| % File src/library/stats/man/Tukey.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2018 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{Tukey} |
| \alias{Tukey} |
| \title{The Studentized Range Distribution} |
| \description{ |
| Functions of the distribution of the studentized range, \eqn{R/s}, |
| where \eqn{R} is the range of a standard normal sample and |
| \eqn{df \times s^2}{df*s^2} is independently distributed as |
| chi-squared with \eqn{df} degrees of freedom, see \code{\link{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) |
| } |
| \alias{ptukey} |
| \alias{qtukey} |
| \arguments{ |
| \item{q}{vector of quantiles.} |
| \item{p}{vector of probabilities.} |
| \item{nmeans}{sample size for range (same for each group).} |
| \item{df}{degrees of freedom for \eqn{s} (see below).} |
| \item{nranges}{number of \emph{groups} whose \bold{maximum} range is |
| considered.} |
| \item{log.p}{logical; if TRUE, probabilities p are given as log(p).} |
| \item{lower.tail}{logical; if TRUE (default), probabilities are |
| \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.} |
| } |
| \details{ |
| 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. |
| } |
| \value{ |
| \code{ptukey} gives the distribution function and \code{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 \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. |
| } |
| \source{ |
| \code{qtukey} is in part adapted from Odeh and Evans (1974). |
| } |
| \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. |
| \emph{Journal of Statistical Computation and Simulation}, \bold{30}, |
| 1--15. |
| \doi{10.1080/00949658808811082}. |
| |
| Odeh, R. E. and Evans, J. O. (1974). |
| Algorithm AS 70: Percentage Points of the Normal Distribution. |
| \emph{Applied Statistics}, \bold{23}, 96--97. |
| \doi{10.2307/2347061}. |
| } |
| \seealso{ |
| \link{Distributions} for standard distributions, including |
| \code{\link{pnorm}} and \code{\link{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: |
| \donttest{summary(abs(.95 - ptukey(qtt, 2, df = 2:11)))} |
| } |
| \keyword{distribution} |