The **Wilcoxon signed-rank test** is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to estimate whether their population mean ranks differ e.g it is a paired difference test. It can be applied as an alternative to the paired Student’s t-test also known as “t-test for matched pairs” or “t-test for dependent samples” when the distribution of the difference between the two samples’ means cannot be assumed to be normally distributed. A Wilcoxon signed-rank test is a nonparametric test that can be used to determine whether two dependent samples were selected from populations having the same distribution. This test can be divided into two parts:

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

### 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 i.e. a theoretical value. In R Language one can perform this test very easily.

#### Implementation in R

To perform 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:**

Here, let’s use an example data set containing the weight of 10 rabbits. Let’s know if the median weight of the rabbit differs from 25g?

`# R program to illustrate ` `# one-sample Wilcoxon signed-rank test ` ` ` `# The data set ` `set` `.seed(` `1234` `) ` `myData ` `=` `data.frame( ` ` ` `name ` `=` `paste0(rep(` `"R_"` `, ` `10` `), ` `1` `:` `10` `), ` ` ` `weight ` `=` `round` `(rnorm(` `10` `, ` `30` `, ` `2` `), ` `1` `) ` `) ` ` ` `# Print the data ` `print` `(myData) ` ` ` `# One-sample wilcoxon test ` `result ` `=` `wilcox.test(myData$weight, mu ` `=` `25` `) ` ` ` `# Printing the results ` `print` `(result) ` |

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**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

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

If one wants to test whether the median weight of the rabbit is less than 25g (one-tailed test), then the code will be:

`# R program to illustrate ` `# one-sample Wilcoxon signed-rank test ` ` ` `# The data set ` `set` `.seed(` `1234` `) ` `myData ` `=` `data.frame( ` ` ` `name ` `=` `paste0(rep(` `"R_"` `, ` `10` `), ` `1` `:` `10` `), ` ` ` `weight ` `=` `round` `(rnorm(` `10` `, ` `30` `, ` `2` `), ` `1` `) ` `) ` ` ` `# One-sample wilcoxon test ` `wilcox.test(myData$weight, mu ` `=` `25` `, ` ` ` `alternative ` `=` `"less"` `) ` ` ` `# Printing the results ` `print` `(result) ` |

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**Output:**

Wilcoxon signed rank test with continuity correction data: myData$weight V = 55, p-value = 0.9979 alternative hypothesis: true location is less than 25

Or, If one wants to test whether the median weight of the rabbit is greater than 25g (one-tailed test), then the code will be:

`# R program to illustrate ` `# one-sample Wilcoxon sign-rank test ` ` ` `# The data set ` `set` `.seed(` `1234` `) ` `myData ` `=` `data.frame( ` ` ` `name ` `=` `paste0(rep(` `"R_"` `, ` `10` `), ` `1` `:` `10` `), ` ` ` `weight ` `=` `round` `(rnorm(` `10` `, ` `30` `, ` `2` `), ` `1` `) ` `) ` ` ` `# One-sample wilcoxon test ` `wilcox.test(myData$weight, mu ` `=` `25` `, ` ` ` `alternative ` `=` `"greater"` `) ` ` ` `# Printing the results ` `print` `(result) ` |

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**Output:**

Wilcoxon signed rank test with continuity correction data: myData$weight V = 55, p-value = 0.002897 alternative hypothesis: true location is greater than 25

### 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 data are not normally distributed.

#### 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:**

Here, let’s use an example data set, which contains the weight of 10 rabbits before and after the treatment. We want to know, if there is any significant difference in the median weights before and after treatment?

`# R program to illustrate ` `# Paired Samples Wilcoxon Test ` ` ` `# The data set ` `# Weight of the rabbit before treatment ` `before <` `-` `c(` `190.1` `, ` `190.9` `, ` `172.7` `, ` `213` `, ` `231.4` `, ` ` ` `196.9` `, ` `172.2` `, ` `285.5` `, ` `225.2` `, ` `113.7` `) ` ` ` `# Weight of the rabbit after treatment ` `after <` `-` `c(` `392.9` `, ` `313.2` `, ` `345.1` `, ` `393` `, ` `434` `, ` ` ` `227.9` `, ` `422` `, ` `383.9` `, ` `392.3` `, ` `352.2` `) ` ` ` `# Create a data frame ` `myData <` `-` `data.frame( ` ` ` `group ` `=` `rep(c(` `"before"` `, ` `"after"` `), each ` `=` `10` `), ` ` ` `weight ` `=` `c(before, after) ` `) ` ` ` `# Print all data ` `print` `(myData) ` ` ` `# Paired Samples Wilcoxon Test ` `result ` `=` `wilcox.test(before, after, paired ` `=` `TRUE) ` ` ` `# Printing the results ` `print` `(result) ` |

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**Output:**

group weight 1 before 190.1 2 before 190.9 3 before 172.7 4 before 213.0 5 before 231.4 6 before 196.9 7 before 172.2 8 before 285.5 9 before 225.2 10 before 113.7 11 after 392.9 12 after 313.2 13 after 345.1 14 after 393.0 15 after 434.0 16 after 227.9 17 after 422.0 18 after 383.9 19 after 392.3 20 after 352.2 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.

If one wants to test whether the median weight before treatment is less than the median weight after treatment, then the code will be:

`# R program to illustrate ` `# Paired Samples Wilcoxon Test ` ` ` `# The data set ` `# Weight of the rabbit before treatment ` `before <` `-` `c(` `190.1` `, ` `190.9` `, ` `172.7` `, ` `213` `, ` `231.4` `, ` ` ` `196.9` `, ` `172.2` `, ` `285.5` `, ` `225.2` `, ` `113.7` `) ` ` ` `# Weight of the rabbit after treatment ` `after <` `-` `c(` `392.9` `, ` `313.2` `, ` `345.1` `, ` `393` `, ` `434` `, ` ` ` `227.9` `, ` `422` `, ` `383.9` `, ` `392.3` `, ` `352.2` `) ` ` ` `# Create a data frame ` `myData <` `-` `data.frame( ` ` ` `group ` `=` `rep(c(` `"before"` `, ` `"after"` `), each ` `=` `10` `), ` ` ` `weight ` `=` `c(before, after) ` `) ` ` ` `# Paired Samples Wilcoxon Test ` `result ` `=` `wilcox.test(weight ~ group, ` ` ` `data ` `=` `myData, ` ` ` `paired ` `=` `TRUE, ` ` ` `alternative ` `=` `"less"` `) ` ` ` `# Printing the results ` `print` `(result) ` |

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**Output:**

Wilcoxon signed rank test data: weight by group V = 55, p-value = 1 alternative hypothesis: true location shift is less than 0

Or, If one wants to test whether the median weight before treatment is greater than the median weight after treatment, then the code will be:

`# R program to illustrate ` `# Paired Samples Wilcoxon Test ` ` ` `# The data set ` `# Weight of the rabbit before treatment ` `before <` `-` `c(` `190.1` `, ` `190.9` `, ` `172.7` `, ` `213` `, ` `231.4` `, ` ` ` `196.9` `, ` `172.2` `, ` `285.5` `, ` `225.2` `, ` `113.7` `) ` ` ` `# Weight of the rabbit after treatment ` `after <` `-` `c(` `392.9` `, ` `313.2` `, ` `345.1` `, ` `393` `, ` `434` `, ` ` ` `227.9` `, ` `422` `, ` `383.9` `, ` `392.3` `, ` `352.2` `) ` ` ` `# Create a data frame ` `myData <` `-` `data.frame( ` ` ` `group ` `=` `rep(c(` `"before"` `, ` `"after"` `), each ` `=` `10` `), ` ` ` `weight ` `=` `c(before, after) ` `) ` ` ` `# Paired Samples Wilcoxon Test ` `result ` `=` `wilcox.test(weight ~ group, ` ` ` `data ` `=` `myData, ` ` ` `paired ` `=` `TRUE, ` ` ` `alternative ` `=` `"greater"` `) ` ` ` `# Printing the results ` `print` `(result) ` |

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**Output:**

Wilcoxon signed rank test data: weight by group V = 55, p-value = 0.0009766 alternative hypothesis: true location shift is greater than 0

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