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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1415-1430

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\centerline{\large\bf OPTIMAL DESIGNS FOR GENERALIZED LINEAR MODELS}
\vspace{2pt} \centerline{\large\bf WITH MULTIPLE DESIGN VARIABLES}
\vspace{.25cm}
\author{Min Yang,  Bin Zhang and Shuguang Huang}
\vspace{.4cm} \centerline{\it University of Missouri, University of
Alabama-Birmingham and} \centerline{\it
 Precision Therapeutics, Inc.} \vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:}
 Binary response experiments are common in
scientific studies. However, the study of optimal designs in this
area is in a very underdeveloped stage. Sitter and Torsney (1995a)
studied optimal designs for binary response experiments with two
design variables. In this paper, we consider a general situation
with multiple design variables. A novel approach is proposed to
identify optimal designs for  the commonly used multi-factor
logistic and probit models. We give explicit formulas for a large
class of optimal designs, including $D$-, $A$-, and $E$-optimal
designs.  In addition, we identify the general structure of optimal
designs, which has a relatively simple format. This property makes
it feasible to solve seemingly intractable problems. This result can
also be applied in a multi-stage approach.
\par
\vspace{9pt} \noindent {\it Key words and phrases:} $A$-optimality,
$D$-optimality, $E$-optimality, Loewner ordering, logistic model,
probit model. \par
\end{quotation}
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