\HeaderA{Tukey}{The Studentized Range Distribution}{Tukey}
\aliasA{ptukey}{Tukey}{ptukey}
\aliasA{qtukey}{Tukey}{qtukey}
\keyword{distribution}{Tukey}
\begin{Description}\relax
Functions on the distribution of
the studentized range, \eqn{R/s}{}, where \eqn{R}{} is the range of a
standard normal sample of size \eqn{n}{} and \eqn{s^2}{} is independently
distributed as chi-squared with \eqn{df}{} degrees of freedom, see
\code{\LinkA{pchisq}{pchisq}}.
\end{Description}
\begin{Usage}
\begin{verbatim}
ptukey(q, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
qtukey(p, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{nmeans}] sample size for range (same for each group).
\item[\code{df}] degrees of freedom for \eqn{s}{} (see below).
\item[\code{nranges}] number of \emph{groups} whose \bold{maximum} range is
considered.
\item[\code{log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \eqn{n_g =}{ng =}\code{nranges} is greater than one, \eqn{R}{} is
the \emph{maximum} of \eqn{n_g}{ng} groups of \code{nmeans}
observations each.
\end{Details}
\begin{Value}
\code{ptukey} gives the distribution function and \code{qtukey} its
inverse, the quantile function.
\end{Value}
\begin{Note}\relax
A Legendre 16-point formula is used for the integral of \code{ptukey}.
The computations are relatively expensive, especially for
\code{qtukey} which uses a simple secant method for finding the
inverse of \code{ptukey}.
\code{qtukey} will be accurate to the 4th decimal place.
\end{Note}
\begin{References}\relax
Copenhaver, Margaret Diponzio and Holland, Burt S. (1988)
Multiple comparisons of simple effects in
the two-way analysis of variance with fixed effects.
\emph{Journal of Statistical Computation and Simulation}, \bold{30}, 1--15.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{pnorm}{pnorm}} and \code{\LinkA{qnorm}{qnorm}} for the corresponding
functions for the normal distribution.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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)))
\end{ExampleCode}
\end{Examples}

