Mixture Experiments: ILL-Conditioning and Quadratic Model Specification

Well-conditioned models are important, particularly for practitioners who work with regression models for mixture experiments where parameter estimates are individually meaningful. In this article we investigate conditioning in second-order mixture models, using variance inflation factors, maximum and minimum eigenvalues of the information matrix and condition numbers to assess conditioning. A range of equivalent mixture models that lie “between” the Scheffé model (S-model) and the Kronecker model (K-model) is examined, and pseudocomponent transformations for lower bounds (L-pseudocomponents) and upper bounds (U-pseudocomponents) are also discussed. We prove that the maximum eigenvalue for the information matrix for the K-model is always smaller than that for any other model in the above range. We recommend in practice the use of the K-model, to reduce ill-conditioning, and the appropriate use of pseudocomponents.