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
ptukey(q, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE) qtukey(p, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
- vector of quantiles.
- vector of probabilities.
- sample size for range (same for each group).
- degrees of freedom for \(s\) (see below).
- number of groups whose maximum range is considered.
- logical; if TRUE, probabilities p are given as log(p).
- logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
If \(n_g =\)
nranges is greater than one, \(R\) is
the maximum of \(n_g\) groups of
ptukey gives the distribution function and
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
A Legendre 16-point formula is used for the integral of
The computations are relatively expensive, especially for
qtukey which uses a simple secant method for finding the
qtukey will be accurate to the 4th decimal place.
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. Odeh, R. E. and Evans, J. O. (1974) Algorithm AS 70: Percentage Points of the Normal Distribution. Applied Statistics 23, 96--97.
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)))