Bayesian Designs for Hierarchical Linear Models

Two Bayesian optimal design criteria for hierarchical linear models are discussed - the psi_beta criterion for the estimation of individual-level parameters beta, and the psi_theta criterion for the estimation of hyper-parameters theta. While the psi_beta criterion involves only the specification of the treatments, the psi_theta criterion involves the specification of both the treatments and the covariates. We focus on a specific case in which all subjects receive the same set of treatments and the covariates are independent of treatments. We obtain the explicit structure of psi_beta and psi_theta optimal continuous (approximate) designs for both the situation of independent random effects and some special situations of correlated random effects. Through examples and simulations we then compare psi_beta and psi_theta optimal designs under more general scenarios of correlated random effects. While orthogonal designs are often psi_beta optimal even when the random effects are correlated, psi_theta optimal designs tend to be non-orthogonal and unbalanced. In our study of the robustness of psi_beta and psi_theta optimal designs, both types of designs are found to be insensitive to various specifications of the response errors and the variances of the random effects. However, they are sensitive to the specifications of the signs of the correlations of the random effects, especially the psi_theta optimal designs. Resulting implications for practical applications are discussed