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Statistica Sinica 21 (2011), 1571-1589
doi:10.5705/ss.2008.336





A THEORY ON CONSTRUCTING $\bm{2^{n-m}}$ DESIGNS WITH

GENERAL MINIMUM LOWER ORDER CONFOUNDING


Pengfei Li$^1$, Shengli Zhao$^2$ and Runchu Zhang$^{3,4}$


$^1$University of Alberta, $^2$Qufu Normal University,
$^3$Northeast Normal University and $^4$Nankai University


Abstract: When designing an experiment, it is important to choose a design that is optimal under model uncertainty. The general minimum lower-order confounding (GMC) criterion can be used to control aliasing among lower-order factorial effects. A characterization of GMC via complementary sets was considered in Zhang and Mukerjee (2009a); however, the problem of constructing GMC designs is only partially solved. We provide a solution for two-level factorial designs with $n$ factors and $N=2^{n-m}$ runs subject to a restriction on $(n,N)$: $5N/16+1\leq n\leq N-1$. The construction is quite simple: every GMC design, up to isomorphism, consists of the last $n$ columns of the saturated $2^{(N-1)-(N-1-n+m)}$ design with Yates order. In addition, we prove that GMC designs differ from minimum aberration designs when $(n,N)$ satisfies either of the following conditions: (i) $5N/16+1\leq n\leq N/2-4$, or (ii) $n \ge N/2$, $4\leq
n+2^r-N\leq 2^{r-1}-4$ with $r\ge 4$.



Key words and phrases: Aliased effect-number pattern, effect hierarchy principle, fractional factorial design, minimum aberration, resolution, wordlength pattern, Yates order.

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