A New Approach to the Nonparametric Behrens-Fisher Problem With Compatible Confidence Intervals

Area of Science:

• Statistics
• Nonparametric statistics
• Hypothesis testing

Background:

• The Behrens-Fisher problem traditionally assumes equal variances, limiting its application.
• Existing nonparametric methods may struggle with unequal distribution functions and small sample sizes.
• Accurate statistical inference is crucial in diverse fields, including clinical trials.

Purpose of the Study:

• To introduce a novel nonparametric method for the Behrens-Fisher problem accommodating unequal distribution functions.
• To test the null hypothesis concerning the Mann-Whitney effect, .
• To develop range-preserving compatible confidence intervals with improved coverage.

Main Methods:

• The proposed test utilizes the ratio of the true variance of the Mann-Whitney effect estimator to its theoretical maximum, based on the Birnbaum-Klose inequality.
• No restrictions are imposed on the underlying data distributions, except for trivial one-point distributions.
• Simulations were conducted to evaluate type-I error rates and confidence interval coverage under various conditions.

Main Results:

• The new method effectively controls the type-I error rate across different scenarios, including small and unbalanced sample sizes.
• It demonstrates superior type-I error control compared to the Brunner-Munzel test, particularly at stringent significance levels (e.g., $\alpha =0.005$).
• The constructed confidence intervals exhibit enhanced coverage accuracy relative to those compatible with the Brunner-Munzel test.

Conclusions:

• The proposed nonparametric test offers a robust and accurate solution for the Behrens-Fisher problem with unequal distributions.
• It provides a valuable alternative to existing methods, especially when strict error control is required.
• The method's practical utility is highlighted through its application in a clinical trial example.