MVN Q-Test I: A Bootstrap-Based Implementation in R

In 2023, a multivariate normality test based on a chi-square approximation was developed. This method assumes independence among Gaussian random variables, and defines the test statistic, denoted by Q, as the sum of squared values. This study aims to develop R scripts that implement the Q-test for multivariate normality using either the Shapiro-Wilk W statistic (QSWa) or the Shapiro-Francia W’ statistic (QSFa). A bootstrap version of the Q-test (QSWb and QSFb), which does not assume independence, is also included. Additionally, it incorporates Royston’s H-test. The use of the scripts is illustrated with a sample of 50 participants assessed on a variable across four yearly administrations. The sampling distribution generated by the bootstrap method differs from the chi-square distribution and corresponds to a generalized chi-square distribution—namely, the distribution of a sum of squares of correlated variables. This distribution is less peaked and has a heavier right tail than the chi-square distribution. It is concluded that the bootstrap approach is conservative under the null hypothesis of multivariate normality; however, it is theoretically more appropriate than the chi-square approximation. To approximate the distributions of the two versions of the Q-test, it is recommended that the z or z’ values set to zero in the calculation of the Q statistic not be subtracted when determining the degrees of freedom in the chi-square approximation. Moreover, a significance level of 10% is suggested for the bootstrap approach, rather than the conventional 5%.