Wilcoxon Rank Sum and Signed Rank Tests
Performs one- and two-sample Wilcoxon tests on vectors of data; the latter is also known as Mann-Whitney test.
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, ...)
- numeric vector of data values. Non-finite (e.g., infinite or missing) values will be omitted.
- an optional numeric vector of data values: as with
xnon-finite values will be omitted.
- a character string specifying the alternative
hypothesis, must be one of
"less". You can specify just the initial letter.
- a number specifying an optional parameter used to form the null hypothesis. See Details.
- a logical indicating whether you want a paired test.
- a logical indicating whether an exact p-value should be computed.
- a logical indicating whether to apply continuity correction in the normal approximation for the p-value.
- a logical indicating whether a confidence interval should be computed.
- confidence level of the interval.
- a formula of the form
lhs ~ rhswhere
lhsis a numeric variable giving the data values and
rhsa factor with two levels giving the corresponding groups.
- an optional matrix or data frame (or similar: see
model.frame) containing the variables in the formula
formula. By default the variables are taken from
- an optional vector specifying a subset of observations to be used.
- a function which indicates what should happen when
the data contain
NAs. Defaults to
- further arguments to be passed to or from methods.
The formula interface is only applicable for the 2-sample tests.
x is given, or if both
y are given
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
y are given and
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
differ by a location shift of
mu and the alternative is that
they differ by some other location shift (and the one-sided
"greater" is that
x is shifted to the right
By default (if
exact is not specified), an exact 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
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
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 two-sample 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
If exact 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. These are continuity-corrected for the interval but
not the estimate (as the correction depends on the
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.
A list with class
- the value of the test statistic with a name describing it.
- the parameter(s) for the exact distribution of the test statistic.
- the p-value for the test.
- the location parameter
- a character string describing the alternative hypothesis.
- the type of test applied.
- a character string giving the names of the data.
- a confidence interval for the location parameter.
(Only present if argument
conf.int = TRUE.)
- an estimate of the location parameter.
(Only present if argument
conf.int = TRUE.)
"htest"containing the following components:
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 $m(m+1)/2$ 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 Mann-Whitney test.
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).
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 (one-sample), 68--75 (two-sample). Or second edition (1999).
require(graphics) ## 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))