Two-stage central composite designs with dispersion effects

Abstract

Traditional response surface methods assume homogeneous variance throughout the design region. To estimate a second order location model, a central composite design is often used. In situations with few design variables, it is possible to have replications at the design locations. Mays (1996) illustrated that an equal replicate central composite design is not optimal if the true heterogeneous variances are known.

However, the true heterogeneous variances are rarely known. Hence the variance structure must be estimated, and a weighted least squares type approach is beneficial. This analysis considers a two-stage experimental design procedure in which the first stage estimates the variance structure, and then the second stage augments this first stage design with additional runs that produces the most efficient total design. The integrated variance optimality criterion is used to analyze the two-stage procedure and compare it to a standard equal replicate one-stage design analyzed by ordinary least squares (ignore the heterogeneous variance) and analyzed by weighted least squares. The analysis involves simulation of the two-stage design results, and indicates that in most dispersion effects cases the application of the two-stage procedure to central composite designs is superior to the equal replicate one-stage design.

Keywords:

• Integrated variance optimality
• axial distance
• likelihood Function
• information matrix