wilcox.test
Wilcoxon Rank Sum and Signed Rank Tests
Performs one and twosample Wilcoxon tests on vectors of data; the latter is also known as ‘MannWhitney’ test.
 Keywords
 htest
Usage
wilcox.test(x, ...)
"wilcox.test"(x, y = NULL, alternative = c("two.sided", "less", "greater"), mu = 0, paired = FALSE, exact = NULL, correct = TRUE, conf.int = FALSE, conf.level = 0.95, ...)
"wilcox.test"(formula, data, subset, na.action, ...)
Arguments
 x
 numeric vector of data values. Nonfinite (e.g., infinite or missing) values will be omitted.
 y
 an optional numeric vector of data values: as with
x
nonfinite values will be omitted.  alternative
 a character string specifying the alternative
hypothesis, must be one of
"two.sided"
(default),"greater"
or"less"
. You can specify just the initial letter.  mu
 a number specifying an optional parameter used to form the null hypothesis. See ‘Details’.
 paired
 a logical indicating whether you want a paired test.
 exact
 a logical indicating whether an exact pvalue should be computed.
 correct
 a logical indicating whether to apply continuity correction in the normal approximation for the pvalue.
 conf.int
 a logical indicating whether a confidence interval should be computed.
 conf.level
 confidence level of the interval.
 formula
 a formula of the form
lhs ~ rhs
wherelhs
is a numeric variable giving the data values andrhs
a factor with two levels giving the corresponding groups.  data
 an optional matrix or data frame (or similar: see
model.frame
) containing the variables in the formulaformula
. By default the variables are taken fromenvironment(formula)
.  subset
 an optional vector specifying a subset of observations to be used.
 na.action
 a function which indicates what should happen when
the data contain
NA
s. Defaults togetOption("na.action")
.  ...
 further arguments to be passed to or from methods.
Details
The formula interface is only applicable for the 2sample tests.
If only x
is given, or if both x
and y
are given
and paired
is TRUE
, a Wilcoxon signed rank test of the
null that the distribution of x
(in the one sample case) or of
x  y
(in the paired two sample case) is symmetric about
mu
is performed.
Otherwise, if both x
and y
are given and paired
is FALSE
, a Wilcoxon rank sum test (equivalent to the
MannWhitney test: see the Note) is carried out. In this case, the
null hypothesis is that the distributions of x
and y
differ by a location shift of mu
and the alternative is that
they differ by some other location shift (and the onesided
alternative "greater"
is that x
is shifted to the right
of y
).
By default (if exact
is not specified), an exact pvalue
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 conf.int
is true), a nonparametric
confidence interval and an estimator for the pseudomedian (onesample
case) or for the difference of the location parameters xy
is
computed. (The pseudomedian of a distribution $F$ is the median
of the distribution of $(u+v)/2$, where $u$ and $v$ are
independent, each with distribution $F$. If $F$ is symmetric,
then the pseudomedian and median coincide. See Hollander & Wolfe
(1973), page 34.) Note that in the twosample case the estimator for
the difference in location parameters does not estimate the
difference in medians (a common misconception) but rather the median
of the difference between a sample from x
and a sample from
y
.
If exact pvalues are available, an exact confidence interval is
obtained by the algorithm described in Bauer (1972), and the
HodgesLehmann estimator is employed. Otherwise, the returned
confidence interval and point estimate are based on normal
approximations. These are continuitycorrected for the interval but
not the estimate (as the correction depends on the
alternative
).
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
 statistic
 the value of the test statistic with a name describing it.
 parameter
 the parameter(s) for the exact distribution of the test statistic.
 p.value
 the pvalue for the test.
 null.value
 the location parameter
mu
.  alternative
 a character string describing the alternative hypothesis.
 method
 the type of test applied.
 data.name
 a character string giving the names of the data.
 conf.int
 a confidence interval for the location parameter.
(Only present if argument
conf.int = TRUE
.)  estimate
 an estimate of the location parameter.
(Only present if argument
conf.int = TRUE
.)
"htest"
containing the following components:
Note
The literature is not unanimous about the definitions of the Wilcoxon rank sum and MannWhitney 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 SPLUS does not, giving a value which is larger by $m(m+1)/2$ for a first sample of size $m$. (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
(x[i], y[j])
for which y[j]
is not greater than
x[i]
, the most common definition of the MannWhitney test.
Warning
This function can use large amounts of memory and stack (and even
crash R if the stack limit is exceeded) if exact = TRUE
and
one sample is large (several thousands or more).
References
David F. Bauer (1972), Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687690.
Myles Hollander and Douglas A. Wolfe (1973), Nonparametric Statistical Methods. New York: John Wiley & Sons. Pages 2733 (onesample), 6875 (twosample). Or second edition (1999).
See Also
wilcox_test
in package
\href{https://CRAN.Rproject.org/package=#1}{\pkg{#1}}coincoin for exact, asymptotic and Monte Carlo
conditional pvalues, including in the presence of ties.
kruskal.test
for testing homogeneity in location
parameters in the case of two or more samples;
t.test
for an alternative under normality
assumptions [or large samples]
Examples
library(stats)
require(graphics)
## Onesample 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
## Twosample 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))