so the upper tail suffers from
cancellation and a warning will be given when this is likely to be
significant.
For central qt
, a C translation of
Hill, G. W. (1970) Algorithm 396: Student's tquantiles.
Communications of the ACM, 13(10), 619–620.
altered to take account of
Hill, G. W. (1981) Remark on Algorithm 396, ACM Transactions on
Mathematical Software, 7, 250–1.
The noncentral case is done by inversion.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
The New S Language.
Wadsworth & Brooks/Cole. (Except noncentral 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
require(graphics)
1  pt(1:5, df = 1)
qt(.975, df = c(1:10,20,50,100,1000))
tt < seq(0, 10, length.out = 21)
ncp < seq(0, 6, length.out = 31)
ptn < outer(tt, ncp, function(t, d) pt(t, df = 3, ncp = d))
t.tit < "Noncentral 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 = "noncentrality parameter",
zlab = "Pr(T <= t)")
plot(function(x) dt(x, df = 3, ncp = 2), 3, 11, ylim = c(0, 0.32),
main = "Noncentral t  Density", yaxs = "i")
ptBet < function(t, n) {
x < n/(n + t^2)
r < pb < pbeta(x, n/2, 1/2) / 2
pos < t > 0
r[pos] < 1  pb[pos]
r
}
x < seq(5, 5, by = 1/8)
nu < 3:10
pt. < outer(x, nu, pt)
ptB < outer(x, nu, ptBet)
stopifnot(all.equal(pt., ptB, tolerance = 1e15))