What’s one-sample Wilcoxon signed rank test?

The one-sample Wilcoxon signed rank test is a non-parametric alternative to 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. theoretical value).

Note that, the data should be distributed symmetrically around the median. In other words, there should be roughly the same number of values above and below the median.

Research questions and statistical hypotheses

Typical research questions are:

1. whether the median ($$m$$) of the sample is equal to the theoretical value ($$m_0$$)?
2. whether the median ($$m$$) of the sample is less than to the theoretical value ($$m_0$$)?
3. whether the median ($$m$$) of the sample is greater than to the theoretical value($$m_0$$)?

In statistics, we can define the corresponding null hypothesis ($$H_0$$) as follow:

1. $$H_0: m = m_0$$
2. $$H_0: m \leq m_0$$
3. $$H_0: m \geq m_0$$

The corresponding alternative hypotheses ($$H_a$$) are as follow:

1. $$H_a: m \ne m_0$$ (different)
2. $$H_a: m > m_0$$ (greater)
3. $$H_a: m < m_0$$ (less)

Note that:

• Hypotheses 1) are called two-tailed tests
• Hypotheses 2) and 3) are called one-tailed tests

Visualize your data and compute one-sample Wilcoxon test in R

Install ggpubr R package for data visualization

You can draw R base graphs as described at this link: R base graphs. Here, we’ll use the ggpubr R package for an easy ggplot2-based data visualization

# Install
if(!require(devtools)) install.packages("devtools")
devtools::install_github("kassambara/ggpubr")
• Or, install from CRAN as follow:
install.packages("ggpubr")

R function to compute one-sample Wilcoxon test

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

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

• 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”.

1. Prepare your data as specified here: Best practices for preparing your data set for R

2. Save your data in an external .txt tab or .csv files

3. Import your data into R as follow:

# If .txt tab file, use this
# Or, if .csv file, use this
my_data <- read.csv(file.choose())

Here, we’ll use an example data set containing the weight of 10 mice.

We want to know, if the median weight of the mice differs from 25g?

set.seed(1234)
my_data <- data.frame(
name = paste0(rep("M_", 10), 1:10),
weight = round(rnorm(10, 20, 2), 1)
)

# Print the first 10 rows of the data
head(my_data, 10)
   name weight
1   M_1   17.6
2   M_2   20.6
3   M_3   22.2
4   M_4   15.3
5   M_5   20.9
6   M_6   21.0
7   M_7   18.9
8   M_8   18.9
9   M_9   18.9
10 M_10   18.2
# Statistical summaries of weight
summary(my_data$weight)  Min. 1st Qu. Median Mean 3rd Qu. Max. 15.30 18.38 18.90 19.25 20.82 22.20  • Min.: the minimum value • 1st Qu.: The first quartile. 25% of values are lower than this. • Median: the median value. Half the values are lower; half are higher. • 3rd Qu.: the third quartile. 75% of values are higher than this. • Max.: the maximum value Visualize your data using box plots library(ggpubr) ggboxplot(my_data$weight,
ylab = "Weight (g)", xlab = FALSE,
ggtheme = theme_minimal())

Compute one-sample Wilcoxon test

We want to know, if the average weight of the mice differs from 25g (two-tailed test)?

# One-sample wilcoxon test
res <- wilcox.test(my_data$weight, mu = 25) # Printing the results res   Wilcoxon signed rank test with continuity correction data: my_data$weight
V = 0, p-value = 0.005793
alternative hypothesis: true location is not equal to 25
# print only the p-value
res$p.value [1] 0.005793045 The p-value of the test is 0.005793, which is less than the significance level alpha = 0.05. We can reject the null hypothesis and conclude that the average weight of the mice is significantly different from 25g with a p-value = 0.005793. Note that: • if you want to test whether the median weight of mice is less than 25g (one-tailed test), type this: wilcox.test(my_data$weight, mu = 25,
alternative = "less")
• Or, if you want to test whether the median weight of mice is greater than 25g (one-tailed test), type this:
wilcox.test(my_data\$weight, mu = 25,
alternative = "greater")

One-sample t-test (parametric)

Infos

This analysis has been performed using R software (ver. 3.2.4).