\name{wilcox.test}
\alias{wilcox.test}
\alias{wilcox.test.default}
\alias{wilcox.test.formula}
\concept{Mann-Whitney Test}
\title{Wilcoxon Rank Sum and Signed Rank Tests}
\description{
  Performs one and two sample Wilcoxon tests on vectors of data; the
  latter is also known as \sQuote{Mann-Whitney} test.
}
\usage{
wilcox.test(x, \dots)

\method{wilcox.test}{default}(x, y = NULL,
            alternative = c("two.sided", "less", "greater"),
            mu = 0, paired = FALSE, exact = NULL, correct = TRUE,
            conf.int = FALSE, conf.level = 0.95, \dots)

\method{wilcox.test}{formula}(formula, data, subset, na.action, \dots)
}
\arguments{
  \item{x}{numeric vector of data values.  Non-finite (e.g. infinite or
    missing) values will be omitted.}
  \item{y}{an optional numeric vector of data values.}
  \item{alternative}{a character string specifying the alternative
    hypothesis, must be one of \code{"two.sided"} (default),
    \code{"greater"} or \code{"less"}.  You can specify just the initial
    letter.}
  \item{mu}{a number specifying an optional parameter used to form the
    null hypothesis.  See Details.}
  \item{paired}{a logical indicating whether you want a paired test.}
  \item{exact}{a logical indicating whether an exact \eqn{p}-value
    should be computed.}
  \item{correct}{a logical indicating whether to apply continuity
    correction in the normal approximation for the \eqn{p}-value.}
  \item{conf.int}{a logical indicating whether a confidence interval
    should be computed.}
  \item{conf.level}{confidence level of the interval.}
  \item{formula}{a formula of the form \code{lhs ~ rhs} where \code{lhs}
    is a numeric variable giving the data values and \code{rhs} a factor
    with two levels giving the corresponding groups.}
  \item{data}{an optional matrix or data frame (or similar: see
    \code{\link{model.frame}}) containing the variables in the
    formula \code{formula}.  By default the variables are taken from
    \code{environment(formula)}.}
  \item{subset}{an optional vector specifying a subset of observations
    to be used.}
  \item{na.action}{a function which indicates what should happen when
    the data contain \code{NA}s.  Defaults to
    \code{getOption("na.action")}.}
  \item{\dots}{further arguments to be passed to or from methods.}
}
\details{
  The formula interface is only applicable for the 2-sample tests.

  If only \code{x} is given, or if both \code{x} and \code{y} are given
  and \code{paired} is \code{TRUE}, a Wilcoxon signed rank test of the
  null that the distribution of \code{x} (in the one sample case) or of
  \code{x - y} (in the paired two sample case) is symmetric about
  \code{mu} is performed.

  Otherwise, if both \code{x} and \code{y} are given and \code{paired}
  is \code{FALSE}, a Wilcoxon rank sum test (equivalent to the
  Mann-Whitney test: see the Note) is carried out.  In this case, the
  null hypothesis is that the distributions of \code{x} and \code{y}
  differ by a location shift of \code{mu} and the alternative is that
  they differ by some other location shift (and the one-sided
  alternative \code{"greater"} is that \code{x} is shifted to the right
  of \code{y}).

  By default (if \code{exact} is not specified), an exact \eqn{p}-value
  is computed if the samples contain less than 50 finite values and
  there are no ties.  Otherwise, a normal approximation is used.

  Optionally (if argument \code{conf.int} is true), a nonparametric
  confidence interval and an estimator for the pseudomedian (one-sample
  case) or for the difference of the location parameters \code{x-y} is
  computed.  (The pseudomedian of a distribution \eqn{F} is the median
  of the distribution of \eqn{(u+v)/2}, where \eqn{u} and \eqn{v} are
  independent, each with distribution \eqn{F}.  If \eqn{F} is symmetric,
  then the pseudomedian and median coincide.  See Hollander & Wolfe
  (1973), page 34.)  If exact \eqn{p}-values are available, an exact
  confidence interval is obtained by the algorithm described in Bauer
  (1972), and the Hodges-Lehmann estimator is employed.  Otherwise, the
  returned confidence interval and point estimate are based on normal
  approximations.
  
  With small samples it may not be possible to achieve very high
  confidence interval coverages. If this happens a warning will be given
  and an interval with lower coverage will be substituted.
}
\value{
  A list with class \code{"htest"} containing the following components:
  \item{statistic}{the value of the test statistic with a name
    describing it.}
  \item{parameter}{the parameter(s) for the exact distribution of the
    test statistic.}
  \item{p.value}{the \eqn{p}-value for the test.}
  \item{null.value}{the location parameter \code{mu}.}
  \item{alternative}{a character string describing the alternative
    hypothesis.}
  \item{method}{the type of test applied.}
  \item{data.name}{a character string giving the names of the data.}
  \item{conf.int}{a confidence interval for the location parameter.
    (Only present if argument \code{conf.int = TRUE}.)}
  \item{estimate}{an estimate of the location parameter.
    (Only present if argument \code{conf.int = TRUE}.)}
}
\note{
  The literature is not unanimous about the definitions of the Wilcoxon
  rank sum and Mann-Whitney tests.  The two most common definitions
  correspond to the sum of the ranks of the first sample with the
  minimum value subtracted or not: \R subtracts and S-PLUS does not,
  giving a value which is larger by \eqn{m(m+1)/2} for a first sample
  of size \eqn{m}.  (It seems Wilcoxon's original paper used the
  unadjusted sum of the ranks but subsequent tables subtracted the
  minimum.)

  \R's value can also be computed as the number of all pairs
  \code{(x[i], y[j])} for which \code{y[j]} is not greater than
  \code{x[i]}, the most common definition of the Mann-Whitney test.
}
\section{Warning}{
  This function can use large amounts of memory and stack (and even
  crash \R if the stack limit is exceeded) if \code{exact = TRUE} and
  one sample is large (several thousands or more). 
}
\references{
  David F. Bauer (1972),
  Constructing confidence sets using rank statistics.
  \emph{Journal of the American Statistical Association}
  \bold{67}, 687--690.

  Myles Hollander & Douglas A. Wolfe (1973),
  \emph{Nonparametric statistical inference}.
  New York: John Wiley & Sons.
  Pages 27--33 (one-sample), 68--75 (two-sample).\cr
  Or second edition (1999).
}
\seealso{
  \code{\link{psignrank}}, \code{\link{pwilcox}}.

  \code{\link[exactRankTests]{wilcox.exact}}
  in \pkg{exactRankTests} covers much of the same ground, but also
  produces exact \eqn{p}-values in the presence of ties.

  \code{\link[coin]{wilcox_test}} in package \pkg{coin} for exact and
  approximate \emph{conditional} \eqn{p}-values for the Wilcoxon tests.
  
  \code{\link{kruskal.test}} for testing homogeneity in location
  parameters in the case of two or more samples;
  \code{\link{t.test}} for an alternative under normality
  assumptions [or large samples]
}
\examples{
## One-sample test.
## Hollander & Wolfe (1973), 29f.
## Hamilton depression scale factor measurements in 9 patients with
##  mixed anxiety and depression, taken at the first (x) and second
##  (y) visit after initiation of a therapy (administration of a
##  tranquilizer).
x <- c(1.83,  0.50,  1.62,  2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)
wilcox.test(x, y, paired = TRUE, alternative = "greater")
wilcox.test(y - x, alternative = "less")    # The same.
wilcox.test(y - x, alternative = "less",
            exact = FALSE, correct = FALSE) # H&W large sample
                                            # approximation

## Two-sample test.
## Hollander & Wolfe (1973), 69f.
## Permeability constants of the human chorioamnion (a placental
##  membrane) at term (x) and between 12 to 26 weeks gestational
##  age (y).  The alternative of interest is greater permeability
##  of the human chorioamnion for the term pregnancy.
x <- c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)
y <- c(1.15, 0.88, 0.90, 0.74, 1.21)
wilcox.test(x, y, alternative = "g")        # greater
wilcox.test(x, y, alternative = "greater",
            exact = FALSE, correct = FALSE) # H&W large sample
                                            # approximation

wilcox.test(rnorm(10), rnorm(10, 2), conf.int = TRUE)

## Formula interface.
boxplot(Ozone ~ Month, data = airquality)
wilcox.test(Ozone ~ Month, data = airquality,
            subset = Month \%in\% c(5, 8))
}
\keyword{htest}
