

Wilcoxon {base}                              R Documentation

_D_i_s_t_r_i_b_u_t_i_o_n _o_f _t_h_e _W_i_l_c_o_x_o_n _R_a_n_k _S_u_m _S_t_a_t_i_s_t_i_c

_D_e_s_c_r_i_p_t_i_o_n_:

     These functions provide information about the distribu-
     tion of the Wilcoxon rank sum statistic obtained from
     samples with size `m' and `n', respectively.  `dwilcox'
     gives the density, `pwilcox' gives the distribution
     function, `qwilcox' gives the quantile function, and
     `rwilcox' generates random deviates.

_U_s_a_g_e_:

     dwilcox(x, m, n)
     pwilcox(q, m, n)
     qwilcox(p, m, n)
     rwilcox(nn, m, n)

_A_r_g_u_m_e_n_t_s_:

     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.  Must be positive integers
          less than 50.

_D_e_t_a_i_l_s_:

     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 num-
     ber 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', respec-
     tively.

_A_u_t_h_o_r_(_s_)_:

     Kurt Hornik hornik@ci.tuwien.ac.at

_S_e_e _A_l_s_o_:

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

_E_x_a_m_p_l_e_s_:

     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")

