Construction And Properties For Blocked Fractional Factorial Designs

Factorial experiments have wide applications in many diverse areas of human investigation. In general such an experiment has an output variable which is dependent on several controllable or input variables. These input variables are called factors. For each factor there are two or more possible settings known as levels. Any combination of the levels of all the factors under consideration is called treatment combination or run.Usually we wish to perform an experiment which considers n factors (vari-ables), which have s1,……,sn levels, respectively. This design is called symmetrical if s1=…=sn=s; otherwise it is called asymmetrical or mixed-level. A full factorial design would run the experiment at every possible combination of fac-tor level settings (treatment or run). The number of runs grows rapidly as the number of factors increases. So full factorial designs are rarely used in practice for large n (say, n≧7). For replacement, fractional factorial (FF) designs, which consist of a subset or fraction of full factorial designs, are commonly used. FF designs can broadly be classified into two types:regular and non-regular. An FF design is called regular if it can be constructed through defining relations among factors. On the contrary, the designs that do not possess this property are called non-regular designs. In most part of this thesis we consider regular two-level designs.FF designs involve a completely random allocation of the selected treatment combinations to the experimental units. This kind of allocation is appropriate only if the experimental units are homogeneous. In fact, such homogeneity may not always be guaranteed especially when the size of the experiment is relatively large. A practical design strategy is to partition the experimental units into homogeneous groups, known as blocks, and restrict randomization separately to each block. Examples of blocks include days, weeks, batches, lots, etc. If blocking to be effective, the units should be arranged so that the within-block variation is much smaller than the between-block variation.In regular blocked designs, arranging a 2n-m design into 2r blocks is equivalent to selecting r independent defining words for the 2r-1 block effects each con-founded with 2m treatment effects. This kind of designs is denoted by 2n-m:2r. In 2n-m:2r designs we use Ai,0 and Ai,1 to denote the number of defining words of length i involving only treatment factors and the number of block-defining words containing i treatment factors respectively. For 2n-m:2r designs, we restate the following generally recognizing assumptions:(1) Block-by-treatment interactions are assumed to be negligible.(2) Although block effects are existing, the purpose in a blocked design is to estimate the effects of treatment factors, and the estimations of block effects are not interested to us.(3) A1,0= A2,0= A0,1= A1,1= 0.(4) Block main effect and block-by-block interaction or interaction of some more block effects are the same important (Bisgaard 1994,example 1).But sometimes the assumption (4) rules out. Let b1,b2,……,br be r block fac-tors, that is, block main effects, the products of these block main effects generated every order block effects. However, block main effects and block-by-block effects or higher-order block effects may not be the same important. In a design, for example, if b1 represents two suppliers and b2 represents the day or night shift, b3 represent two workers, it would be hard to give a meaning for block-by-block effects b1b2,b1b3,b2b3 or three-block interaction b1b2b3,So, it seems reasonable to-assume that b1,b2 are more significant or more important than b1b2 (Sitter et al.1997, p.389 and Wu and Hamada 2000, p.131), and it is also reasonable to assume that b1b2 is more significant or more important than b1b2b3. As the hierar-chy principle in treatment effects, we may apply the hierarchy principle to block effects. For this reason, three-order or higher-order block effects can be supposed negligible in a blocked design. So we give the assumption(4') We may neglect three-order or higher-order block effects.In this thesis, we first use the assumptions (1), (2), (3), (4) for the first kind of blocking designs, and then, we use (1), (2), (3), (4') for the second kind of blocking designs.There are three type criteria in regular blocked designs. The first type is based on minimum abberation (MA) criterion. MA criterion has been applied to blocked fractional factorial designs. It can be applied to the treatment and block wordlength patterns separately. But, MA designs with respect to one wordlength pattern may not have MA with respect to the other wordlength pattern. One approach, as done by Sun et al (1997) and Mukerjee and Wu (1999), is to introduce the concept of admissible blocking schemes, however, it is often to have too many admissible designs. Another approach is to combine the treatment and block wordlength patterns into one single wordlength pattern, we call this as combined wordlength pattern.There are four blocked minimum aberration (BMA) criterions for blocked 2n-m designs, the four criterions are based on the four combined word-length patterns respectively. Each criterion is to minimize the terms sequentially in the corresponding combined word-length patterns, and the optimal designs under each criterion are called BMA (Chen et al.2006) designs. The four combined word-length patterns was proposed by Sitter et al. (1997), Chen and Cheng (1999), Zhang and Park (2000) and Cheng and Wu (2002), Cheng and Wu (2002).The second type is based on clear effects (CE) criterion. Up to now there are not many papers to specialize this criterion on selecting optimal blocked designs, but the CE criterion has been become a viewpoint of evaluating a. blocking scheme (see Wu and Hamada (2000)). Therefore we still treat it as a. special type of criterion in blocking case.The third type is based on maximum estimation capacity (MEC) criterion. Also, by now only a few papers utilize this criterion to discuss the selection of optimal blocking schemes. However, since there are several results which use this criterion to study regular designs without blocking (e.g., Sun (1993), Cheng and Mukerjee (1998)), using the criterion to study blocked designs may become a potential research topic. Hence we should pay an attention on it.We note the following fact:in many cases, the existing criteria lead to dif-ferent optimal designs. One natural question is that, among so many different criteria, under a common accepted principle, say, the effect hierarchy principle, which one is the best? For any set of parameters, whether one of existing criteria can really give an optimal design under the above principle? What is the most essential difference between the existing criteria? How to consider the basic infor-mation of confounding contained in the subgroup G, generated by both treatment and block factor words? Is there a more reasonable criterion which can really reflect the effect hierarchy principle in the blocking case? In this thesis we try answering these questions. Recently, Zhang, Li, Zhao and Ai (2008, hereafter called ZLZA) introduced a new aliasing pattern in 2n-m designs, called aliased effect-number pattern (AENP) and based on the AENP proposed a new criterion, general minimum lower order confounding (denoted by GMC) criterion. They proved that, under effect hierar-chy principle, the GMC criterion has much better performances than the MA and CE criteria at finding optimal regular designs. Zhang and Mukerjee (2009) later gave a further characterization to the GMC criterion via complementary sets.In the chapter 1 to chapter 5 of this thesis we introduce a new aliasing pattern fitting for assessing regular blocked designs, blocked aliased-effect number pattern (denoted by BAP). We also give some properties about construction of blocked designs. Based on the BAP, we propose two new criteria, B1-GMC and B2-GMC, respectively for blocking Kinds 1 and 2. Next we give some comparisons with the four existing criteria of MA-based type and CE criterion, and make a comparison between B1-GMC and B2-GMC criteria in addition. We also give some relationships between blocked designs under the MEC criterion and the BAP. The B1-GMC and B2-GMC designs of 16-,32-,64-,128-run and comparisons with existing criteria are tabulated.In chapter 6 we give some discussion about blocked non-regular designs.Fractional factorial split-plot (FFSP) design is a special kind of blocked de-sign. FFSP designs have been widely used in industrial experiments where the levels of some factors in an experiment are difficult or expensive to change. In this situation, a FFSP design involving a two-phase randomization is provided as a preferred option. In FFSP designs, the factors with hard-to-change levels are called whole plot (WP) ones, and the factors with relatively-easy-to-change levels are called subplot (SP) ones. Box and Jones (1992) gave an excellent discussion on this kind of designs.Up to now, the most existing results of choosing optimal FFSP designs are of MA type. Huang, Chen and Voelkel (1998) adopted the MA-FFSP criterion to choose an optimal regular two-level FFSP design for a thin-film coating ex-periment. Bingham and Sitter (1999) developed a new sequential construction method and compiled a catalog of MA two-level FFSP designs with 8 and 16 runs via primarily algorithmic approaches. Continuing with the two-level case, Bingham and Sitter (2001) listed MA-FFSP designs with up to 32 runs. In ad-dition, split-plot designs do not have the interchangeability between WP factors and SP factors so that there frequently exist several non-isomorphic FFSP designs which have MA. To overcome this problem, Mukerjee and Fang (2002) explored a criterion of minimum secondary aberration (MSA), denoted as MA-MSA-FFSP criterion, which significantly narrows the class of competing non-isomorphic MA designs and often yields a unique optimal design of MA type. Ai and Zhang (2006) constructed MA-MSA-FFSP designs in terms of consulting designs. Yang, Zhang and Liu (2007) constructed this kind of designs with weak MA. With a consideration on clear effects, Yang, Li, Liu and Zhang (2006) and Zi, Zhang and Liu (2006) had further investigations. Another line on the study of FFSP designs focused on D-optimal criterion such as, for example, Goos and Vandebroek (2001, 2003), Goos (2006) and Jones and Goos (2009).However, in many cases, an MA-MSA-FFSP design is not real optimal in the sense of effect estimation. Such a fact motivates us to establish some new but better criterion for selecting optimal FFSP designs.In chpter 7 we extend the GMC theory proposed by Zhang, Li, Zhao and Ai (2008, hereafter denoted as ZLZA) to FFSP designs to give a GMC-FFSP criterion. We establish a new criterion in split-plot designs, and then make some comparisons between the GMC-FFSP criterion and MA-MSA-FFSP criterion. At last, in this chapter, we describe an algorithm to search GMC-FFSP and MA-MSA-FFSP designs. Optimal 32-run split-plot designs under the two different criteria up to 14 factors are completely tabulated.Robust parameter design (or parameter design) which was first proposed by Taguchi (1987) is a statistical and engineering methodology that aims at reduc-ing the performance variation of a product or process by appropriately choosing the setting of its control factors so as to make it less sensitive to noise variation. The factors in parameter design experiments are divided into two types:control factors and noise factors. Control factors are variables whose values can be ad-justed but remain fixed once they are chosen. Control factors can include reaction temperature and time, and type and concentration of catalyst. By contrast, noise factors are variables whose levels are hard to control during the normal process or use conditions. They include variation in product and process parameters, environmental variation, load factors, user conditions, degradation, etc. In ro-bust parameter designs the control-by-noise interactions are crucial in achieving robustness. Thus this type of two-factor interactions must be placed in the same category of importance as the main effects. This obviously violates the effect hierarchy principle.Split-plot designs can be used to study robust parameter designs (Bingham and Sitter 2003). In a split-plot design, we can chose whole-plot factors as control factors and sub-plot factors as noise factors; if necessary, we can also chose whole-plot factors as noise factors and sub-plot factors as control factors. For this reason, robust parameter design can also be treated as one kind of blocked design.Taguchi (1987) proposed crossed array in robust parameter design. However, the crossing of the orthogonal arrays in a product array often results in an ex-orbitant number of runs and, moreover, several degrees of freedom are used for estimating higher-order interactions.As an alternative, we use a single array for both the control and noise factors. In single array, the required run size can be much smaller. This was proposed by Borkowski and Lucas (1991), and was discussed by Box and Jones (1992), Lucas (1994), Montgomery (1991,2001), Myers (1991), Shoemaker et al. (1991), Welch et al. (1990), Welch and Sacks (1991)and Wu and Zhu (2003). Cross arrays can be considered as a special kind of single arrays.In chapter 8, we discuss the estimation of lower-order effects in robust pa-rameter designs. Firstly, we study the estimation of lower-order effects in 2n-m designs; then we consider effects estimation, especially, the estimation of control-by-noise 2fi's in cross array and single array; at last, we propose a new criterion in single array for robust parameter designs and give some comparisons with the existing criterion in robust parameter designs.