Statistica Sinica 15(2005), 165-176

OPTIMAL DESIGNS FOR AN ADDITIVE QUADRATIC

MIXTURE MODEL INVOLVING THE AMOUNT OF

MIXTURE

C. Q. Zhang$^1$, L. Y. Chan$^2$, Y. N. Guan$^3$, K. H. Li$^4$ and T. S. Lau$^4$

$^1$Guangzhou University, $^2$The University of Hong Kong, $^3$Northeastern

University, and $^4$The Chinese University of Hong Kong

Abstract: This paper is concerned with $D$- and $A$-optimal designs for a quadratic additive model for experiments with mixtures, in which the response depends not only on the relative proportions but also on the actual amounts of the mixture components. It is found that the origin and vertices of the simplex are support points of these optimal designs, and when the number of mixture components increases, other support points shift gradually from barycentres of depth 1 to barycentres of higher depths. It is shown that the $D$-optimal designs have high efficiency in terms of $A$-optimality, and vice versa.

Key words and phrases: A-optimal design, additive model, D-optimal design, experiments with mixtures, mixture amount.