Journal of Statistical Planning and Inference 143 (2013) 809–817
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Partially replicated two-level fractional factorial designs via semifoldover Zujun Ou a, Hong Qin b,n, Xu Cai a a b
College of Physical Science and Technology, Central China Normal University, Wuhan 430079, China Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
a r t i c l e i n f o
abstract
Article history: Received 11 December 2011 Received in revised form 21 August 2012 Accepted 22 October 2012 Available online 2 November 2012
Most fractional factorial designs have no replicated points and thus do not provide a reliable estimate for experimental error. The objective of this paper is to study the issue of partially replicated two-level fractional factorial (FF) designs, thereby allowing for the unbiased estimation of the experimental error while maintaining the orthogonality of the main effects. Through the tool of indicator function and the idea of semifoldover, we propose two simple and effective techniques to produce designs with partially replicated points in general two-level FF designs, whether they are regular or not. The related properties of constructed partially replicated designs are investigated. Our results indicate that partially replicated FF are competitive in practice. & 2012 Elsevier B.V. All rights reserved.
Keywords: Indicator function Semifoldover Partially replicated designs
1. Introduction During the initial stages of experimentation, unreplicated two-level factorial designs are commonly used to identify important or active effects, due mainly to their cost-effectiveness. But statistical inference based on such unreplicated experiments typically is a challenge, because there are no pure replicates for estimating the experimental error variance. Obviously, the fully replicated design is a simple resolution to provide pure replicates, but it often leads to a costly experiment. Thus designs with partial duplication may provide promising alternatives, not only offering pure replicates, but also providing cost savings. From the duplicate observations, it would be possible to obtain an estimate of the between duplicates variance. Comparing to the design with replications, the analysis methods for unreplicated data may perform unsatisfactorily in identifying truly active effects, particularly when the effect sparsity principle does not hold. This is due mainly to the lack of a replication-based estimate of the error variance. Partially replicated designs usually work well regardless of the effect sparsity, see Liao and Chai (2009). On the other hand, a partially replicated design can save runs compared to the fully replicated design. It also provides more power in identifying truly active effects than the unreplicated design, even without the effect sparsity assumption. Recently, there has been considerable interest in studying design issues involving factorial experiments with partial replication. Some results have been given by Dykstra (1959), and Pigeon and McAllister (1989). However, these approaches have some limitations such as the requirement of a significant number of runs in the experimental process and the loss of orthogonality of main effects. Most recently, Liao and Chai (2004) recognized the partially replicated two-level factorial design presented by Pigeon and McAllister (1989) as one composed of three different fractions, belonging to a family of
n
Corresponding author. E-mail address:
[email protected] (H. Qin).
0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2012.10.010
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regular 2nk designs with the same defining relations, and a duplicate of one of the three fractions. Subsequently, Liao and Chai (2009) considered the construction of parallel-flats designs with two identical parallel flats that allow estimation of a set of specified possibly active effects and the pure error variance, and Tsai and Liao (2011) extended the results in Liao and Chai (2004, 2009) to 2n1 3n2 mixed factorial experiments. Liau (2008) discussed partial duplication mainly in two-level regular design and proposed two techniques based on the idea of semifolding to produce orthogonal main-effect design with partially duplication. A new class of partially replicated two-level FF designs based on Hadamad matrices was proposed in Pigeon and Lupinaci (2008). Dasgupta et al. (2010) provided some partially replicated 16-run designs with m factors, where 4 rm r 11. Foldover is a technique which reverses signs of one or more factors in the original design and adds the new runs to the original design. The new design, called the (full) foldover design, is used to de-alias effects. Although foldover designs can de-alias many effects, they involve twice the original runs and so it will be much more efficient to do a partial foldover. One of the partial foldover designs is the semifoldover design. Semifoldover designs are obtained by reversing signs of one or more factors in the original design, but added only half of the new runs to the original design. Thus, semifoldover designs save half of the original runs compared to full foldover designs, and are more valuable sometimes. Huang et al. (2008) discussed some properties of semifolding designs of two-level regular designs. Balakrishnan and Yang (2009) investigated various semifoldover designs that are obtained from a general two-level FF design with the use of indicator functions. The present paper aims at obtaining further results. We extend the findings in Liau (2008) to general two-level factorials. First, unlike Liao and Chai (2004, 2009) and Liau (2008), the results of this paper are suitable for general twolevel factorials, whether regular or not. In addition, our results are more flexible than the ones in Pigeon and Lupinaci (2008), which are only suitable for two-level designs based on Hadamard matrices. Second, a class of partially replicated designs are constructed via semifoldover method, which is adequately discussed in Huang et al. (2008) and Balakrishnan and Yang (2009) only for unreplicated designs. Third, Theorems 2–7 and their corollaries provide the conditions of the partially replicated design constructed in this paper be a resolution III or higher design when the resolution of the original design is III or higher, and Theorems 8 and 9 provide the properties for the constructed designs. Finally, using the effective tool of indicator function, some results reported in this paper reduce to that of Liau (2008), where it is only for fo and z (see Section 2) being main effects or two-factor interactions. The paper is organized as follows. In Section 2, we first introduce indicator functions and study them for partial replicated designs. Section 3 gives our main results. In this section, we propose two techniques to produce designs with replicated points in two-level FF designs via semifoldover method, and then investigate the properties of the resulting designs. Finally, we make some concluding remarks in Section 4. 2. Indicator function and indicator function of partially replicated designs Let D be a 2s full factorial design, each factor only with two levels, a high level denoted by þ1 and a low level denoted by 1, respectively. The design points of D are just the solutions of the polynomial system {x21 ¼ 1,x22 ¼ 1, . . . ,x2s ¼ 1}. An N runs unreplicated fractional factorial design F is an any subset of D. The indicator function f(x) of design F is defined in Fontana et al. (2000) as a function on D such that ( 1 if x 2 F , f ðxÞ ¼ 0 if x 2 DF : Following Fontana et al. (2000), the indicator function f(x) of design F can be uniquely represented by the polynomial function defined on D as X f ðxÞ ¼ ba xa , ð1Þ a2L
where L is the set of all binary s tuples a, that is, L ¼ fa ¼ ða1 , a2 , . . . , as Þ9ai ¼ 1 or 0 for i ¼ 1,2, . . . ,sg,xa ¼ xa11 xa22 xas s and 1X a ba ¼ s x : ð2Þ 2 x2F About the applications in fractional designs of indicator function, one can refer to Balakrishnan and Yang (2006a,b, 2009) and Ou and Qin (2010). Note that b0 ¼ N=2s , where N is the run size of F . In other words, b0 is just the ratio between the number of points of F and the number of points of D. The coefficients ba of f(x) satisfy 9ba =b0 9 r1. A design is a regular design if and only if 9ba =b0 9 ¼ 1 for any ba a0 (Fontana et al., 2000; Ye, 2003). A word or a word of the design F is defined as the term with nonzero coefficient (except the constant) in the indicator function f(x) of design F . Following Li et al. (2003), the length of a P word xa is defined as JxJ ¼ JaJ þ ð19ba =b0 9Þ, where JaJ ¼ si ¼ 1 ai represents the number of letters in the word xa . The length of the shortest word of f(x) is called as the generalized resolution of design F in the literature. Supposing that the foldover design is obtained by reversing the signs of r factors of the original design F , 1 rr r s, let W r0 be the set of all xa in (1) such that the corresponding ba a0 and xa contains 0 or even number of the r foldover factors,
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and W r1 be the set of all xa in (1) satisfy the corresponding ba a0 and xa contains odd number of the r foldover factors. Then, the indicator function of f(x) in (1) can be rewritten as X X f ðxÞ ¼ ba xa þ ba xa : ð3Þ a2W r0
a2W r1
Let F z ¼ e ¼ fx 2 F 9z ¼ eg, where e 2 f1,1g, and z ¼ xa11 xa22 xas s for some nonzero a ¼ ða1 , a2 , . . . , as Þ 2 L, that is, z is a main effect or an interaction of design F . Note that we should choose z such that z is not a word of f(x). Otherwise, F z ¼ e ¼ F or F z ¼ e ¼ f, and F z ¼ e is not a half fraction of F . Let F ðrÞ be the foldover design of F obtained by reversing the signs of any r factors xi1 ,xi2 , . . . ,xir and F ðrÞz ¼ e ¼ fx 2 F ðrÞ9z ¼ eg. In this case, we say that the fractions are obtained by foldover on fo ¼ xi1 xi2 . . . xir factors and subset on z, as the notations in Mee and Peralta (2000). Now we propose a class of partially replicated designs constructed via semifoldover method as follows. Let F be a N runs unreplicated fractional factorial design with s two-level factors, F z ¼ e1 and F ðrÞz ¼ e2 are obtained by the method described as above, where e1 ,e2 2 f1,1g, and z ¼ xa11 xa22 xas s for some nonzero a ¼ ða1 , a2 , . . . , as Þ 2 L. Denote by F ðfo,z,e1 ,e2 Þ the combined fractions of F , F z ¼ e1 and F ðrÞz ¼ e2 , that is, 0 1 F B Fz ¼ e C F ðfo,z,e1 ,e2 Þ ¼ @ A, 1 F ðrÞz ¼ e2 then F ðfo,z,e1 ,e2 Þ is a 2N runs design with at least N=2 pairs of duplicates. Example 1. Consider the design F with seven two-level factors and eight runs, these runs are shown in the first part of Table 1 (Run No. 1–8). If we take z ¼ x1 , fo ¼ x1 x2 , e1 ¼ 1, e2 ¼ 1, then we can obtain the partially replicated design F ðfo,z,1,1Þ by the above construction method. From Table 1, one can find that F ðfo,z,1,1Þ has four pairs of duplicates (Run No. 5–12). The following theorem gives the indicator function of F ðfo,z,e1 ,e2 Þ, and the related properties of F ðfo,z,e1 ,e2 Þ can be effectively deduced on its indicator function. Theorem 1. Let the indicator function of F be as in (3), h1 ðxÞ, h2 ðxÞ, h3 ðxÞ and h4 ðxÞ be the indicator functions of F ðfo,z,1,1Þ, F ðfo,z,1,1Þ, F ðfo,z,1,1Þ and F ðfo,z,1,1Þ, respectively. Then we have X X X h1 ðxÞ ¼ 2 ba xa þ ba xa þ z ba xa , ð4Þ a2W r0
a2W r1
a2W r1
Table 1 F ðfo,z,1,1Þ. F Run no.
x1
x2
x3
x4
x5
x6
x7
1 2 3 4 5 6 7 8
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
Fz ¼ 1 Run no. 9 10 11 12
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
F ðrÞz ¼ 1 Run no. 13 14 15 16
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h2 ðxÞ ¼ 2
X
ba xa þ
2W r0
a
h3 ðxÞ ¼ 2
X
a
ba xa þ
a2W r0
h4 ðxÞ ¼ 2
X
X
ba xa z
2W r1
X
ba xa þ
a
X a
ba xa ,
ð5Þ
ba xa ,
ð6Þ
ba xa :
ð7Þ
a2W r1
ba xa þz
a2W r1
2W r0
X X
a2W r0
ba xa z
2W r1
X
a2W r0
Proof. We only prove Eq. (4) and the rest can be proved by the same way. Let f 1 ðxÞ and f 2 ðxÞ is the indicator function of F z ¼ 1 and F ðrÞz ¼ 1 , respectively. Then, following Lemma 2 in Ye (2003), the indicator function h1 ðxÞ of F ðfo,z,1,1Þ can be expressed as h1 ðxÞ ¼ f ðxÞ þf 1 ðxÞ þf 2 ðxÞ:
ð8Þ
Note that the indicator function of F z ¼ 1 is 1 ð1 þzÞf ðxÞ 2 1 X 1 X 1 X 1 X ¼ ba xa þ ba xa þ z ba xa þ z ba xa : 2 a2W r 2 a2W r 2 a2W r 2 a2W r
f 1 ðxÞ ¼
0
1
0
ð9Þ
1
Li et al. (2003) showed that the indicator function of F ðrÞ is X X ba xa ba xa : f 3 ðxÞ ¼ a2W r0
a2W r1
So the indicator function of F ðrÞz ¼ 1 is 1 ð1zÞf 3 ðxÞ 2 1 X 1 X 1 X 1 X ba xa ba xa z ba xa þ z ba xa : ¼ 2 a2W r 2 a2W r 2 a2W r 2 a2W r
f 2 ðxÞ ¼
0
1
0
ð10Þ
1
Substituting (3), (9) and (10) into (8), the result follows obviously and completes the proof.
&
Remark 1. Theorem 1 shows that the words in h1 ðxÞ and h2 ðxÞ, respectively, are the same. Thus, the two designs F ðfo,z,1,1Þ and F ðfo,z,1,1Þ have the same alias structure. Similarly, it is true for F ðfo,z,1,1Þ and F ðfo,z,1,1Þ. Therefore, we only consider the two designs F ðfo,z,1,1Þ and F ðfo,z,1,1Þ henceforth. 3. Main results In this section, we study an orthogonal main-effect plan with partial replicated runs obtained from a two-level resolution III:a or higher factorial design F , where 0 r a o 1, that is F can be a regular or nonregular design. We examine when a design with partial replicated runs can be an orthogonal main-effect plan. In Sections 3.1 and 3.2, we respectively consider partial replicated designs obtained from F ðfo,z,1,1Þ and F ðfo,z,1,1Þ, and provide some sufficient conditions for partial replicated designs to be an orthogonal main-effect plan in Section 3.3. Since indicator function provides the unified presentation of regular and nonregular designs, so our results are suitable for general design, whether regular or not. Moreover, both fo and z can be main effects or high-order interactions (see Theorems 2–9 and their corollaries), where Liau (2008) is only for fo and z to be main effects or two-factor interactions. Therefore, our results in this paper are more general and flexible than the ones in Liau (2008) and Pigeon and Lupinaci (2008). 3.1. F ðfo,z,1,1Þ We first consider the case of subset on a main effect. The following theorem provides the conditions of F ðfo,z,1,1Þ being a resolution III:a design when F is a design of resolution III:a. Theorem 2. Let F be a two-level s-factor design of resolution III:a. For any positive integer 1 rr r s, let fo ¼ xi1 xi2 . . . xir and z ¼ xh , where 1r i1 o o ir rs and 1r h rs. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r1 containing xh is four or higher in the indicator function f(x) of F . Particularly, if fo ¼ xh xj ð1 r j, h rs,jahÞ and xh xj always appears the same word in f(x), then we can replicate the original design N points. Proof. Since F is a design of resolution III:a, then the wordlength of each word xa 2 W r0 ðor W r1 Þ is three or higher in f(x). P Therefore we only need consider the third term z a2W r ba xa in h1 ðxÞ. Note that z ¼ xh , so the wordlength of each word in 1 P a z a2W r ba x is three or higher if and only if the wordlength of each word xa 2 W r1 containing xh is four or higher in f(x). 1
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Particularly, if fo ¼ xh xj and xh xj always appears the same word in f(x), then W r1 is an empty set. It means that each word P 2 W r1 with ba ¼ 0 and the indicator function of F reduce to f ðxÞ ¼ a2W r ba xa . Moreover, the indicator function of 0 P F ðfo,z,1,1Þ becomes h1 ðxÞ ¼ 2 a2W r ba xa . It implies that F ðfo,z,1,1Þ is the duplication of F . Hence proved. & xa
0
Partial replicated designs F ðfo,z,1,1Þ can also obtained by subsetting on an interaction. The following Theorem 3 provides the conditions of F ðfo,z,1,1Þ, where F ðfo,z,1,1Þ is a resolution III:a design when z is a two-factor interaction. Theorem 3. Let F be a two-level s-factor design of resolution III:a. For any positive integer 1r r rs, let fo ¼ xi1 xi2 . . . xir and z ¼ xh xj , where 1 ri1 o o ir rs and 1 r ho j r s. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r1 containing xh xj is five or higher in f(x). Particularly, if r ¼2 and fo ¼ xh xj then F ðfo,z,1,1Þ always is a design of resolution III:a with N=2 pairs of duplicates. Furthermore, if xh xj always appears the same word in f(x), then we can replicate the original design N points. Proof. If fo ¼ xi1 xi2 . . . xir and z ¼ xh xj , then the first result is obviously whenever the following three cases: (i) h or j 2 fi1 ,i2 , . . . ,ir g, (ii) h,j 2 fi1 ,i2 , . . . ,ir g, (iii) both h and j are not in the set fi1 ,i2 , . . . ,ir g. Particularly, if r ¼2 and fo ¼ xh xj , then xh and xj does not simultaneous appear in the same word xa 2 W r1 , then F ðfo,z,1,1Þ always is a design of resolution III:a with P N=2 pairs of duplicates. Furthermore, if xh xj always appears the same word in f(x), then f ðxÞ ¼ a2W r ba xa , and similar to the 0 proof of Theorem 2, we can obtain the result obviously. & The following Theorem 4 provides the conditions of F ðfo,z,1,1Þ, where F ðfo,z,1,1Þ is a resolution III:a design when z is a three-factor interaction. Theorem 4. Let F be a two-level s-factor design of resolution III:a. For any positive integer 1r r rs, let fo ¼ xi1 xi2 . . . xir and z ¼ xh xj xk , where 1 r i1 o oir r s and 1r h oj ok rs. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r1 containing xh xj xk is six or higher and the wordlength of each word xa 2 W r1 containing two elements of fxh ,xj ,xk g is four or higher in f(x). Proof. If fo ¼ xi1 xi2 . . . xir and z ¼ xh xj xk , then the result is obviously whenever the following three cases: (i) h ðor j,or kÞ 2 fi1 ,i2 , . . . ,ir g, (ii) h,j ðor h,k, or k,jÞ 2 fi1 ,i2 , . . . ,ir g, (iii) h, j and k are all not in the set fi1 ,i2 , . . . ,ir g. & As the direct conclusions of Theorem 4, the following Corollaries 1 and 2 consider subsetting on a three-factor interaction that contains the foldover factor(s) in fo, respectively. Corollary 1. Let F be a two-level s-factor design of resolution III:a. Let fo ¼ xh and z ¼ xh xj xk , where 1 rh o j ok r s. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r1 containing xh xj xk is six or higher and the wordlength of each word xa 2 W r1 containing xh xj or xh xk is four or higher in f(x). Corollary 2. Let F be a two-level s-factor design of resolution III:a. Let fo ¼ xh xj and z ¼ xh xj xk , where 1 r h oj o k rs. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r1 containing xh xj xk is six or higher and the wordlength of each word xa 2 W r1 containing xh xj or xh xk is four or higher in f(x). Particularly, if xh and xj always appear simultaneous in the same word in f(x), then we can replicate the original design N points. When we subset on a four or higher factors interaction z, it is more complicated than Theorems 2–4. But if we subset on z ¼ x1 x2 . . . xs , then we have the following result. Theorem 5. Let F be a two-level s-factor design of resolution III:a. For any positive integer 1r r rs, let fo ¼ xi1 xi2 . . . xir and z ¼ x1 x2 . . . xs , where 1 r i1 o oir r s and s Z 6. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of the longest word xa 2 W r1 is at most s3 in f(x). 3.2. F ðfo,z,1,1Þ Since the third summation formula of the indicator function h3 ðxÞ of F ðfo,z,1,1Þ in (6) contains b0 z, we only consider z is a three or higher factor interaction for F ðfo,z,1,1Þ. We first consider the case of subset on a three-factor interaction. The following theorem provides the necessary and sufficient conditions such that F ðfo,z,1,1Þ is a resolution III:a design when F is a design of resolution III:a. Theorem 6. Let F be a two-level s-factor design of resolution III:a. For any positive integer 1r r rs, let fo ¼ xi1 xi2 . . . xir and z ¼ xh xj xk , where 1r i1 o oir rs and 1 r ho j o kr s. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r0 containing xh xj xk is six or higher and the wordlength of each word xa 2 W r0 containing two elements of fxh ,xj ,xk g is four or higher in f(x). Proof. If fo ¼ xi1 xi2 . . . xir and z ¼ xh xj xk , then the result is obviously whenever the following three cases: (i) h ðor j,or kÞ 2 fi1 ,i2 , . . . ,ir g, (ii) h,j ðor h,k, or k,jÞ 2 fi1 ,i2 , . . . ,ir g, (iii) h, j and k are all not in the set fi1 ,i2 , . . . ,ir g. & As the direct conclusions of Theorem 6, the following Corollaries 3–5 consider subsetting on a three-factor interaction that contains the foldover factor(s) in fo, respectively.
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Corollary 3. Let F be a two-level s-factor design of resolution III:a. Let fo ¼ xh and z ¼ xh xj xk , where 1 r ho j o kr s. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 W r0 containing xj xk is four or higher. Proof. If fo ¼ xh , then xh only appears in the word xa 2 W r1 . From Theorem 6, we can obtain the result.
&
Corollary 4. Let F be a two-level s-factor design of resolution III:a. Let fo ¼ xh xj and z ¼ xh xj xk , where 1 rh oj ok rs. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if either the wordlength of each word xa 2 W r0 containing xh xj is four or higher or xh and xj do not always appear simultaneous in the same word of f(x). Proof. If fo ¼ xh xj and xh , xj does not always appear simultaneous in the same word, then any word xa 2 W r0 does not always contain xh and xj. From Theorem 6, we can obtain the result. & Corollary 5. Let F be a two-level s-factor design of resolution III:a. Let fo ¼ z ¼ xh xj xk , where 1 rh oj ok rs. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of each word xa 2 Wr 0 containing two elements of fxh ,xj ,xk g is four or higher in f(x). Proof. If fo ¼ xh xj xk , then any word xa 2 W r0 does not simultaneous contain xh ,xj ,xk . Following Theorem 6, we can obtain the result. & Similarly to Theorem 5, if we subset on z ¼ x1 x2 . . . xs , then we can obtain the following result whose proof is obviously. Theorem 7. Let F be a two-level s-factor design of resolution III:a. For any positive integer 1 rr r s, let fo ¼ xi1 xi2 . . . xir and z ¼ x1 x2 . . . xs , where 1 r i1 o o ir r s and s Z6. Then F ðfo,z,1,1Þ is a design of resolution III:a with N=2 pairs of duplicates if and only if the wordlength of the longest word xa 2 W r0 is at most s3 in f(x). Liau (2008) considered various semifoldover designs obtained from regular designs. Here, we present an illustrative example to demonstrate how the results in the preceding sections can be applied to get partial replicated designs F ðfo,z,1,1Þ and F ðfo,z,1,1Þ with resolution III:a from a nonregular design in detail. Example 2. Consider a six-factor non-regular design F 1 with the runs in Table 2. The indicator function of F 1 is f ðxÞ ¼ 14 þ 18 x1 x4 x5 þ 18 x2 x3 x6 18 x1 x5 x6 18 x2 x3 x4 18 x2 x5 x6 18 x1 x3 x6 18 x2 x4 x5 18 x1 x3 x4 þ 14x1 x2 x3 x4 x5 x6 : From the indicator function f(x) of design F 1 , we know that its resolution is 3.5. (i) If z ¼ x1 and fo ¼ x2 , then W 11 ¼ fx2 x3 x6 ,x2 x3 x4 ,x2 x5 x6 ,x2 x4 x5 ,x1 x2 x3 x4 x5 x6 g and the indicator function h1 ðxÞ of F 1 ðfo,z,1,1Þ is h1 ðxÞ ¼ 12 þ 14 x1 x4 x5 þ 18 x2 x3 x6 14 x1 x3 x4 14 x1 x3 x6 14 x1 x5 x6 18 x2 x3 x4 18 x2 x5 x6 18 x2 x4 x5 18 x1 x2 x3 x4 þ 18 x1 x2 x3 x6 18 x1 x2 x4 x5 18 x1 x2 x5 x6 þ 14 x2 x3 x4 x5 x6 þ 14 x1 x2 x3 x4 x5 x6 , and F 1 ðfo,z,1,1Þ is a design of resolution 3.5 with eight pairs of duplicates. (ii) If z ¼ x1 x2 and fo ¼ x3 , then the indicator function h1 ðxÞ of F 1 ðfo,z,1,1Þ is h1 ðxÞ ¼ 12 þ 14 x1 x4 x5 14 x1 x5 x6 14 x2 x3 x4 14 x2 x5 x6 14 x2 x4 x5 14 x1 x3 x4 þ 14 x3 x4 x5 x6 þ 14 x1 x2 x3 x4 x5 x6 , and F 1 ðfo,z,1,1Þ also is a design of resolution 3.5 with eight pairs of duplicates. Table 2 A 16-run non-regular resolution 3.5 design F 1 . x1
x2
x3
x4
x5
x6
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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(iii) If z ¼ x1 x2 x3 and fo ¼ x5 , then the indicator function h1 ðxÞ of F 1 ðfo,z,1,1Þ is h1 ðxÞ ¼ 12 þ 18 x1 x4 x5 þ 14 x2 x3 x6 14 x1 x3 x4 14 x1 x3 x6 18 x1 x5 x6 14 x2 x3 x4 18 x2 x5 x6 18 x2 x4 x5 þ 14 x4 x5 x6 18 x1 x3 x4 x5 18 x1 x3 x5 x6 18 x2 x3 x5 x6 þ 18 x2 x3 x4 x5 þ 14 x1 x2 x3 x4 x5 x6 , and F 1 ðfo,z,1,1Þ also is a design of resolution 3.5 with eight pairs of duplicates. (iv) If z ¼ x3 x4 x5 and fo ¼ x3 x5 , then the indicator function h3 ðxÞ of F 1 ðfo,z,1,1Þ is h3 ðxÞ ¼ 12 þ 18 x1 x4 x5 þ 18 x2 x3 x6 18 x1 x3 x4 18 x1 x3 x6 18 x1 x5 x6 18 x2 x3 x4 18 x2 x5 x6 18 x2 x4 x5 þ 14 x1 x2 x6 þ 14 x3 x4 x5 þ 12 x1 x2 x3 x4 x5 x6 , and F 1 ðfo,z,1,1Þ also is a design of resolution 3.5 with eight pairs of duplicates. 3.3. Properties of F ðfo,z,1,1Þ and F ðfo,z,1,1Þ For any positive integer 1 rr, t r s, let fo ¼ xi1 xi2 . . . xir and z ¼ xj1 xj2 . . . xjt , where 1 r i1 o oir r s and 1 rj1 o ojt rs. Given fo and z, denote y ¼ fxi1 ,xi2 , . . . xir gfxj1 ,xj2 , . . . xjt g, let Z be the submatrix of F z ¼ 1 corresponding to z of order N=2 t, and Y ¼ ðY 01 ,Y 02 Þ0 be the submatrix of F corresponding to y of order N 9y9, where 9y9 is the cardinality of the set y. Then the original design F can be expressed as ! Z Y 1 X1 F¼ , Z Y 2 X 2 where X1 and X2 is the design submatrix of F z ¼ 1 and F z ¼ 1 corresponding to the columns except z, respectively. Under the partition of F as above, some sufficient conditions for partial replicated designs obtained from F ðfo,z,1,1Þ and F ðfo,z,1,1Þ to be orthogonal main-effect plans are given in the following Theorems 8 and 9, respectively. Theorem 8. Let F be an orthogonal main-effect plan with N=2 points at each level of each factor. F ðfo,z,1,1Þ is an orthogonal main-effect plan with 2N points and N=2 pairs of duplicates if one of the following conditions hold: (i) z ¼ xh and z D fo, where h ¼ 1, . . . ,s; (ii) fo ¼ x1 . . . xs , z ¼ xj1 xj2 . . . xjt , where 1 rt r s. Proof. Since F is an orthogonal main-effect plan, thus 0 1 Z 0 Y 1 Z 0 Y 2 Z 0 X 1 Z 0 X 2 2Z 0 Z B Y 0 ZY 0 Z Y 0 Y þ Y 0 Y 0 0 0 Y 1X1 þ Y 2X2 C F F ¼@ 1 A ¼ NIN , 2 1 1 2 2 0 0 0 0 X 1 ZX 2 Z X 1 Y 1 þ X 2 Y 2 X 01 X 1 þ X 02 X 2
ð11Þ
where IN is the N order identity matrix. If z ¼ xh and z D fo, where h ¼ 1, . . . ,s, then ! Z X1 F¼ Z X 2 and F 0F ¼
2Z 0 Z
Z 0 X 1 Z 0 X 2
X 01 ZX 02 Z
X 01 X 1 þ X 02 X 2
! ¼ NIN :
ð12Þ
From (12), it follows that X 01 X 1 þ X 02 X 2 ¼ NIN1 and Z 0 X 1 Z 0 X 2 ¼ 0. Noting that each factor with N=2 points at each level, it follows that Z 0 X 1 þ Z 0 X 2 ¼ 0. Hence we obtain Z 0 X 1 ¼ Z 0 X 2 ¼ 0. On the other hand, 0 1 Z X1 B C B Z X 2 C C ð13Þ F ðfo,z,1,1Þ ¼ B B Z X 1 C, @ A Z X2 therefore, following the facts in (12), we know that 0
F ðfo,z,1,1Þ F ðfo,z,1,1Þ ¼
4Z 0 Z
2Z 0 X 1
2X 01 Z
2ðX 01 X 1 þ X 02 X 2 Þ
! ¼ 2NI2N :
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If fo ¼ x1 . . . xs , then 0
Z B B Z F ðfo,z,1,1Þ ¼ B B Z @ Z
Y1 Y2 Y1 Y 2
1 X1 C X2 C C X1 C A X 2
and 0
4Z 0 Z
2ðZ 0 Y 1 Z 0 Y 2 Þ
B 0 0 F ðfo,z,1,1Þ0 F ðfo,z,1,1Þ ¼ @ 2ðY 1 ZY 2 ZÞ 2ðX 01 ZX 02 ZÞ hence proved.
2ðY 01 Y 1 þ Y 02 Y 2 Þ 2ðX 01 Y 1 þ X 02 Y 2 Þ
2ðZ 0 X 1 Z 0 X 2 Þ
1
2ðY 01 X 1 þ Y 02 X 2 Þ C A ¼ 2NI2N , 2ðX 01 X 1 þ X 02 X 2 Þ
&
Theorem 9. Let F be an orthogonal main-effect plan with N=2 points at each level of each factor. If z ¼ xh , fo ¼ x1 . . . xh1 xh þ 1 . . . xs , where 1 r hr s, then F ðfo,z,1,1Þ is an orthogonal main-effect plan with 2N points and N=2 pairs of duplicates. Proof. If z ¼ xh , fo ¼ x1 . . . xh1 xh þ 1 . . . xs , then ! Z Y1 F¼ Z Y 2 and F 0F ¼
2Z 0 Z Y 01 ZY 02 Z
Z 0 Y 1 Z 0 Y 2 Y 01 Y 1 þ Y 02 Y 2
! ¼ NIN :
ð14Þ
From (14), it follows that Y 01 Y 1 þ Y 02 Y 2 ¼ NIN1 and Z 0 Y 1 Z 0 Y 2 ¼ 0. Since each factor with N=2 points at each level, it follows that Z 0 Y 1 þ Z 0 Y 2 ¼ 0. Hence we obtain Z 0 Y 1 ¼ Z 0 Y 2 ¼ 0. On the other hand, 0 1 Z Y1 B C B Z Y 2 C C, F ðfo,z,1,1Þ ¼ B B Z Y1 C @ A Z Y 2
therefore, following the facts in (14), we know that 0
F ðfo,z,1,1Þ F ðfo,z,1,1Þ ¼ hence proved.
4Z 0 Z
2Z 0 Y 1
2Y 01 Z
2ðY 01 Y 1 þY 02 Y 2 Þ
! ¼ 2NI2N ,
&
Example 2. (Continued) Consider the design F 1 with the runs in Table 2, and F 01 F 1 ¼ 16I16 . (i) (ii) (iii) (iv)
If If If If
z ¼ fo ¼ x1 , then F ðfo,z,1,1Þ0 F ðfo,z,1,1Þ ¼ 32I32 : z ¼ x1 and fo ¼ x1 x2 , we also have F ðfo,z,1,1Þ0 F ðfo,z,1,1Þ ¼ 32I32 : z ¼ x1 and fo ¼ x1 x2 x3 x4 x5 x6 , we also have F ðfo,z,1,1Þ0 F ðfo,z,1,1Þ ¼ 32I32 : z ¼ x1 and fo ¼ x2 x3 x4 x5 x6 , we also have F ðfo,z,1,1Þ0 F ðfo,z,1,1Þ ¼ 32I32 :
Therefore, the designs in (i)–(iv) are all orthogonal main-effect plans. 4. Concluding remarks The objective of this paper is to study the issue of partially replicated two-level FF designs, thereby allowing for the unbiased estimation of the experimental error while maintaining the orthogonality of the main effects. We propose two techniques to produce designs with replicate points in both regular and nonregular two-level FF designs via the tool of indicator function and semifoldover method. The partially replicated factorials are considered on only the class of N=2 pairs of duplicates obtained from an orthogonal main-effect plan with N points in the current study. It would be interesting to investigate the selection of partial replication with any pairs of duplicates. This will be our next research project.
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Acknowledgements The authors would like to thank the referee and Chief Editor for their valuable comments and suggestions that lead to improve the presentation of the paper. This work was partially supported by SRFDP (No. 20090144110002), the National Natural Science Foundation of China (Grant Nos. 11201177 and 11271147). References Balakrishnan, N., Yang, P., 2006a. Classification of three-word indicator functions of two-level factorial designs. Annals of the Institute of Statistical Mathematics 58, 595–608. Balakrishnan, N., Yang, P., 2006b. Connections between the resolutions of general two-level factorial designs. Annals of the Institute of Statistical Mathematics 58, 609–618. Balakrishnan, N., Yang, P., 2009. De-aliasing effects using semifoldover techniques. Journal of Statistical Planning and Inference 139, 3102–3111. Dasgupta, N., Jacroux, M., SahaRay, R., 2010. Partially replicated fractional factorial designs. Metrika 71, 295–311. Dykstra Jr., O., 1959. Partial duplication of factorial experiments. Technometrics 1, 63–75. Fontana, R., Pistone, G., Rogantin, M.P., 2000. Classification of two-level factorial fractions. Journal of Statistical Planning and Inference 87, 149–172. Huang, P.H., Liau, P.H., Liau, C.S., Lee, C.T., 2008. Some properties of semi-folding designs. Communications in Statistics—Theory and Methods 37, 1245–1257. Li, W., Lin, D.K.J., Ye, K.Q., 2003. Optimal foldover plans for non-regular orthogonal designs. Technometrics 45, 347–351. Liao, C.T., Chai, F.S., 2004. Partially replicated two-level fractional factorial designs. The Canadian Journal of Statistics 32, 421–438. Liao, C.T., Chai, F.S., 2009. Design and analysis of two-level factorial experiments with partial replication. Technometrics 51, 66–74. Liau, P.H., 2008. Partial duplication in two-level fractional factorial designs. Statistical Papers 49, 353–361. Mee, R.W., Peralta, M., 2000. Semifolding 2kp designs. Technometrics 42, 122–134. Ou, Z.J., Qin, H., 2010. Some applications of indicator function in two-level factorial designs. Statistics & Probability Letters 80, 19–25. Pigeon, J.G., McAllister, P.R., 1989. A note on partially replicated orthogonal main-effect plan. Technometrics 31, 249–251. Pigeon, J.G., Lupinaci, P.J., 2008. A class of partially replicated two-level fractional factorial designs. Journal of Quality Technology 40, 184–193. Tsai, S.F., Liao, C.T., 2011. Optimal partially replicated parallel-flats design for 2n13n2 mixed factorial experiments. Journal of Statistical Planning and Inference 141, 2803–2810. Ye, K.Q., 2003. Indicator function and its application in two-level factorial designs. Annals of Statistics 31, 984–994.