Wilcoxon rank sum test

Wilcoxon rank sum test, also known as the Mann-Whitney U test, is a non-parametric statistical hypothesis test used when comparing two independent samples to assess whether their population mean ranks differ. It is an alternative to the t-test when the data cannot be assumed to be normally distributed. The test is named after Frank Wilcoxon, who introduced it in 1945, and Henry B. Mann and Donald R. Whitney, who extended the theory in 1947.

Overview

The Wilcoxon rank sum test is based on the ranks of the data rather than their numerical values. This approach makes it more robust to outliers and non-normal distributions, providing a useful tool for analyzing ordinal data or continuous data that does not meet the assumptions required for the t-test. The test is applicable for two independent samples of sizes n1 and n2, and it evaluates whether one sample tends to have higher or lower values than the other.

Procedure

To perform the Wilcoxon rank sum test, the following steps are taken:

1. Combine all observations from both groups into a single dataset.
2. Rank all observations from the smallest to the largest, assigning average ranks in case of ties.
3. Calculate the sum of ranks for each of the two groups.
4. Use the smaller of the two rank sums as the test statistic (W).
5. Determine the significance of the observed W by comparing it to the distribution of W under the null hypothesis, which can be approximated using normal approximation for large samples.

Assumptions

The Wilcoxon rank sum test makes several assumptions:

• The samples are independent.
• The data are ordinal, interval, or ratio.
• The distributions of both groups are similar in shape.

Applications

The Wilcoxon rank sum test is widely used in various fields such as psychology, medicine, and ecology, where the assumptions of parametric tests are not met. It is particularly useful for small sample sizes or for data with outliers that could influence the results of parametric tests.

Comparison with Other Tests

The Wilcoxon rank sum test is often compared to the t-test for independent samples. While the t-test is more powerful for data that meet its assumptions (normal distribution of the differences, homogeneity of variances), the Wilcoxon test provides a non-parametric alternative that is less sensitive to outliers and does not require the data to be normally distributed.

Limitations

While the Wilcoxon rank sum test is versatile, it has limitations:

• It is less powerful than the t-test when the assumptions of the t-test are met.
• The interpretation of significant results can be less straightforward, as it relates to medians rather than means.
• It requires at least 5 observations in each sample to provide reliable results.

See Also

• Kruskal-Wallis test
• Friedman test
• Non-parametric statistics