wilcox.test {stats} | R Documentation

## Wilcoxon Rank Sum and Signed Rank Tests

### Description

Performs one and two sample Wilcoxon tests on vectors of data; the latter is also known as ‘Mann-Whitney’ test.

### Usage

    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, ...)

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

### Arguments

`x` | numeric vector of data values.
`y` | an optional numeric vector of data values.
`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 location parameter.
`paired` | a logical indicating whether you want a paired test.
`exact` | a logical indicating whether an exact p-value should be computed.
`correct` | a logical indicating whether to apply continuity correction in the normal approximation for the p-value.
`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.

### Details

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) 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 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.) 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.

### 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 p-value 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`.)

### References

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.

### See Also

`kruskal.test` for testing homogeneity in location parameters in the case of two or more samples; `t.test` for a parametric alternative under normality assumptions.

### Examples

    ## 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.
    data(airquality)
    boxplot(Ozone ~ Month, data = airquality)
    wilcox.test(Ozone ~ Month, data = airquality,
                subset = Month %in% c(5, 8))

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