wilcox.test | R Documentation |
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, ...)
x |
numeric vector of data values. Non-finite (e.g., infinite or missing) values will be omitted. |
y |
an optional numeric vector of data values: as with |
alternative |
a character string specifying the alternative
hypothesis, must be one of |
mu |
a number specifying an optional parameter used to form the null hypothesis. See ‘Details’. |
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. |
tol.root |
(when |
digits.rank |
a number; if finite, |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
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 |
... |
further arguments to be passed to or from methods. |
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:
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 |
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 |
estimate |
an estimate of the location parameter.
(Only present if argument |
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 subtracted or not: R subtracts and S-PLUS does not, giving a value which is larger by m(m+1)/2 for a first sample of size m. (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. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_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).
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]
require(graphics) ## 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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.