16.5 Paired Wilcoxon Signed Rank Test

Gordon E. Sarty

16. Non-parametric Tests

16.5 Paired Wilcoxon Signed Rank Test

This test is an alternative to the paired sample $t$ -test; it is a hypothesis test about means. It is based on a binomial distribution and we again have two cases, one for small samples and one for large samples.

Case 1. Small samples ( $n < 10$ ).

Test statistic : $w_{s} = \min\left( | \sum \mbox{of + ranks}|, | \sum\mbox{of - ranks}| \right)$

Critical statistic : $w_{\mbox{crit}}$ from the Wilcoxon Signed-Rank Test Critical Values Table for which you need $n$ , $\alpha$ and whether you want a one- or two-tailed test. Reject $H_{0}$ if $w_{s} \leq w_{\mbox{crit}}$ .

Case 2. Large samples ( $n \geq 10$ ).

Test statistic :

$z_{\mbox{test}} = \frac{w_{s} - \frac{n(n+1)}{4}}{\sqrt{\frac{n(n+1)(2n+1)}{24}}}$

Critical statistic : $z_{\mbox{crit}}$ from the t Distribution Table.

Example 16.7 : Using the data given below for numbers of shoplifters at some store for a time before a security guard was hired and after a security guard was hired, decide if the expense of a security guard is worth it.

Here’re the shoplifter data, before and after the hiring of a security guard, combined with some data reduction calculations :

Day (Subject)	Before $x_{b}$	After $x_{a}$	$D=x_{b}-x_{a}$	\|D\|	Rank	Signed Rank
M	7	5	2	2	3.5	3.5
T	2	3	-1	1	1.5	-1.5
W	3	4	-1	1	1.5	-1.5
T	6	3	3	3	5	5
F	5	1	4	4	6	6
S	8	6	2	2	3.5	3.5
S	12	4	8	8	7	7

The data reduction columns include the essential steps of computing the difference $D$ , it’s absolute value $|D|$ , the rank of the absolute value and, finally, the ranks with the sign of $D$ added. It may be useful to order the data, like we did in Example 16.6, to make the ranking easier. As always, the order of the difference, and it’s sign. is important for interpretation and getting the direction of one-tailed tests right. In this case, we would hope that the number of shoplifters would go down after the security guard was hired; a positive difference would be good.

1. Hypothesis.

With the assignment 1 = before and 2 = after :

$H_{0}: \mu_{1} = \mu_{2}$

$H_{1}: \mu_{1} \neq \mu_{2}$

or

$H_{0}: \overline{D} = 0$

$H_{1}: \overline{D} \neq 0$

2. Critical statistic.

From the Wilcoxon Signed-Rank Test Critical Values Table with $\alpha = 0.05$ for a two-tailed test and $n = 7$ find

$w_{\mbox{crit}} = 2$

3. Test statistic. First compute:

$\begin{eqnarray*} | \sum \mbox{ of $+$ ranks} |& = & | 3.5 + 5 + 6 + 3.5 + 7 | = | 25 | = 25 \\ | \sum \mbox{ of $-$ ranks} | & = & | -1.5 - 1.5 | = | -3 | = 3 \end{eqnarray*}$