Large-sample confidence region useful in characterizing the stationary point of a quadratic response surface

A second-order response-surface model is often used to approximate the relationship between a response variable and a set of explanatory variables. The nature of the stationary point of the surface can be determined by considering the eigenvalues of the matrix of the model's second-order terms. Since the elements of this matrix are estimated from the data, however, it follows that the eigenvalues are random variables. Hence the sampling properties of these eigenvalues should be considered in characterizing the nature of the stationary point. In this article, it is shown how a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface. The delta method is used to construct an approximate confidence region for these eigenvalues. Box and Draper (1987) gave a result that simplifies the calculation of such a confidence region for rotatable or nearly rotatable designs. A simulation study was performed to determine and compare the coverage probabilities for the confidence regions for some frequently used experimental designs. For the cases considered, the approximate procedure developed in this article has the desired small-sample properties and is appropriate for rotatable and nonrotatable designs.