Extensions to Confidence Region Calculations For the Path of Steepest Ascent - ASQ

Extensions to Confidence Region Calculations For the Path of Steepest Ascent

The method of steepest ascent plays an important role in response surface methodology by giving a direction for further experimentation after an initial factorial design. To determine whether the path of steepest ascent has been determined precisely enough, the fraction of all possible directions that are included in the confidence cone'around the fitted path of steepest ascent can be calculated. In this paper, extensions to the existing method are made to cover heterogeneous error variances, nonorthogonal designs and generalized linear models. We also propose a design criterion for augmenting a given experimental design in problems for which the path of steepest ascent has not been determined precisely enough.

Key Words: Augmenting Design, Generalized Least Squares, Response Surface Methodology.

By EWA M. SZTENDUR and NEIL T. DIAMOND, Victoria University of Technology, P.O. Box 14428 MCMC Melbourne, Australia, 8001

Introduction

IN the initial stage of a response surface study, a first order design is used to fit a linear model,


where y is the response and are the coded levels of the experimental factors. If the linear model fits well, the path of steepest ascent giving the maximum predicted increase in the response can be calculated. Experimentation is conducted along the path of steepest ascent until curvature is detected.

If curvature is present, a second order experiment is conducted, a second order model


is fitted, and the results are summarized using contour diagrams and canonical analysis (see, for example, Box and Draper (1987, chapters 10 and 11)).

Box (1955) and Box and Draper (1987, pp. 190–194) presented a method for determining whether the path of steepest ascent has been determined precisely enough (see also Myers and Montgomery (1995, pp. 194–198)). The method gives the fraction of all possible directions that are included in the confidence cone around the fitted path of steepest ascent. If this proportion is small enough, then we can say that the path has been determined precisely enough and experiments along the path can be conducted. Otherwise, further experiments to improve the precision of the path need to be run. This method applies to orthogonal designs with uncorrelated errors and homogeneous error variances. However, in practice there are many situations where the design will not be orthogonal nor the error variances homogeneous. One example would be in generalized linear models, which are being increasingly used.

The purpose of this paper is to develop a method that would apply more generally than the current method. Another important aspect is to develop a method for augmenting the original design in situations where the path of steepest ascent has not been determined precisely enough.

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