Optimal Designs for Mixed-Effects Models with Random Nested Factors

The problem of experimental design for the purpose of estimating the fixed effects and the variance components corresponding to random nested factors is a widely applicable problem in industry. Random nested factors arise from quantity designations such as lot or batch and from sampling and measurement procedures. We introduce a new class of designs, called assembled designs, where all the nested factors are nested under the treatment combinations of the crossed factors. We provide parameters and notation for describing and enumerating assembled designs. Using maximum likelihood estimation and the D-optimality criterion, we show that, for most practical situations, designs that are as balanced as possible are optimal for estimating both fixed effects and variance components in a mixed-effects model.