Spherical Prediction-Variance Properties of Central Composite and Box—Behnken Designs

For quadratic regression on the hypercube, a single-number criterion, such as a G efficiency that is based on the prediction variance, is often included as one of the criteria when selecting a response surface design, As an alternative to the single-number-criterion approach, the variance dispersion graph, presented by Giovannitti-Jensen and Myers, is a graphical technique for evaluating prediction-variance properties throughout the experimental region. Three properties of interest are the maximum, minimum, and average spherical prediction variances, given the spherical radius. As an alternative to the computer-based approach requiring an optimization algorithm to evaluate these properties, the maximum, minimum, and spherical prediction variances for central composite and Box–Behnken designs can be determined analytically and are functions only of the radius and the design parameters. These functions yield the exact values of the spherical prediction-variance properties of central composite and Box–Behnken designs, thereby removing the need for extensive computing involving algorithms that do not guarantee finding global optima. Results are presented for spherical and cuboidal regions.