How to test Bivariate and Multivariate Normality:

Refer to the post, Data Assumption: Univariate Normality, for general comments about normality.

There are no special tests to screen for either Bivariate or Multivariate normality. The only test I am aware of is the Mardia’s statistic test for multivariate normality. Use univariate screening and while univariate normality does not guarantee multivariate normality, most often multivariate won’t be far off if the univariate screening test was passed.

Even better than univariate tests are bivariate such as a bivariate scatterplot of any pairs of variables in the model.

In addition, save the residuals of all pairwise regressions in the model and subject the residuals to PP/QQ, skewness and kurtosis tests against their univariate heuristics.

Testing normality is also an assumption for ANOVA’s and similar to Regression, we need to look at the residuals (often referred to as “errors” in the ANOVA context). However, with ANOVA we may only have 3 or 4 groups, so we can look at the normality of the DV in each group of the IV (use “split-data function”). When we have many groups, or very few observations in each group (or if we introduce a covariate in ANCOVA), then we need to look at the residuals /errors. In SPSS use the GLM procedures, save the residuals, and plot a QQ-plot where data points should lie on the diagonal to indicate normality.

Note that serious violations of multivariate normality will be flagged by Box’s M test (the multivariate counterpart of Levene’s test of variance equality). As long as violations are due to skewness and not outliers, then most violations are acceptable. Very serious violations can be “fixed” through transformations though this should be administered very carefully through the most appropriate procedures.

Stevens (2002) noted, “Although it is difficult to completely characterize multivariate normality, normality on each of the variables separately is a necessary, but not sufficient, condition for multivariate normality to hold” (p. 262).
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Reference:
Stevens, J. P. (1984), Outliers and influential data points in regression analysis, Psychological Bulletin, 95, 334–344.
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