Optimal Central Composite Designs in the Presence of Dispersion Effects

Abstract

The central composite design (CCD) is often used to provide estimation of a second-order regression model. The CCD involves a possibly fractionated two-level factorial design, axial points, and replicated runs in the design center and is usually used with the assumption of homogeneous variance in the design region. However, when dispersion effects exist, standard CCD's may not be optimal. With a specified number of experimental runs available, the goal is to find the optimal allocation of the runs to the CCD locations. In this paper, several variance structures are considered in conjunction with several scaled degrees of heterogeneity expressed in terms of variance ratios. Optimality criteria based on the determinant of the variance-covariance matrix of coefficients (D-optimality) and on the integrated prediction variance (I-optimality) are used. Two, three, and four variable models are considered, and in each heterogeneous variance structure case, the run allocations that creates the D- and l-optimal designs are presented.