  A constrained multivariate linear model is a multivariate linear model with
the columns of its coefficient matrix constrained to lie in a known subspace.
This class of models includes those typically used to study growth curves and
longitudinal data. Envelope methods have been proposed to improve estimation
efficiency in the class of unconstrained multivariate linear models, but have
not yet been developed for constrained models that we develop in this article.
We first compare the standard envelope estimator based on an unconstrained
multivariate model with the standard estimator arising from a constrained
multivariate model in terms of bias and efficiency. Then, to further improve
efficiency, we propose a novel envelope estimator based on a constrained
multivariate model. Novel envelope-based testing methods are also proposed. We
provide support for our proposals by simulations and by studying the classical
dental data and data from the China Health and Nutrition Survey and a study of
probiotic capacity to reduced Salmonella infection.
