Statistica Sinica 22 (2012), 393-417

doi:http://dx.doi.org/10.5705/ss.2009.319

BAYESIAN DESIGNS FOR HIERARCHICAL

LINEAR MODELS

Qing Liu$^1$, Angela M. Dean$^2$ and Greg M. Allenby$^2$

$^1$University of Wisconsin-Madison and $^2$The Ohio State University

Abstract: Two Bayesian optimal design criteria for hierarchical linear models are discussed - the $\psi_\beta$ criterion for the estimation of individual-level parameters $\bbeta$, and the $\psi_\theta$ criterion for the estimation of hyperparameters $\btheta$. 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.

Key words and phrases: Bayesian design, D-optimality, design robustness, hierarchical linear model, hyperparameter, random effects model.