Multivariate tests for patterned covariance matrices

In many real datasets, the same variables are measured on objects from different groups, and the covariance structure may vary from group to group. Oftentimes, the underlying population covariance matrices are not identical, yet they still have a common basic structure, e.g. there exists some rotation that diagonalizes simultaneously all covariance matrices in the groups, or all covariances can be made congruent by some translation and/or dilation. The purpose of this paper is to show how a test of homoscedasticity can be made more informative by performing a separate check for equality between shapes and equality between orientations of the concentration ellipsoids. This approach, combined with parsimonious parametrization in mixture data modeling, provides a formal hypothesis testing procedure for model assessment.