Mixed Level Fractional Factorial Split-plot Designs Containing Clear Effects

Fractional factorial (FF) designs are commonly used for factorial experiments. Clear effects is a popular optimality criterion for selecting designs. In factorial investigations, especially those involving physical experiments, there are often factors with four levels. Then mixed-level designs are used in the experiments. Such designs can be constructed from two-level designs by the method of replacement, which was first formally introduced by Addelman (1962) and developed by Wu (1989), Wu et al. (1992), Hedayat et al. (1992) and Zhang and Shao (2001). A design with n two-level factors and m four-level factors is usually denoted by 2ⁿ4^m. Zhao and Zhang (2008) gave a complete classification of the existence of clear two-factor interaction components (2FIC) of 2ⁿ4^m designs..When the levels of some of the factors are difficult to be changed or controlled, it may be impractical or even impossible to perform the experimental runs of FF designs in a completely random order. This motivates us to use fractional factorial split-plot (FFSP) designs to meet the special demands. FFSP designs have received much attention in recent years. If there are both two and four-level factors in an experiment and it is difficult to change or control the levels of some factors, a split-plot 2^{(n₁+n₂)-(k₁+k₂)}4^m design can be used.This paper considers the regular split-plot 2^{(n₁+n₂)-(k₁+k₂)}4^m designs. It consists of two chapters. Chapter 1 introduces the basic definitions related to FF design, optimality criterion and fractional factional split-plot design.Chapter 2 gives a complete classfication of the 2^{(n₁+n₂)-(k₁+k₂)}4^m designs containing various clear effects. Section 2.1 gives a simple summary on the literature. Section 2.2 introduces the notations and definitions of two types of mixed level fractional factorial split-plot designs, 2^{(n₁+n₂)-(k₁+k₂)}4_s¹ designs and 2^{(n₁+n₂)-(k₁+k₂)}4_w¹ designs according to the difference of the four level factor in WP section or in SP section, and gives the concept of three types of two factor interaction components. Sections 2.3 and 2.4, respectively, study resolutionâ…¢andâ…£2^{(n₁+n₂)-(k₁+k₂)}4_s¹ designs and give the sufficient and necessary conditions of such designs containing various clear effects. And Section 2.5 gives the sufficient and necessary conditions for resolutionâ…¢andâ…£2^{(n₁+n₂)-(k₁+k₂)}4_w¹ designs containing various clear effects.