Wilcoxon Signed Rank

STA4173: Biostatistics
Spring 2025

Introduction

Today we have discussed that we turn to nonparametric tests when we do not meet distributional assumptions for t-tests.
If we do not meet the normality assumption for the paired t-test, we turn to the Wilcoxon signed rank.
Like in the dependent t-test, we will analyze the difference between two values.
Like in the Wilcoxon rank sum, we will be analyzing ranks.

Wilcoxon Signed Rank

Before ranking, we will find the difference between the paired observations and eliminate any 0 differences.
- Note 1: elimniating 0 differences is the big difference between the other tests!
- Note 2: because we are eliminating 0 differences, this means that our sample size will update to the number of pairs with a non-0 difference.
When ranking, we the differences are ranked based on the absolute value of the difference.
We also keep the sign of the difference.
- We will have positive ranks and negative ranks.

X	Y	D	\|D\|	Rank
5	8	-3	3	- 1.5
8	5	3	3	+ 1.5
4	4	0	0	———

Wilcoxon Signed Rank

Hypotheses

H_0: M_d = M_0 | H_0: M_d \le M_0 | H_0: M_d \ge M_0
H_1: M_d \ne M_0 | H_1: M_d > M_0 | H_1: M_d < M_0

Test Statistic & p-Value

T_0 = \min(T+,|T_-|) if two-tailed, T_0 = T_+ if left-tailed, and T_0 = |T_-| if right-tailed.
p = (calculated by R :))

Rejection Region

Reject H_0 if p < \alpha.

Conclusion/Interpretation

[Reject or fail to reject] H_0.
There [is or is not] sufficient evidence to suggest [alternative hypothesis in words].

Wilcoxon Signed Rank

We will again use the wilcox.test() function to perform the test,

wilcox.test(dataset$variable1, dataset$variable2,
       alternative = "alternative",
       mu = hypothesized_value,
       paired = TRUE,
       exact = FALSE)

Wilcoxon Signed Rank

A stock analyst believes the median number of shares traded in Walgreens Boots Alliance (WBA) stock is greater than that in McDonald’s (MCD). Test the analyst’s belief at the \alpha=0.05 level of significance.

stocks <- tibble(WBA = c(8.9, 6.3, 6.2, 7.2, 2.8, 3.3, 23.6, 
                         6.0, 15.6, 5.2, 6.3, 10.1, 4.0, 8.4),
                 MCD = c(8.5, 7.6, 8.3, 10.4, 2.5, 2.6, 3.5, 
                         4.7, 9.0, 6.0, 5.6, 5.0, 4.4, 5.6))

Recall the R syntax,

wilcox.test(dataset$variable1, dataset$variable2,
       alternative = "alternative",
       mu = hypothesized_value,
       paired = TRUE,
       exact = FALSE)

Wilcoxon Signed Rank

From the problem statement: A stock analyst believes the median number of shares traded in Walgreens Boots Alliance (WBA) stock is greater than that in McDonald’s (MCD).

wilcox.test(stocks$WBA, stocks$MCD,
       alternative = "greater",
       paired = TRUE, 
       exact = FALSE)


    Wilcoxon signed rank test with continuity correction

data:  stocks$WBA and stocks$MCD
V = 69, p-value = 0.1575
alternative hypothesis: true location shift is greater than 0

Wilcoxon Signed Rank

Hypotheses
- H_0: \ M_{\text{WBA}} \le M_{\text{MCD}} OR M_d \le 0, where d = \text{WBA} - \text{MCD}
- H_1: \ M_{\text{WBA}} > M_{\text{MCD}} OR M_d > 0
Test Statistic and p-Value
- V_0 = 69
- p = 0.158
Rejection Region
- Reject H_0 if p < \alpha; \alpha = 0.05.
Conclusion/Interpretation
- Fail to reject H_0.
- There is not sufficient evidence to suggest that the median stock shares traded is greater for WBA than for MCD.

Wrap Up

Today we reviewed the Wilcoxon signed rank test.
- Nonparametric alternative to the paired t-test.
This completes Module 1.
Next: comparing three or more groups.