Testing Covariance Structure in Multivariate Models: Application to Family Disease Data

Recent interest in modeling multivariate responses for members of groups has emphasized the need for testing goodness of fit. Here we describe a way to test the covariance structure of a multivariate distribution parameterized by a vector θ. The idea is to extend this distribution, the “null” distribution, to a more general distribution that depends on θ, an additional scalar γ, and a specific quadratic function of the response vector chosen to capture features of an alternative covariance structure. When γ = 0, the more general distribution reduces to the null one. Standard likelihood theory yields a score test for γ = 0; that is, a test of fit of the null distribution. The score statistic is the standardized difference between observed and expected values of the quadratic function, where the expectation is taken with respect to the null distribution, with θ replaced by its maximum likelihood estimate. Applying the methods to case-control data on familial cancers of the ovary and breast, we illu...