Wilcoxon Signed-Rank Test Walkthrough Cheat Sheet

Key Facts

• Use the Wilcoxon signed-rank test for paired data when the median difference is being tested and the paired differences are roughly symmetric.
• For each pair, compute the difference as $d_i=x_i-y_i$ or $d_i=\text{after}_i-\text{before}_i$, but keep the same direction throughout the test.
• Remove all observations with $d_i=0$ before ranking, and let $n$ be the number of remaining nonzero differences.
• Rank the absolute differences $|d_i|$ from smallest to largest, using average ranks for ties.
• Compute the positive rank sum $W^+=\sum R_i$ for all $d_i>0$ and the negative rank sum $W^-=\sum R_i$ for all $d_i<0$.
• For a two-sided small-sample test, the test statistic is often $W=\min(W^+,W^-)$, with unusually small $W$ giving evidence against $H_0$.
• For a large-sample approximation, use $z=\frac{W^+-\mu_W}{\sigma_W}$, where $\mu_W=\frac{n(n+1)}{4}$ and $\sigma_W=\sqrt{\frac{n(n+1)(2n+1)}{24}}$ when there are no ties.
• A common effect size is $r=\frac{|z|}{\sqrt{n}}$, where larger values of $r$ indicate stronger practical evidence.

Vocabulary

Paired difference

A paired difference is the value $d_i=x_i-y_i$ computed from two related measurements on the same subject or matched pair.

Signed rank

A signed rank is the rank of $|d_i|$ with the positive or negative sign of the original difference $d_i$ attached.

Positive rank sum

The positive rank sum $W^+$ is the total of the ranks whose original differences satisfy $d_i>0$.

Negative rank sum

The negative rank sum $W^-$ is the total of the ranks whose original differences satisfy $d_i<0$.

Null hypothesis

For the signed-rank test, the null hypothesis usually states that the population median difference is $0$.

Normal approximation

The normal approximation uses $z=\frac{W^+-\mu_W}{\sigma_W}$ to estimate a $p$-value when the sample size is large enough.