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 hyperparameters theta. We focus on a specific case in which all subjects receive the same set of treatments and in which the covariates are independent of treatments. We obtain the explicit structure of psi(beta)- and psi(theta) optimal continuous (approximate) designs for the case of independent random effects, and for some special cases of correlated random effects. Through examples and simulations, we 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 nonorthogonal 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, but sensitive to the specifications of the signs of the correlations of the random effects.