分区组的因析设计的构造与性质
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摘要
因析试验在我们所涉及的各个研究领域都有广泛的应用.一般地,这样的试验有一个输出变量,该输出变量依靠一些可控的输入变量,这些输入变量被称为因子.每一个因子都有两个或更多个水平.所有因子的任一水平组合被称为一个处理组合或安排.通常我们安排一个具有n个因子的试验,这些因子分别有、s1,…,sn个水平.如果s1=,…,=sn=s,该试验称为对称的；否则称为非对称的或混水平的.一个完全的因析试验要安排所有的因子水平组合,而且随着因子个数的增加水平组合的数量增加的特别快.因此当n比较大时(比如n>7)完全因析试验很少使用.取而代之,部分因析(FF)试验被广泛采用,它是由完全因析试验的一部分处理组合构成的. FF设计被分为两大类：正规的和非正规的.一个FF设计称为正规的,如果它是用因子间的定义关系所确定的,否则称为非正规的.在本文大多数情况中,我们只考虑正规的两水平设计.
FF设计是把其处理组合用完全随机的方法分配到所有的试验单元.这种分配方式对所有试验单元是齐性的时候是合适的.事实上这样的齐性并不是总成立的,特别当试验单元的数量比较大的时候.一个合理的对策是将所有试验单元分成一些齐性的组,这些组称为区组,然后在每一个区组中实现随机化.比如,天,星期,批次,堆,等等都可以作为区组.如果试验单元分区组是有效的,则区组内的变差应当明显小于区组间的变差.
在正规区组设计中,分配2n-m设计到2r个区组中等价于选定r个独立的定义字,这些定义字生成2r-1个区组效应,每个区组效应都和2m个处理效应混杂.我们用2n-m:2r表示这样的区组设计.在2n-m:2r设计中,我们用Ai,0和Ai,1分别表示字长为i的只含处理因子的定义字的个数,以及含有区组因子和i个处理因子的定义字的个数.对2n-m:2r设计,我们再陈述一下一般被公认的假设：
(1)区组×处理交互效应是可忽略的.
(2)尽管存在区组效应,但是区组设计的目的是估计处理因子的效应,而对区组效应的估计不感兴趣.
(3)A1,0=A2,0=A0,1=A1,1=0.
(4)区组主效应,区组×区组交互效应,更多的区组因子的交互效应被看成是同等重要的.
但是假设(4)在有些情况下是不合理的.我们用b1,b2,…,b,表示r个区组因子,即区组主效应.这些即区组主效应的乘积生成各阶区组交互效应.区组主效应,区组×区组交互效应,更多的区组因子的交互效应可能不是同等重要的.比如,在一个设计中,b1表示两个批次,b2表示白天和夜间,b3表示两个工人,我们很难给出关于区组效应b1b2,b1b3,b2b3,b1b2b3的解释.因此假设b1,b2比b1b2更显著或更重要是合理的(Sitter et al.1997, p.389 and Wu and Hamada2000,p.131),并且假设b1b2比b1b2b3更重要也是合理的.就像处理效应中有效应排序原理,排序原理在区组效应中也应当成立.因此,在区组设计中,3阶或更高阶的区组效应是可被忽略的.我们有下面的假设：
(4')我们可以忽略3阶或更高阶的区组交互效应.
在本文中,称满足假设(1),(2),(3),(4)的区组设计为第一类区组设计,称满足假设(1),(2),(3),(4')的区组设计为第二类区组设计.
目前,在正规区组设计中,有三类准则.第一类准则是以最小低阶混杂(MA)准则为基础的. MA准则可以被分别应用于区组设计中的处理字长型和区组字长型.可是一种字长型的MA设计在另一种字长型下可能不是MA设计. Sunet al (1997), Mukerjee和Wu(1999)引入可允许区组设计,但是有时得到的可允许区组设计太多.还有一种把处理字长型和区组字长型结合成单一的字长型,我们称为结合字长型.
对于2n-m:2r设计有四种区组最小低阶(BMA)准则,它们分别对应四个不同的结合字长型.每一个准则都是依次最小化相应的字长型中的每一项,并且在每一个准则下最优的设计都被称为BMA(Chen et al.2006)设计.这四个结合字长型是由Sitter et al. (1997), Chen和Cheng (1999), Zhang和Park (2000), Cheng和Wu(2002),Cheng和Wu(2002).提出的.
第二类准则是以纯净效应(CE)准则为基础的.目前很少有文章专门在区组设计中讨论基于CE准则的最优设计,然而在Wu和Hamada (2000)中CE准则已经被应用于区组设计的评价中.因此我们把这个准则看成区组设计中的一种特殊类型的准则.
第三类准则是以最大估计容量(MCE)准则为基础的.目前仅有少数几篇文章利用此准则讨论区组设计.可是在非区组的FF设计中已经有关于这个准则的一些结果(比如,Sun 1993,Cheng和Mukerjee 1998),该准则可能成为区组设计中的一个潜在的研究课题.因此我们将对该准则加以关注.
我们注意下列事实：在很多情况下,现存的这些准则导致不同的最优设计.我们自然要提出一个问题,在一个公认的原则下,例如,效应排序原则下,这些准则中哪一个是最好的?对任意给定的一组参数,这些现存的准则中是否有一个能真正给出效应排序原则下的最优设计?这些现存的准则间本质的区别是什么?怎样考虑由处理定义字和区组定义字生成的定义子群G中所包含的信息?是否存在一个更合理的准则使其能够在区组设计中真实地体现效应排序原则?在本文中,我们试图回答这些问题.
最近,Zhang, Li, Zhao和Ai(2008,简称为ZLZA)在2n-m设计中引进一个新的混杂模型,称为混杂效应个数模型(AENP),在此基础上,提出一个新准则,一般最小低阶混杂(GMC)准则.他们证明了,在效应排序原则下,由GMC准则导致的最优设计比MA,CE准则好很多.接下来,Zhang和Mukerjee (2009)通过补集合的方法对GMC准则做了研究.
在第一章到第五章中,我们引进了评价正规区组设计的新的混杂模型,区组混杂效应个数模型(BAP).我们又给出关于区组设计的构造方面的性质.在BAP的基础上,我们给出两个新准则,B1-GMC准则和B2-GMC准则,其分别对应于我们的第一类区组设计及第二类区组设计.接下来,我们我们对现存的四个BMA准则及CE准则做了比较,并且对B1-GMC准则和B2-GMC准则进行了比较.我们也给出区组设计的MEC准则与BAP的关系.在附表中我们给出16-,32-64-和128-run B1-GMC设计和B2-GMC设计并且和现存的其它准则进行比较.
在第六章中,我们对非正规的区组设计做了讨论.
裂区(FFSP)设计可以看成一种特殊的区组设计FFSP设计在工业的试验中有广泛的应用,其中试验的一些因子的水平难以改变或改变因子水平的费用比较高,而FFSP设计包含两阶段的随机化为此提供了一个好的策略.在FFSP设计中,水平难以改变的因子被称为整区(WP)因子,水平比较容易改变的因子被称为子区(SP)因子.Box和Jones (1992)对这类设计做了很精彩的讨论.
到目前为止,关于选取最优的FFSP设计的结果都是在MA类型的基础上进行的.Huang, Chen和Voelkel (1998)在薄膜覆盖材料的试验中采用了MA-FFSP准则下的最优设计.Bingham和Sitter (1999)给出一系列构造性的方法,并且通过一个基础性的算法得到8-,16-run的两水平的MA-FFSP设计表.也是在两水平的情况下,Bingham和Sitter (2001)给出32-run MA-FFSP设计表.另一方面,裂区设计中WP因子和SP因子是不能互换的,所以常常存在几个非同构的FFSP设计都是MA设计.为了解决这一问题,Mukerjee和Fang(2002)提出最小二级混杂(MSA)准则,称为MA-MSA-FFSP准则,这样明显缩小了非同构的MA设计的选取范围,而且通常给出唯一的MA类型的最优设计.Ai和Zhang(2006)用协同设计的方式构造MA-MSA-FFSP设计.Yang, Zhang和Liu(2007)用弱MA的方法构造这类设计.Yang, Li, Liu和Zhang (2006)和Zi, Zhang和Liu(2006)讨论了这类设计中的纯净效应.FFSP设计的另一研究方向是关于D-最优设计,比如,Goos和Vandebroek (2001,2003), Goos (2006)和Jones和Goos (2009).
然而,在许多情况下,MA-MSA-FFSP设计在效应估计时不一定是最好的设计.因此我们试图构造更好的准则来挑选FFSP设计.
在第七章中,我们把ZLZA的准则推广到FFSP设计中,给出GMC-FFSP准则.我们建立了一个新准则,并且对GMC-FFSP准则和MA-MSA-FFSP准则进行比较.接下来,我们给出搜索GMC-FFSP设计和MA-MSA-FFSP设计的算法.最后给出直到14个因子的32-run的最优设计表.
Taguchi (1987)提出稳健参数设计(或参数设计),它是统计学及工程中的一种方法.这种设计通过适当选取控制因子的水平组合使其对噪声因子的变化不灵敏,从而达到减小生产过程所带来的变差.参数设计试验中的因子被分成两种类型:控制因子和噪声因子.控制因子是一组变量,其取值可以被调整,但是一旦被选取后就固定下来.控制因子可以是,反映的温度,时间,催化剂的类型,浓度.相反地,噪声因子也是一组变量,其水平组合在正常的生产过程和使用条件下难以控制.噪声因子可以是生产过程中的变差,过程参数,环境变差,负载因子,使用者的条件,材料性能的退化等等.控制×噪声交互效应是达到稳健性的关键环节,这种类型的2fi's应当与主效应具有同等的重要性,因而在这里违反了效应排序原则.
裂区设计可以用于稳健参数设计中(Bingham和Sitter 2003).在裂区设计中,我们可以选择整区因子作为控制因子而子区因子作为噪声因子；也可以选择整区因子作为噪声因子而子区因子作为控制因子.所以稳健参数设计也可以被看成一种区组设计.
Taguchi (1987)将乘积表用于稳健参数设计.可是乘积表在试验中常常导致run的个数比较大,并且有一部分自由度用来估计高阶效应.
作为替代方案,我们选择把控制因子和噪声因子安排在同一个单一表中.在单一表中,所需的run的个数要小许多.这个方案已经在许多文章中被讨论过,比如,Borkowski和Lucas (1991),Box和Jones (1992), Lucas (1994), Montgomery (1991,2001), Myers (1991), Shoemaker et al. (1991), Welch et al. (1990), Welch和Sacks (1991)以及Wu和Zhu(2003).乘积表可以看成单一表的一种特殊情况.
在第八章中,我们讨论参数设计中关于低阶效应的估计.首先我们研究在2-m中低阶效应的估计；然后我们讨论乘积表和单一表中的效应的估计,特别地,关于控制×噪声交互效应的估计；最后我们在单一表中给出一个新的准则并且与现存的准则做了比较.
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.
FF设计是把其处理组合用完全随机的方法分配到所有的试验单元.这种分配方式对所有试验单元是齐性的时候是合适的.事实上这样的齐性并不是总成立的,特别当试验单元的数量比较大的时候.一个合理的对策是将所有试验单元分成一些齐性的组,这些组称为区组,然后在每一个区组中实现随机化.比如,天,星期,批次,堆,等等都可以作为区组.如果试验单元分区组是有效的,则区组内的变差应当明显小于区组间的变差.
在正规区组设计中,分配2n-m设计到2r个区组中等价于选定r个独立的定义字,这些定义字生成2r-1个区组效应,每个区组效应都和2m个处理效应混杂.我们用2n-m:2r表示这样的区组设计.在2n-m:2r设计中,我们用Ai,0和Ai,1分别表示字长为i的只含处理因子的定义字的个数,以及含有区组因子和i个处理因子的定义字的个数.对2n-m:2r设计,我们再陈述一下一般被公认的假设：
(1)区组×处理交互效应是可忽略的.
(2)尽管存在区组效应,但是区组设计的目的是估计处理因子的效应,而对区组效应的估计不感兴趣.
(3)A1,0=A2,0=A0,1=A1,1=0.
(4)区组主效应,区组×区组交互效应,更多的区组因子的交互效应被看成是同等重要的.
但是假设(4)在有些情况下是不合理的.我们用b1,b2,…,b,表示r个区组因子,即区组主效应.这些即区组主效应的乘积生成各阶区组交互效应.区组主效应,区组×区组交互效应,更多的区组因子的交互效应可能不是同等重要的.比如,在一个设计中,b1表示两个批次,b2表示白天和夜间,b3表示两个工人,我们很难给出关于区组效应b1b2,b1b3,b2b3,b1b2b3的解释.因此假设b1,b2比b1b2更显著或更重要是合理的(Sitter et al.1997, p.389 and Wu and Hamada2000,p.131),并且假设b1b2比b1b2b3更重要也是合理的.就像处理效应中有效应排序原理,排序原理在区组效应中也应当成立.因此,在区组设计中,3阶或更高阶的区组效应是可被忽略的.我们有下面的假设：
(4')我们可以忽略3阶或更高阶的区组交互效应.
在本文中,称满足假设(1),(2),(3),(4)的区组设计为第一类区组设计,称满足假设(1),(2),(3),(4')的区组设计为第二类区组设计.
目前,在正规区组设计中,有三类准则.第一类准则是以最小低阶混杂(MA)准则为基础的. MA准则可以被分别应用于区组设计中的处理字长型和区组字长型.可是一种字长型的MA设计在另一种字长型下可能不是MA设计. Sunet al (1997), Mukerjee和Wu(1999)引入可允许区组设计,但是有时得到的可允许区组设计太多.还有一种把处理字长型和区组字长型结合成单一的字长型,我们称为结合字长型.
对于2n-m:2r设计有四种区组最小低阶(BMA)准则,它们分别对应四个不同的结合字长型.每一个准则都是依次最小化相应的字长型中的每一项,并且在每一个准则下最优的设计都被称为BMA(Chen et al.2006)设计.这四个结合字长型是由Sitter et al. (1997), Chen和Cheng (1999), Zhang和Park (2000), Cheng和Wu(2002),Cheng和Wu(2002).提出的.
第二类准则是以纯净效应(CE)准则为基础的.目前很少有文章专门在区组设计中讨论基于CE准则的最优设计,然而在Wu和Hamada (2000)中CE准则已经被应用于区组设计的评价中.因此我们把这个准则看成区组设计中的一种特殊类型的准则.
第三类准则是以最大估计容量(MCE)准则为基础的.目前仅有少数几篇文章利用此准则讨论区组设计.可是在非区组的FF设计中已经有关于这个准则的一些结果(比如,Sun 1993,Cheng和Mukerjee 1998),该准则可能成为区组设计中的一个潜在的研究课题.因此我们将对该准则加以关注.
我们注意下列事实：在很多情况下,现存的这些准则导致不同的最优设计.我们自然要提出一个问题,在一个公认的原则下,例如,效应排序原则下,这些准则中哪一个是最好的?对任意给定的一组参数,这些现存的准则中是否有一个能真正给出效应排序原则下的最优设计?这些现存的准则间本质的区别是什么?怎样考虑由处理定义字和区组定义字生成的定义子群G中所包含的信息?是否存在一个更合理的准则使其能够在区组设计中真实地体现效应排序原则?在本文中,我们试图回答这些问题.
最近,Zhang, Li, Zhao和Ai(2008,简称为ZLZA)在2n-m设计中引进一个新的混杂模型,称为混杂效应个数模型(AENP),在此基础上,提出一个新准则,一般最小低阶混杂(GMC)准则.他们证明了,在效应排序原则下,由GMC准则导致的最优设计比MA,CE准则好很多.接下来,Zhang和Mukerjee (2009)通过补集合的方法对GMC准则做了研究.
在第一章到第五章中,我们引进了评价正规区组设计的新的混杂模型,区组混杂效应个数模型(BAP).我们又给出关于区组设计的构造方面的性质.在BAP的基础上,我们给出两个新准则,B1-GMC准则和B2-GMC准则,其分别对应于我们的第一类区组设计及第二类区组设计.接下来,我们我们对现存的四个BMA准则及CE准则做了比较,并且对B1-GMC准则和B2-GMC准则进行了比较.我们也给出区组设计的MEC准则与BAP的关系.在附表中我们给出16-,32-64-和128-run B1-GMC设计和B2-GMC设计并且和现存的其它准则进行比较.
在第六章中,我们对非正规的区组设计做了讨论.
裂区(FFSP)设计可以看成一种特殊的区组设计FFSP设计在工业的试验中有广泛的应用,其中试验的一些因子的水平难以改变或改变因子水平的费用比较高,而FFSP设计包含两阶段的随机化为此提供了一个好的策略.在FFSP设计中,水平难以改变的因子被称为整区(WP)因子,水平比较容易改变的因子被称为子区(SP)因子.Box和Jones (1992)对这类设计做了很精彩的讨论.
到目前为止,关于选取最优的FFSP设计的结果都是在MA类型的基础上进行的.Huang, Chen和Voelkel (1998)在薄膜覆盖材料的试验中采用了MA-FFSP准则下的最优设计.Bingham和Sitter (1999)给出一系列构造性的方法,并且通过一个基础性的算法得到8-,16-run的两水平的MA-FFSP设计表.也是在两水平的情况下,Bingham和Sitter (2001)给出32-run MA-FFSP设计表.另一方面,裂区设计中WP因子和SP因子是不能互换的,所以常常存在几个非同构的FFSP设计都是MA设计.为了解决这一问题,Mukerjee和Fang(2002)提出最小二级混杂(MSA)准则,称为MA-MSA-FFSP准则,这样明显缩小了非同构的MA设计的选取范围,而且通常给出唯一的MA类型的最优设计.Ai和Zhang(2006)用协同设计的方式构造MA-MSA-FFSP设计.Yang, Zhang和Liu(2007)用弱MA的方法构造这类设计.Yang, Li, Liu和Zhang (2006)和Zi, Zhang和Liu(2006)讨论了这类设计中的纯净效应.FFSP设计的另一研究方向是关于D-最优设计,比如,Goos和Vandebroek (2001,2003), Goos (2006)和Jones和Goos (2009).
然而,在许多情况下,MA-MSA-FFSP设计在效应估计时不一定是最好的设计.因此我们试图构造更好的准则来挑选FFSP设计.
在第七章中,我们把ZLZA的准则推广到FFSP设计中,给出GMC-FFSP准则.我们建立了一个新准则,并且对GMC-FFSP准则和MA-MSA-FFSP准则进行比较.接下来,我们给出搜索GMC-FFSP设计和MA-MSA-FFSP设计的算法.最后给出直到14个因子的32-run的最优设计表.
Taguchi (1987)提出稳健参数设计(或参数设计),它是统计学及工程中的一种方法.这种设计通过适当选取控制因子的水平组合使其对噪声因子的变化不灵敏,从而达到减小生产过程所带来的变差.参数设计试验中的因子被分成两种类型:控制因子和噪声因子.控制因子是一组变量,其取值可以被调整,但是一旦被选取后就固定下来.控制因子可以是,反映的温度,时间,催化剂的类型,浓度.相反地,噪声因子也是一组变量,其水平组合在正常的生产过程和使用条件下难以控制.噪声因子可以是生产过程中的变差,过程参数,环境变差,负载因子,使用者的条件,材料性能的退化等等.控制×噪声交互效应是达到稳健性的关键环节,这种类型的2fi's应当与主效应具有同等的重要性,因而在这里违反了效应排序原则.
裂区设计可以用于稳健参数设计中(Bingham和Sitter 2003).在裂区设计中,我们可以选择整区因子作为控制因子而子区因子作为噪声因子；也可以选择整区因子作为噪声因子而子区因子作为控制因子.所以稳健参数设计也可以被看成一种区组设计.
Taguchi (1987)将乘积表用于稳健参数设计.可是乘积表在试验中常常导致run的个数比较大,并且有一部分自由度用来估计高阶效应.
作为替代方案,我们选择把控制因子和噪声因子安排在同一个单一表中.在单一表中,所需的run的个数要小许多.这个方案已经在许多文章中被讨论过,比如,Borkowski和Lucas (1991),Box和Jones (1992), Lucas (1994), Montgomery (1991,2001), Myers (1991), Shoemaker et al. (1991), Welch et al. (1990), Welch和Sacks (1991)以及Wu和Zhu(2003).乘积表可以看成单一表的一种特殊情况.
在第八章中,我们讨论参数设计中关于低阶效应的估计.首先我们研究在2-m中低阶效应的估计；然后我们讨论乘积表和单一表中的效应的估计,特别地,关于控制×噪声交互效应的估计；最后我们在单一表中给出一个新的准则并且与现存的准则做了比较.
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.
引文
1. Ai, M. and He, S., (2005), Theory of optimal blocking for fractional factorial split-plot designs. Sci. China Ser. A 48(5),649-656.
2. Ai, M. and Zhang, R., (2004), Multistratum fractional factorial split-plot designs with minimum aberration and maximum estimation capacity. Statist. Probab. Lett. 69,161-170.
3. Ai, M. and Zhang, R., (2006), Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs. Sci. China Ser. A 49(4),494-512.
4. Ai, M. and Zhang, R., (2004), sn-m designs containing clear main effects or clear two-factor interactions. Statist. Probab. Lett.69,151-160.
5. Addelman, S. (1962a), orthogonal main-effect plans for asymmetrical factorial ex-periments. Technornetrics 4,21-46.
6. Addelman, S. (1962b), Symmetrical and asymmetrical fractional factorial plans. Technometrics 4,47-58.
7. Bingham, D. and Mukerjee, R. (2006), Detailed wordlength pattern of regular frac-tional factorial split-plot designs in terms of complementary sets. Discrete Math. 306,1522-1533.
8. Bingham, D. and Sitter, R. R. (1999a), Minimum aberration two-level fractional factorial split-plot designs. Technometrics 41,62-70.
9. Bingham, D. and Sitter, R. R. (1999b), Some theoretical results for fractional fac-torial split-plot designs. Ann. Statist.27,1240-1255.
10. Bingham, D. and Sitter, R. R. (2001), Design issues in fractional factorial split-plot experiments. J. Quality Technol.33,2-15.
11. Bingham, D. and Sitter, R. R., (2003), Fractional factorial split-plot designs for robust parameter experiments. Technometrics 45,80-89.
12. Bingham, D., Schoen, E. D. and Sitter, R. R. (2004), Designing fractional factorial split-plot experiments with few whole-plot factors. Appl. Statist.53,325-339.
13. Bingham, D., Schoen, E. D. and Sitter, R. R. (2005), Corrigendum:Designing fractional factorial split-plot experiments with few whole-plot factors. Appl. Statist. 54,955-958.
14. Bisgaard, S. (1994). A note on the definition for blocked 2n-p designs. Technomet-rics 36,308-311.
15. Bisgaard, S. (2000), The design and analysis of 2k-p × 2q-r split-plot experiments. J. Quality Technol.32,39-56.
16. Bisgaard, S. and Steinberg, D. M. (1997), The design and analysis of 2k-p × s prototype experiments. Technometrics 39,52-62.
17. Box, G. E. P. and Hunter, J. S. (1961), The 2k-p fractional factorial designs. Tech-nometrics 3,311-351.
18. Box, G. E. P., J. S. Hunter, and W. G. Hunter (2005). Statistics for Experimenters. 2nd Ed. New York:Wiley.
19. Box, G. E. P. and Jones, S. (1992), Split-plot designs for robust product experi-mentation. Appl. Statist.19,3-26.
20. Chen, B. J., Li, P. F., Liu, M. Q., Zhang, R. C. (2006). Some results on blocked regular 2-level fractional factorial designs with clear effects. J. Statist. Plann. Inference,136,4436-4449.
21. Chen, H., Cheng, C.-S. (1999). Theory of optimal blocking of 2n-m designs. Ann. Statist.27,1948-1973.
22. Chen, H. G. and Hedayat, A. S. (1996),2n-l designs with weak minimum aberra-tion. Ann. Statist.24,2536-2548.
23. Chen, H. G. and Hedayat, A. S. (1998),2n-m designs with resolution III or IV containing clear two-factor interactions. J. Statist. Plann. Inference 75,147-158.
24. Chen, J., (1992), Some results on 2n-k fractional factorial designs and search for minimum aberration designs. Ann. Statist.20,2124-2141.
25. Chen, J. (1998). Intelligent search for 213-6 and 214-7 minimum aberration designs. Statistica Sinica 8,1265-1270.
26. Chen, J., Sun, D. X. and Wu, C. F. J. (1993), A catalogue of two-level and three-level fractional factorial designs with small runs, Internat Statist. Rev.61,131-145.
27. Chen, J. and Wu, C. J. F. (1991), Some results on sn-k fractional factorial designs with minimum aberration or optimal moments. Ann. Statist.20,1028-1041.
28. Cheng, C. S. and Mukerjee, R. (1998), Regular fractional factorial designs with minimum aberration and maximum estimation capacity. Ann. Statist.26,2289-2300.
29. Cheng, C. S. and R. Mukerjee (2001). Blocked regular fractional factorial designs with maximum estimation capacity. Ann. Statist.29,530-548.
30. Cheng, C. S., Steinberg, D. M. and Sun, D. X. (1999), Minimum aberration and model robustness for two-level factorial designs. J. Roy. Statist. Soc. Ser. B 61, 85-93.
31. Cheng, C. S. and B. Tang (2005). A general theory of minimum aberration and its applications. Ann. Statist.33,944-958.
32. Cheng, S. W., Wu, C. F. J. (2002). Choice of optimal blocking schemes in two-level and three level designs. Technometrics 44.269-277.
33. Cornell, J. A. (1988), Analyzing data from mixture experiments containing process variables:a split-plot approach. J. Quality Technol.20,2-33.
34. Dey, A. (1985). Orthogonal Fractional Factorial Designs. New York:Halsted Press.
35. Dey, A. and R. Mukerjee (1999). Fractional Factorial Plans. New York:Wiley.
36. Fang, K. T. and Mukerjee, R. (2000), A connection between uniformity and aber-ration in regular fractions of two-level factorials. Biometrika 87,173-198.
37. Fries, A. and Hunter, W. G. (1980), Minimum aberration 2k-p designs. Techno-metrics 22,601-608.
38. Hickernell, F. J. and Liu, M. Q. (2002), Uniform designs limit aliasing. Biometrika 89,893-904.
39. Goos, P. (2002), The Optimal Design of Blocked and Split-Plot Experiments, New York:Springer-Verlag.
40. Goos P. (2006), Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments. Statist Neerlandica,60,361-378
41. Goos P, Vandebroek M. (2001), Optimal Split-Plot Designs. J Quality Technol, 33,435-450
42. Goos P, Vandebroek M. (2003), D-Optimal Split-Plot Designs with Given Numbers and Sizes of Whole Plots. Technometrics,45,235-245
43. Jacroux, M. (2004). A modified minimum aberration criterion for selecting regular 2m-k fractional factorial designs, J. Statist. Plann. Inference 126,325-336.
44. Jones B, Goos P. (2009), D-optimal design of split-split-plot experiments. Biometrika, 96,67-82
45. Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999), Orthogonal Arrays:Theory and Applications. Springer, New York.
46. Huang, P., Chen, D. and Voelkel, J. (1998), Minimum aberration two-level split-plot designs. Technometrics 40,314-326.
47. Ke, W., Tang, B. and Wu, H. (2005), Compromise plans with clear two-factor interactions. Statist. Sinica 15,709-715.
48. Kempthorne, O.,1952. Design and Analysis of Experiments, New york:John Wiley Sons.
49. Kulahci, M., Ramirez, J. and Tobias, R. (2006), Split-plot fractional designs:Is minimum aberration enough? J. Quality Technol,38,56-64.
50. Letsinger, J. D., Myers, R. H. and Lentner, M. (1996), Response surface methods for bi-randomization structures. J. Quality Technol.28,381-397.
51. Li, P. F., Chen, B. J., Liu, M. Q., Zhang, R. C. (2006). A note on minimum aberration and clear criteria. Statistics and Probability Letters 76:1007-1011.
52. Loeppky, J. L. and Sitter, R. R. (2002), Analyzing unreplicated blocked or split-plot fractional factorial designs. J. Quality Technol.34,229-243.
53. Ma, C.X., Fang, K. T. and Lin, D. K. J. (2001), On the isomorphism of fractional factorial designs. J. Complexity 17,86-97.
54. Mcleod R G. (2008), Optimal block sequences for blocked fractional factorial split-plot designs. J Statist Plan Inference,138,2563-2573
55. Mcleod R G, Brewster J F. (2004), The design of blocked fractional factorial split-plot experiments. Technometrics,46,135-146
56. McLeod, R.G. and Brewster, J.F. (2006), Blocked fractional factorial split-plot experiments for robust parameter design. J. Quality Technol.38,267-279.
57. Montgomery, D. C. (2001), The Design and Analysis of Experiments, (5th ed.). John Wiley & Sons, Inc.
58. Mukerjee, R., L. Y. Chan, and K. T. Fang (2000). Regular fractions of mixed factorials with maximum estimation capacity. Statistica Sinica 10,1117-1132.
59. Mukerjee, R. and Fang, K. T. (2002), Fractional factorial split-plot designs with minimum aberration and maximum estimation capacity. Statist. Sinica 12,885-903.
60. Mukerjee, R., Wu, C. F. J. (1999) Blocking in regular fractional factorials:A projective geometric approach. Annals of Statistics, Vol.27, No.4,1256-1271.
61. Mukerjee, R. and Wu, C. F. J. (2001), minimum aberration designs for mixed factorials in terms of complementary sets. Statist. Sinica 11,225-239.
62. Mukerjee, R. and Wu, C. F. J. (2006) A Modern Theory of Factorial designs. Springer Series in Statistics, Springer Science+Business Media Inc., New York.
63. Plotkin, M. (1960), Binary codes with specified minimum distance. IEEE Trans. Inform. Theory 6,445-450.
64. Ren, J.B. and Zhang, R.C. (2008). Optimal selection of two-level regular single arrays for robust parameter experiments, submitted.
65. Shoemaker, A.C., Tsui, K. L. and Wu, C. F. J. (1991), Economical experimentation methods for robust design, Technometrics 33,415-427.
66. Sitter, R.R., Chen, J. and Feder, M. (1997), Fractional resolution and minimum aberration in blocked 2n-k designs. Technometrics 39,382-390.
67. Sun, D. X. (1993), Estimation Capacity and Related Topics in Experimental De-signs. Ph.D. Dissertation, University of Waterloo, Waterloo.
68. Sun, D. X., Wu, C. F. J., Chen, Y. Y. (1997). Optimal blocking schemes for 2n and 2n-P designs. Technometrics 39,298-307.
69. Taguchi, G. (1986), Introduction to Quality Engineering:Designing Quality into Products and Processes, Tokyo:Asian Productivity Organization.
70. Tang. B. (2006), Construction results on minimum aberration blocking schemes for 2m designs. J. Statist. Plann. Inference, available online.
71. Tang, B., Ma, F., Ingram, D. and Wang, H. (2002), Bounds on the maximum number of clear two-factor interactions for 2m-p designs of resolution III and IV. Canad. J. Statist.30,127-136.
72. Tang, B. and Wu, C. F. J. (1996), Characterization of minimum aberration 2n-k designs in terms of their complementary designs. Ann. Statist.24,2549-2559.
73. Welch, W. J., Yu, T. K., Kang, S. M. and Sacks, J. (1990), Computer experiments for quality control by parameter design. J. Quality Technol.22,38-45.
74. Wu, C. F. J. (1989), Construction of 2m4n via a group scheme. Ann. Statist.17, 1880-1885.
75. Wu, C. F. J. and Chen, Y. (1992). A graph-aided method for planning two-level experiments when certain interactions are importaint. Technometrics 34,162-175.
76. Wu, C. F. J. and Hamada, M. S. (2000), Experiments:Planning, Analysis and Parameter Design Optimization (Wiley, New York).
77. Wu, H. and Wu, C. F. J. (2002), Clear two-factor interactions and minimum aber-ration. Ann. Statist.30,1496-1511.
78. Wu, C. F. J. and Zhang, R. C. (1993), Minimum aberration designs with two-level and four-level factors. Biometrika 80,203-209.
79. Wu, C. F. J., Zhang, R. C. and Wang, R. G. (1992), Construction of asymmetrical orthogonal arrays of the type OA(sk, (sr1)n1…(srl)nl). Statist. Sinica 2,203-219.
80. Wu, C. F. J. and Zhu, Y. (2003), Optimal selection of single arrays for parameter design experiments. Statist. Sinica 13,1179-1199.
81. Xu, H. (2006), Blocked regular fractional factorial designs with minimum aberra-tion. Annals of statistics 2006, Vol.34, No.5,2534-2553.
82. Xu, H., Lau, S., (2006), Minimum aberration blocking schemes for two-and three-level fractional factorial designs. J. Statist. Plann. Inference 136,4088-4118.
83. Yang, G. J., Liu, M. Q., Zhang, R. C.(2005), Weak minimum aberration and maxi-mum number of clear two-factor interactions in 2Ⅳm-p designs [J]. Science in China, Ser A 48(11),1479-1487.
84. Yang, G. J., Liu, M. Q.(2006), A note on the lower bound on maximum number of clear two-factor interactions for 2Ⅲm-p and 2Ⅳm-p designs [J]. Communications in Statistics,35(5),849-860.
85. Yang, G. J., Liu, M. Q., Zhang, R. C.(2006), A note on 2Ⅳm-p designs with the maximum number of clear two-factor interactions [J]. ACTA Mathematica Scientia, 26 A(7),1153-1158.
86. Yang, J. F., Li, P. F., Liu, M. Q. and Zhang, R. C. (2006), 2(n1+n2)-(k1+k2) fractional factorial split-plot designs containing clear effects. J. Statist. Plann. Inference, 136,4450-4458.
87. Yang J F, Zhang R C, Liu M Q. (2007), Construction of fractional factorial split-plot designs with weak minimum aberration, Statist & Prob Letters,77,1567-1573
88. Zhang R C, Cheng Y. (2010), General minimum lower order confounding designs: An overview and a construction theory. J Statist Plan Inference,140,1719-1730.
89. Zhang, R.C. and Shao, Q. (2001), Minimum aberration (S2)Sn-k designs. Statist. Sinica 11,213-223.
90. Zhang, R. C., Li, P., Zhao, S. L. and Ai, M. Y. (2008), A general minimum lower-order confounding criterion for two-level regular designs. Statist. Sinica 18,1689-1705.
91. Zhang, R. C., Mukerjee, R.(2009) Characterization of general minimum lower order confounding via complementary. Statist Sinica,19,363-375.
92. Zhang, R. C., Park, D. K., (2000). Optimal blocking of two-level fractional factorial designs. J. Statist. Plann. Inference 91,107-121.
93. Zhao, S. L. (2006), Optimal Criteria and Construction for Two Classes of Regular Factorial Designs. unpublished doctorial thesis, Department of Statistics, School of Mathematical sciences, Nankai University.
94. Zhu, Y. (2000), A Theory of Experimental Design for Multiple Groups of Factors. Unpublished doctoral thesis, University of Michigan, Department of Statistics.
95. Zhu, Y. (2003). Structure function for aliasing patterns in 2l-n 2 design with multiple groups of factors. Ann. Statist.31,995-1011.
96. Zhu, Y. and C. F. J. Wu (2007). Structure function for general sl-m design and blocked regular (sl-m,sr) design and their complementary designs. Statistica Sinica,17,1237-1257.
97. Zi, X. M., Zhang, R. C. and Liu, M. Q. (2006), Bounds on the maximum numbers of clear two-factor interactions for 2(n1+n2)-(k1+k2) fractional factorial split-plot designs. Sci. China Ser. A 49(12),1816-1829.
2. Ai, M. and Zhang, R., (2004), Multistratum fractional factorial split-plot designs with minimum aberration and maximum estimation capacity. Statist. Probab. Lett. 69,161-170.
3. Ai, M. and Zhang, R., (2006), Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs. Sci. China Ser. A 49(4),494-512.
4. Ai, M. and Zhang, R., (2004), sn-m designs containing clear main effects or clear two-factor interactions. Statist. Probab. Lett.69,151-160.
5. Addelman, S. (1962a), orthogonal main-effect plans for asymmetrical factorial ex-periments. Technornetrics 4,21-46.
6. Addelman, S. (1962b), Symmetrical and asymmetrical fractional factorial plans. Technometrics 4,47-58.
7. Bingham, D. and Mukerjee, R. (2006), Detailed wordlength pattern of regular frac-tional factorial split-plot designs in terms of complementary sets. Discrete Math. 306,1522-1533.
8. Bingham, D. and Sitter, R. R. (1999a), Minimum aberration two-level fractional factorial split-plot designs. Technometrics 41,62-70.
9. Bingham, D. and Sitter, R. R. (1999b), Some theoretical results for fractional fac-torial split-plot designs. Ann. Statist.27,1240-1255.
10. Bingham, D. and Sitter, R. R. (2001), Design issues in fractional factorial split-plot experiments. J. Quality Technol.33,2-15.
11. Bingham, D. and Sitter, R. R., (2003), Fractional factorial split-plot designs for robust parameter experiments. Technometrics 45,80-89.
12. Bingham, D., Schoen, E. D. and Sitter, R. R. (2004), Designing fractional factorial split-plot experiments with few whole-plot factors. Appl. Statist.53,325-339.
13. Bingham, D., Schoen, E. D. and Sitter, R. R. (2005), Corrigendum:Designing fractional factorial split-plot experiments with few whole-plot factors. Appl. Statist. 54,955-958.
14. Bisgaard, S. (1994). A note on the definition for blocked 2n-p designs. Technomet-rics 36,308-311.
15. Bisgaard, S. (2000), The design and analysis of 2k-p × 2q-r split-plot experiments. J. Quality Technol.32,39-56.
16. Bisgaard, S. and Steinberg, D. M. (1997), The design and analysis of 2k-p × s prototype experiments. Technometrics 39,52-62.
17. Box, G. E. P. and Hunter, J. S. (1961), The 2k-p fractional factorial designs. Tech-nometrics 3,311-351.
18. Box, G. E. P., J. S. Hunter, and W. G. Hunter (2005). Statistics for Experimenters. 2nd Ed. New York:Wiley.
19. Box, G. E. P. and Jones, S. (1992), Split-plot designs for robust product experi-mentation. Appl. Statist.19,3-26.
20. Chen, B. J., Li, P. F., Liu, M. Q., Zhang, R. C. (2006). Some results on blocked regular 2-level fractional factorial designs with clear effects. J. Statist. Plann. Inference,136,4436-4449.
21. Chen, H., Cheng, C.-S. (1999). Theory of optimal blocking of 2n-m designs. Ann. Statist.27,1948-1973.
22. Chen, H. G. and Hedayat, A. S. (1996),2n-l designs with weak minimum aberra-tion. Ann. Statist.24,2536-2548.
23. Chen, H. G. and Hedayat, A. S. (1998),2n-m designs with resolution III or IV containing clear two-factor interactions. J. Statist. Plann. Inference 75,147-158.
24. Chen, J., (1992), Some results on 2n-k fractional factorial designs and search for minimum aberration designs. Ann. Statist.20,2124-2141.
25. Chen, J. (1998). Intelligent search for 213-6 and 214-7 minimum aberration designs. Statistica Sinica 8,1265-1270.
26. Chen, J., Sun, D. X. and Wu, C. F. J. (1993), A catalogue of two-level and three-level fractional factorial designs with small runs, Internat Statist. Rev.61,131-145.
27. Chen, J. and Wu, C. J. F. (1991), Some results on sn-k fractional factorial designs with minimum aberration or optimal moments. Ann. Statist.20,1028-1041.
28. Cheng, C. S. and Mukerjee, R. (1998), Regular fractional factorial designs with minimum aberration and maximum estimation capacity. Ann. Statist.26,2289-2300.
29. Cheng, C. S. and R. Mukerjee (2001). Blocked regular fractional factorial designs with maximum estimation capacity. Ann. Statist.29,530-548.
30. Cheng, C. S., Steinberg, D. M. and Sun, D. X. (1999), Minimum aberration and model robustness for two-level factorial designs. J. Roy. Statist. Soc. Ser. B 61, 85-93.
31. Cheng, C. S. and B. Tang (2005). A general theory of minimum aberration and its applications. Ann. Statist.33,944-958.
32. Cheng, S. W., Wu, C. F. J. (2002). Choice of optimal blocking schemes in two-level and three level designs. Technometrics 44.269-277.
33. Cornell, J. A. (1988), Analyzing data from mixture experiments containing process variables:a split-plot approach. J. Quality Technol.20,2-33.
34. Dey, A. (1985). Orthogonal Fractional Factorial Designs. New York:Halsted Press.
35. Dey, A. and R. Mukerjee (1999). Fractional Factorial Plans. New York:Wiley.
36. Fang, K. T. and Mukerjee, R. (2000), A connection between uniformity and aber-ration in regular fractions of two-level factorials. Biometrika 87,173-198.
37. Fries, A. and Hunter, W. G. (1980), Minimum aberration 2k-p designs. Techno-metrics 22,601-608.
38. Hickernell, F. J. and Liu, M. Q. (2002), Uniform designs limit aliasing. Biometrika 89,893-904.
39. Goos, P. (2002), The Optimal Design of Blocked and Split-Plot Experiments, New York:Springer-Verlag.
40. Goos P. (2006), Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments. Statist Neerlandica,60,361-378
41. Goos P, Vandebroek M. (2001), Optimal Split-Plot Designs. J Quality Technol, 33,435-450
42. Goos P, Vandebroek M. (2003), D-Optimal Split-Plot Designs with Given Numbers and Sizes of Whole Plots. Technometrics,45,235-245
43. Jacroux, M. (2004). A modified minimum aberration criterion for selecting regular 2m-k fractional factorial designs, J. Statist. Plann. Inference 126,325-336.
44. Jones B, Goos P. (2009), D-optimal design of split-split-plot experiments. Biometrika, 96,67-82
45. Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999), Orthogonal Arrays:Theory and Applications. Springer, New York.
46. Huang, P., Chen, D. and Voelkel, J. (1998), Minimum aberration two-level split-plot designs. Technometrics 40,314-326.
47. Ke, W., Tang, B. and Wu, H. (2005), Compromise plans with clear two-factor interactions. Statist. Sinica 15,709-715.
48. Kempthorne, O.,1952. Design and Analysis of Experiments, New york:John Wiley Sons.
49. Kulahci, M., Ramirez, J. and Tobias, R. (2006), Split-plot fractional designs:Is minimum aberration enough? J. Quality Technol,38,56-64.
50. Letsinger, J. D., Myers, R. H. and Lentner, M. (1996), Response surface methods for bi-randomization structures. J. Quality Technol.28,381-397.
51. Li, P. F., Chen, B. J., Liu, M. Q., Zhang, R. C. (2006). A note on minimum aberration and clear criteria. Statistics and Probability Letters 76:1007-1011.
52. Loeppky, J. L. and Sitter, R. R. (2002), Analyzing unreplicated blocked or split-plot fractional factorial designs. J. Quality Technol.34,229-243.
53. Ma, C.X., Fang, K. T. and Lin, D. K. J. (2001), On the isomorphism of fractional factorial designs. J. Complexity 17,86-97.
54. Mcleod R G. (2008), Optimal block sequences for blocked fractional factorial split-plot designs. J Statist Plan Inference,138,2563-2573
55. Mcleod R G, Brewster J F. (2004), The design of blocked fractional factorial split-plot experiments. Technometrics,46,135-146
56. McLeod, R.G. and Brewster, J.F. (2006), Blocked fractional factorial split-plot experiments for robust parameter design. J. Quality Technol.38,267-279.
57. Montgomery, D. C. (2001), The Design and Analysis of Experiments, (5th ed.). John Wiley & Sons, Inc.
58. Mukerjee, R., L. Y. Chan, and K. T. Fang (2000). Regular fractions of mixed factorials with maximum estimation capacity. Statistica Sinica 10,1117-1132.
59. Mukerjee, R. and Fang, K. T. (2002), Fractional factorial split-plot designs with minimum aberration and maximum estimation capacity. Statist. Sinica 12,885-903.
60. Mukerjee, R., Wu, C. F. J. (1999) Blocking in regular fractional factorials:A projective geometric approach. Annals of Statistics, Vol.27, No.4,1256-1271.
61. Mukerjee, R. and Wu, C. F. J. (2001), minimum aberration designs for mixed factorials in terms of complementary sets. Statist. Sinica 11,225-239.
62. Mukerjee, R. and Wu, C. F. J. (2006) A Modern Theory of Factorial designs. Springer Series in Statistics, Springer Science+Business Media Inc., New York.
63. Plotkin, M. (1960), Binary codes with specified minimum distance. IEEE Trans. Inform. Theory 6,445-450.
64. Ren, J.B. and Zhang, R.C. (2008). Optimal selection of two-level regular single arrays for robust parameter experiments, submitted.
65. Shoemaker, A.C., Tsui, K. L. and Wu, C. F. J. (1991), Economical experimentation methods for robust design, Technometrics 33,415-427.
66. Sitter, R.R., Chen, J. and Feder, M. (1997), Fractional resolution and minimum aberration in blocked 2n-k designs. Technometrics 39,382-390.
67. Sun, D. X. (1993), Estimation Capacity and Related Topics in Experimental De-signs. Ph.D. Dissertation, University of Waterloo, Waterloo.
68. Sun, D. X., Wu, C. F. J., Chen, Y. Y. (1997). Optimal blocking schemes for 2n and 2n-P designs. Technometrics 39,298-307.
69. Taguchi, G. (1986), Introduction to Quality Engineering:Designing Quality into Products and Processes, Tokyo:Asian Productivity Organization.
70. Tang. B. (2006), Construction results on minimum aberration blocking schemes for 2m designs. J. Statist. Plann. Inference, available online.
71. Tang, B., Ma, F., Ingram, D. and Wang, H. (2002), Bounds on the maximum number of clear two-factor interactions for 2m-p designs of resolution III and IV. Canad. J. Statist.30,127-136.
72. Tang, B. and Wu, C. F. J. (1996), Characterization of minimum aberration 2n-k designs in terms of their complementary designs. Ann. Statist.24,2549-2559.
73. Welch, W. J., Yu, T. K., Kang, S. M. and Sacks, J. (1990), Computer experiments for quality control by parameter design. J. Quality Technol.22,38-45.
74. Wu, C. F. J. (1989), Construction of 2m4n via a group scheme. Ann. Statist.17, 1880-1885.
75. Wu, C. F. J. and Chen, Y. (1992). A graph-aided method for planning two-level experiments when certain interactions are importaint. Technometrics 34,162-175.
76. Wu, C. F. J. and Hamada, M. S. (2000), Experiments:Planning, Analysis and Parameter Design Optimization (Wiley, New York).
77. Wu, H. and Wu, C. F. J. (2002), Clear two-factor interactions and minimum aber-ration. Ann. Statist.30,1496-1511.
78. Wu, C. F. J. and Zhang, R. C. (1993), Minimum aberration designs with two-level and four-level factors. Biometrika 80,203-209.
79. Wu, C. F. J., Zhang, R. C. and Wang, R. G. (1992), Construction of asymmetrical orthogonal arrays of the type OA(sk, (sr1)n1…(srl)nl). Statist. Sinica 2,203-219.
80. Wu, C. F. J. and Zhu, Y. (2003), Optimal selection of single arrays for parameter design experiments. Statist. Sinica 13,1179-1199.
81. Xu, H. (2006), Blocked regular fractional factorial designs with minimum aberra-tion. Annals of statistics 2006, Vol.34, No.5,2534-2553.
82. Xu, H., Lau, S., (2006), Minimum aberration blocking schemes for two-and three-level fractional factorial designs. J. Statist. Plann. Inference 136,4088-4118.
83. Yang, G. J., Liu, M. Q., Zhang, R. C.(2005), Weak minimum aberration and maxi-mum number of clear two-factor interactions in 2Ⅳm-p designs [J]. Science in China, Ser A 48(11),1479-1487.
84. Yang, G. J., Liu, M. Q.(2006), A note on the lower bound on maximum number of clear two-factor interactions for 2Ⅲm-p and 2Ⅳm-p designs [J]. Communications in Statistics,35(5),849-860.
85. Yang, G. J., Liu, M. Q., Zhang, R. C.(2006), A note on 2Ⅳm-p designs with the maximum number of clear two-factor interactions [J]. ACTA Mathematica Scientia, 26 A(7),1153-1158.
86. Yang, J. F., Li, P. F., Liu, M. Q. and Zhang, R. C. (2006), 2(n1+n2)-(k1+k2) fractional factorial split-plot designs containing clear effects. J. Statist. Plann. Inference, 136,4450-4458.
87. Yang J F, Zhang R C, Liu M Q. (2007), Construction of fractional factorial split-plot designs with weak minimum aberration, Statist & Prob Letters,77,1567-1573
88. Zhang R C, Cheng Y. (2010), General minimum lower order confounding designs: An overview and a construction theory. J Statist Plan Inference,140,1719-1730.
89. Zhang, R.C. and Shao, Q. (2001), Minimum aberration (S2)Sn-k designs. Statist. Sinica 11,213-223.
90. Zhang, R. C., Li, P., Zhao, S. L. and Ai, M. Y. (2008), A general minimum lower-order confounding criterion for two-level regular designs. Statist. Sinica 18,1689-1705.
91. Zhang, R. C., Mukerjee, R.(2009) Characterization of general minimum lower order confounding via complementary. Statist Sinica,19,363-375.
92. Zhang, R. C., Park, D. K., (2000). Optimal blocking of two-level fractional factorial designs. J. Statist. Plann. Inference 91,107-121.
93. Zhao, S. L. (2006), Optimal Criteria and Construction for Two Classes of Regular Factorial Designs. unpublished doctorial thesis, Department of Statistics, School of Mathematical sciences, Nankai University.
94. Zhu, Y. (2000), A Theory of Experimental Design for Multiple Groups of Factors. Unpublished doctoral thesis, University of Michigan, Department of Statistics.
95. Zhu, Y. (2003). Structure function for aliasing patterns in 2l-n 2 design with multiple groups of factors. Ann. Statist.31,995-1011.
96. Zhu, Y. and C. F. J. Wu (2007). Structure function for general sl-m design and blocked regular (sl-m,sr) design and their complementary designs. Statistica Sinica,17,1237-1257.
97. Zi, X. M., Zhang, R. C. and Liu, M. Q. (2006), Bounds on the maximum numbers of clear two-factor interactions for 2(n1+n2)-(k1+k2) fractional factorial split-plot designs. Sci. China Ser. A 49(12),1816-1829.