Wilcoxon Rank Sum Test

1. Overview

Wilcoxon Rank Sum Test is equivalent to the Mann-Whitney U Test. Both tests compare the medians of two independent populations, do not assume the distribution of the data, and do not require equal sample sizes.

2. Test Procedure

Consider two independent samples $X=\{x_1,x_2,\dots ,x_m\}$ and $Y=\{y_1,y_2,\dots ,y_n\}$. Use the Wilcoxon Rank Sum Test to test if the medians of two populations are equal under a given significance level $\alpha$.

Given the null hypothesis $H_0:\text{median}(X)=\text{median}(Y)$, and alternative hypothesis $H_1:\text{median}(X)\ne \text{median}(Y)$.

Mix the two samples and rank the values, sharing the rank for tied values.

$$\begin{array}{ccc}\text{Pop.}& \text{Rank}& \text{Rank Sum}\\ X& r(X_1)r(X_2)\cdots r(X_m)& W_1={\sum }_{i=1}^{m}r(X_i)\\ Y& r(Y_1)r(Y_2)\cdots r(Y_n)& W_2={\sum }_{i=1}^{n}r(Y_i)\end{array}$$

Conclude that

$$W_1+W_2=\frac{(m+n)(m+n+1)}{2}.$$

Define the test statistic $W$ to be the one with smaller sample size. Assume $m\le n$, so $W=W_1$.

Then the rejection region is

$$\mathcal{RR}=\left\{W\le w_{\alpha /2}^L(m,n)\right\}\cup \left\{W\ge w_{\alpha /2}^U(m,n)\right\},$$

where the lower critical value $w_{\alpha /2}^L(m,n)$ and upper critical value $w_{\alpha /2}^U(m,n)$ should be determined from the Wilcoxon Rank Sum Table.

• For a two-tailed test, choose $\mathcal{RR}=\left\{W\le w_{\alpha /2}^L\right\}\cup \left\{W\ge w_{\alpha /2}^U\right\}$.
• For a one-tailed test:
  • If $H_a:\text{median}(X)>\text{median}(Y)$, then $\mathcal{RR}=\left\{W\ge w_{\alpha }^U\right\}$.
  • If $H_a:\text{median}(X)<\text{median}(Y)$, then $\mathcal{RR}=\left\{W\le w_{\alpha }^L\right\}$.

3. Normal Approximation

Theorem (Normal approximation of Wilcoxon Rank Sum Test). For large sample sizes ($m,n\ge 10$), the Wilcoxon Rank Sum Test can be approximated by a normal distribution:

$$\begin{array}{rl}{W}_1& \stackrel{\mathcal{L}|H_0}{⟶}\mathcal{N}\left(\frac{m(m+n+1)}{2},\frac{mn(m+n+1)}{12}\right),\\ W_2& \stackrel{\mathcal{L}|H_0}{⟶}\mathcal{N}\left(\frac{n(m+n+1)}{2},\frac{mn(m+n+1)}{12}\right).\end{array}$$

Either $W_1$ or $W_2$ can be used as the test statistic regardless of the sample sizes, though $W_1$ may be more intuitive with respect to the hypothesis.

$$Z=\frac{W-\frac{m(m+n+1)}{2}}{\sqrt{\frac{mn(m+n+1)}{12}}}\stackrel{H_0}{⟶}\mathcal{N}(0,1).$$

• For the two-tailed test, the rejection region is $\mathcal{RR}=\left\{|Z|\ge z_{\alpha /2}\right\}$.
• For the one-tailed test:
  • If $H_a:\text{median}(X)>\text{median}(Y)$, then $\mathcal{RR}=\left\{Z\ge z_{\alpha }\right\}$.
  • If $H_a:\text{median}(X)<\text{median}(Y)$, then $\mathcal{RR}=\left\{Z\le -z_{\alpha }\right\}$.

Related Notes

• Mann-Whitney U Test
• Wilcoxon Signed Rank Test
• Nonparametric Tests Overview