Optimal Response Surface Designs in the Presence of Dispersion Effects

Abstract

Standard response surface design procedures typically assume homogeneous variance throughout the design region, and the designs used in practice usually reflect this assumption. With two design variables, a replicated two-level factorial design is D- and l-optimal for a first-order or interaction model, and a balanced three-level factorial design is l-optimal for a second-order model. However, in many situations the assumption of homogeneous variance is not appropriate. For an identified heterogeneous variance structure and level of dispersion, the goal is to find the optimal design and to illustrate the relative inefficiency of the standard replicated 22 factorial design when used with a first-order or interaction model and of the standard balanced three-level factorial design when used with a second-order model. Some useful alternative designs are presented that could be used when dispersion effects exist. Presented with the optimal designs are the efficiencies of the standard designs relative to these optimal designs, and these efficiencies reflect the loss of effectiveness of the standard design when the dispersion effects are ignored. The results are such that they can be extended to heterogeneous variance structures and experiment sizes not presented, and general recommendations are made as to what types of designs are preferred in different heterogeneous variance situations.