TDist {stats}

R Documentation

The Student t Distribution

Description

Density, distribution function, quantile function and random generation for the t distribution with df degrees of freedom (and optional non-centrality 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 non-integer). df = Inf is allowed.
ncp	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).
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.

Note

Setting ncp = 0 is not equivalent to omitting ncp. R uses the non-centrality functionality whenever ncp is specified which provides continuous behavior at ncp = 0.

Source

The central dt is computed via an accurate formula provided by Catherine Loader (see the reference in dbinom).

For the non-central case of dt, 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 non-central case of pt based on a C translation of

Lenth, R. V. (1989). Algorithm AS 243 — Cumulative distribution function of the non-central t distribution, Applied Statistics 38, 185–189.

For central qt, a C translation of

Hill, G. W. (1970) Algorithm 396: Student's t-quantiles. 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 non-central 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 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

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, 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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