In this paper we study hypotheses related to certain hierarchical dependence structure (such that the interclass and intraclass correlations belong to a commutative quadratic subspace) in doubly multivariate models. The Rao score test and likelihood ratio test statistics are derived, as well as the exact distribution of the likelihood ratio test. Simulation studies show the advantage of the Rao score test over the likelihood ratio test in the speed of convergence to the limiting chi-square distribution, while both proposed tests are competitive from the point of view of their power. The results presented are applied to real data example.