On Using Asymptotic Critical Values in Testing for Multivariate Normality

Many multivariate statistical methods call upon the assumption of multivariate normality. However, many researchers fail to test this assumption. This omission could be due either to ignorance of the existence of tests of multivariate normality or confusion about which test to use. Although at least 50 tests of multivariate normality exist, relatively little is known about the power of these procedures. The purpose of this study was to examine the power (in small sample situations) of 8 promising tests of multivariate normality with a Monte Carlo study. Ten thousand data sets were generated from several multivariate distributions. The test statistic for each procedure was calculated and compared with the appropriate asymptotic critical value. The number of rejections of the null hypothesis of multivariate normality was tabled for each situation. No single test was found to be the most powerful in all situations. The use of the Henze-Zirkler test is recommended as a formal test of multivariate normality. Supplementary procedures such as Mardia’s skewness and kurtosis measures and the chi-square plot are also recommended for diagnosing possible deviations from normality.