Introduction:

The Mann-Whitney U test is a powerful tool used to compare the central tendency of two populations based on samples of unrelated data. While the Student’s t-test is commonly employed in such scenarios, authors are often unaware of its limitations when the variances differ between the underlying populations. In this article, we will explore the advantages of the Mann-Whitney U test and why it can be a more reliable choice when variances are unequal.

Table of Contents

## The Pitfall of Unequal Variances

When the variances of the populations being compared are unequal, the Student’s t-test may produce inaccurate results. In contrast, the Mann-Whitney U test offers a viable alternative in these situations. This test ensures that the Type I error rate, a crucial factor in statistical analysis, remains close to the nominal 5% value.

### Unleashing the Power of the Mann-Whitney U Test

Even when variances are identical, the Mann-Whitney U test performs as effectively as the Student’s t-test in terms of controlling Type I errors. Moreover, its power is similar to that of the t-test, proving its competence in controlling both Type I and Type II error rates when the underlying distributions are normal. Several studies support these findings, highlighting the reliability and robustness of the Mann-Whitney U test (e.g., Moser et al. 1989; Moser and Stevens 1992; Coombs et al. 1996).

### Handling Variances Effectively

To make an informed decision on whether to employ the unequal variance t-test or the Student’s t-test, it is advisable to conduct an initial test for homogeneity of variance. However, relying on the outcome of this test to choose the subsequent analysis can lead to less effective control of Type I error rates. Therefore, it is generally recommended to perform the unequal variance t-test consistently, unless there are logical, physical, or biological grounds to assume that the variances are likely identical for the populations under study.

### The Impact of Normality Assumptions

While the Mann-Whitney U test has proven effective in controlling Type I errors when variances are unequal, it may not be more reliable than the Student’s t-test if the assumption of normality in the underlying populations is violated. However, Zimmerman and Zumbo (1993) argue that performing the unequal variance t-test on ranked data can yield results comparable to the Mann-Whitney U test, even when variances are unequal. Moreover, there are alternative tests that exhibit greater robustness to non-normality (e.g., Coombs et al. 1996; Keselman et al. 2004), but the unequal variance t-test remains a recommended choice due to its combination of performance and ease of use.

### Nomenclature and References

The unequal variance t-test is primarily known by this name in the literature. However, it is also referred to as the Welch test, the Welch Approximate Degrees of Freedom (APDF) test, or the Smith/Welch/Satterthwaite test. The decision to pool variances becomes crucial not only when comparing two groups but also when analyzing multiple groups in an analysis of variance. Consideration should be given to this decision in constructing randomization tests as well.

## In Conclusion: A Step-by-Step Summary

When comparing the central tendency of two populations based on samples of unrelated data, the unequal variance t-test is the preferred choice over the Student’s t-test or the Mann-Whitney U test. Begin by examining the distributions of the two samples graphically. If non-normality is evident, rank the data before performing the unequal variance t-test. Draw conclusions based on this test alone, ignoring other simultaneous tests performed by statistical packages. When presenting the outcome, provide a suitable reference for the adoption of the test, including its formulation, mean, variance, number of samples, calculated tâ€² value, degrees of freedom (v), and P value.

References