Introduction to the Mann-Whitney U-test: An alternative to the independent T-test in R

In statistics, hypothesis tests are a crucial tool for determining whether differences between groups are significant. A widely used test for investigating differences between two independent groups is the independent T-test. However, this test assumes that the data is normally distributed and that the variances in the groups are equal. If these assumptions are not met, the Mann-Whitney U test (also known as the Wilcoxon rank sum test) can provide a non-parametric alternative.

In this blog post, I will introduce you to the Mann-Whitney U test and show you how to use it in R. We will discuss the prerequisites of the test and give practical examples of how to perform the test in R.

What is the Mann-Whitney U test?

The Mann-Whitney U test is a non-parametric test used to investigate whether two independent samples come from populations with different distributions. It tests the null hypothesis that the distributions of the two groups are the same. In contrast to the T-test, the Mann-Whitney U-test does not require a normal distribution assumption, which makes it particularly useful if the data is asymmetrically distributed or contains outliers.

Requirements for the Mann-Whitney U test

Before using the Mann-Whitney U test, the following requirements should be met:

- Independence: The observations in the two groups must be independent of each other.

- Ordinal scale or continuous data: The data must be measured on an ordinal scale or higher.

Installing and loading the required packages

To perform the Mann-Whitney U test in R, you need the stats package, which is already included in the basic installation of R. You can simply load it with the following command:

# Paket laden
library(stats)

However, if you want to work with extended outputs, the coin package can also be helpful:

# Installieren und Laden des coin Pakets
install.packages("coin")
library(coin)

Example data set

Let's assume we have a data set df that contains the weight and gender of 100 people. The Mann-Whitney U-test is used to check whether the weight differs significantly between the two genders.

# Beispiel-Datensatz erstellen
set.seed(123)
df <- data.frame(
  Geschlecht = factor(sample(c("Männlich", "Weiblich"), 100, replace = TRUE)),
  Gewicht = rnorm(100, mean = 70, sd = 15)
)

Carrying out the Mann-Whitney U test in R

To perform the Mann-Whitney U test in R, we use the wilcox.test function. Suppose we want to investigate whether the weight differs significantly between the two sexes:

# Mann-Whitney-U-Test
wilcox.test(Gewicht ~ Geschlecht, data = df)

This command displays the test result, the U-value and the p-value, which you can use to assess whether there is a significant difference between the groups.

Interpretation of the result

The test result could look like this:

Wilcoxon rank sum test with continuity correction

data:  Gewicht by Geschlecht
W = 1245, p-value = 0.045
alternative hypothesis: true location shift is not equal to 0

- W-value: This is the U-value (also known as W) of the test.

- p-value: The p-value indicates whether the difference is significant. In this case, the p-value is 0.045, which indicates that there is a significant difference in weight between the genders at a significance level of 5 % (α = 0.05).

- Alternative hypothesis: The output shows that the alternative hypothesis is that the distribution of the two groups is different.

Extended edition with coin

If you need more precise results and confidence intervals, you can use the coin package, which offers an extended version of the Mann-Whitney U test:

# Erweiterter Mann-Whitney-U-Test mit coin
wilcox_test(Gewicht ~ Geschlecht, data = df, distribution = "exact")

Conclusion

The Mann-Whitney U test is a powerful alternative to the independent T test when the assumptions of normal distribution or variance homogeneity are not met. In R, the test is quick and easy to use, and it provides a robust method for analyzing differences between groups in non-normally distributed data.

Whether for research, clinical studies or other statistical analyses, the Mann-Whitney U test should be part of your statistical toolbox, especially if you are working with non-normally distributed data.