A general approach to confidence regions for optimal factor levels of response surfaces

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.