Experimental designs for mean and variance estimation in variance components models

Abstract

For a mixed linear model E( Y ) = X α , V( dY ) = V, with V a known function of an unknown vector parameter θ , under normality assumptions, maximum likelihood can be employed to estimate α and θ simultaneously by iterative procedures. If the design is factorial, an important practical problem is the choice of the number of observations for each level of the random factors (including error). In this paper we modify logdet of the exptected Fisher information matrix, introducing weights to allow for different emphasis on estimation of mean and variance, and choose this as our optimality criterion. We show that for the case of only one random factor plus error, balanced designs are optimal, and we find the optimal number of replications. We also consider the possibility that the costs of replications and of different levels of the random effect may differ, and find the expression of the optimal replication number, keeping the total cost of the experiment fixed. Not surprisingly, this expression depends on the variance ratio, which is unknown, thus a pseudo-Bayesian approach is required.