TDist {stats} | R Documentation |

Density, distribution function, quantile function and random
generation for the t distribution with `df`

degrees of freedom
(and optional non-centrality parameter `ncp`

).

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)

`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. For `qt` only values of at least one are
currently supported. |

`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 <= x], otherwise, P[X > x]. |

The *t* distribution with `df`

*= n* degrees of
freedom has density

*f(x) = Gamma((n+1)/2) / (sqrt(n pi) Gamma(n/2)) (1 + x^2/n)^-((n+1)/2)*

for all real *x*.
It has mean *0* (for *n > 1*) and
variance *n/(n-2)* (for *n > 2*).

The general *non-central* *t*
with parameters *(df,Del)* `= (df, ncp)`

is defined as the distribution of
*T(df, Del) := (U + Del) / (Chi(df) / sqrt(df)) *
where *U* and *Chi(df)* are independent random
variables, *U ~ N(0,1)*, and
*Chi(df)^2*
is chi-squared, see Chisquare.

The most used applications are power calculations for *t*-tests:

Let *T= (mX - m0) / (S/sqrt(n))*
where
*mX* is the `mean`

and *S* the sample standard
deviation (`sd`

) of *X_1,X_2,...,X_n* which are i.i.d.
*N(mu,sigma^2)*.
Then *T* is distributed as non-centrally *t* with
`df`

*= n-1*
degrees of freedom and **n**on-**c**entrality **p**arameter
`ncp`

*= (mu - m0) * sqrt(n)/sigma*.

`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.

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*.

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 != 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.

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.

`df`

for the F distribution.

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)) image(tt,ncp,ptn, zlim=c(0,1),main=t.tit <- "Non-central t - Probabilities") 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")

[Package *stats* version 2.5.0 Index]