Testing the assumption of multivariate normality

When researchers rely on multivariate data, they often assume the data follow a multinormal distribution. Checking that assumption is less straightforward than it sounds. This article looks at ways to measure how far a set of multivariate observations departs from multinormality, and where the usual tools fall short.

Mardia's tests of multivariate skewness and kurtosis are the most familiar approach. They let you test null hypotheses that agree with the assumption of multinormality. The problem shows up once a null hypothesis is rejected: the researcher is left knowing that something is wrong without any sense of which regions of the data are responsible for the deviation.

To address that gap, Alexander von Eye and G. Anne Bogat propose two additional procedures, a sector test and an omnibus test. The sector test pinpoints subspaces holding more or fewer cases than you would expect if the distribution really were multinormal. The omnibus test answers the broader question, namely whether the hypothesis of a multinormal distribution can be held overall.

The authors put all of these methods to work, applying Mardia's tests alongside their new sector and omnibus tests to data drawn from a study of the parenting abilities of adolescents.