stats (version 3.3)

TDist: 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<="" code="">. 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, $P[X > x]$.

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.

encoding

UTF-8

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, C code contributed by Claus Ekstrøm{Ekstroem} 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.

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 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.

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${} = (\mu - \mu_0) \sqrt{n}/\sigma$.

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

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")