What statistical tests validate multivariate normality assumptions?

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Validating the assumption of multivariate normality is crucial in many multivariate statistical analyses, as it ensures that the results from techniques such as multivariate regression, factor analysis, or MANOVA are reliable. Several statistical tests are used to assess multivariate normality:

1. Mardia’s Test: This is a widely used method to assess multivariate normality. It consists of two components: skewness and kurtosis. Mardia’s multivariate skewness test examines the asymmetry of the data distribution, while the kurtosis test looks at the "tailedness" of the distribution. A non-significant result for both tests suggests the data follows a multivariate normal distribution.
2. Shapiro-Wilk Test: Although this test is typically used for univariate normality, it can be applied to each variable individually to assess its normality. For multivariate data, however, the test might not be as powerful as others, especially when there are correlations between variables.
3. Anderson-Darling Test: This test extends the Kolmogorov-Smirnov test to assess whether the data follows a specific distribution, such as a multivariate normal distribution. It is particularly useful for datasets with moderate sample sizes.
4. Q-Q Plot and P-P Plot: Quantile-Quantile (Q-Q) and Probability-Probability (P-P) plots are visual tools that can provide a quick assessment of normality. If the data points lie approximately along a straight line, it indicates that the data likely follows a multivariate normal distribution.
5. Hartley’s Test: This test is used to assess the homogeneity of variances and covariances, which is related to multivariate normality. Non-homogeneous variances or covariances may indicate deviations from normality.
6. Box’s M Test: This test evaluates the equality of covariance matrices across groups. Significant results suggest the data does not meet the assumption of multivariate normality.

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