Wilcoxon Rank Sum and Signed Rank Tests

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

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")
#> 
#> 	Exact Wilcoxon signed rank test
#> 
#> data:  x and y
#> V = 40, p-value = 0.01953
#> alternative hypothesis: true mu is greater than 0
#> 
wilcox.exact(y - x, alternative = "less")    # The same.
#> 
#> 	Exact Wilcoxon signed rank test
#> 
#> data:  y - x
#> V = 5, p-value = 0.01953
#> alternative hypothesis: true mu is less than 0
#> 

## 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
#> 
#> 	Exact Wilcoxon rank sum test
#> 
#> data:  x and y
#> W = 35, p-value = 0.1272
#> alternative hypothesis: true mu is greater than 0
#> 

## Formula interface.
data(airquality)
boxplot(Ozone ~ Month, data = airquality)

wilcox.exact(Ozone ~ Month, data = airquality,
            subset = Month %in% c(5, 8))
#> 
#> 	Exact Wilcoxon rank sum test
#> 
#> data:  Ozone by Month
#> W = 127.5, p-value = 6.109e-05
#> alternative hypothesis: true mu is not equal to 0
#> 


# 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)
#> 
#> 	Exact Wilcoxon signed rank test
#> 
#> data:  y and x
#> V = 5, p-value = 0.03906
#> alternative hypothesis: true mu is not equal to 0
#> 95 percent confidence interval:
#>  -0.786 -0.010
#> sample estimates:
#> (pseudo)median 
#>          -0.46 
#> 

# 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)
#> 
#> 	Exact Wilcoxon rank sum test
#> 
#> data:  y and x
#> W = 15, p-value = 0.2544
#> alternative hypothesis: true mu is not equal to 0
#> 95 percent confidence interval:
#>  -0.76  0.15
#> sample estimates:
#> difference in location 
#>                 -0.305 
#>