Split-Plot and Multi-Stratum Designs for Statistical Inference

ABSTRACT

It is increasingly recognized that many industrial and engineering experiments use split-plot or other multi-stratum structures. Much recent work has concentrated on finding optimum, or near-optimum, designs for estimating the fixed effects parameters in multi-stratum designs. However, often inference, such as hypothesis testing or interval estimation, will also be required and for inference to be unbiased in the presence of model uncertainty 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 to obtain multi-stratum designs. It is shown that these designs have better properties for performing inference than standard optimum designs. Compound criteria, which combine the inference criteria with traditional point estimation criteria, are also used and the designs obtained are shown to compromise between point estimation and inference. Designs are obtained for two real split-plot experiments and an illustrative split–split-plot structure. Supplementary materials for this article are available online.

KEYWORDS:

• A-optimality
• D-optimality
• Hard-to-change factor
• Hard-to-set factor
• Mixed model
• Response surface
• Split–split-plot design

Supplementary Materials

suppMSPE.pdf a pdf file containing tables showing the designs obtained, which are not in the main article, and the algorithm for Example 2.

codeMSPE.rar a zipped folder containing R code for all of the examples given in the article.

Acknowledgments

The authors gratefully acknowledge financial support from UNESP grant numbers PDI/028900413/PROPG-CDC and PDI/828900413/PROPG-CDC and from FAPESP grant number 2014/01818-0.