D-optimal designs of mean-covariance models for longitudinal data

Longitudinal data analysis has been very common in various fields. It is important in longitudinal studies to choose appropriate numbers of subjects and repeated measurements and allocation of time points as well. Therefore, existing studies proposed many criteria to select the optimal designs. However, most of them focused on the precision of the mean estimation based on some specific models and certain structures of the covariance matrix. In this paper, we focus on both the mean and the marginal covariance matrix. Based on the mean-covariance models, it is shown that the trick of symmetrization can generate better designs under a Bayesian D-optimality criterion over a given prior parameter space. Then, we propose a novel criterion to select the optimal designs. The goal of the proposed criterion is to make the estimates of both the mean vector and the covariance matrix more accurate, and the total cost is as low as possible. Further, we develop an algorithm to solve the corresponding optimization problem. Based on the algorithm, the criterion is illustrated by an application to a real dataset and some simulation studies. We show the superiority of the symmetric optimal design and the symmetrized optimal design in terms of the relative efficiency and parameter estimation. Moreover, we also demonstrate that the proposed criterion is more effective than the previous criteria, and it is suitable for both maximum likelihood estimation and restricted maximum likelihood estimation procedures.