Response Surface Design and Analysis in the Presence of Restricted Randomization

Abstract

Practical restrictions on randomization are commonplace in industrial experiments due to the presence of hard-to-change or costly-to-change factors. Employing a split-plot design structure minimizes the number of required experimental settings for the hard-to-change factors. In this research, we propose classes of equivalent estimation second-order response surface split-plot designs for which the ordinary least squares estimates of the model are equivalent to the generalized least squares estimates. Designs that possess the equivalence property enjoy the advantages of best linear unbiased estimates and design selection that is robust to model misspecification and independent of the variance components. We present a generalized proof of the equivalence conditions that enables the development of several systematic design construction strategies and provides the ability to verify numerically that a design provides equivalent estimates, resulting in a broad catalog of designs. We explore the construction of balanced and unbalanced split-plot versions of the central composite and Box-Behnken designs. In addition, we illustrate the utility of numerical verification in generating D-optimal and minimal point designs, including split-plot versions of the Notz, Hoke, Box and Draper, and hybrid designs. Finally, we consider the practical implications of analyzing a near-equivalent design when a suitable equivalent design is not available. By simulation, we compare methods of estimation to provide a practitioner with guidance on analysis alternatives when a best linear unbiased estimator is not available. Our goal throughout this research is to develop practical experimentation strategies for restricted randomization that are consistent with the philosophy of traditional response surface methodology.