Proposition of bootstrap tests for comparisons between two independent mean vectors in high dimensionality - Carvalho Nascimento, Miguel; Costa Braz, Lúcia Helena; Furtado Ferreira, Daniel (Brazilian Journal of Biometrics, 2025)
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.