I-Optimal Design of Mixture Experiments

ABSTRACT

In mixture experiments, the factors under study are proportions of the ingredients of a mixture. The special nature of the factors necessitates specific types of regression models, and specific types of experimental designs. Although mixture experiments usually are intended to predict the response(s) for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients, little research has been done concerning their I-optimal design. This is surprising given that I-optimal designs minimize the average variance of prediction and, therefore, seem more appropriate for mixture experiments than the commonly used D-optimal designs, which focus on a precise model estimation rather than precise predictions. In this article, we provide the first detailed overview of the literature on the I-optimal design of mixture experiments and identify several contradictions. For the second-order and the special cubic model, we present continuous I-optimal designs and contrast them with the published results. We also study exact I-optimal designs, and compare them in detail to continuous I-optimal designs and to D-optimal designs. One striking result of our work is that the performance of D-optimal designs in terms of the I-optimality criterion very strongly depends on which of the D-optimal designs is considered. Supplemental materials for this article are available online.

Keywords:

• D-optimality
• Moments matrix
• Scheffé model
• Simplex-lattice designs
• Simplex-centroid designs
• V-optimality

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Correction

Supplementary Materials

The supplementary materials include one pdf file containing the analytical expressions for the I-optimal weights derived by Laake (1975) for the second-order model, the special cubic model and the full cubic model, the analytical expressions for the weights obtained by Liu and Neudecker (1995) for the qth degree model, and our results for the qth degree model which have been published in Goos and Syafitri (2014).

Acknowledgment

Peter Goos is with KU Leuven, and the Department of Engineering Management, Universiteit Antwerpen. Bradley Jones is with the SAS Institute Inc., and Universiteit Antwerpen. Utami Syafitri is with the Department of Engineering Management, Universiteit Antwerpen.

Funding

The authors acknowledge the financial support of the Fonds voor Wetenschappelijk Onderzoek - Vlaanderen. The third author was financially supported by the General Directorat of Higher Education of the Indonesian Ministry of National Education.