## Split-Plot and Multi-Stratum Designs for Statistical Inference

**Steve Gilmour (University of Southampton, UK)**

It is increasingly realized that many industrial and engineering experiments use split-plot or other multi-stratum structures. Much recent work has concentrated on finding optimal, or near-optimal, designs for estimating the fixed effects parameters in multi-stratum designs. However often inference, such as hypothesis tests or interval estimation, will also be required and valid inference requires pure error estimates of the variance components. Most optimal designs provide few, if any, pure error degrees of freedom. Gilmour and Trinca (2012) introduced design optimality criteria for inference in the context of completely randomized and block designs. Here these criteria are used stratum-by-stratum in order to obtain multi-stratum designs. It is shown that these designs have better properties for performing inference. Compound criteria, which combine the inference criteria with traditional point estimation criteria are also used and the designs obtained are shown to give good compromise between point estimation and inference. Designs are obtained for two real split-plot experiments and an illustrative split-split-plot structure.