Optimal designs for mixed-effects models with two random nested factors

The main objective of this paper is to provide experimental designs for the estimation of fixed effects and two variance components, in the presence of nested random effects. Random nested factors arise from quantity designations such as lot or batch, and from sampling and measurement procedures. We introduce a general class of designs for mixed-effects models with random nested factors, called assembled designs, where the nested factors are nested under the treatment combinations of the crossed factors. We provide parameters and notation for describing and enumerating assembled designs. Conditions for existence and uniqueness of D-optimal assembled designs for the case of two variance components are presented. Specifically, we show that, in most practical situations, designs that are most balanced (i.e., where the samples are distributed as uniformly as possible among batches) result in D-optimal designs for maximum likelihood estimation.