Multivariate Normality Tests (Mardia, Henze-Zirkler, Royston)

Multivariate normality tests (Mardia, Henze-Zirkler, Royston) assess whether several continuous variables jointly follow a multivariate normal distribution, using multivariate skewness and kurtosis, characteristic functions, and combined Shapiro-Wilk statistics.

다변량 정규성 검정(Mardia, Henze-Zirkler, Royston)은 여러 연속 변수가 함께 다변량 정규분포를 따르는지를 다변량 왜도·첨도, 특성함수, 결합된 Shapiro-Wilk 통계량으로 판정하는 검정입니다.

Before running MANOVA or SEM, how can I check whether my variables are multivariate normal?

MANOVA 나 구조방정식을 돌리기 전에 변수들이 다변량 정규분포를 따르는지 어떻게 확인하나요?

Mardia's tests check multivariate skewness and kurtosis, the Henze-Zirkler test compares the empirical and normal characteristic functions, and Royston's test combines the Shapiro-Wilk statistics of each variable while adjusting for their correlations. A small p value in any of them indicates a departure from multivariate normality, and a chi-square Q-Q plot of squared Mahalanobis distances shows where the departure lies.

When to use Multivariate Normality Tests (Mardia, Henze-Zirkler, Royston)

• You plan MANOVA, discriminant analysis or maximum-likelihood SEM and need to check the multivariate normality assumption.
• You want to choose between normal-theory and robust estimators for a set of continuous variables.
• You want to screen for multivariate outliers using squared Mahalanobis distances.

When to use something else

• You only have one variable; use Shapiro-Wilk or another univariate normality test.
• The variables are ordinal or binary; use polychoric-based or robust methods rather than normality tests.

Assumptions and requirements

• Rows are independent observations.
• Variables are continuous and the covariance matrix is non-singular.
• Sample size exceeds the number of variables plus two.

What the analysis reports

• Mardia skewness chi-square (with small-sample version when n < 20) and kurtosis z with p values
• Henze-Zirkler statistic with its log-normal p value
• Royston H with equivalent degrees of freedom and p value
• Per-variable Shapiro-Wilk W and p
• Chi-square Q-Q correlation and flagged multivariate outliers

How to report it

Multivariate normality was assessed with Mardia's skewness, chi2([df]) = [value], p = [value], Mardia's kurtosis, z = [value], p = [value], and the Henze-Zirkler test, HZ = [value], p = [value].

Related methods

• Shapiro-Wilk Test
• MANOVA
• Covariance-Based SEM (CB-SEM)

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