V- and D-optimal population designs for the simple linear regression model with a random intercept term

In this paper V- and D-optimal population designs for the simple linear regression model with a random intercept term are considered. This is done with special reference to longitudinal data, that is data measured repeatedly at specified time points. Individual designs comprising up to k+1 distinct and equally spaced values of the explanatory variable are assumed to be available. The problem of constructing a population design which allocates weights to these individual designs in such a way that the variances associated with the mean marginal responses at a given vector of time points are in some sense minimized is addressed. V-optimal designs are obtained and a geometric approach to confirm the global optimality or otherwise of these designs is introduced. The study is extended to the D-optimal case. It is noted that the V- and D-optimal population designs are robust to the value of the intraclass correlation coefficient.