Rank sum test

Rank sum test is a non-parametric statistical hypothesis test used to compare the distributions of two independent samples. It is particularly useful when the assumption of normality cannot be met for the data being analyzed. The rank sum test is often referred to by specific names depending on the context or slight variations in the method, such as the Mann-Whitney U test or the Wilcoxon rank-sum test.

Overview

The rank sum test operates under the null hypothesis that the two samples come from the same distribution. Unlike parametric tests, which rely on the data following a specific distribution (usually normal distribution), rank sum tests only require the data to be ordinal or continuous. This makes the rank sum test a robust alternative to the t-test when the assumptions of the latter cannot be satisfied.

Procedure

The basic steps of conducting a rank sum test are as follows:

1. Combine all observations from both groups into a single dataset.
2. Rank all observations from the smallest to the largest. In the case of ties (identical values), assign to each tied value the average of the ranks they would have received had they not been tied.
3. Calculate the sum of ranks for each of the original groups.
4. Use the rank sums to calculate the test statistic. The specific form of the statistic depends on the version of the test being used (e.g., Mann-Whitney U or Wilcoxon rank-sum).
5. Determine the significance of the observed statistic under the null hypothesis, often through comparison with tabulated values or computation of a p-value.

Applications

Rank sum tests are widely used in various fields such as medicine, psychology, and ecology, where the assumptions of parametric tests are often violated. They are particularly favored for small sample sizes or when the data exhibit skewness or outliers.

Comparison with Other Tests

The rank sum test is often compared to the t-test, which is used for similar purposes but under the assumption of normally distributed data. While the t-test is more powerful under its assumptions, the rank sum test provides a more flexible and robust alternative in many practical situations.

Limitations

While the rank sum test is versatile, it is not without limitations. It is less powerful than parametric tests when their assumptions are met, and its non-parametric nature means it can only assess differences in median rather than mean values. Additionally, it does not provide information about the magnitude of the difference between groups, only the likelihood that a difference exists.

See Also

• Non-parametric statistics
• Mann-Whitney U test
• Wilcoxon signed-rank test
• Kruskal-Wallis test