Wilcoxon Signed Rank Test in R Programming

The Wilcoxon signed rank test is a non-parametric method used to compare two related groups. It works well when we have matched data or repeated measurements from the same group and want to see if there is a meaningful difference between them. This test is often used as an alternative to the paired t-test, especially when the data does not follow a normal distribution.

In simple terms, the Wilcoxon test helps us find out if the differences between two sets of related data are likely due to chance or if they suggest a real change.

Types of Wilcoxon Signed-Rank Test in R

This test can be divided into two parts: 

• One-Sample Wilcoxon Signed Rank Test
• Paired Samples Wilcoxon Test

1. One-Sample Wilcoxon Signed Rank Test

The one-sample Wilcoxon signed-rank test is a non-parametric alternative to a one-sample t-test , when the data cannot be assumed to be normally distributed. It’s used to determine whether the median of the sample is equal to a known standard value (theoretical value) .

Implementation in R

To perform a one-sample Wilcoxon test, R provides a function wilcox.test() that can be used as follow:

Syntax: wilcox.test(x, mu = 0, alternative = "two.sided")

Parameters: 

• x: a numeric vector containing your data values
• mu: the theoretical mean/median value. Default is 0 but you can change it.
• alternative: the alternative hypothesis. Allowed value is one of “two.sided” (default), “greater” or “less”.

Example: Is the median weight of the rabbit differs from 25g? 

set.seed(1234)
myData <- data.frame(
  name = paste0("R_", 1:10),
  weight = round(rnorm(10, 30, 2), 1)
)

print(myData)

result <- wilcox.test(myData$weight, mu = 25)

print(result)

Output: 

name weight
1 R_1 27.6
2 R_2 30.6
3 R_3 32.2
4 R_4 25.3
5 R_5 30.9
6 R_6 31.0
7 R_7 28.9
8 R_8 28.9
9 R_9 28.9
10 R_10 28.2

Wilcoxon signed rank test with continuity correction

data: myData$weight
V = 55, p-value = 0.005793
alternative hypothesis: true location is not equal to 25

Since the p-value of the test is 0.005793, which is less than the significance level alpha = 0.05, we can reject the null hypothesis and conclude that the average weight of the rabbit is significantly different from 25g.

To test whether the median is less than 25g,

wilcox.test(myData$weight, mu = 25, alternative = "less")

Output: 

Wilcoxon signed rank exact test

data: myData$weight
V = 55, p-value = 0.9979
alternative hypothesis: true location is less than 25

To test whether the median is greater than 25g,

wilcox.test(myData$weight, mu = 25, alternative = "greater")

Output: 

Wilcoxon signed rank exact test

data: myData$weight
V = 55, p-value = 0.002897
alternative hypothesis: true location is less than 25

2. Paired Samples Wilcoxon Test in R

The paired samples Wilcoxon test is a non-parametric alternative to paired t-test used to compare paired data. It’s used when you have two related measurements for the same subjects, like before and after a treatment. It compares whether there is a significant shift in the median.

Implementation in R

To perform Paired Samples Wilcoxon-test, the R provides a function wilcox.test() that can be used as follow:

Syntax: wilcox.test(x, y, paired = TRUE, alternative = "two.sided")

Parameters: 

• x, y: numeric vectors
• paired: a logical value specifying that we want to compute a paired Wilcoxon test
• alternative: the alternative hypothesis. Allowed value is one of “two.sided” (default), “greater” or “less”.

Example: Compare weights before and after treatment

before <- c(190.1, 190.9, 172.7, 213, 231.4,
            196.9, 172.2, 285.5, 225.2, 113.7)
            
after <- c(392.9, 313.2, 345.1, 393, 434,
           227.9, 422, 383.9, 392.3, 352.2)

result <- wilcox.test(before, after, paired = TRUE)

print(result)

Output: 

Wilcoxon signed rank test

data: before and after
V = 0, p-value = 0.001953
alternative hypothesis: true location shift is not equal to 0

In the above output, the p-value of the test is 0.001953, which is less than the significance level alpha = 0.05. We can conclude that the median weight of the mice before treatment is significantly different from the median weight after treatment with a p-value = 0.001953.

To check if weight before treatment is less than after: 

wilcox.test(before, after, paired = TRUE, alternative = "less")

Output: 

Wilcoxon signed rank test

data: weight by group
V = 55, p-value = 0.0009766
alternative hypothesis: true location shift is less than 0

To check if weight before treatment is greater than after:

wilcox.test(before, after, paired = TRUE, alternative = "greater")

Output: 

Wilcoxon signed rank test

data: weight by group
V = 55, p-value = 1
alternative hypothesis: true location shift is greater than 0

In this article, we performed the Wilcoxon Signed Rank Test in R for both one-sample and paired sample scenarios, along with practical examples and interpretation of results.