Rethinking the Optimal Response Surface Design for a First-Order Model with Two-Factor Interactions, When Protecting against Curvature

Abstract

There are common choices for best response surface designs for the three-, four-, and five-factor first-order model with all two-factor interactions when the goal is to precisely estimate the model and a measure of experimental error, while protecting against possible curvature from quadratic effects. When the design size is to be kept small, the statistics literature has long recommended the 23 and 25−1 factorial designs with added center runs for this purpose for the three- and five- factor cases, and a Plackett–Burman design with center runs is common for the four- factor case. In this article we evaluate these designs against all other competitors using formal criteria for three objectives. In addition to D-optimality, two new criteria are proposed to measure the precision of the pure error estimate and the ability to detect lack-of-fit from missing model terms. A Pareto frontier approach is used to conduct a formal search for optimal designs based on simultaneously considering the three criteria of interest. Candidate designs with their robustness to weight specifications and sensitivity to different scalings and forms of desirability functions are explored. For different numbers of factors, the standard designs have mixed performance. The Pareto front approach offers insights into how this problem might be more thoroughly examined. A new graphical summary, the synthesized efficiency plot, shows relative performance of designs over user-specified weightings of the different criteria.