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Before MANOVA, I need to test multivariate normality.
Then I tried MVN::mvn() function in R, output as below:
Error in FUN(newX[, i], ...) : sample size must be between 3 and 5000

tt.png

Based on the qq-plot, I don't think data fit multivariate normality.

As my sample size always more than 3000, among Mardia’s test, Henze-Zirkler’s test, Royston’s test, Doornik-Hansen’s test, E-statistic, which one is best to test multivariate normality?

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  • $\begingroup$ You don't need a "best" test. Just test a random subsample of 3K observations. If you're right--and it looks that way--the test will reject normality. This is of little interest, though, because what you do afterward is the key. The departures from normality are curious, because they indicate clustering of values around $3$ and $6$ in the squared Mahalanobis distance. You need to investigate that and decide to what extent this clustering might affect whatever analyses you are interested in. $\endgroup$
    – whuber
    May 13, 2020 at 18:14
  • $\begingroup$ @whuber,based the above plot,is data multivariate normality? $\endgroup$
    – kittygirl
    May 13, 2020 at 18:48

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A suggestion: "A Powerful Test for Multivariate Normality". To quote from the Introduction:

...topic of testing for normality, Thode [23] reviewed more than thirty formal procedures proposed specifically for testing normality. Briey, in terms of power performance against a broad range of alternatives, the Shapiro-Wilk (SW) test [20] is the benchmark of omnibus tests for univariate data [3, 6, 23]. For testing multivariate normality, the Henze-Zirkler (HZ) test [13] is recommended by Thode [23, pp. 220]. In many practical applications, researchers often prefer to use tests that are both informative and easy to understand [4]. Although generally quite powerful as a multivariate test, the HZ test has the drawback of not being as easy to understand as the simple skewness or kurtosis based tests, or as the SW test which is known to many researchers and is generally powerful to detect outliers or influential observations as well as to skewed distributions. To the best of our knowledge, there is no known test that is both informative and has competitive power in all dimensions. In this paper, we introduce a simple informative test that is easy to understand and to implement in all dimensions. Simulations studies indicate that the new test has very competitive power compared to the HZ test and other best known tests in both univariate and multivariate cases.

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