Proposal of multivariate multiple comparison tests with the control treatment
Main Article Content
Abstract
Problems involving comparisons of treatment effectiveness for multivariate responses are common in various fields of knowledge. Typically, methods for comparing vectors of means use the Bonferroni inequality to construct conservative tests, avoiding the complexities of the exact distribution of the maximum test statistic T2 max, the maximum of Hotelling’s T2. In high-dimensional scenarios, traditional methods are not viable as they depend on the inverse of the sample covariance matrix, which becomes singular. To address this problem, Dempster’s trace criterion can be used, and a second alternative is Ahmad’s test statistic Tig, however, in both cases, the Bonferroni inequality is used. Another issue is that both in the estimation process and in the exact distribution of statistics for multiple comparison tests, there is a need to deal with sophisticated and complex numerical methods. These facts make these approximations not readily usable. To try to overcome these challenges, this work proposes multivariate multiple comparison tests with the control treatment using the nonparametric bootstrap method. The performance of the tests was evaluated through experimentwise type I error rates (EER) and power in different scenarios using Monte Carlo simulation and the R program. The results showed that for homoscedastic scenarios, the proposed bootstrap test ATB showed more effective control of EER, in addition to having higher power, regardless of whether the distribution was normal or not, in both low and high-dimensional contexts. Thus, the ATB test proved to be the most recommended alternative in these situations. For heteroscedastic scenarios, it was not possible to identify a clearly superior test, but in several circumstances, the proposed bootstrap tests demonstrated superior performance compared to their respective asymptotic versions.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
References
1. Ahmad, M. R. A unified approach to testing mean vectors with large dimensions. AStA Advances in Statistical Analysis 103, 593–618 (2018).
2. Ahmad, M. R. Multiple comparisons of mean vectors with large dimension under general conditions. Journal of Statistical Computation and Simulation 89, 1044–1059 (2019).
3. Box, G. P. A general distribution theory for a class of likelihood criteria. Biometrika 36, 317–346 (1949).
4. Bretz, F, Hothorn, T &Westfall, P. Multiple comparisons using R 182 (Chapman and Hall, USA, 2011).
5. Dean, A & Voss, D. Design and analysis of experiments 740 (Springer, New Jersey, 1999).
6. Dean, A, Voss, D & Draguljić, D. Design and analysis of experiments 840 (Springer, New York, 2016).
7. Dempster, A. P. A high dimensional two sample significance test. The Annals of Mathematical Statistics 29, 995–1010 (1958).
8. Dempster, A. P. A significance test for the separation of two highly multivariate small samples. Biometrics 16, 41–50 (1960).
9. Gentle, J. E. Random number generation and Monte Carlo methods 2nd ed., 381 (Springer, New York, 2003).
10. Hinkelman, K & Kempthorne, O. Design and analysis of experiments 2nd ed., 632 (JohnWiley and Sons, New Jersey, 2008).
11. Hochberg, Y & Tamhane, A. C. Multiple comparisons procedures 450 (John Wiley and Sons, Canadian, 1987).
12. Hsu, J. C. Multiple comparisons - theory and methods 277 (Chapman and Hall, USA, 1999).
13. Hyodo, M, Takahashi, S & Nishiyama, T. Multiple comparisons among mean vectors when the dimension is larger than the total sample size. Communications in Statistics - Simulation and Computation 43, 2283–2306 (2014).
14. Kakizawa, Y. Multiple comparisons of several heteroscedastic multivariate populations. Statistics and Probability Letters 78, 1328–1338 (2008).
15. Kakizawa, Y. Multiple comparisons of several homoscedastic multivariate populations. Annals of the Institute of Statistical Mathematics 61, 1–26 (2009).
16. Machado, A. A., Silva, J. G. C., Demétrio, C. G. & Ferreira, D. F. in Reunião anual da região brasileira da sociedade internacional de biometria 50, 290 (Simpósio de estatística aplicada a experimentação agronômica, Londrina, 2005).
17. Manly, B. F. J. Randomization, bootstrap and Monte Carlo methods in biology 2nd ed., 356 (University of Otag, New Zealand, 1997).
18. Nascimento, M. C., Braz, L. H. C. & Ferreira, D. F. Proposition of Bootstrap Tests for Comparisons Between Two Independent Mean Vectors in High Dimensionality. Brazilian Journal of Biometrics 43, 1–21. https://doi.org/10.28951/bjb.v43i3.772 (2025).
19. Nishiyama, T, Hyodo,M& Seo, T. Recent developments of multivariate multiple comparisons among mean vectors. SUT Journal of Mathematics 50, 247–270 (2014).
20. Nóbrega, R. S. A. Efeito de sistemas de uso da terra na Amazônia sobre atributos do solo, ocorrência,
eficiência e diversidade de bactérias que nodulam caupi [Vigna unguiculata (L.) Walp] Doutorado (Universidade Federal de Lavras, Lavras, MG, 2006).
21. Oliveira, I. R. C. & Ferreira, D. F. Multivariate extension of chi-squared univariate normality test. Journal of Statistical Computation and Simulation 80, 513–526 (2010).
22. Royston, J. P. Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Applied Statistics - Journal of the Royal Statistical Society - Series C 32, 121–133 (1983).
23. Santos, E. N. F. & Ferreira, D. F. Multivariate multiple comparisons by bootstrap and permutation tests. Revista Brasileira de Biometria 30, 381–400 (2012).
24. Seo, T, Mano, S & Fujikoshi, Y. A generalized Tukey conjecture for multiple comparisons among mean vectors. Journal of the American Statistical Association 89, 676–679 (1994).
25. Seo, T & Nishiyama, T. On the conservative simultaneous confidence procedures for multiple comparisons among mean vectors. Journal of Statistical Planning and Inference 138, 3448–3456 (2008).
26. Staffa, S. J. & Zurakowski, D. Strategies in adjusting for multiple comparisons. A primer for pediatric surgeons. Journal of Pediatric Surgery 55, 1699–1705 (2020).
27. Takahashi, S, Masashi, H., Takahiro, N. & Pavlenko, T. Multiple comparisons procedures for high-dimensional data and their robustness under non-normality. Journal of the Japanese Society of Computational Statistics 26, 71–82 (2013).
28. Westfall, P. H. On using the bootstrap for multiple comparisons. Journal of Biopharmaceutical Statistics 21, 1187–1205 (2011).
29. Westfall, P. H. & Young, S. S. p Value adjustments for multiple tests in multivariate binomial models. Journal of the American Statistical Association 84, 780–786 (1989).