Some Theoretical Issues Of Blocked Designs And Compromise Designs

As an important branch of statistics, experimental design has received increasing attentions from the statisticians. Design of factorial experiment is an important part of experiment design, Plenty of research achievements focusing on designs of facto-rial experiment have been proposed for decades, including how to economically and efficiently select optimal fractional factorial design is the main concern of statisticians.According to different measuring standards, statisticians proposed various op-timality criteria for choosing fractional factorial designs. A few of famous criteria are maximum resolution (denoted as MR in short) criterion (Box & Hunter[5]), mini-mum aberration (MA) criterion (Fries & Hunter[25]),clear effects (CE) criterion (Wu& Chen1471), maximum estimation capacity (MEC) criterion (Sun[38]). These criteria are based on effect hierarchy principle (EHP, Wu & Hamada[48]), which implies lower order effects are more important than higher order effects, and same order effects are equally important. Obviously, a good fractional factorial design tends to make the lower order effects being not confounding with other effects or being confounding with other effects in quite slight degree. Though these optimality criteria follow the idea of EHP, their distinct parameters of confounding probably lead to completely different optimal fractional factorial designs.Still based on the idea of EHP, Zhang et al.[62]proposed general minimum con-founding (denoted as GMC in short) criterion. This criterion focuses on regular two-level fractional factorial designs, and its core point is the concept named as aliased effect number pattern (AENP). First, for a regular design, any two factor effects are either orthogonal or completely aliased, so it has a relatively simple confounding struc-ture. AENP contains the basic information of all factor effects confounding with other factor effects on different serious confounding degrees, and sufficiently and directly reflects the confounding relationship between different orders of factor effects. More-over, the theoretical results shows that most of the existed criterion can be expressed as a specific function of AENP. Therefore, GMC criterion embodies the idea of EHP in a more explicit and elaborate manner, comparing with the existed criteria.As the research continues, AENP and GMC criterion have been widely devel-oped. The main works are: Zhang & Mukerjee[63] generalized the GMC criterion in situation with factors having prime or prime order levels, and stated constructions of GMC design using complementary set theory; Zhang & Mukerjee1641 extended AENP and GMC criterion to blocked designs, Wei, Li & Zhang1441, Zhang, Li & Wei[61],Tan &Zhang[39] and Zhao et al.[67] made important achievements on construction of B-GMC,B1-GMC and B2-GMC; Wei, Yang Li & Zhang1451 and Ren, Li & Zhang1361 discussed the usage of GMC criterion in split-plot designs and robust parameter designs; Cheng& Zhang[21] and Zhou & Zhang1711 extended the AENP and GMC criterion to non-regular fractional factorial designs; Zhou, Balakrishnan & Zhang1701 applied the idea of AENP for studying optimal arrangement of factors; and so on. Furthermore, GMC criterion work has attracted extensive concern in the field of experimental design, sev-eral statist participate in the study, and Cheng[16] also introduces GMC criterion with recognition.The above works derive from deep understanding of the idea of AENP which possesses explicit expression and widely application and makes GMC theory become a system with increasing improvement. Using AENP, as an efficient tool for study with compatibility, a lot of new problems can be explored. In this article, we discuss optimal arrangements of factors in blocked designs, constructions of clear compromise design and extension of the idea of AENP in compromise designs.First, in Chapter 1, we review the development of GMC theory, and give an overview of the relationship between GMC criterion and other optimal design crite-ria. Chapter 2 provides definitions and basic notations used in this article, which are essential for the statements and proofs of theoretical result. The following chapters address the works we have done.In Chapter 3, we address blocked factor aliased effect number pattern (denoted as B-F-AENP in short) and its application in optimal arrangement of factors in B1-GMC designs. Zhou, Balakrishnan & Zhang[70] first present factor aliased effect number pattern (F-AENP) criterion for unblocked 2n-m designs. However, for the blocked case,the existence of block factors makes situations that treatment effects being confounded more complicate. We partition effects involving one treatment factors into four classes:g-, b-, m- and ?-class. Then, in each of the four classes of effects, we consider the confounding situations of each order of effects involving this treatment factor, and we give definition of B-F-AENP criterion accordingly. Based on the B-F-AENP criterion,we can rank columns in any 2m-n : 2s designs. Section 3.3 studies calculation on the B-F-AENP of columns in B1-GMC designs, in two cases with N/2 + 1?n?N-1 and 5N/16 + 1?n?N/2, the techniques are partitions of treatment columns and the definition of 2fi's of class i. Section 3.4 gives out the B-F-AENP rank of columns in B1-GMC designs, and the computation results imply that if 5N/16 + 1?n?N-1 then the B-F-AENP rank of columns in a 2n-m : 2s B1-GMC design is the same as the F-AENP rank of columns in a unblocked 2n-m GMC design. Several applications of B-F-AENP of B1-GMC designs, including strategies of arranging factors, are provided in Section 3.5. As a final illustration, we catalogue the sequential q(= n-m) optimal columns of 16-run, 32-run and 64-run B1-GMC designs, see Tables A1.1.1-A1.1.3 in the Appendix.In Chapter 4, we study compromise design containing clear specified effects and its construction. Four classes of compromise plans proposed by Addelman[2] and Sun[38] only focus on situations that all the main effects are clear, moreover they only consider designs of resolution IV. In this chapter, we extend the study range, and re-define four classes of compromise designs with ?1 which denotes the specified sets of effects: (1). {G1,G1×G1},(2). {G1,G1×G1,G2×G2},(3). {G1,G1×G1,G1×G2},(4).{G1,G1×G2}. We call them as clear compromise design (denoted as CCD in short) if effects in ?1 are all clear. CCDs can be designs with resolution ? and ?,details on the existence of CCDs are addressed in Section 4.2, Section 4.3 and Section 4.4 provide theoretical results on the constructions of CCDs.However, not all CCDs exits for all kinds of parameters. So, it's necessary to de-fine a "good" compromise design for any kind of parameter. In Chapter 5, we provide a more general criterion for ranking compromise designs, that is partial-aliased effect number, denoted as P-AENP in short. P-AENP criterion quantifies the aliased severity degrees of specified effects with any order. Here, the specified set of effects, ?1, has much more general form than the four types of compromise design as presented in Chapter 4. Based on the P-AENP criterion, we can define optimal compromise design(OCD) and provide constructions of OCDs. In Section 5.2, we construct OCDs of class three with ?{G1}=1,2. Further studies on OCDs would be our future work to be done.