Generalized Ridge Analysis Under Linear Restrictions, With Particular Applications to Mixture Experiments Problems

To investigate the behavior of a response y over a specified region of interest by fitting a secondorder response surface, standard ridge analysis provides a way of following the locus of, for example, a maximum response, moving outward from the origin of the predictor variable space. Because this approach does not require one to view the fitted regression surface as a whole, this important technique may be applied even when visualization of the surface is difficult in several dimensions. The ridge trace view enables practitioners to assess and understand the typically complex interplay between the input variables as the response improves. To explore a subspace defined by a linear restriction on the predictors, a situation discussed infrequently in the literature and never in the context of mixture experiments, we show how a modification of ridge regression can be used generally to investigate second-order mixture surfaces with many ingredients, particularly when the experimental mixture space is itself limited by further linear equalities in addition to the mixture requirement. In some cases, the ridge origin need not be moved into the mixture space to achieve the desired results, and any form of the second-order fitted model, whether of Scheffé type, Kronecker type, or something in between, can be accommodated.