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.

vector of probabilities.

number of observations. If length(n) > 1, the length is taken to be the number required.

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, $P[X > x]$.

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.

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.

Examples

Run this code

# NOT RUN {
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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