TITLE:
MVN Q-Test II: A Comparison of the MVN H-Test with the Chi-Square Approximation and Bootstrap Versions of the Q-Test
AUTHORS:
José Moral de la Rubia
KEYWORDS:
Multivariate Normality, Parametric Bootstrap, Non-Parametric Bootstrap, Bootstrap P-Value, Bootstrap Power
JOURNAL NAME:
Journal of Data Analysis and Information Processing,
Vol.13 No.4,
September
26,
2025
ABSTRACT: In a previous article, an R script was developed and divided into three parts to implement the multivariate normality (MVN) Q-test based on both the chi-square approximation and the bootstrap approach, using either the Shapiro-Wilk W statistic (QSWa and QSWb) or the Shapiro-Francia W’ statistic (QSFa and QSFb). Royston’s H-test was included as a supplementary MVN test. The aim of this study is to compare the hit rate and statistical power of the four Q-test variants and the H-test using 200 samples drawn from multivariate standard normal distributions and 200 samples from multivariate t-distributions with five degrees of freedom. The simulations vary in sample size (50, 75, 100, 125, 150, 200, 250, and 500), number of variables (from 2 to 6), and homogeneous inter-variable correlation (0, 0.3, 0.5, 0.7, and 0.9). The H-test outperformed QSWb and QSFb, but not QSWa in the multivariate normal samples or QSFa in the multivariate t-distribution samples. QSFb performed better than QSWb. It is concluded that the bootstrap approach is conservative under the null hypothesis of multivariate normality. However, when the assumption of independence is violated, the bootstrap approach is theoretically more appropriate than QSWa and QSFa. A 10% significance level is recommended for QSFb in terms of hit rate, but in terms of statistical power, only when rejecting the null hypothesis.