A Monte Carlo comparison of the Type I and Type II error rates of tests of multivariate normality

Many multivariate statistical methods call upon the assumption of multivariate normality (MVN). However, many researchers fail to test this assumption. This omission could be due to either ignorance of the existence of tests of MVN or confusion about which test to use. Although at least 50 tests of MVN exist, relatively little is known about the power of these procedures. The purpose of this study was to examine the power of 13 promising tests of MVN 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 critical value. The number of rejections of the null hypothesis of MVN 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 MVN. Supplementary procedures such as Mardia's skewness and kurtosis measures and the chi-square plot are also recommended for diagnosing possible deviations from normality.

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