Wilcoxon Signed Rank Test

1. Applications

Wilcoxon Signed Rank Test does not assume the distribution of the data.

Applications:

• Single population quantile or median test.
• Paired population median test.

2. Test Procedure

Given $n$ data points $x_1,x_2,\dots ,x_n$, a null hypothesis $H_0:\text{median}=m_0$, and an alternative hypothesis $H_1:\text{median}\ne m_0$, Wilcoxon Signed Rank Test is used to test the null hypothesis under a given significance level $\alpha$.

1. Go through the data points and remove ties, i.e. data points equal to $m_0$. Redefine the sample size as $n^{\prime }$.

2. Compute the difference between the data points and the hypothesized median $D_i=x_i-m_0$. Rank the absolute values of $D_i$: $r_i=\text{rank}(|D_i|)$. If ranks are tied, average the ranks.

   $$\begin{array}{ccccc}i& 1& 2& \cdots & n\\ x_i& x_1& x_2& \cdots & x_n\\ D_i=x_i-m_0& D_1& D_2& \cdots & D_n\\ r_i=\text{rank}(|D_i|)& r_1& r_2& \cdots & r_n\end{array}$$

3. Compute $W_{+}$ as the sum of ranks of positive differences, and $W_{-}$ as the sum of ranks of negative differences:

   $$W_{+}=\sum _{i=1}^n{r}_i\mathbb{I}(D_i>0),W_{-}=\sum _{i=1}^n{r}_i\mathbb{I}(D_i<0).$$

   Then

   $$W_{+}+W_{-}=\frac{n(n+1)}{2}.$$

4. Define the test statistic:

   $$W=\min \left(W^{+},W^{-}\right).$$

5. Compute the rejection region. Since $W$ always has a left-tail rejection region, we always use $\le$:

   $$\mathcal{RR}=\{W\le w_{\alpha /2}(n)\},$$

   where $w_{\alpha /2}(n)$ should be determined from the Wilcoxon Signed Rank Table.

   • For a two-tailed test, choose $w_{\alpha /2}$.
   • For a one-tailed test, choose $w_{\alpha }$.

6. If $W_{\text{obs}}\in \mathcal{RR}$, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

3. Normal Approximation

Theorem (Normal approximation of Wilcoxon Signed Rank Test). For large sample sizes ($n\ge 15$), the Wilcoxon Signed Rank Test can be approximated by a normal distribution:

$$W=\min \left(W^{+},W^{-}\right)\stackrel{\mathcal{L}|H_0}{⟶}\mathcal{N}\left(\frac{n(n+1)}{4},\frac{n(n+1)(2n+1)}{24}\right).$$

Thus, we can compute the z-score and p-value for the test statistic $W$:

$$Z=\frac{W-\frac{n(n+1)}{4}}{\sqrt{\frac{n(n+1)(2n+1)}{24}}}\stackrel{H_0}{⟶}\mathcal{N}(0,1).$$

• For a two-tailed test, the rejection region is $\mathcal{RR}=\{Z\le -z_{\alpha /2}\}$.
• For a one-tailed test, the rejection region is $\mathcal{RR}=\{Z\le -z_{\alpha }\}$.

4. Example

Example (Paired product quality comparison). A survey was conducted to compare the quality of two products A and B, with $n=5$ samples: $A_1,B_1,\cdots ,A_5,B_5$. The null hypothesis is that the quality of two products are the same: $H_0:\text{median}(A)=\text{median}(B)$. The alternative hypothesis is that the quality of product A is not the same as B: $H_a:\text{median}(A)\ne \text{median}(B)$.

Solution

Given the data, compute the differences $D_i=A_i-B_i$ and the ranks of the absolute differences:

$$\begin{array}{cccccc}i& 1& 2& 3& 4& 5\\ D_i=A_i-B_i& 6& 3& 2& -1& 5\\ |r_i|& 5& 3& 2& 1& 4\end{array}$$

Then compute the sum of ranks of positive differences $W^{+}=5+3+2+4=14$, and the sum of ranks of negative differences $W^{-}=1$. The test statistic is $W=\min (14,1)=1$.

From the Wilcoxon Signed Rank Table, the rejection region for $n=5$ and $\alpha =0.1$ is

$$\mathcal{RR}=\{W\le w_{\alpha /2}(5)=1\}.$$

Since $W_{\text{obs}}=1\in \mathcal{RR}$, we reject the null hypothesis.

Related Notes

• Sign Test
• Wilcoxon Rank Sum Test
• Nonparametric Tests Overview