Tukey {stats} | R Documentation |

## The Studentized Range Distribution

### Description

Functions of the distribution of the studentized range, *R/s*,
where *R* is the range of a standard normal sample and
*df*s^2* is independently distributed as
chi-squared with *df* degrees of freedom, see `pchisq`

.

### Usage

ptukey(q, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
qtukey(p, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)

### Arguments

`q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`nmeans` |
sample size for range (same for each group). |

`df` |
degrees of freedom for *s* (see below). |

`nranges` |
number of *groups* whose **maximum** range is
considered. |

`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]*. |

### Details

If *ng =*`nranges`

is greater than one, *R* is
the *maximum* of *ng* groups of `nmeans`

observations each.

### Value

`ptukey`

gives the distribution function and `qtukey`

its
inverse, the quantile function.

### Note

A Legendre 16-point formula is used for the integral of `ptukey`

.
The computations are relatively expensive, especially for
`qtukey`

which uses a simple secant method for finding the
inverse of `ptukey`

.
`qtukey`

will be accurate to the 4th decimal place.

### References

Copenhaver, Margaret Diponzio and Holland, Burt S. (1988)
Multiple comparisons of simple effects in
the two-way analysis of variance with fixed effects.
*Journal of Statistical Computation and Simulation*, **30**, 1–15.

### See Also

`pnorm`

and `qnorm`

for the corresponding
functions for the normal distribution.

### Examples

if(interactive())
curve(ptukey(x, nm=6, df=5), from=-1, to=8, n=101)
(ptt <- ptukey(0:10, 2, df= 5))
(qtt <- qtukey(.95, 2, df= 2:11))
## The precision may be not much more than about 8 digits:
summary(abs(.95 - ptukey(qtt,2, df = 2:11)))

[Package

*stats* version 2.5.0

Index]