mann_whitney_test_pv: Wilcoxon-Mann-Whitney _U_ test

In DiscreteTests: Vectorised Computation of P-Values and Their Supports for Several Discrete Statistical Tests

Wilcoxon-Mann-Whitney U test

Description

mann_whitney_test_pv() performs an exact or approximate Wilcoxon-Mann-Whitney U test about the location shift between two independent groups when the data is not necessarily normally distributed. In contrast to stats::wilcox.test(), it is vectorised and only calculates p-values. Furthermore, it is capable of returning the discrete p-value supports, i.e. all observable p-values under a null hypothesis. Multiple tests can be evaluated simultaneously.

Usage

mann_whitney_test_pv(
  x,
  y,
  mu = 0,
  alternative = "two.sided",
  exact = NULL,
  correct = TRUE,
  digits_rank = Inf,
  simple_output = FALSE
)

Arguments

x, y	numerical vectors forming the samples to be tested or lists of numerical vectors for multiple tests.
mu	numerical vector or single number of hypothesised location shift(s).
alternative	character vector that indicates the alternative hypotheses; each value must be one of "two.sided" (the default), "less" or "greater".
exact	single logical value that indicates whether p-values are to be calculated by exact computation (TRUE) or by a continuous approximation (FALSE). NULL (the default) is allowed (see details).
correct	either a single logical value that indicates if a continuity correction in the normal approximation is to be applied (TRUE; the default) or not (FALSE), or a single integer between 0 and 3 specifying both that a continuity correction should be used and the number of terms of an Edgeworth expansion for a more accurate normal approximation. Ignored, if exact = TRUE.
digits_rank	single number giving the significant digits used to compute ranks for the test statistics.
simple_output	logical value that indicates whether an R6 class object, including the tests' parameters and support sets, i.e. all observable p-values under each null hypothesis, is to be returned (see below).

Details

We use a test statistic called the Wilcoxon Rank Sum Statistic, defined by

U = \sum_{i = 1}^{n_X}{rank(X_i)} - \frac{n_X(n_X + 1)}{2},

where rank(X_i) is the rank of X_i in the concatenated sample of X and Y, and n_X and n_Y are the respective sizes of the samples X and Y. Note that U can range from 0 to n_X \cdot n_Y. This is the same statistic used by stats::wilcox.test() and whose distribution is accessible with pwilcox. This is also the statistic defined by the two given references. Note, however, that it is not what is called the Mann-Whitney U Statistic in the (English-language) Wikipedia article (as of February 12, 2026). The latter is defined as, using our notation, \min(U, n_X \cdot n_Y - U). Using the Wikipedia notation, the Wilcoxon Rank Sum Statistic is U_2.

The parameters x, y, mu and alternative are vectorised. If x and y are lists, they are replicated automatically to have the same lengths. In case x or y are not lists, they are added to new ones, which are then replicated to the appropriate lengths. This allows multiple hypotheses to be tested simultaneously.

In the presence of ties, computation of the Edgeworth series (up to correct = 3) is not possible. Therefore, numeric values of correct are ignored and only a continuity correction is performed.

By setting exact = NULL, exact computation is performed only if both samples sizes in a test setting are lower than or equal to 200. Otherwise, p-values are computed by normal approximation.

If digits_rank = Inf (the default), rank() is used to compute ranks for the tests statistics instead of rank(signif(., digits_rank))

Value

If simple.output = TRUE, a vector of computed p-values is returned. Otherwise, the output is a DiscreteTestResults R6 class object, which also includes the p-value supports and testing parameters. These have to be accessed by public methods, e.g. ⁠$get_pvalues()⁠.

References

Mann, H. D. & Whitney, D. R. (1947). On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other. Ann. Math. Statist., 18(1), pp. 50-60. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/aoms/1177730491")}

Hollander, M. & Wolfe, D. (1973). Nonparametric Statistical Methods. Third Edition. New York: Wiley. pp. 115-135. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/9781119196037")}

See Also

stats::wilcox.test(), pwilcox, wilcox_test_pv()

Examples

# Constructing
set.seed(1)
r1 <- rnorm(100)
r2 <- rnorm(100, 1)

# Exact two-sided p-values and their supports
results_ex  <- mann_whitney_test_pv(r1, r2)
print(results_ex)
results_ex$get_pvalues()
results_ex$get_pvalue_supports()

# Normal-approximated one-sided p-values ("less") and their supports
results_ap  <- mann_whitney_test_pv(r1, r2, alternative = "less", exact = FALSE)
print(results_ap)
results_ap$get_pvalues()
results_ap$get_pvalue_supports()

DiscreteTests documentation built on Sept. 2, 2026, 9:06 a.m.