### Abstract

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.

Original language | English (US) |
---|---|

Pages | 425-435 |

Number of pages | 11 |

Volume | 32 |

No | 4 |

Specialist publication | Technometrics |

DOIs | |

State | Published - Jan 1 1990 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

### Cite this

*Technometrics*,

*32*(4), 425-435. https://doi.org/10.1080/00401706.1990.10484729

}

*Technometrics*, vol. 32, no. 4, pp. 425-435. https://doi.org/10.1080/00401706.1990.10484729

**Large-sample confidence region useful in characterizing the stationary point of a quadratic response surface.** / Carter, Walter H.; Chinchilli, Vernon M.; Campbell, Eleanor D.

Research output: Contribution to specialist publication › Article

TY - GEN

T1 - Large-sample confidence region useful in characterizing the stationary point of a quadratic response surface

AU - Carter, Walter H.

AU - Chinchilli, Vernon M.

AU - Campbell, Eleanor D.

PY - 1990/1/1

Y1 - 1990/1/1

N2 - 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.

AB - 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.

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U2 - 10.1080/00401706.1990.10484729

DO - 10.1080/00401706.1990.10484729

M3 - Article

AN - SCOPUS:0025517392

VL - 32

SP - 425

EP - 435

JO - Technometrics

JF - Technometrics

SN - 0040-1706

ER -