Statistica Sinica 22 (2012), 869-883

doi:http://dx.doi.org/10.5705/ss.2010.040

MINIMAL SUFFICIENT CONFOUNDING INFORMATION

AMONG MAIN EFFECTS AND TWO-FACTOR

INTERACTIONS

Jianwei Hu$^{1}$ and Runchu Zhang$^{2,3}$

$^1$Central China Normal University, $^2$Nankai University

and $^3$Northeast Normal University

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).

Key words and phrases: Aliased effect-number pattern, information matrix, maximum estimation capacity, minimal sufficient confounding information, minimum aberration.