A General Approach to Ridge Analysis With Confidence Intervals

Abstract

In response-surface methodology, ridge analysis is a graphical technique for interpreting response surfaces, particularly those of three or more dimensions. Standard ridge-analysis techniques employ a second-order polynomial regression model. An approach to ridge analysis is presented here that generalizes its current scope. This approach allows the use of a general regression model for the response surface. Within the class of linear models, conservative or approximate simultaneous confidence intervals about the optimal mean responses, for varying radii, can be obtained. These simultaneous confidence intervals form a guidance band to aid the experimenter in determining optimal levels of operation. A variety of explanatory variable constraints, including mixture constraints, can also be imposed. For the standard quadratic model, these confidence intervals can be easily modified to obtain confidence bounds about the maximum (or minimum) eigenvalues of B, the matrix of second-order regression coefficients. Such bounds are useful for assessing the optimality of a stationary point on the underlying response surface. Proofs and computational details are available from the author's technical report.

KEY WORDS:

• Confidence band
• Constrained optimization
• Eigenvalues
• Mixture experiments
• Response-surface methodology
• Rotatability