Proposition of Bootstrap Tests for Comparisons Between Two Independent Mean Vectors in High Dimensionality
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Abstract
Inference regarding the comparison of mean vectors between two independent populations is of great interest in applied fields, especially in scenarios where high-dimensional data analyses are common. In low-dimensional cases with the multivariate Behrens-Fisher problem, there are numerous solutions, but most test statistics have asymptotic distributions. In multivariate procedures, a problem arises when the number of variables, p, is greater than or equal to the sample size, n. In this case, it is not possible to use the few existing methods, as they rely on the inverse of the sample covariance matrix, which cannot be obtained in this situation (p ≥ n) since the covariance matrix is singular. In most cases, asymptotic tests are very liberal, particularly in small samples and specifically in multivariate cases when the dimensionality is high. The bootstrap method is one of the main computationally intensive methods, where its key advantage is that it does not require knowledge of the population probability distribution. Additionally, when the conditions assumed for the application of a test are violated, the bootstrap makes the problem extremely simple to address. Based on this, the present study aimed to propose multivariate comparison tests between two independent mean vectors: the Ahmad Bootstrap Test (ABT) and the Hyodo, Takahashi,and Nishiyama Bootstrap Test (HTNBT), in high-dimensional settings, for balanced or unbalanced, nonnormal and normal data, under the multivariate Behrens-Fisher problem. The performance of these tests was evaluated and compared with tests indicated by the literature, namely Hotelling’s T2, the modified Nel and Merwe (MNV) test proposed by Krishnamoorthy and Yu, the test proposed by Ahmad (AT), and the test proposed by Hyodo, Takahashi, and Nishiyama (HTNT), using Monte Carlo simulation. Power and Type I error rate were considered as evaluation measures. Comparisons were conducted in various scenarios, such as cases of homoscedasticity and heteroscedasticity of covariance matrices, in low and high dimensionality for multivariate normal, t with 3 degrees of freedom, and uniform (0, 1) distributions. In other words, scenarios in which the conditions assumed for the application of most tests are violated. The results showed that the ATB test was generally robust and consistent compared to its competitors in most evaluated situations, while the HTNTB test was strongly conservative and had low power.
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References
1. Ahmad, M. R. A unified approach to testing mean vectors with large dimensions. AStA Advances in Statistical Analysis 103, 593–618 (2018). https://doi.org/10.1007/s10182-018-00343-z
2. Bennett, B. M.Note on a solution of generalized Behrens-Fisher problem. Annals of the Institute of Statistical Mathematics 2, 87–90 (1951). https://doi.org/10.17654/TS054010021
3. Dempster, A. P. A high dimensional two sample significance test. The Annals of Mathematical Statistics 29, 995–1010 (1958). DOI: 10.1214/aoms/1177706437
4. Dempster, A. P. A significance test for the separation of two highly multivariate small samples. Biometrics 16, 41–50 (1960).
5. Ferreira, D. F. Estatística multivariada 3rd ed., 622 (Editora UFLA, Lavras, 2018).
6. Gebert, D. M. P. Uma solução via bootstrap paramétrico para o problema de Behrens-Fisher multivariado. Universidade Federal de Lavras, 120 (2014).
7. 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). https://doi.org/10.1080/03610918.2012.762387
8. James, G. S. Tests of linear hypotheses in univariate and multivariate analysis when the ratios of the populations variances are unknown. Biometrika 41, 19–43 (1954). https://doi.org/10.1093/biomet/41.1-2.19
9. Johansen, S. The Welch-James approximation to the distribution of the residual sum of squares in a weighted linear regression. Biometrika 67, 85–92 (1980). https://doi.org/10.1093/biomet/67.1.85
10. Krishnamoorthy, K. & Yu, J. Modified Nel and Van der Merwe test for the multivariate Behrens-Fisher problem. Statistics and Probability Letters 15, 161–169 (2004). https://doi.org/10.1016/j.spl.2003.10.012
11. Nel, D. G. & Van der Merwe, C. A. A solution to the multivariate Behrens-Fisher problem. Communication in Statistics - Theory and Methods 15, 3719–3735 (1986). https://doi.org/10.1080/03610928608829342
12. Nishiyama, T, Hyodo,M& Seo, T. Recent developments of multivariate multiple comparisons among mean vectors.. SUT Journal of Mathematics 50, 247–270 (2014). DOI: 10.55937/sut/1424793883
13. 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). https://doi.org/10.1080/00949650902731377
14. Silva, R. B. V., Ferreira, D. F. & Nogueira, D. A. Robustness of asymptotic and bootstrap tests for multivariate homogeneity of covariance matrices. Ciência e Agrotecnologia 32, 157–166 (2008).
15. 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). http://dx.doi.org/10.5183/jjscs.1211001_202
16. Yao, Y. An approximate degrees of freedomsolution to the Behrens-Fisher problem. Biometrika 52, 139–147 (1965). https://doi.org/10.1093/biomet/52.1-2.139