Wilcoxon Rank Sum and Signed Rank Tests
Description
Performs one and twosample Wilcoxon tests on vectors of data; the latter is also known as ‘MannWhitney’ test.
Usage
1 2 3 4 5 6 7 8 9 10 
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 
alternative 
a character string specifying the alternative
hypothesis, must be one of 
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 
data 
an optional matrix or data frame (or similar: see

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 
... 
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 "htest"
containing the following components:
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 
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 
estimate 
an estimate of the location parameter.
(Only present if argument 
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).
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.
References
David F. Bauer (1972), Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687–690.
Myles Hollander and Douglas A. Wolfe (1973),
Nonparametric Statistical Methods.
New York: John Wiley & Sons.
Pages 27–33 (onesample), 68–75 (twosample).
Or second edition (1999).
See Also
psignrank
, pwilcox
.
wilcox_test
in package
\CRANpkgcoin 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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  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))

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