Construction Theory On General Minimum Lower Order Confounding Designs

In the scientific research and industrial practice, people always learn the results of some matter or the characteristics of some thing from experimentations. The design of experiments is an efficient procedure for planning experiments so that the data obtained can be analyzed to yield valid and objective conclusions.Since R.A. Fisher founded the modern statistics, experimental design has played an important role and been widely applied in many fields of scientific investigations, such as agriculture, chemical manufacturing, medicine and other high-tech industries. Wu and Hamada. (2000) classified these experimental problems into five categories according to their objectives:(i) treatment comparison, (ii) variable screening, (iii) response sur-face exploration, (iv) system optimization, and (v) system robustness. Mukerjee and Wu (2006) further pointed out that except for treatment comparison with one-or two-way layouts, these problems involve the study of the effects of multiple input variables on the experimental response. These input variables are called "factors" and the experiments are called "factorial experiments". Each factor must have two or more settings so that the effect of change in factor setting on the response can be explored. These settings are called "levels" of the factor. Any combination of levels of "factors" is known as a "treatment combination". A treatment combination is also called a "run" in industrial experimentation. "Factorial design" concerns the "selection and arrangement" of treat-ment combinations in a factorial experiment. A "full factorial design" involves all the possible treatment combinations. The number of such combinations grows rapidly as the number of factors or levels increases. Thus, the practical solution is to choose an economic and efficient fraction of the full factorial design, which is called "fractional factorial de-sign"The issue of choosing optimal fractional factorial designs has been explored by the statisticians for many years. Quite a few criteria were proposed before 2006, notably those of maximum resolution (MR) (Box and Hunter (1961)), minimum aberration (MA) (Fries and Hunter (1980)), maximum clear effects (for main effects and two-factor inter- actions (2fi's)) (MCE) (Wu and Chen (1992)), and maximum estimation capacity (MEC) (Sun (1993)). For more details, see Wu and Hamada (2000) and Mukerjee and Wu (2006). These criteria are all under the effect hierarchy principle (EHP, Wu and Hamada (2000)), which states that lower order effects are likely more important than higher order effects and same order effects are likely equally important. Thus, good fractional factorial designs should minimize the confounding between lower order effects. However, though the exist-ing criteria are all based on this principle, the optimal designs are sometimes different.The two-level regular fractional factorial designs have simple aliasing structures in that any two effects are either orthogonal or fully aliased. These designs are widely studied and used in various experimentations. Zhang, Li, Zhao and Ai (2008) (ZLZA) introduced an aliased effect-number pattern (AENP) for two-level regular designs, and based on it, they proposed a general minimum lower order confounding (GMC) criterion, the optimal designs under this criterion are called GMC designs. The AENP contains the basic in-formation of all the effects aliased with the other effects at various severity degrees and implies the confounding between different effects more directly than the parameters used in other criteria. Thus, the GMC criterion with the AENP aims at finding optimal de-signs under EHP in a more elaborate and explicit manner. Actually, they also proved that all the existing optimality criteria can be obtained by the functions of AENP. Later, the AENP and the GMC criterion have been developed well and formed GMC theory. The work includes:Zhang and Mukerjee (2009a) considered the general s-level factorials with GMC criterion (s?2, is a prime or prime power), and gave a theory of construction with complementary set, Zhang and Mukerjee (2009b) considered the GMC criterion with complementary set theory applied in block designs, Hu and Zhang (2009) proved that the GMC designs must minimize the number of words with length 3 in the defining relation subgroup, Wei, Yang and Zhang (2010) considered the GMC theory in split-plot designs.In Chapter 1, we give an overview of the GMC theory and the relationship between it and other existing criteria.In practice, experimenters need to choose different optimal designs in different situa-tions. According to the flexibility of the AENP, the GMC theory and the GMC design can well satisfy this point. Thus, the construction of GMC designs is very important. The construction of the optimal designs under other criteria always needs a lot of complicated computer searching. Li, Zhao and Zhang (2009) constructed the two-level regular GMC designs with its number of factors n satisfying 5N/16+1?n?N-1 (N is the number of treatment combinations). Their construction method is very simple, for any n satisfying the condition, its corresponding GMC design just consists of the last n columns from the saturated design in Yates order. This method avoids the computer searching, which is an outstanding advantage of the GMC theory. So, a question is that whether the GMC designs with other parameters can also be obtained by a similar way.We consider the construction of the GMC designs with 9N/32+1?n?5N/16 and N/4+1?n?9N/32 respectively in Chapter 2. Chen and Cheng (2004,2006) expounded the doubling theory in finite projection geometry in design language. In Section 2.2, we further develop this theory and utilize it for constructing the GMC designs. Block and Mee (2003) defined the two-level resolution IV second order saturated design (SOS de-sign), and proved that any non-SOS design must be a projection of some SOS design. According to the results in finite projection geometry, the SOS designs with factor num-ber N/2,5N/16,9N/32,17N/64,33N/128,...,N/4+1 can be obtained by the doubling method, and the ones with factor number N/2 and 5N/16 are respectively unique up to isomorphism. Thus any GMC design with 9N/32+1?n?5N/16 must be projected from one of these two SOS designs. In Section 2.3, through studying the projections of these two SOS designs, we show that the GMC designs with 9N/32+1?n?5N/16 must be pro-jections of the SOS design with factor number 5N/16 and give the construction method. However. with the factor number approaching to N/4+1, the SOS designs with the same factor number are no longer unique and may be more and more (Block (2003)). Thus rank-ing the projections of all those SOS designs needs lots of work. For simplifying the problem we first construct a class of SOS designs with some specialized structure. This kind of de-signs is proved to be unique up to isomorphism. Based on that, we directly construct the GMC designs with N/4+1?n?9N/32 by using doubling theory and this kind of SOS de-signs, and show that any GMC design with N/4+1?n?9N/32 is different from the MA design with the same design parameters. As same as the case of 5N/16+1?n?N-1, any GMC design with 9/32N+1?n?5N/16 or N/4+1?n?9N/32 consists of the last n columns from the saturated design in some specialized rechanged Yates order (RC Yates order). In Section 2.5, we introduce the construction method in terms of the RC Yates order. The "small" SOS designs with n<N/4+1 can no longer be obtained by doubling. In Section 2.6, we construct a class of "small" SOS designs and study their properties, which are helpful for solving the problem of constructing the GMC designs with n<N/4+1.In Chapter 3, we further investigate the aliasing structures of these constructed GMC designs and show their properties, which can be helpful for experimenters to use them in practice. In Section 3.2, we propose the concept of second order degree of freedom and prove that the GMC designs with N/4+1?n?N/2 process the maximum second order degrees of freedom. In many cases, one need to add a follow-up experiment to de-alias the confounding between some effects. One strategy is "foldover" planning. We discuss the optimal foldover planning for the GMC designs in Section 3.3.In the past years, nonregular designs have been noticed by researchers because of their economy and flexible run sizes. These designs include the Plackett-Burman designs (Plackett and Burman (1946)), some Hadamard matrices (Hedayat and Wallis (1978)) and many other symmetrical and asymmetrical orthogonal arrays as described in Dey and Mukerjee (1999), Hedayat, Sloane and Stufken (1999) and Wu and Hamada (2000). Different from two-level regular designs, for nonregular designs, there are at least two effects aliased with each other partially. Such an aliasing structure is called complex aliasing (Hamada and Wu (1992)). Because of it, nonregular designs are traditionally used for screening main effects. However, Hamada and Wu (1992) showed that some interactions could also be detected in such designs, which led to a growth of interest in them. Especially, a natural research topic is to follow the parallel work of optimal selection in the regular case. Recently, there has been a. lot of work on optimality criteria for nonregular designs. including Deng and Tang (1999) proposed the generalized resolution and minimum G-aberration (MGA) criterion, Tang and Deng (1999) proposed the minimum G2-aberration (MG2A) criterion and Xu and Wu (2001) proposed the generalized minimum aberration (GMA) criterion. It is worthy of notice that these criteria do not work out consistent rank sometimes and may fail to find good designs under EHP.We extend the GMC theory to nonregular designs in Chapter 4, and show the root cause of the contradictions between the existing criteria. In Section 4.2, we introduce a generalized aliased effect-number pattern (GAENP) and a generalized GMC (GGMC) criterion. The GGMC criterion can be used for all kinds of orthogonal designs, including various nonregular, symmetrical and asymmetrical designs and taking regular designs as its special case. We study the relationship between GGMC and other existing criteria in Section 4.3. In Section 4.4, we consider geometrical isomorphism with the. GAENP. In Section 4.5, we consider the application of GGMC in practice. We first give an algorithm for comparing the GAENP of two designs, by using this algorithm, we obtain the GGMC projections of the OA(18,2137) design in Table 7C.2 of Wu and Hamada (2000), and compare them with the ones obtained under other criteria. We also get the GGMC designs with run number 16 and 20. As same as the AENP and the GMC criterion of regular cases, the GAENP and the GGMC criterion also have good flexibility. We give some modifications according to the experimenters'motivations in Section 4.6, which select more appropriate GGMC designs. Finally, we further illustrate the GGMC criterion by a practical example.In Chapter 5, we summarize the main results and the key technical approaches, and suggest some interesting problems worthy of further investigation.