A General Approach to Confidence Regions for Optimal Factor Levels of Response Surfaces

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 Research42, 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, Biometrika41, 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.

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