Confidence regions for the stationary point of a quadratic response surface based on the asymptotic distribution of its MLE

Response surface methodology aims at finding the combination of factor levels which optimizes a response variable. A second order polynomial model is typically employed to make inference on the stationary point of the true response function. A suitable reparametrization of the polynomial model, where the coordinates of the stationary point appear as the parameter of interest, is used to derive unconstrained confidence regions for the stationary point. These regions are based on the asymptotic normal approximation to the sampling distribution of the maximum likelihood estimator of the stationary point. A simulation study is performed to evaluate the coverage probabilities of the proposed confidence regions. Some comparisons with the standard confidence regions due to Box and Hunter are also showed.

[1]  Russell C. H. Cheng,et al.  Optimal Designs for the Evaluation of an Extremum Point , 2001 .

[2]  F. Graybill,et al.  Matrices with Applications in Statistics. , 1984 .

[3]  M. Deaton,et al.  Response Surfaces: Designs and Analyses , 1989 .

[4]  W. H. Carter,et al.  Confidence regions for constrained optima in response-surface experiments. , 1983, Biometrics.

[5]  L. Piccinato,et al.  Likelihood and bayesian approaches to inference for the stationary point of a quadratic response surface , 2008 .

[6]  R. H. Myers,et al.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[7]  Douglas C. Montgomery,et al.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[8]  John J. Peterson.  A General Approach to Ridge Analysis With Confidence Intervals , 1993 .

[9]  R. Gunst.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[10]  Russell C. H. Cheng,et al.  Designs for estimating an extremal point of quadratic regression models in a hyperball , 2003 .

[11]  George E. P. Box,et al.  A CONFIDENCE REGION FOR THE SOLUTION OF A SET OF SIMULTANEOUS EQUATIONS WITH AN APPLICATION TO EXPERIMENTAL DESIGN , 1954 .

[12]  Raymond H. Myers,et al.  Confidence intervals and an improved ridge analysis of response surfaces , 1986 .

[13]  G. Seber,et al.  Nonlinear Regression: Seber/Nonlinear Regression , 2005 .

[14]  N. Khanna,et al.  Assessing the Precision of Turning Point Estimates in Polynomial Regression Functions , 2007 .

[15]  S. Bisgaard,et al.  Standard errors for the eigenvalues in second-order response surface models , 1996 .

[16]  G. Box,et al.  On the Experimental Attainment of Optimum Conditions , 1951 .

[17]  D. D. Mason,et al.  An Investigation of Some of the Relationships between Copper, Iron, and Molybdenum in the Growth and Nutrition of Lettuce: I. Experimental Design and Statistical Methods for Characterizing the Response Surface1, 2 , 1957 .

[18]  A. Atkinson,et al.  Optimum design: 2000 , 2001 .

[19]  Vernon M. Chinchilli,et al.  A larage-sample confidence region useful in characterizing the stationary point of a quadratic response surface , 1990 .

[20]  Werner G. Müller,et al.  Another view on optimal design for estimating the point of extremum in quadratic regression , 1997 .

[21]  V. Sambucini.  A reference prior for the analysis of a response surface , 2007 .

[22]  A Large-Sample Confidence Region Useful , 1990 .

[23]  Louis Coroller,et al.  New Prediction Interval and Band in the Nonlinear Regression Model: Application to Predictive Modeling in Foods , 2010, Commun. Stat. Simul. Comput..

[24]  J. Vila,et al.  Optimal designs based on exact confidence regions for parameter estimation of a nonlinear regression model , 2007 .

[25]  Dennis K. J. Lin,et al.  STATISTICAL INFERENCE FOR RESPONSE SURFACE OPTIMA , 2006 .

[26]  C. Anderson‐Cook.  Response Surfaces, Mixtures, and Ridge Analyses , 2008 .

[27]  Ping Xu,et al.  Medium optimization by combination of response surface methodology and desirability function: an application in glutamine production , 2007, Applied Microbiology and Biotechnology.

[28]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[29]  John J. Peterson,et al.  A General Approach to Confidence Regions for Optimal Factor Levels of Response Surfaces , 2002, Biometrics.

[30]  André I. Khuri,et al.  Response Surface Methodology and Related Topics , 2007 .

[31]  Douglas M. Bates,et al.  Nonlinear Regression Analysis and Its Applications , 1988 .