A Hierarchy of Relationships Between Covariance Matrices

Abstract

In multivariate methods involving several populations, such as discriminant analysis or MANOVA, equality of all covariance matrices is a frequent assumption. If a test for equality of the covariance matrices suggests that this assumption does not hold, the usual reaction is to estimate the covariance matrices individually in each group. For k populations and p variables this means that the number of parameters estimated increases by (k−1)p(p−1)/2, which is quadratic in p. In many practical applications (as in the example given in section 4), this is not satisfactory, for two reasons: First, the k covariance matrices, although not being identical, may exhibit some common structure. Second, in parametric model fitting, the “principle of parsimony” (Dempster, 1972, p. 157) suggests that parameters should be introduced sparingly and only when the data indicate that they are needed.