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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 869-883

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2010.040

\vspace{3pc} \normalsize \centerline{\large\bf MINIMAL SUFFICIENT
CONFOUNDING INFORMATION} \vspace{2pt} \centerline{\large\bf  AMONG
MAIN EFFECTS AND TWO-FACTOR} \vspace{2pt} \centerline{\large\bf
INTERACTIONS} \vspace{.25cm}
\author{Jianwei Hu$^{1}$ and
Runchu Zhang$^{2,3}$} \vspace{.4cm} \centerline{\it $^1$Central
China Normal University, $^2$Nankai University} \centerline{\it and
$^3$Northeast Normal University} \vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:} For two-level regular designs, we obtain
the structures of Fisher information matrices for estimating main
effects and two-factor interactions (2fi's). Based on these results,
we propose the definition of minimal sufficient confounding
information among
 main effects and 2fi's. As an application, we
demonstrate that minimum aberration (MA) designs must be
(M,S)-optimal designs for two-level regular designs. In addition, we
show that sequentially minimizing $M_{{(1,2)}_1},M_{{(2,2)}_2}$ and
$M_{{(2,2)}_1}$, as the core of the minimum M-aberration criterion
proposed by Zhu and Zeng (2005),
 is equivalent to
sequentially minimizing word length pattern $A_3$ and $A_4$. In
particular, we show that sequentially minimizing $A_3$ and $A_4$ is
equivalent to sequentially maximizing the first two components of
the maximum estimation capacity, $E_1(d)$ and $E_2(d)$, defined in
Cheng, Steinberg, and Sun (1999).
\par \vspace{9pt} \noindent {\it
Key words and phrases:} Aliased effect-number pattern, information
matrix, maximum estimation capacity,  minimal sufficient confounding
information, minimum aberration. \par
\end{quotation}
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