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A General Approach to Confidence Regions for Optimal Factor Levels of Response Surfaces
Author(s) -
Peterson John J.,
Cahya Suntara,
Castillo Enrique
Publication year - 2002
Publication title -
biometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.298
H-Index - 130
eISSN - 1541-0420
pISSN - 0006-341X
DOI - 10.1111/j.0006-341x.2002.00422.x
Subject(s) - confidence region , confidence interval , constraint (computer aided design) , quadratic equation , mathematics , coverage probability , bounded function , confidence distribution , mathematical optimization , statistics , computer science , mathematical analysis , geometry
Summary. For a response surface experiment, an approximate hypothesis test and an associated confidence region is proposed for the minimizing (or maximizing) factor‐level configuration. Carter et al. (1982, Cancer Research 42 , 2963–2971) show that confidence regions for optimal conditions provide a way to make decisions about therapeutic synergism. The response surface may be constrained to be within a specified, bounded region. These constraint regions can be quite general. This allows for more realistic constraint modeling and a wide degree of applicability, including constraints occurring in mixture experiments. The usual assumption of a quadratic model is also generalized to include any regression model that is linear in the model parameters. An intimate connection is established between this confidence region and the Box–Hunter (1954, Biometrika 41 , 190–199) confidence region for a stationary point. As a byproduct, this methodology also provides a way to construct a confidence interval for the difference between the optimal mean response and the mean response at a specified factor‐level configuration. The application of this confidence region is illustrated with two examples. Extensive simulations indicate that this confidence region has good coverage properties.