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

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.

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 non-central 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 non-zero 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 non-central 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 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; doi:10.2307/2347693 .

This computes the lower tail only, so the upper tail currently 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; doi:10.1145/355598.355599 .

altered to take account of

Hill, G. W. (1981) Remark on Algorithm 396, ACM Transactions on Mathematical Software, 7, 250–1; doi:10.1145/355945.355956 .

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

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 <- "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")

## Relation between F_t(.) = pt(x, n) and pbeta():
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)
## matplot(x, pt., type = "l")
stopifnot(all.equal(pt., ptB, tolerance = 1e-15))