Wilcoxon {base}R Documentation

Distribution of the Wilcoxon Rank Sum Statistic

Description

Density, distribution function, quantile function and random generation for the distribution of the Wilcoxon rank sum statistic obtained from samples with size m and n, respectively.

Usage

dwilcox(x, m, n, log = FALSE)
pwilcox(q, m, n, lower.tail = TRUE, log.p = FALSE)
qwilcox(p, m, n, lower.tail = TRUE, log.p = FALSE)
rwilcox(nn, m, n)

Arguments

x, q vector of quantiles.
p vector of probabilities.
nn number of observations to generate.
m, n numbers of observations in the first and second sample, respectively.
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Details

This distribution is obtained as follows. Let x and y be two random, independent samples of size m and n. Then the Wilcoxon rank sum statistic is the number of all pairs (x[i], y[j]) for which y[j] is not greater than x[i]. This statistic takes values between 0 and m * n, and its mean and variance are m * n / 2 and m * n * (m + n + 1) / 12, respectively.

Value

dwilcox gives the density, pwilcox gives the distribution function, qwilcox gives the quantile function, and rwilcox generates random deviates.

Author(s)

Kurt Hornik hornik@ci.tuwien.ac.at

See Also

dsignrank etc, for the one-sample Wilcoxon rank statistic.

Examples

x <- -1:(4*6 + 1)
fx <- dwilcox(x, 4, 6)
all(fx == dwilcox(x, 6, 4))
Fx <- pwilcox(x, 4, 6)
all(abs(Fx - cumsum(fx)) < 10 * .Machine$double.eps)

layout(rbind(1,2),width=1,heights=c(3,2))
plot(x, fx,type='h', col="violet",
     main= "Probabilities (density) of Wilcoxon-Statist.(n=6,m=4)")
plot(x, Fx,type="s", col="blue",
     main= "Distribution of Wilcoxon-Statist.(n=6,m=4)")
abline(h=0:1, col="gray20",lty=2)
layout(1)# set back

N <- 200
hist(U <- rwilcox(N, m=4,n=6), breaks=0:25 - 1/2, border="red", col="pink",
     sub = paste("N =",N))
mtext("N * f(x),  f() = true ``density''", side=3, col="blue")
 lines(x, N*fx, type='h', col='blue', lwd=2)
points(x, N*fx, cex=2)

## Better is a Quantile-Quantile Plot
qqplot(U, qw <- qwilcox((1:N - 1/2)/N, m=4,n=6),
       main = paste("Q-Q-Plot of empirical and theoretical quantiles",
                     "Wilcoxon Statistic,  (m=4, n=6)",sep="\n"))
n <- as.numeric(names(print(tU <- table(U))))
text(n+.2, n+.5, labels=tU, col="red")
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