D-Optimal and D-Efficient Equivalent-Estimation Second-Order Split-Plot Designs

Abstract

Industrial experiments often involve factors that are hard to change or costly to manipulate and thus make it undesirable to use a complete randomization. In such cases, the split-plot design structure is a cost-efficient alternative that reduces the number of independent settings of the hard-to-change factors. In general, model estimation for split-plot designs requires the use of generalized least squares (GLS). However, for some split-plot designs (including not only classical agricultural split-plot designs, but also some second-order split-plot response surface designs), ordinary least squares (OLS) estimates are equivalent to GLS estimates. These designs are called equivalent-estimation designs and offer the advantage that estimation of the factor effects does not require estimation of the variance components in the split-plot model. As an alternative to these equivalent-estimation designs, one can use D-optimal designs that guarantee efficient estimation of the fixed effects of the statistical model that is appropriate given the split-plot structure. We explore the relationship between equivalent-estimation and D-optimal split-plot designs for a second-order response surface model and propose an algorithm for generating D-efficient equivalent-estimation split-plot designs. This approach allows for a flexible choice of the number of hard-to-change factors, the number of easy-to-change factors, the number of whole plots, and the total sample size.