UC Riverside

Many large scale longitudinal cohort studies have been carried out in different fields of science. Such studies need a careful planning at the design stage to achieve precise estimates of model parameters. This thesis presents the application of optimal design theory in a longitudinal study with two cohorts of n subjects each. For each subject, the observations are taken at three different time points denoted by (-1, ai, 1), where -1 < ai < 1 (i = 1, 2). Our class of longitudinal cohort designs is {(-1, a1, 1) (-1, a2, 1)}; -1 < a1 < a2 < 1. Optimal cohort designs for linear mixed effects models with a random intercept and a random slope are computed analytically with respect to the D-, A-, and E-optimality criteria. The results are demonstrated by optimality regions. We also compare between cohort designs with equidistant and non-equidistant time points. We have learned that when the covariance of the random effects satisfies certain conditions, the design with equidistant time points is preferred. However, under certain cases, for example, the third case stated in Theorem 3.1, the design with non-equidistant time points is better. We propose a new iterative method for computing the Restricted Maximum likelihood (REML) estimators of the variance components in the linear mixed effects models using three criterion functions l*, Delta, and P. Two simulated data sets and one observed facial growth data are used to illustrate our method.