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\noindent Statistica Sinica {\bf 13}(2003), 709-725

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\centerline{\large\bf A NOTE ON OPTIMAL MIXTURE AND MIXTURE}
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\centerline{\large\bf AMOUNT DESIGNS}
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\author{Berthold Heiligers and Ralf-Dieter Hilgers}
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\centerline{\it University of Magdeburg and RWTH-Aachen}
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\begin{quotation}
\noindent {\it Abstract:}
In many applications of mixture experiments in medicine or
 biology, for example, not only the proportions of the involved
 mixture ingredients, but also their total amount is of particular
 interest. This calls for designs in mixture amount models, which
 are obtained from classical mixture setups by including terms
 capturing the total amount, simultaneously dropping the
 sidecondition on the proportions to sum up to one. While
 design optimality usually depends sensitively on the underlying
 model, we establish here a close relation between admissible
 mixture, and admissible mixture amount designs in additive and
 homogeneous models. This particularly allows to obtain
 $D$-, $A$- and $V$-optimal mixture amount
 from optimal mixture designs, and vice versa. We present some
 examples for Becker's and Scheff\'e's mixture models.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Admissibility, $A$-optimality, approximate design, Becker models,
complete class, $D$-optimality, $I$-optimality, mixture amount experiments,
mixture experiments, $V$-optimality, tic-type polynomials.\par
\end{quotation}
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