The Relative Power of the Wilcoxon-Mann-Whitney Test and Student t Test Under Simple Bounded Transformations

Area of Science:

• Statistics
• Statistical Methods
• Data Analysis

Background:

• Parametric tests, such as the Student t test, can be less powerful than nonparametric rank-based methods (e.g., Wilcoxon-Mann-Whitney test) when dealing with heavy-tailed distributions.
• Heavy-tailed distributions (e.g., exponential, Cauchy, mixed-normal) are common in various scientific fields and are susceptible to the influence of outliers.

Purpose of the Study:

• To investigate the underlying reasons for the power advantage of nonparametric rank-based tests in heavy-tailed distributions.
• To explore alternative data transformation methods that could improve the power of parametric tests.

Main Methods:

• Computer simulations were employed to compare statistical test performance.
• Random samples were drawn from heavy-tailed distributions.
• Sample values were transformed into bounded sets, including a set with random components, and then analyzed using the Student t test.

Main Results:

• The power advantage of rank-based tests stems from reducing the influence of outliers through rank transformation.
• Applying the Student t test to transformed data (bounded sets, not ranks) significantly increased its power compared to using original data.
• The transformed-data Student t test achieved power comparable to the Wilcoxon-Mann-Whitney test in most simulations.

Conclusions:

• Transforming data to bounded sets, even without preserving order, can mitigate outlier effects and enhance parametric test power.
• This transformation offers a viable alternative to rank-based methods for improving statistical power with heavy-tailed data.
• The findings suggest that reducing outlier influence is key to improving test performance in such distributions.