Tukey
The Studentized Range Distribution
Functions of the distribution of the studentized range, \(R/s\),
where \(R\) is the range of a standard normal sample and
\(df \times s^2\) is independently distributed as
chi-squared with \(df\) degrees of freedom, see pchisq.
- Keywords
- distribution
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 \(P[X \le x]\), otherwise, \(P[X > x]\).
Details
If \(n_g =\)nranges is greater than one, \(R\) is
the maximum of \(n_g\) groups of nmeans
observations each.
Value
ptukey gives the distribution function and qtukey its
inverse, the quantile function.
The length of the result is the maximum of the lengths of the numerical arguments. The other numerical arguments are recycled to that length. Only the first elements of the logical arguments are used.
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). Computation of the distribution of the maximum studentized range statistic with application to multiple significance testing of simple effects. Journal of Statistical Computation and Simulation, 30, 1--15. 10.1080/00949658808811082.
Odeh, R. E. and Evans, J. O. (1974). Algorithm AS 70: Percentage Points of the Normal Distribution. Applied Statistics, 23, 96--97. 10.2307/2347061.
See Also
Distributions for standard distributions, including
pnorm and qnorm for the corresponding
functions for the normal distribution.
Examples
library(stats)
# NOT RUN {
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:
# }
# NOT RUN {
summary(abs(.95 - ptukey(qtt, 2, df = 2:11)))
# }