wilcox.test {stats}R Documentation

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, ...)

## Default S3 method:
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,
            tol.root = 1e-4, digits.rank = Inf, ...)

## S3 method for class 'formula'
wilcox.test(formula, data, subset, na.action = na.pass, ...)



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 x non-finite values will be omitted.


a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "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.


(when conf.int is true:) a positive numeric tolerance, used in uniroot(*, tol=tol.root) calls.


a number; if finite, rank(signif(r, digits.rank)) will be used to compute ranks for the test statistic instead of (the default) rank(r).


a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done, see Examples.


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 environment(formula).


an optional vector specifying a subset of observations to be used.


a function which indicates what should happen when the data contain NAs.


further arguments to be passed to or from methods. For the formula method, this includes arguments of the default method, but not paired.


The formula interface is only applicable for the 2-sample 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 Mann-Whitney 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 one-sided alternative "greater" is that x is shifted to the right of y).

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.

For stability reasons, it may be advisable to use rounded data or to set digits.rank = 7, say, such that determination of ties does not depend on very small numeric differences (see the example).

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 x-y 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 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 y.

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 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.

When x (and y if applicable) are valid, the function now always returns, also in the conf.int = TRUE case when a confidence interval cannot be computed, in which case the interval boundaries and sometimes the estimate now contain NaN.


A list with class "htest" containing the following components:


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 mu.


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.)


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).


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 (m(m+1)/2 for a first sample of size m) subtracted or not: R subtracts. 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.


David F. Bauer (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687–690. doi:10.1080/01621459.1972.10481279.

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).

See Also

psignrank, pwilcox.

wilcox_test in package coin for exact, asymptotic and Monte Carlo conditional p-values, 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]


## 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

## Formula interface to one-sample and paired tests

depression <- data.frame(first = x, second = y, change = y - x)
wilcox.test(change ~ 1, data = depression)
wilcox.test(Pair(first, second) ~ 1, data = depression)

## 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))

## accuracy in ties determination via 'digits.rank':
wilcox.test( 4:2,      3:1,     paired=TRUE) # Warning:  cannot compute exact p-value with ties
wilcox.test((4:2)/10, (3:1)/10, paired=TRUE) # no ties => *no* warning
wilcox.test((4:2)/10, (3:1)/10, paired=TRUE, digits.rank = 9) # same ties as (4:2, 3:1)

[Package stats version 4.4.0 Index]