The Experimental Planning Of Fractional Factorial Designs

Design of experiments is one of the most important branches of statistics. Thefields of its application are not just limited to agriculture, but also including industry,biology, economics, finances, meteorology, astronautics and so on, relating to all as-pects of scientific researches. It greatly and continuously promotes the development ofmodern statistics.With the development of science and technology, more factors are concerned inall kinds of scientific experiments. Hence, factorial experiment and its analysis are paidmore attention by the researchers and experimenters. Because of economy and otherreasons, how to obtain suitable designs, especially the optimal designs, is significantlyconcerned by the researchers. A design, involving n factors, is denoted as s1×...×sn,where sipresents the number of levels of the i-th factor. Hence, there are s1··· sntreatment combinations in a full factorial design at least. In practices, full factorialdesign are not tolerant to the experimenters and researches. Hence, fractional factorialdesign is introduced and applied. It is a fraction of a full factorial design. Especially,if s1=...=sn=s, it is called a symmetrical factorial; otherwise it is called an asym-metrical factorial. The frequently used experimental design involves block factorialdesign, symmetrical design, asymmetrical design, split-plot design, robust parameterdesign, and so on.Optimally designing factorial experiments is the most popular and important is-sue, and has received significant attention in the last five decades. Quite a few criteriafor selecting optimal designs have been proposed. Among them, the maximum reso-lution (MR), minimum aberration (MA), clear efects (CE) and maximum estimationcapacity (MEC) criteria become popular ones. The above criteria and their correspond-ing optimal designs are widely accepted and applied. Recently, a new criterion and itsoptimal designs, general minimum lower order confounding (GMC), was proposed andexpanded by Zhang, Li, Zhao and Ai (2008), are paid more attention. Many literaturedevoted to studying and popularizing these criteria. In2008, Zhang, Li, Zhao and Ai introduced a new pattern called aliased efectnumber pattern (AENP) for classifying two-level regular2n mfractional factorial de-signs with N=2n mtreatment combinations, n factors, q=n m independent columnsand N1aliased sets. And based on AENP, they proposed a general minimum lowerorder confounding (GMC) criterion for selecting optimal designs. The design selectedby the new criterion is called a GMC design. Further, they proved that every existingcriterion can be expressed by optimizing a specific function of the AENP, which meansthat the AENP can be used to describe and investigate the existing criteria. As the ex-plicit expression and widely application of AENP, all the results, which are obtainedby AENP and its core idea, construct the GMC theory. In the past few years, the devel-opment of this theory is very fast, many significant results are obtained. For examples,Zhang and Mukerjee (2009a) characterized the GMC designs via complementary sets;Zhang and Mukerjee (2009b) extended it to the case of block designs; Li, Zhao andZhang (2011), Zhang and Cheng (2010) and Cheng and Zhang (2010) constructed allthe GMC2n mdesigns for N/4+1≤n≤N1; Wei, Li and Zhang (2013) and Li,Wei and Zhang (2011) respectively proposed B1-GMC and B2-GMC criteria in blockfactorial design; Zhao, Li, Zhang and Karunamum (2013) completely constructed the2n m:2rB1-GMC designs for5N/16+1≤n≤N1; Wei, Yang, Li and Zhang(2010) and Ren and Zhang (2012) extended it to the cases of split-plot designs androbust parameter designs, and so on.Once the experimenters decide to use one selected design to plan a specified ex-periment, as the columns of designs play diferent roles, how to suitably assign thefactors to the columns is a key issue and meaningful. There are few researches inthis field, hence, experimenters always randomly assign the factors to columns. How-ever, in many practical experiments, experimenters have a prior knowledge about theimportance ordering of factors and would prefer to pay more attention to estimating ef-ficiently the main efects and the associated interactions of the most important factors.For this reason, the determination of optimal designs that are suitable for this situationbecome of interest from both theoretical and practical viewpoints. This important issue attracts us to start our initial research, and gradually extendsour research fields. The present dissertation is devoted to the following fields:(1) Proposing a new concept, factor aliased efect number pattern (F-AENP), tomeasure and rank the columns of a regular fractional factorial2n mdesign;(2) Utilizing the special constructions of GMC designs proposed by Li, Zhao andZhang (2011), we give the F-AENP’s mathematical result of the GMC2n mdesignswith5N/16+1≤n≤N1and their columns’ ordering. This result can be optimallyused when the experimenters have the importance of factors ordering and just care themain efects and two-factor interactions;(3) Utilizing the results of F-AENPs in GMC2n mdesigns, a procedure for select-ing n m sequentially best columns from these designs for assigning n m orderly mostimportant factors is also given. And listing the n m columns with N=16,32,64;(4) To the SOS designs, proposed by Zhang and Cheng (2010) and Cheng andZhang (2010) and used to construct GMC2n mdesigns with N/4+1≤n≤5N/16, wepresent a concise approach to directly obtain the SOS and GMC designs from Yatesorder;(5) For a given fractional factorial design, its AENP and F-AENPs are respectivelyused to measure and compare design and columns, and derives from the same idea, weshow the theoretical relationship between AENP and F-AENP;(6) Li, Zhao and Zhang (2011) ideally obtained the GMC2n mdesigns with5N/16+1≤n≤N1, and proved that these GMC designs can be selected byjust considering the first two terms of AENP, we prove that all the N1aliased setsof every specified GMC design can be classified into n m classes, and present themathematical results of AENP’s first two terms;(7) Exploring the construction of the GMC2n mdesigns with n≤N/4+1;(8) To select the optimal design, where interest centers primarily around thelower-order efects involving a few important factors, we collaboratively propose anew pattern, called individual-word length pattern (I-WLP), and completely obtain the optimal designs with N=16in lists;(9) Under MA criterion, studying the application of I-WLP in foldover designs.