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Design for Hyperparameter Estimation in Linear Models

Optimal design for the joint estimation of the mean and covariance matrix of the random effects in hierarchical linear models is discussed. A criterion is derived under a Bayesian formulation which requires the integration over the prior distribution of the covariance matrix of the random effects. A theoretical optimal design structure is obtained for the situation where the random effects have equal variances and zero covariances. For other situations, optimal designs are obtained through computer search. It is shown that orthogonal designs, if they exist, are optimal under a main effects model with independent random effects. When the random effects are believed to be correlated, it is shown by example that nonorthogonal designs tend to be more efficient than orthogonal designs. In addition, design robustness is studied under various prior mean specifications of the random effects covariance matrix.