Abstract
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.
Similar content being viewed by others
References
Bates, D.M., Watts, D.G.: Nonlinear Regression Analysis and Its Applications. Wiley, New York (1988)
Bisgaard, S., Ankenman, B.: Standard errors for the eigenvalues in second-order response surface models. Technometrics 38, 238–246 (1996)
Box, G.E.P., Draper, N.R.: Response Surfaces, Mixtures, and Ridge Analyses, 2nd edn. Wiley, New York (2007)
Box, G.E.P., Hunter, J.S.: A confidence region for the solution of a set of simultaneous equations with an application to experimental design. Biometrika 41, 190–199 (1954)
Box, G.E.P., Wilson, K.B.: On the experimental attainment of optimum conditions. J. R. Stat. Soc. B 13, 1–45 (1951)
Carter, W.H., Chinchilli, V.M., Myers, R.H., Campbell, E.D.: Confidence intervals and an improved ridge analysis of response surfaces. Technometrics 28, 339–346 (1986)
Carter, W.H., Chinchilli, V.M., Campbell, E.D.: A large-sample confidence region useful in characterizing the stationary point of a quadratic response surface. Technometrics 32, 425–435 (1990)
Cheng, R.C.H., Melas, V.B., Pepelyshev, A.N.: Optimal designs for the evaluation of an extremum point. In: Atkinson, A., Bogacka, B., Zhigljavsky, A. (eds.) Optimum Design 2000, pp. 15–24. Kluwer, Dordrecht (2001)
Fedorov, V.V., Müller, W.G.: Another view on optimal design for estimating the point of extremum in quadratic regression. Metrika 46, 147–157 (1997)
Gauchi, J.P., Vila, J.P., Coroller, L.: New prediction confidence intervals and band in the nonlinear regression model: Application to the predictive modelling in food. Commun. Stat. Simul. C 39(2), 322–334 (2010)
Graybill, F.A.: Matrices with Applications in Statistics, 2nd edn. Wadsworth, Belmont (1983)
Hader, R.J., Harward, M.E., Mason, D.D., Moore, D.P.: 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 surface. Soil. Sci. Soc. Am. J. 21, 59–64 (1957)
Khuri, A.I., Cornell, J.A.: Response Surfaces: Design and Analyses, 2nd edn. rev. and expanded. Dekker, New York (1996)
Li, J., Ma, C., Ma, Y., Li, Y., Zhou, W., Xu, P.: Medium optimization by combination of response surface methodology and desirability function: an application in glutamine production. Appl. Microbiol. Biot. 74, 563–571 (2007)
Lin, D.K.J., Peterson, J.J.: Statistical inference for response surface optima. In: Khuri, A.I. (ed.) Response Surface Methodology and Related Topics, Chap. 4. World Scientific, Singapore (2006)
Melas, V.B., Pepelyshev, A.N., Cheng, R.C.H.: Designs for estimating an extremal point of quadratic regression models in a hyperball. Metrika 58, 193–208 (2003)
Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M.: Response Surface Methodology: Process and Produce Optimization Using Designed Experiments, 3rd edn. Wiley, New York (2009)
Peterson, J.J.: A general approach to ridge analysis with confidence intervals. Technometrics 35, 204–214 (1993)
Peterson, J.J., Cahya, S., del Castillo, E.: A general approach to confidence regions for optimal factor levels of response surfaces. Biometrics 58, 422–431 (2002)
Plassmann, F., Khanna, N.: Assessing the precision of turning point estimates in polynomial regression functions. Econom. Rev. 26(5), 503–528 (2007)
R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2009). ISBN3-900051-07-0, URL http://www.R-project.org
Sambucini, V.: A reference prior for the analysis of a response surface. J. Stat. Plan. Inference 137, 1119–1128 (2007)
Sambucini, V., Piccinato, L.: Likelihood and Bayesian approaches to inference for the stationary point of a quadratic response surface. Can. J. Stat. 36, 223–238 (2008)
Seber, G.A.F., Wild, C.J.: Nonlinear Regression. Wiley, New York (1989)
Stablein, D.M., Carter, W.H., Wampler, G.L.: Confidence regions for constrained optima in response surface experiments. Biometrics 39, 759–763 (1983)
Verzani, J.: An introduction to gWidgets. R News 7(3), 26–33 (2007)
Vila, J.P., Gauchi, J.P.: Optimal designs based on exact confidence regions for parameter estimation of a nonlinear regression model. J. Stat. Plan. Inference 137, 2935–2953 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sambucini, V. Confidence regions for the stationary point of a quadratic response surface based on the asymptotic distribution of its MLE. Stat Comput 22, 739–751 (2012). https://doi.org/10.1007/s11222-010-9202-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-010-9202-3