Performs one and two sample Wilcoxon tests on vectors of data for possibly tied observations.

```
# S3 method for default
wilcox.exact(x, y = NULL, alternative = c("two.sided", "less", "greater"),
mu = 0, paired = FALSE, exact = NULL,
conf.int = FALSE, conf.level = 0.95, …)
# S3 method for formula
wilcox.exact(formula, data, subset, na.action, …)
```

x

numeric vector of data values.

y

an optional numeric vector of data values.

alternative

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 location parameter.

paired

a logical indicating whether you want a paired test.

exact

a logical indicating whether an exact p-value should be computed.

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`

where `lhs`

is a numeric variable giving the data values and `rhs`

a factor
with two levels giving the corresponding groups.

data

an optional data frame containing the variables in the model 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 to
`getOption("na.action")`

.

…

further arguments to be passed to or from methods.

A list with class `"htest"`

containing the following components:

the value of the test statistic with a name describing it.

the p-value for the test.

this gives the probability of observing the test
statistic itself (called `point-prob`

).

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`

.)

Hodges-Lehmann estimate of the location parameter.
(Only present if argument `conf.int = TRUE`

.)

This version computes exact conditional (on the data) p-values and quantiles using the Shift-Algorithm by Streitberg & R\"ohmel for both tied and untied samples.

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 median of `x`

(in the one sample case) or of
`x-y`

(in the paired two sample case) equals `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) is carried out. In this case, the null hypothesis
is that the location of the distributions of `x`

and `y`

differ by `mu`

.

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 for the median (one-sample case) or for the
difference of the location parameters `x-y`

is computed. If
exact p-values are available, an exact confidence interval is obtained
by the algorithm described in Bauer (1972). Otherwise, an asymptotic
confidence interval is returned.

Myles Hollander & Douglas A. Wolfe (1973),
*Nonparametric statistical inference*.
New York: John Wiley & Sons.
Pages 27--33 (one-sample), 68--75 (two-sample).

David F. Bauer (1972),
Constructing confidence sets using rank statistics.
*Journal of the American Statistical Association*
**67**, 687--690.

Cyrus R. Mehta & Nitin R. Patel (2001),
*StatXact-5 for Windows.*
Manual, Cytel Software Cooperation, Cambridge, USA

`perm.test`

for the one and two sample permutation test.

# NOT RUN { ## 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.exact(x, y, paired = TRUE, alternative = "greater") wilcox.exact(y - x, alternative = "less") # The same. ## 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.exact(x, y, alternative = "g") # greater ## Formula interface. data(airquality) boxplot(Ozone ~ Month, data = airquality) wilcox.exact(Ozone ~ Month, data = airquality, subset = Month %in% c(5, 8)) # Hollander & Wolfe, p. 39, results p. 40 and p. 53 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.exact(y,x, paired=TRUE, conf.int=TRUE) # Hollander & Wolfe, p. 110, results p. 111 and p. 126 x <- c(0.8, 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.exact(y,x, conf.int=TRUE) # }