| % File src/library/stats/man/SignRank.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{SignRank} |
| \alias{SignRank} |
| \alias{dsignrank} |
| \alias{psignrank} |
| \alias{qsignrank} |
| \alias{rsignrank} |
| \title{Distribution of the Wilcoxon Signed Rank Statistic} |
| \description{ |
| Density, distribution function, quantile function and random |
| generation for the distribution of the Wilcoxon Signed Rank statistic |
| obtained from a sample with size \code{n}. |
| } |
| \usage{ |
| dsignrank(x, n, log = FALSE) |
| psignrank(q, n, lower.tail = TRUE, log.p = FALSE) |
| qsignrank(p, n, lower.tail = TRUE, log.p = FALSE) |
| rsignrank(nn, n) |
| } |
| \arguments{ |
| \item{x, q}{vector of quantiles.} |
| \item{p}{vector of probabilities.} |
| \item{nn}{number of observations. If \code{length(nn) > 1}, the length |
| is taken to be the number required.} |
| \item{n}{number(s) of observations in the sample(s). A positive |
| integer, or a vector of such integers.} |
| \item{log, 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]}.} |
| } |
| \value{ |
| \code{dsignrank} gives the density, |
| \code{psignrank} gives the distribution function, |
| \code{qsignrank} gives the quantile function, and |
| \code{rsignrank} generates random deviates. |
| |
| The length of the result is determined by \code{nn} for |
| \code{rsignrank}, and is the maximum of the lengths of the |
| numerical arguments for the other functions. |
| |
| The numerical arguments other than \code{nn} are recycled to the |
| length of the result. Only the first elements of the logical |
| arguments are used. |
| } |
| \details{ |
| This distribution is obtained as follows. Let \code{x} be a sample of |
| size \code{n} from a continuous distribution symmetric about the |
| origin. Then the Wilcoxon signed rank statistic is the sum of the |
| ranks of the absolute values \code{x[i]} for which \code{x[i]} is |
| positive. This statistic takes values between \eqn{0} and |
| \eqn{n(n+1)/2}, and its mean and variance are \eqn{n(n+1)/4} and |
| \eqn{n(n+1)(2n+1)/24}, respectively. |
| |
| If either of the first two arguments is a vector, the recycling rule is |
| used to do the calculations for all combinations of the two up to |
| the length of the longer vector. |
| } |
| \author{Kurt Hornik; efficiency improvement by Ivo Ugrina.} |
| \seealso{ |
| \code{\link{wilcox.test}} to calculate the statistic from data, find p |
| values and so on. |
| |
| \link{Distributions} for standard distributions, including |
| \code{\link{dwilcox}} for the distribution of \emph{two-sample} |
| Wilcoxon rank sum statistic. |
| } |
| \examples{ |
| require(graphics) |
| |
| par(mfrow = c(2,2)) |
| for(n in c(4:5,10,40)) { |
| x <- seq(0, n*(n+1)/2, length.out = 501) |
| plot(x, dsignrank(x, n = n), type = "l", |
| main = paste0("dsignrank(x, n = ", n, ")")) |
| } |
| \dontshow{ |
| p <- c(1, 1, 1, 2, 2:6, 8, 10, 11, 13, 15, 17, 20, 22, 24, |
| 27, 29, 31, 33, 35, 36, 38, 39, 39, 40) |
| stopifnot(round(dsignrank(0:56, n = 10)* 2^10) == c(p, rev(p), 0), |
| qsignrank((1:16)/ 16, n = 4) == c(0:2, rep(3:7, each = 2), 8:10)) |
| } |
| } |
| \keyword{distribution} |