wilcox.exact {exactRankTests}

Wilcoxon Rank Sum and Signed Rank Tests

Description

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

Usage

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

Arguments

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 NAs. Defaults to getOption("na.action").

...

further arguments to be passed to or from methods.

Details

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.

Value

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

statistic

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

p.value

the p-value for the test.

pointprob

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

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

Hodges-Lehmann 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.

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

See Also

perm.test for the one and two sample permutation test.

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.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)