An Application of the Theory of Spherical Distributions in Multiple Mean Comparison

  • Jiajuan LiangEmail author
  • Man-Lai Tang
  • Jing Yang
  • Xuejing Zhao


Multiple normal mean comparison without the equal-variance assumption is frequently encountered in medical and biological problems. Classical analysis of variance (ANOVA) requires the assumption of equal variances across groups. When variations across groups are found to be different, classical ANOVA method is essentially inapplicable for multiple mean comparison. Although various approximation methods have been proposed to solve the classical Behrens-Fisher problem, there exists computational complexity in approximating the null distributions of the proposed tests. In this paper we employ the theory of spherical distributions to construct a class of exact F-tests and a simple generalized F-test for multiple mean comparison. The methods in this paper actually provide a simple exact solution and a simple approximate solution to the classical Behrens-Fisher problem in the case of balanced sample designs. A simple Monte Carlo study shows that the recommended tests have fairly good power performance. An analysis on a real medical dataset illustrates the application of the new methods in medicine.



The authors would like to thank Prof. Zengrong Sun and her research associates in Tianjin Medical University, China, for providing the real medical data in gene comparisons under different experimental conditions.


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Jiajuan Liang
    • 1
    Email author
  • Man-Lai Tang
    • 2
  • Jing Yang
    • 3
  • Xuejing Zhao
    • 4
  1. 1.University of New HavenWest HavenUSA
  2. 2.Hang Seng Management CollegeHong KongChina
  3. 3.Tianjin Medical UniversityTianjinChina
  4. 4.Lanzhou UniversityLanzhouChina

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