Mixed two- and four-level fractional factorial split-plot designs with clear effects

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Abstract

Mixed-level designs have become widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. It is highly to know when a mixed-level FFSP design with resolution III or IV has clear effects. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and four-level factors to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples.

Introduction

Mixed-level fractional factorial designs are commonly used in the experiments. One of the methods to obtain such designs is replacement, which was first formally introduced by Addelman (1962) and then developed by Wu (1989), Wu et al. (1992), Hedayat et al. (1992) and Zhang and Shao (2001). When the levels of some factors are difficult to be changed or controlled, it may be impractical or even impossible to perform the experimental runs in a completely random order (Bisgaard and Steinberg, 1997). Then fractional factorial split-plot (FFSP) designs can be utilized to meet the special demands. For development on FFSP designs we refer to Huang et al. (1998), Bingham and Sitter (1999), Mukerjee and Fang (2002), Yang et al. (2006) and Zi et al. (2006). If there are both two-level and four-level factors in an experiment and it is difficult to change or control the levels of some factors, then a mixed-level split-plot design with both two-level and four-level factors can be used.

Clear effects (Wu and Chen, 1992) is a popular optimality criterion for selecting designs. Under the assumption that interactions involving three or more factors are absent, a clear effect can be estimated unbiasedly. Recent results on clear effects criterion include Chen and Hedayat (1998), Tang et al. (2002), Ai and Zhang (2004), Yang et al. (2005), Chen et al. (2006), Yang and Liu (2006), Zhao and Zhang, 2008a, Zhao and Zhang, 2008b and Zhao et al. (2008). A natural and interesting question is when a split-plot design with both two-level and four-level factors can have clear effects. This paper gives the necessary and sufficient conditions for such designs to have various clear effects. Section 2 introduces the notation and definitions. Section 3 gives the main results.

Section snippets

Notation and definitions

A regular 2nk design D with N=2nk runs and n factors is determined by k independent defining words. The group generated by the k words is called the defining contrast subgroup of D, denoted by G. Any word except I, the grand mean, in G is called a defining word. The number of letters in a defining word is called its length. The resolution of D is defined as the length of the shortest defining word. A main effect or two-factor interaction of D is said to be clear if it is not aliased with any

Main results

First, we focus on studying the necessary and sufficient conditions for the existence of clear effects of 2III(n1+n2)(k1+k2)4w1 designs.

Theorem 1

There exist 2III(n1+n2)(k1+k2)4w1 designs containing clear four-level WP main effect if and only if n12p121 and n22p22p12.

Proof

Suppose that a 2III(n1+n2)(k1+k2)4w1 design D is determined by C in (1) with d1d2d3=I and the four-level WP factor A={d1,d2,d3} is clear. Without loss of generality, suppose {d1,d2,d3}={a1,a2,a1a2}. Let H(a3,,ap1)={α1,,α2p121},H

Acknowledgements

The authors would like to thank the associate editor and the referee for their valuable comments to improve the paper. This work was partially supported by the NNSF of China Grants 10826059, 10901092, 11171188 and 11171165, SRRF of Shandong Province of China Grant BS2011SF006 and the NSF of Shandong Province of China Grant Q2007A05.

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