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The problem of testing the intercept and slope parameters of doubly multivariate linear models with site-dependent covariates using Rao's score test (RST) is studied. The RST statistic is developed for a block exchangeable covariance structure on the error vector under the assumption of multivariate normality. We compare our developed RST statistic with the likelihood ratio test (LRT) statistic. Monte Carlo simulations indicate that the RST statistic is much more accurate than its counterpart LRT statistic and it takes significantly less computation time than the LRT statistic. The proposed method is illustrated with an example of multiple response variables measured on multiple trees in a single plot in an agricultural study.

We generalize the theory of linear models for doubly multivariate data from matrix-variate normally distributed errors to matrix-variate skew normally distributed errors. In addition, we assume that the covariance matrix $$\varvec{\varOmega }$$ defining the location-scale matrix-variate skew normal distribution has block compound symmetry structure. We derive the maximum likelihood estimators of the model’s parameters; the Fisher information matrix for the direct, working, and centered parametrizations; Rao’s score tests and likelihood ratio tests for model building tests of hypotheses; and a hypothesis test for the centered intercept. A profiling argument is used to reduce the dimensionality of the optimization method used to obtain the maximum likelihood estimators and a comprehensive discussion of initial values is provided. Finally, we provide a real-world example to illuminate these derivations and apply a goodness-of-fit test to validate the distributional assumption.