Response Surface Designs in Flexible Regions

Abstract

When both variance and bias errors are present, the choice of an experimental design to fit a response surface depends on the choice of the region of interest. Box and Draper (1959, 1963) and Draper and Lawrence (1965a, 1966) investigated spherical and cuboidal regions, respectively. Both of these regions can, however, be regarded as special cases of a more general flexible region family, depending on a single parameter. In this manner, the sensitivity of the design criterion can be studied and more general results can be obtained. The problem is set out here for k predictor variables, and specific designs are provided for k = 2 predictor variables when variance and bias errors are of roughly the same order. A broad rule of thumb that emerges from this work is that, for practical situations involving both variance and bias errors, and for designs of the composite type used for fitting a second-order surface, the design points should be near the boundary of the region of interest (over which a uniform interest is assumed) and about three center points should be used. The foregoing rule is specific to the application described. The theoretical framework provided in this article, however, is very general and can be used to provide guidance for any specified type of design choice under any specified set of conditions.

Key Words:

• Alphabetic optimality
• Composite design
• Variance and bias errors