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Optimal blocking and foldover plans for nonregular two-level designs
Journal of Statistical Planning and Inference 141 (2011) 1635–1645
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Review
Optimal blocking and foldover plans for nonregular two-level designs Zujun Ou a,b, Hong Qin b,, Hongyi Li c a
College of Mathematics and Computer Science, Jishou University, Jishou 416000, China Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China c Normal College, Jishou University, Jishou 416000, China b
a r t i c l e in f o
abstract
Article history: Received 29 May 2010 Received in revised form 8 December 2010 Accepted 8 December 2010 Available online 17 December 2010
This paper discusses the issue of choosing optimal designs when both blocking and foldover techniques are simultaneously employed to nonregular two-level fractional factorial designs. By using the indicator function, the treatment and block generalized wordlength patterns of the combined blocked design under a general foldover plan are defined. Some general properties of combined block designs are also obtained. Our results extend the findings of Ai et al. (2010) from regular designs to nonregular designs. Based on these theoretical results, a catalog of optimal blocking and foldover plans in terms of the generalized aberration criterion for nonregular initial design with 12, 16 and 20 runs is tabulated, respectively. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: Combined blocked design Generalized minimum aberration Generalized resolution Indicator function
Contents 1. 2. 3. 4. 5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635 Indicator function and related criteria for blocked designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636 Foldover plans for blocked designs and related criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638 Characterization of optimal blocking and foldover plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1639 Optimal blocking and foldover plans for 12, 16 and 20-run designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1641 5.1. Optimal blocking and foldover plans for 16-run designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1641 5.2. Optimal blocking and foldover plans for 12 and 20-run designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Acknowdgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644
1. Introduction Recently, there has been considerable interest in exploring the potential application of the foldover technique to fractional factorial designs. In order to break the aliasing in two-level designs, a standard strategy is to add a foldover of the initial design by reversing the signs of one or more of its factors. A foldover plan refers to the collection of columns whose signs are reversed in the foldover design. Box et al. (1978) firstly discussed the foldover plan for reversing only one factor to de-alias the specific factor from all other factors. Montgomery and Runger (1996) studied foldover plan for resolution IV designs. Li and Mee Corresponding author.
E-mail address: [email protected] (H. Qin). 0378-3758/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.12.008
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(2002) and Li and Lin (2003), respectively, proposed the concept of optimal foldover plan for combined designs which consist of the initial design and the foldover design, and by an exhausted search, gave all optimal foldover plans for regular two-level fractional factorial designs, in terms of the aberration criterion of the combined design. Li et al. (2003) extended optimal foldover plans to nonregular two-level factorial designs in terms of the generalized minimum aberration criterion of combined designs. Ye and Li (2003) studied the theoretical properties of the foldover design and the resulting combined design under a general foldover plan. In order to control systematic noises, blocking is a commonly used technique in experiments. For a given blocked regular two-level design, Li and Jacroux (2007) provided the optimal treatment foldover plans by an algorithm under two proposed optimality criteria. Ai et al. (2010) considered the optimal plans for regular two-level designs when both blocking and foldover techniques are employed and provided some theoretical insight into the relationships between an initial design and the resulting combined blocked design under a general foldover plan. This paper is intended to compliment the work of Ai et al. (2010) by simultaneously considering optimal blocking and foldover plans for nonregular two-level designs via the tool of indicator function. Indicator function is an effective tool in studying two-level factorial designs. Fontana et al. (2000) firstly introduced indicator function to study fractional factorial designs with no replicates. Recently, many papers focus on the application of indicator function in factorial designs. Among these papers are Ye (2003), Balakrishnan and Yang (2006a,b), Balakrishnan and Yang (2009), Ou and Qin (2010). In particular, Cheng et al. (2004) discussed the optimal blocking criteria for nonregular two-level designs via the tool of indicator function. Our research focuses on the optimal plans for nonregular two-level designs when both blocking and foldover techniques are simultaneously employed. Some general theoretical results are obtained and described. This paper is organized as follows. In Section 2, some preliminary concepts of the indicator function of unblocked or blocked two-level factorial designs are introduced. Some related optimality criteria based on indicator function are also introduced in this section. In Section 3, the indictor function and related optimality criteria of combined blocked design are obtained under a general foldover plan. The relationships between the treatment and block generalized wordlength patterns of an initial design and its combined blocked design are obtained in Section 4. Based on these theoretical results, a catalog of optimal blocking and foldover plans in terms of generalized aberration is, respectively, tabulated and compared for 12-run, 16-run, 20-run nonregular initial factorial designs in Section 5. Some concluding remarks are given in Section 6. 2. Indicator function and related criteria for blocked designs Let D be a 2s full factorial design with levels being 1 and 1 throughout this paper. The design points of D are the solutions of the polynomial system fx21 1 ¼ 0,x22 1 ¼ 0, . . . ,x2s 1 ¼ 0g. A n runs fractional factorial design F is an any subset of D without any restriction on its run size. We firstly introduce indicator functions of an unblocked two-level design presented by Fontana et al. (2000) and Ye (2003) as follows. Definition 1. Let D be a 2s design. The indicator function f ðxÞ of its fraction F is a function defined on D such that ( rx if x 2 F , f ðxÞ ¼ 0 if x 2 DF ,
ð1Þ
where rx is the number of replicates of point x in design F . ¨ Using the theory of Grobner basis and algebraic geometry, Fontana et al. (2000) and Ye (2003) showed that the indicator function of F has a unique polynomial representation as follows: X bI XI ðxÞ, ð2Þ f ðxÞ ¼ I2P
Q where XI ðxÞ ¼ i2I xi is defined on D for I 2 P and P is the collection of all subsets of f1,2, . . . ,sg. The coefficient of f ðxÞ in (2), bI, can be calculated by the formula bI ¼
1X XI ðxÞ: 2s x2F
ð3Þ
In particular, bf ¼ n=2s , where n is the run size of F . This implies that bf is just the ratio between the number of points of F and the number of points of D. The coefficients of indicator functions satisfy jbI =bf jr 1 for any I 2 P such that bI a0. Fontana et al. (2000) showed that a two-level factorial design F is a regular design if and only if jbI =bf j ¼ 1 for any I 2 P such that bI a0, otherwise F is a nonregular design. Moreover, F is an orthogonal array of strength t if and only if all the coefficients of the indicator function up to the order t are zero, that is, bI = 0 for any I 2 P such that 1 rjIjr t, where bI is the coefficient of XI ðxÞ of the indicator function f ðxÞ. Those polynomial terms XI ðxÞ with nonzero coefficients are called as words. The wordlength of the word XI ðxÞ is defined as jIj þ 1jbI =bf j, where jIj is the cardinality of the set I. The integer part of wordlength is just the number of letters in the word, while the degree of aliasing is presented by the fractional part. The resolution R of F refers to the smallest wordlength of the words in f ðxÞ.
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For i= 1, y, s, define Ai ðF Þ ¼
X jIj ¼ i,I2P
bI bf
!2 ,
then the vector ðA1 ðF Þ, . . . ,As ðF ÞÞ is called generalized wordlength pattern by Ye (2003). The generalized minimum aberration (GMA) criterion due to Ye (2003) sequentially minimizes ðA1 ðF Þ, . . . ,As ðF ÞÞ. Now we introduce the definition of indicator function of a blocked two-level design. Arranging a n runs and s two-level factors design F into 2q blocks is equivalent to selecting q independent factors out of the s independent factors as the q blocking factors, referred to as a blocking plan. Considering that the block factors should be orthogonal to treatment factors, we assign a set of q columns which no words in f ðxÞ are solely associated with these columns as the blocking factors, following the line of Cheng et al. (2004). In particularly, any q columns in an orthogonal array of strength t, where qr t, are eligible for constructing 2q blocks. In general, consider a blocked design with p treatment factors and q block factors, where p+ q= s and 1 r qr p1. One can start with an n ðp þ qÞ two-level design F and assign eligible q columns fxB1 , . . . ,xBq g as block factors, and assign the reminding p columns fx1 , . . . ,xs gfxB1 , . . . ,xBq g as treatment factors, where 1r B1 oB2 o . . . o Bq r s. This blocked design is formally denoted by ðn,2p : 2q Þ design F b . Under the assumption that the interactions between the q block factors are as important as the main effects of block factors as usual, therefore we denote xB1 xB2 ,xB1 xB3 , . . . ,xB1 xB2 xBq as xBq þ 1 , . . . ,xB2q 1 . Following Cheng et al. (2004), the indicator function for nonregular two-level design with blocks can be obtained by replacing xB1 xB2 ,xB1 xB3 , . . . ,xB1 xB2 xBq by xBq þ 1 , . . . ,xB2q 1 as follows: X X bI XI ðxÞ þ bI XI ðxÞ, ð4Þ fb ðxÞ ¼ I2P t
I2P b
where P t is the collection of all subsets of f1,2, . . . ,sgfB1 , . . . ,Bq g and P b ¼ PP t ¼ fI1 [ I2 jI1 2 P t ,I2 fB1 , . . . ,Bq g and I2 afg. The resulting polynomial function is referred to as blocked indicator function. The values of xBq þ 1 , . . . ,xB2q 1 depend on the value of xB1 , . . . ,xBq . Based on the usual assumptions that the block-by-treatment interactions are negligible and that the interactions between block factors are as important as the main effects of block factors, every word XI ðxÞ,I 2 P b represents a treatment effect confounded with a block effect. In a blocked indicator function, we can see that there are two types of polynomial terms with nonzero coefficients (i.e., words), one involving treatment factors only for the words XI ðxÞ,I 2 P t , and the other involving both block and treatment factors for the words XI ðxÞ,I 2 P b . We call the former pure-type words and the latter mixed-type words following the language of Cheng et al. (2004). The wordlength of the two type words is defined as follows. Definition 2. Suppose fb ðxÞ is the blocked indicator function of blocked design F b . Then the wordlength L(XI) of the word XI ðxÞ is defined as 8 bI > > if I 2 P t , jIj þ 1 > b < f ð5Þ LðXI Þ ¼ b > > > jI1 jþ 2 I if Ið ¼ I1 [ I2 Þ 2 P b : : bf The treatment resolution Rt is defined as the smallest wordlength of pure-type words and the block resolution Rb as the smallest wordlength of mixed-type words. Definition 3. Suppose fb ðxÞ is the blocked indicator function of blocked design F b . For j =1, y, p, define !2 X bI Aj,0 ðF b Þ ¼ bf jIj ¼ j,I2P t
and Aj,1 ðF b Þ ¼
X jI1 j ¼ j,I2P b
bI bf
!2 ,
then the vectors W t ðF b Þ ¼ ðA1,0 ðF b Þ, . . . ,Ap,0 ðF b ÞÞ and W b ðF b Þ ¼ ðA1,1 ðF b Þ, . . . ,Ap,1 ðF b ÞÞ are called treatment generalized wordlength pattern and block generalized wordlength pattern of F b , respectively. For a given design F , a GMA blocking plan is sequentially to minimize the components in W b ðF b Þ. Since there are two wordlength patterns for blocked designs, one usually combines the treatment generalized wordlength pattern and block generalized wordlength pattern into a combined sequence based on some ordering scheme for comparing and selecting optimal blocked designs, and then sequentially minimizes this sequence among all blocked designs with the same parameters. Based on the two wordlength patterns defined above, Cheng et al. (2004) suggested two optimality criteria for nonregular two-level designs below, which follows the line of Cheng and Wu (2002) for regular two-level designs. The
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generalized wordlength patterns for the blocked nonregular designs F b are defined as W1 ðF b Þ ¼ ðA3,0 ðF b Þ,A4,0 ðF b Þ,A2,1 ðF b Þ,A5,0 ðF b Þ,A6,0 ðF b Þ,A3,1 ðF b Þ,A7,0 ðF b Þ, . . .Þ, b
b
ð6Þ
b
where Ai,1 ðF Þ is inserted between A2i,0 ðF Þ and A2i þ 1,0 ðF Þ, and W2 ðF b Þ ¼ ðA3,0 ðF b Þ,A2,1 ðF b Þ,A4,0 ðF b Þ,A5,0 ðF b Þ,A3,1 ðF b Þ,A6,0 ðF b Þ,A7,0 ðF b Þ, . . .Þ,
ð7Þ
where Ai,1 ðF b Þ is inserted between A2i1,0 ðF b Þ and A2i,0 ðF b Þ. The GMA blocked designs can be obtained by sequentially minimizing the wordlength patterns W1 or W2. A catalog of the GMA blocking plans for some nonregular two-level designs with small runs can be found in Cheng et al. (2004). Although the orderings are different, it is essential that the number {Aj,1} is always put behind the number {Aj,0} for j =1, y, p, based on the popular effect hierarchy principle (Wu and Hamada, 2000).
3. Foldover plans for blocked designs and related criteria Let B ¼ f1,2, . . . ,sgfB1 , . . . ,Bq g and O ¼ fðgt , gb Þjgt ¼ ðgi1 , . . . , gip Þ,ij 2 B,1 rj rp, gb ¼ ðgB1 , . . . , gBq Þg, where gt and gb are the p- and q-dimensional row vectors with elements 0 or 1, respectively. Then any g ¼ ðgt , gb Þ 2 O defines a pair of treatment and block foldover plan for design F b , where gt and gb , respectively, represent the foldover plans of the treatment factors and blocking factors. For convenience, the factors corresponding to 0 are called unfoldover factors with respect to the foldover plan, and those corresponding to 1 are referred to as foldover factors. For a foldover plan g ¼ ðgt , gb Þ, the combined blocked design, denoted by F b ðgÞ, consisting of the initial design F b and its foldover design F bg , that is, F b ðgÞ ¼
Fb F bg
! :
Noting that the combined blocked design F b ðgÞ is blocked into 2q blocks of block size 2n/2q by the same block factors used in the initial design. For any g ¼ ðgt , gb Þ 2 O, let m1 and m2 be the number of nonzero elements of gt and gb , respectively. In particular, when m1 +m2 = 0, g ¼ ðgt , gb Þ is called the null foldover plan, the combined blocked design reduces to two replicates of the blocked design F b , when m1 + m2 = s, g ¼ ðgt , gb Þ is called the full foldover plan. For any g ¼ ðgt , gb Þ 2 O, let Egt ¼ fij : gij ¼ 1g, Egb ¼ fBj :
gBj ¼ 1g and Eg ¼ Egt [ Egb . The following theorem gives the indicator function of the combined blocked design F b ðgÞ. Theorem 1. Suppose fb ðxÞ is the blocked indicator function of blocked design F b as defined in (4). Then for any g ¼ ðgt , gb Þ 2 O, the indicator function Fb ðxÞ of the combined blocked design F b ðgÞ can be expressed as follows: X X Fb ðxÞ ¼ 2 bI XI ðxÞ þ 2 bI XI ðxÞ, ð8Þ I2P 2t
I2P 2b
where P 2t ¼ fIjI 2 P t ,jI \ Egt j is0 or even} and P 2b ¼ fIjI 2 P b ,jI \ Eg j is0 or even}. Proof. Let P 1t ¼ fIjI 2 P t ,jI \ Egt j is oddg and P 1b ¼ fIjI 2 P b ,jI \ Eg j is oddg, fb, g ðxÞ be the indicator function of F bg , respectively. Then fb ðxÞ and fb, g ðxÞ can be, respectively, expressed as follows: X X X X fb ðxÞ ¼ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ ð9Þ I2P 2t
I2P 1t
I2P 2b
I2P 1b
and fb, g ðxÞ ¼
X
bI XI ðxÞ
I2P 2t
X
bI XI ðxÞ þ
I2P 1t
X
bI XI ðxÞ
I2P 2b
X
bI XI ðxÞ:
Combining (9) and (10), we have X X Fb ðxÞ ¼ fb ðxÞ þ fb, g ðxÞ ¼ 2 bI XI ðxÞ þ 2 bI XI ðxÞ, I2P 2t
which completes the proof.
&
For j =1, y, p, define Aj,0 ðF b ðgÞÞ ¼
X jIj ¼ j,I2P 2t
bI bf
!2
ð10Þ
I2P 1b
I2P 2b
ð11Þ
Z. Ou et al. / Journal of Statistical Planning and Inference 141 (2011) 1635–1645
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and X
Aj,1 ðF b ðgÞÞ ¼
jI1 j ¼ j,I2P 2b
bI bf
!2 ,
then the vectors ðA1,0 ðF b ðgÞÞ, . . . ,Ap,0 ðF b ðgÞÞÞ and ðA1,1 ðF b ðgÞÞ, . . . ,Ap,1 ðF b ðgÞÞÞ are the treatment generalized wordlength pattern and block generalized wordlength pattern of the combined blocked design F b ðgÞ, respectively. A foldover plan g is called W1 (or W2) optimal foldover plan, if it sequentially minimizes the generalized wordlength pattern of the combined blocked designs as defined in (6) (or (7)). 4. Characterization of optimal blocking and foldover plans From the point of blocking, the combined design can be considered to be clearly classified into the initial design and the foldover design since the sequential nature of a foldover plan, i.e., there exists an additional blocking factor which takes the value 1 for the first half and + 1 for the other half. This blocking factor is called the implicit blocking factor, denoted by xb . Therefore, the combined design is blocked into 2q + 1 blocks of block size n/2q. Without loss of generality, it is assumed that the implicit blocking factor xb is in the last column of the combined blocked designs. Its design matrix has the form ! F b 1n b F ðgÞ ¼ , F bg 1n where 1n is the n-dimensional column vector of ones. The following theorem gives the indicator function of the combined blocked design F b ðgÞ. Theorem 2. Suppose fb ðxÞ is the blocked indicator function of blocked design F b as defined in (4). Then for any g ¼ ðgt , gb Þ 2 O, the indicator function Fb ðx,xb Þ of the combined blocked design F b ðgÞ can be expressed as follows: X X X X Fb ðx,xb Þ ¼ bI XI ðxÞ þ bI XI ðxÞxb bI XI ðxÞxb bI XI ðxÞ, ð12Þ I2P 2t
I2P 2b
I2P 1t
I2P 1b
where P it and P ib are defined as in Theorem1 for i =1, 2. Proof. Since the implicit blocking factor xb takes the value 1 for the initial design F b , the indicator function of the first half of F b ðgÞ can be expressed as follows: 0 1 X X X X 1 1 bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞA fb ðx,xb Þ ¼ ð1xb Þfb ðxÞ ¼ @ 2 2 I2P 2t I2P 1t I2P 2b I2P 1b 0 1 X X X 1 @X ð13Þ xb bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞA: 2 2 1 2 1 I2P t
I2P t
I2P b
I2P b
Moreover, the implicit blocking factor xb takes the value + 1 for the foldover design F b ðgÞ, it implies that the implicit blocking factor xb always exchanges its sign in any foldover plan g ¼ ðgt , gb Þ 2 O, that is, F b ðgÞ is obtained by the foldover plan g ¼ ðgt , gb ,1Þ. Therefore the indicator function of the second half of F b ðgÞ can be expressed as follows: 0 1 0 1 X X X X X X X 1 @X 1 fb, g ðx,xb Þ ¼ bI XI ðxÞ bI XI ðxÞ þ bI XI ðxÞ bI XI ðxÞA þ xb @ bI XI ðxÞ bI XI ðxÞ þ bI XI ðxÞ bI XI ðxÞA: 2 2 2 1 2 1 2 1 2 1 I2P t
I2P t
I2P b
I2P b
I2P t
I2P t
I2P b
I2P b
ð14Þ Noting that Fb ðx,xb Þ ¼ fb ðx,xb Þ þ fb, g ðx,xb Þ, combing (13) and (14), (12) follows.
&
Theorem 3. Suppose fb ðxÞ is the blocked indicator function of blocked design F b as defined in (4). Then for 1 rj rp, we have the following relationships: (i) Aj,0 ðF b ðgÞÞ ¼ Aj,0 ðF b ðgÞÞ; (ii) Aj,1 ðF b ðgÞÞ ¼ Aj,1 ðF b Þ þAj,0 ðF b ÞAj,0 ðF b ðgÞÞ.
Proof. From Definition 3 and (12), we have !2 X bI Aj,0 ðF b ðgÞÞ ¼ ¼ Aj,0 ðF b ðgÞÞ, bf 2 jIj ¼ j,I2P t
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hence (i) follows. Similarly, Aj,1 ðF b ðgÞÞ ¼
X jI1 j ¼ j,I2P 2b
bI bf
!2 þ
X jIj ¼ j,I2P 1t
bI bf
!2 þ
X jI1 j ¼ j,I2P 1b
bI bf
!2 ¼
X jI1 j ¼ j,I2P b
bI bf
!2 þ
X jIj ¼ j,I2P t
bI bf
!2
X
jIj ¼ j,I2P 2t
bI bf
!2
¼ Aj,1 ðF b Þ þ Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ ¼ Aj,1 ðF b Þ þ Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ, which completes the proof.
&
Remark 1. From (i) of Theorem 3, one can find that the treatment generalized wordlength pattern of F b ðgÞ is the same as the one of F b ðgÞ. It implies that the treatment generalized wordlength pattern is independent of the consideration of the implicit blocking factor in the combined blocked design. Remark 2. Ai et al. (2010) provided a similar relationship in Theorem 3 for regular two-level designs. In fact, if F is a regular two-level design, then the results of Theorem 3 cover the corresponding results of Ai et al. (2010) as a special case. However, as pointed in Ai et al. (2010), the treatment and block wordlength pattern of F b ðgÞ are independent of the choice of the block foldover plan when F is a regular two-level design. This may be invalid when F is nonregular, because it is dependent of both the choice of the block factors and the foldover plan gb for the block factors. For a fixed blocked design F b , if there exists a foldover plan for which the corresponding combined blocked design has GMA under the criterion of W1 or W2, then it is called a generalized minimum aberration foldover plan for the combined blocked design. Although the orderings of the generalized treatment and block wordlength patterns vary with the optimality criteria W1 and W2, it is essential that the number Aj,1 is always put behind the number Aj,0 for j= 1, 2, y, p, therefore we have the following theorem. Theorem 4. For a given blocked design F b , a foldover plan g has generalized minimum aberration for the combined blocked design F b ðgÞ if and only if it sequentially minimizes Aj,0 ðF b ðgÞÞ for j= 1, 2, y, p. From Theorem 4, we know that for a fixed initial blocked design, the GMA foldover plan is just the GMA foldover plan for the unblocked case. Therefore, we need only consider the GMA foldover plans for the unblocked cases. Some examples can be found in Li et al. (2003). A blocking plan for a given two-level design F , is said to have GMA if it sequentially minimizes the generalized block wordlength pattern W b ðF b Þ of the blocked design F b . A blocking plan for the design F and a fixed foldover plan g have GMA if it sequentially minimizes the block wordlength pattern W b ðF b ðgÞÞ of the combined blocked design F b ðgÞ. Noting that the P Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ ¼ jIj ¼ j,I2P 1 ðbI =bf Þ2 are fixed in this case, therefore from (ii) of Theorem 3, we have the following t conclusion. Theorem 5. For a two-level design F with a fixed foldover plan, a blocking plan has GMA if and only if it has GMA for F without consideration of the foldover plans. A pair of blocking and foldover plans is said to be optimal if it together ensures that the combined blocked design of a twolevel design has GMA. By combining Theorems 4 and 5, we have the following result. Theorem 6. For a two-level design F , a pair of blocking and foldover plans has GMA for the combined blocked design F b ðgÞ if and only if the foldover plan has GMA for the design F without consideration of the blocking plans and the blocking plan has GMA for F without consideration of the foldover plans. Remark 3. Ai et al. (2010) also provided the similar results corresponding to Theorems 4–6 for two-level regular design in terms of minimum aberration criterion. Example 1. Consider a nonregular design F listed in Table 1, where n =16 and s =6. The indicator function of the unblocked design F is f ðx1 ,x2 ,x3 ,x4 ,x5 ,x6 Þ ¼ 14 ð1x1 x3 x4 12 x3 x5 x6 12 x4 x5 x6 þ 12 x1 x3 x5 x6 þ 12x1 x4 x5 x6 þ 12 x2 x3 x5 x6 12 x2 x4 x5 x6 þ 12 x1 x2 x3 x5 x6 12x1 x2 x4 x5 x6 Þ: If we assign the second column as blocking factor and the first, sixth columns as foldover plan, respectively, then F is blocked into two blocks of size eight, and the corresponding blocked indicator function of F b is fb ðx1 ,xB ,x3 ,x4 ,x5 ,x6 Þ ¼ 14 ð1x1 x3 x4 12 x3 x5 x6 12 x4 x5 x6 þ 12 x1 x3 x5 x6 þ 12x1 x4 x5 x6 þ 12 xB x3 x5 x6 12 xB x4 x5 x6 þ 12 x1 xB x3 x5 x6 12x1 xB x4 x5 x6 Þ,
Z. Ou et al. / Journal of Statistical Planning and Inference 141 (2011) 1635–1645
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Table 1 F. Run
x1
x2
x3
x4
x5
x6
Block II
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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
B1 B1 B1 B1 B2 B2 B2 B2 B1 B1 B1 B1 B2 B2 B2 B2
Table 2 Optimal blocking and foldover plans for (12,2p: 21) designs. p
Design
3
4.1
4
5.1
5
6.1
6
7.1
7
8.1
8
9.1
9
10.1
Rt
Rb III.67
(0)
(0.333, 0.222)
1
IV.67
III.67
(0, 0.111)
(0.667, 0.889, 0)
2/1
IV.67
III.67
(0, 0.556, 0)
(1.111, 2.222, 0, 0.444)
1/6
IV.67
III.67
(0, 1.667, 0, 0)
(1.667, 4.444, 0.889, 0.889, 0.111)
1
IV.67
III.67
(0, 3.889, 0, 0.444, 0)
(2.333, 7.778, 2.222, 2.667, 0.778)
1
IV.67
III.67
(0, 7.778, 0, 1.778, 0, 0.111)
(3.111, 12.444, 4.444, 7.111, 3.111)
1
IV.67
III.67
(0, 14, 0, 5.333, 0, 1)
(4, 18.667, 8, 16, 9.333)
Foldover plan
Blocking plan
gf gf gf gf gf gf gf
1
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
which is obtained by replacing x2 in f(x1, x2, x3, x4, x5, x6) with xB. From Theorem 2, we can obtain the indicator function of the combined blocked design F b ðgÞ as follows: Fb ðx1 ,xB ,x3 ,x4 ,x5 ,x6 ,xb Þ ¼ 14 ð1 þx1 x3 x4 xb þ 12 x1 x3 x5 x6 þ 12 x1 x4 x5 x6 þ 12 x3 x5 x6 xb þ 12x4 x5 x6 xb þ 12 x1 xB x3 x5 x6 12 x1 xB x4 x5 x6 12 xB x3 x5 x6 xb þ 12xB x4 x5 x6 xb Þ, where xb is the implicit blocking factor. From fb(x1, xB, x3, x4, x5, x6), we know that Rt = III and Rb = IV.5, Wt ðF b Þ ¼ ð0,0,1:5,0:5,0Þ and Wb ðF b Þ ¼ ð0,0,0:5,0:5,0Þ for the n blocked design. From Fb(x1, xB, x3, x4, x5, x6, xb), we have Rt = IV.5 and Rb =IV, Wt ðF b ðgÞÞ ¼ ð0,0,0,0:5,0Þ and Wb ðF b ðgÞÞ ¼ ð0,0,2,0:5,0Þ for the combined blocked design. Here the combined blocked design F b ðgÞ is a (16, 25: 22) blocked design. One can find that Aj,1 ðF b ðgÞÞ ¼ Aj,1 ðF b Þ þ Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ for j =1, y, 5, which is indicated by part (ii) of Theorem 3. Moreover, the blocking plan and foldover plan considered above is the optimal GMA blocking plan and foldover plan for F , respectively. Therefore, from Theorem 6, the pair of blocking and foldover plans considered above is optimal for design F . 5. Optimal blocking and foldover plans for 12, 16 and 20-run designs In this section, we present some optimal blocking and foldover plans for two-level designs with 12-run, 16-run and 20-run based on the generalized minimum aberration criterion discussed in the previous section. We only focus on the blocked designs with Rt Z 3 and Rb Z 3. Specifically, the restriction Rt Z 3 ensures the orthogonality among treatment main effects, and the restriction Rb Z 3 guarantees that block effects are orthogonal to treatment main effects. A necessary condition for Rb Z 3 is that n/2q must be an even number (Cheng et al., 2004). Therefore, we only consider the case of one or two block factors for 16-run designs and one block factor for 12 and 20-run designs. 5.1. Optimal blocking and foldover plans for 16-run designs Tables 3 and 4 give the optimal blocking and foldover plans for two-level designs as given in Sun et al. (2002). The first column of each table gives the number of treatment factors p, and the second column of each table gives the design index which is corresponding to the index number used by Sun et al. (2002). The third and fourth columns show the optimal
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Table 3 Optimal blocking and foldover plans for (16,2p:21) designs. p
Design
2
Foldover plan
Blocking plan
Rt
Rb
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
Rank
3.1 3.2 3.3
3 0 3
1 1 1
III
(1)
III.5
(0.25)
3 1 2
4.1 4.3 4.4 4.5
3 0 4 4
4 1 1 1
IV IV.5 III.5
(0) (0) (0) (0)
(0, 1) (0, 0) (0, 0.5) (0.25, 0.25)
3 1 2 4
4
5.1 5.2 5.3 5.5 5.6 5.7 5.8 5.10 5.11
3, 5 3 4 5 1, 5 5 5 5 1, 2
2 4 5 1 4 1 1 1 1
III IV V IV.5 III.5 IV.5 III.5 III.5 III.5
(0, (0, (0, (0, (0, (0, (0, (0, (0,
0) 0) 0) 0) 0) 0) 0) 0) 0)
(1, 2, 0) (0, 1, 0) (0, 0, 1) (0, 0.5, 0.5) (0.5, 1.5, 0) (0, 1, 0) (0.25, 0.5, 0.25) (0.5, 0.5, 0) (0.5, 1, 0)
9 4 1 2 8 3 5 6 7
5
6.1 6.2 6.3 6.4 6.6 6.7 6.9 6.12 6.13 6.18 6.19
3, 3, 3, 3, 3, 1, 1, 6 6 5, 2,
5, 6 5 6 6 6 6 6
1 6 1 4 4 2 4 1 1 1 1
IV IV
IV.5
III IV III IV III.5 IV III.5 IV.5 IV.5 III.5 IV.5
(0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0,
1, 0) 1, 0) 0, 0) 0, 0) 0, 0) 0.5, 0) 0.5, 0) 1, 0) 0, 0) 0, 0) 0.5, 0)
(2, 4, 0, 0) (0, 2, 0, 0) (1, 1, 0, 1) (0, 2, 1, 0) (0.25, 1.75, 0.75, 0.25) (0, 2, 0.5, 0) (0.25, 1.75, 0.25, 0.25) (0, 2, 0, 0) (0, 2, 1, 0) (1, 1, 0, 1) (0, 2, 0.5, 0)
19 9 5 2 3 7 10 8 1 4 6
7.1 7.2 7.3 7.4 7.5 7.7 7.8 7.9 7.10
3, 3, 3, 3, 3, 1, 1, 1, 1,
5, 5, 5, 5, 5, 7 2, 7 2,
6 6 7 7 7
1 7 2 4 6 6 6 2 1
IV IV IV IV IV IV IV IV IV.5
III IV III III IV III.5 III.5 IV III
(0, (0, (0, (0, (0, (0, (0, (0, (0,
3, 3, 1, 1, 1, 1, 1, 2, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0,
0) 0) 0) 0) 0) 0) 0) 0) 0)
(3, 8, 0, 0, 1) (0, 4, 0, 0, 0) (1, 4, 0, 0, 1) (1, 3, 1, 1, 0) (0, 4, 2, 0, 0) (0.5, 3.5, 1.5, 0.5, 0) (0.75, 3.5, 1, 0.5, 0.25) (0, 4, 1, 0, 0) (1, 3, 1, 1, 0)
9 3 8 7 1 4 5 2 6
8.1 8.2 8.3 8.5 8.7 8.8 8.9 8.10
3, 3, 3, 3, 1, 1, 1, 1,
5, 5, 5, 5, 2, 8 2, 2,
6 6, 8 6 7, 8 3, 8 3, 8 8
8 7 7 3 7 2 3 7
IV IV IV IV IV IV IV.5 IV
IV III IV III III.5 III III III.5
(0, (0, (0, (0, (0, (0, (0, (0,
7, 3, 3, 2, 3, 2, 2, 3,
0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 1, 0, 0, 1, 0,
0) 0) 0) 0) 0) 0) 0) 0)
(0, 7, 0, 0, 0) (1, 6, 2, 2, 1) (0, 7, 4, 0, 0) (1, 6, 3, 2, 0) (0.75, 6.25, 2.5, 1.5, 0.75) (1, 6, 3, 1, 1) (1, 6, 3, 2, 0) (1, 6, 2, 2, 1)
2 8 1 4 6 5 3 7
8
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
3, 3, 3, 3, 3, 1, 1, 1,
5, 5, 5, 5, 5, 2, 2, 8,
6, 6, 6, 7, 7, 3, 3, 9
9 8, 9 8 8 8, 9 9 9
8 4 8 2 1 8 9 2
IV IV IV IV IV IV IV IV
III III III III III III.5 III.5 III
(0, (0, (0, (0, (0, (0, (0, (0,
7, 5, 5, 3, 5, 7, 5, 5,
0, 0, 0, 4, 0, 0, 0, 2,
0, 2, 2, 0, 2, 0, 2, 0,
0, 0, 0, 0, 0, 0, 0, 0,
(1, 10, 4, 4, 3) (2, 9, 4, 6, 2) (1, 10, 6, 4, 1) (1, 10, 8, 0, 3) (2, 8, 5, 8, 0) (1, 10, 4, 4, 3) (1.5, 9.5, 5, 5, 1.5) (1, 10, 6, 2, 3)
5 8 2 1 7 4 6 3
9
10.1 10.2 10.3 10.6
3, 3, 3, 1,
5, 5, 5, 2,
6, 6, 6, 3,
9, 10 8, 9, 10 8, 9 10
8 1 6 10
IV IV IV IV
III III III III.5
(0, (0, (0, (0,
10, 0, 4, 0, 1) 9, 0, 6, 0, 0) 9, 0, 6, 0, 0) 9, 0, 6, 0, 0)
(2, 14, 8, 12, 6) (3, 13, 6, 15, 7) (2, 14, 9, 12, 4) (2.25, 13.75, 8.25, 12.75, 4.75)
2 4 1 3
10
11.1 11.2
3, 5, 6, 9, 10 3, 5, 6, 9, 10, 11
4 8
IV IV
III III
(0, 16, 0, 12, 0, 3) (0, 16, 0, 12, 0, 3)
(3, 20, 12, 24, 16) (3, 19, 13, 27, 13)
2 1
11
12.1 12.5 12.8
3, 5, 6, 9, 10, 12 1, 2, 3, 10, 11, 12 1, 2, 10, 11, 12
12 1 1
IV IV IV
III III III
(0, 26, 0, 24, 0, 13) (0, 25.5, 0, 25.5, 0, 11.5) (0, 25, 0, 27, 0, 10)
(0, 4, 26, 0, 25) (4, 25.5, 19.5, 51, 29.5) (4, 26, 19, 50, 31)
3 2 1
12
13.1
3, 5, 6, 9, 10, 12
2
IV
III
(0, 38, 0, 52, 0, 33)
(5, 34, 28, 88, 62)
1
3
6
7
6 3
7 7
IV.5 IV.5 IV
0) 0) 0) 0) 0) 0) 0) 0)
Z. Ou et al. / Journal of Statistical Planning and Inference 141 (2011) 1635–1645
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Table 4 Optimal blocking and foldover plans for (16,2p:22) designs. p
Design
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
Rank
Blocking plan
4.1 4.2 4.3 4.4
3 4 0 4
1, 1, 1, 1,
4 2 2 2
III.5 III
3
5.1 5.2 5.5 5.6 5.7 5.10 5.11
3, 5 3 5 1, 5 5 5 1, 2
2, 4, 1, 1, 1, 1, 1,
4 5 2 4 2 2 2
III IV IV.5 III.5 III.5 III.5 III.5
(0) (0) (0) (0) (0) (0) (0)
4
6.1 6.2 6.3 6.6 6.7 6.10 6.17 6.18 6.22 6.23 6.25 6.26
3, 3, 3, 3, 1, 1, 5, 5, 1, 1, 1, 1,
5, 6 5 6 6 6 6 6 6 2, 3 2 2 2
1, 2, 1, 1, 2, 1, 5, 1, 1, 3, 1, 1,
6 6 4 4 5 6 6 2 2 4 2 2
IV
III III III III III.5 III III.5 III.5 III.5 III.5 III.5 III.5
(0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0,
1) 0) 0) 0) 0) 1) 1) 0) 0) 0.25) 1) 0)
(6, 0, 0) (1, 2, 0) (2, 0, 1) (1.5, 1, 0.5) (1, 2, 0) (4, 0, 0) (2, 0, 0) (2, 0, 1) (1.5, 1.5, 0) (1.5, 1, 0.25) (3, 0, 0) (3, 0, 0)
5
7.2 7.3 7.4 7.8 7.9
3, 3, 3, 1, 1,
5, 5, 5, 2, 7
6 7 7 7
1, 2, 4, 1, 2,
7 4 6 6 7
IV
IV
III III III III III.5
(0, (0, (0, (0, (0,
1, 0, 0, 0, 1,
0) 0) 0) 0) 0)
(2, 4, 0, 0) (3, 3, 0, 1) (2, 4, 1, 0) (2.5, 3.5, 0.5, 0.5) (2, 4, 0, 0)
5 3 1 2 4
6
8.1 8.2 8.5 8.7 8.10
3, 3, 3, 1, 1,
5, 5, 5, 2, 2,
6 6, 8 7, 8 3, 8 8
1, 6, 3, 1, 1,
8 7 8 7 7
IV IV VI IV IV
III III III III III
(0, (0, (0, (0, (0,
3, 1, 0, 1, 1,
0, 0, 0, 0, 0,
0) 0) 1) 0) 0)
(3, 8, 0, 0, 1) (3, 8, 2, 0, 1) (3, 8, 3, 0, 0) (3.5, 7, 2, 1, 0.5) (4, 6, 2, 2, 0)
3 2 1 4 5
7
9.1 9.2 9.4 9.5 9.8
3, 3, 3, 3, 1,
5, 5, 5, 5, 8,
6, 6, 7, 7, 9
2, 4, 2, 1, 2,
8 8 4 6 4
IV IV IV IV IV
III III III III III
(0, (0, (0, (0, (0,
3, 2, 1, 3, 2,
0, 0, 2, 0, 1,
0, 1, 0, 0, 0,
(5, (5, (5, (6, (5,
4 2 1 5 3
8
10.1 10.2 10.6
3, 5, 6, 9, 10 3, 5, 6, 8, 9, 10 1, 2, 3, 10
4, 8 1, 6 1, 10
IV IV IV
III III III
(0, 5, 0, 2, 0, 0) (0, 5, 0, 2, 0, 0) (0, 5, 0, 2, 0, 0)
(7, 18, 10, 12, 7) (8, 16, 9, 16, 6) (7.5, 17, 9.5, 14, 6.5)
1 3 2
9
11.1 11.4 11.7
3, 5, 6, 9, 10 1, 2, 3, 11 1, 2, 3, 10, 11
4, 8 1, 10 2, 10
IV IV IV
III III III
(0, 9, 0, 6, 0, 0) (0, 10, 0, 4, 0, 1) (0, 9.5, 0, 5, 0, 0.5)
(9, 27, 18, 27, 21) (10, 24, 18, 32, 18) (9.5, 25.5, 18, 29.5, 19.5)
1 3 2
10
12.1 12.2 12.5
3, 5, 6, 9, 10, 12 3, 5, 6, 8, 9, 12 1, 2, 3, 10, 11, 12
4, 9 1, 6 1, 6
IV IV IV
III III III
(0, 16, 0, 12, 0, 3) (0, 18, 0, 8, 0, 5) (0, 17, 0, 10, 0, 4)
(12, 36, 30, 60, 48) (13, 32, 32, 64, 42) (12.5, 34, 31, 62, 45)
1 3 2
11
13.1
3, 5, 6, 9, 10, 12
2, 12
IV
III
(0, 26, 0, 24, 0, 13)
(15, 48, 48, 112, 102)
1
2
9 8, 9 8 8, 9
Rt
Rb
Foldover plan
III.5
IV IV
IV IV
0) 0) 0) 0) 0)
(1) (1) (0) (0.5)
3 4 1 2
(3, 0) (0, 1) (0, 1) (2, 0) (0.5, 0.5) (1, 0) (1.5, 0)
7 2 1 6 3 4 5
12, 4, 4, 3) 12, 5, 4, 2) 12, 6, 2, 3) 9, 6, 6, 0) 12, 5, 3, 3)
12 2 6 3 1 11 9 5 4 8 10 7
foldover plan and blocking plan. For ease of comparison and selection of optimal designs, the treatment generalized wordlength pattern (GTWP) and the block generalized wordlength pattern (GBWP) of the optimal combined blocked design are listed in the fifth and sixth columns, respectively. Note that only some initial components of GTWP and GBWP are given for the sake of brevity. In order to compare combined blocked designs in terms of their aberration, we adopt the combined wordlength pattern W2 to rank-order these designs which are suggested by Cheng et al. (2004). According to the combined wordlength pattern W2 defined in (7), the rank of each design is given in the last column. Certainly, other criteria based on different combined wordlength patterns can be adopted to rank-order designs. From Tables 3 and 4, one can easily find that the optimal foldover plan g of some designs depend on the block foldover plan gb , which is different from the ones as shown in Ai et al. (2010), the optimal foldover plan is independent of the choice of block foldover plan for regular designs. For example, the optimal blocking plan is 1 for design 5.11 in Table 3, the optimal foldover plan is 1, 2 which is dependent on the block foldover plan for this design.
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Table 5 Optimal blocking and foldover plans for (20,2p:21) designs. p
Design
2 3
3.3 4.3
4
5.11
5
6.74
6
7.469
7
8.1540
7
8.1601
8
9.2444
8
9.2477
9
10.2389
10
11.1913
11
12.1292
12
13.728
13
14.320
14
15.92
Foldover plan
Blocking plan
3
1 1
gf gf gf gf gf gf gf gf gf gf gf gf gf gf
Rt
Rb
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
III.8 III.8
(0)
(0.04) (0.12, 0.08) (0.24, 0.32, 0)
1
IV.8
III.8
(0, 0.04)
1/2
IV.8
III.8
(0, 0.2, 0)
(0.4, 0.8, 0.48, 0.32)
5/1
IV.4
III.8
(0, 0.92, 0, 0)
(0.6, 1.92, 1.28, 0.64, 0.04)
8/1
IV.4
III.8
(0, 2.68, 0, 0.16, 0)
(0.84, 3.44, 2.72, 1.6, 0.28)
5/2
IV.4
III.8
(0, 3, 0, 0.16, 0)
(0.84, 3.76, 1.76, 1.92, 0.28)
6/3
IV.4
III.8
(0, 4.4, 0, 0.96, 0, 0.04)
(1.12, 6.72, 3.84, 5.12, 1.44)
3/1
IV.4
III.8
(0, 6, 0, 0.64, 0, 0.04)
(1.12, 7.04, 3.2, 3.52, 2.08)
2/1
IV.4
III.8
(0, 10.8, 0, 1.92, 0, 0.36)
(1.44, 10.56, 6.08, 8.32, 5.92)
1/3
IV.4
III.4
(0, 14.16, 0, 7.36, 0, 3.08)
(2.12, 14.72, 10.24, 21.44, 11.6)
1/2
IV.4
III.4
(0, 20.24, 0, 21.44, 0, 7.88)
(2.84, 19.92, 16.96, 39.36, 23.6)
5/3
IV.4
III.4
(0, 30.04, 0, 41.28, 0, 27.16)
(3.6, 25.92, 24, 73.28, 47.84)
1/3
IV.4
III.4
(0, 42.36, 0, 79.68, 0, 67.8)
(4.4, 33.76, 34.24, 120.48, 88.16)
15/3
IV.4
III.4
(0, 59.24, 0, 139.84, 0, 157.24)
(5.24, 42.56, 48, 189.12, 152.44)
5.2. Optimal blocking and foldover plans for 12 and 20-run designs Tables 2 and 5 give the optimal blocking and foldover plans for two-level designs which also considered by Cheng et al. (2004). They only considered the optimal block plans based on the combined wordlength patterns W1 or W2. From Tables 2 and 5, we can find that the full foldover plan gf is the unique optimal foldover plan for these designs except design 3.3. This is indicated by Theorem 6 that the full foldover plan is the unique optimal foldover plan these designs as shown in Li et al. (2003). The fourth column of Tables 2 and 5 gives the optimal blocking plan for both this paper and Cheng et al. (2004). For example, the optimal blocking plan is 2 of this paper and 1 of Cheng et al. (2004) for design 6.1 as denoted by 2/1 in Table 2. If there is only one number in the fourth column, it implies that the optimal blocking plan of us is the same as the one of Cheng et al. (2004). 6. Concluding remarks In this paper, using the effective tool of indicator function and based on the assumption of the implicit block effect is significant, optimal blocking and foldover plan of nonregular two-level designs are studied. It is shown that, for a two-level design F , a pair of blocking and foldover plans has GMA for the combined blocked design F b ðgÞ if and only if the blocking plan has GMA for F without consideration of the foldover plans and the foldover plan has GMA for design F without consideration of the blocking plans. All of these GMA results hold true for any generalized aberration criterion for blocked designs which orders Aj,1 behind Aj,0. These results extend the ones of Ai et al. (2010) from regular designs to nonregular designs.
Acknowledgements The authors would like to thank the Referees and the Executive 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 No. 10671080), NCET (No. 06-672), Scientific Research Plan Item of Hunan Provincial Department of Education (No. 10C1091), the Innovation Program and Independent Research Project Funded by Central China Normal University. References Ai, M.Y., Xu, X., Wu, C.F.J., 2010. Optimal blocking and foldover plans for regular two-level designs. Statist. Sinica 20, 513–536. Balakrishnan, N., Yang, P., 2006a. Classification of three-word indicator functions of two-level factorial designs. Ann. Inst. Statist. Math. 58, 595–608. Balakrishnan, N., Yang, P., 2006b. Connections between the resolutions of general two-level factorial designs. Ann. Inst. Statist. Math. 58, 609–618. Balakrishnan, N., Yang, P., 2009. De-aliasing effects using semifoldover techniques. J. Statist. Plann. Inference 139, 3102–3111. Box, G.E.P., Hunter, W.G., Hunter, J.S., 1978. Statistics for Experiments. John Wiley and Sons, New York. Cheng, S.W., Li, W., Ye, K.Q., 2004. Blocked nonregular two-level factorial designs. Technometrics 46, 269–279. Cheng, S.W., Wu, C.F.J., 2002. Choice of optimal blocking schemes in two-level and three-level designs. Technometrics 44, 269–277. Fontana, R., Pistone, G., Rogantin, M.P., 2000. Classification of two-level factorial fractions. J. Statist. Plann. Inference 87, 149–172. Li, F., Jacroux, M., 2007. Optimal foldover plans for blocked 2m-k fractional factorial designs. Statist. Plann. Inference 137, 2439–2452. Li, H., Mee, R.W., 2002. Better foldover fractions for resolution III 2k-p designs. Technometrics 44, 278–283. Li, W., Lin, D.K.J., 2003. Optimal foldover plans for two-Level fractional factorial designs. Technometrics 45, 142–149. Li, W., Lin, D.K.J., Ye, K.Q., 2003. Optimal foldover plans for non-regular orthogonal designs. Technometrics 45, 347–351.
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Montgomery, D.C., Runger, G.C., 1996. Foldover of 2k-p resolution IV experimental designs. J. Qual. Technol. 28, 446–450. Ou, Z.J., Qin, H., 2010. Some applications of indicator function in two-level factorial designs. Statist. Probab. Lett. 80, 19–25. Sun, D.X., Li, W., Ye, K.Q. 2002. An algorithm for sequentially constructing non-isomorphic orthogonal designs and its applications. Technical report SUNYSBAMS-02-13, Dept. of Applied Math. Statistics, SUNY Stony Brook. Wu, C.F.J., Hamada, M., 2000. Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley, New York. Ye, K.Q., 2003. Indicator function and its application in two-level factorial designs. Ann. Statist. 31, 984–994. Ye, K.Q., Li, W., 2003. Some properties of blocked and unblocked foldover of 2k-p designs. Statist. Sinica 13, 403–408.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Review
Optimal blocking and foldover plans for nonregular two-level designs Zujun Ou a,b, Hong Qin b,, Hongyi Li c a
College of Mathematics and Computer Science, Jishou University, Jishou 416000, China Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China c Normal College, Jishou University, Jishou 416000, China b
a r t i c l e in f o
abstract
Article history: Received 29 May 2010 Received in revised form 8 December 2010 Accepted 8 December 2010 Available online 17 December 2010
This paper discusses the issue of choosing optimal designs when both blocking and foldover techniques are simultaneously employed to nonregular two-level fractional factorial designs. By using the indicator function, the treatment and block generalized wordlength patterns of the combined blocked design under a general foldover plan are defined. Some general properties of combined block designs are also obtained. Our results extend the findings of Ai et al. (2010) from regular designs to nonregular designs. Based on these theoretical results, a catalog of optimal blocking and foldover plans in terms of the generalized aberration criterion for nonregular initial design with 12, 16 and 20 runs is tabulated, respectively. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: Combined blocked design Generalized minimum aberration Generalized resolution Indicator function
Contents 1. 2. 3. 4. 5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635 Indicator function and related criteria for blocked designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636 Foldover plans for blocked designs and related criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638 Characterization of optimal blocking and foldover plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1639 Optimal blocking and foldover plans for 12, 16 and 20-run designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1641 5.1. Optimal blocking and foldover plans for 16-run designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1641 5.2. Optimal blocking and foldover plans for 12 and 20-run designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Acknowdgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644
1. Introduction Recently, there has been considerable interest in exploring the potential application of the foldover technique to fractional factorial designs. In order to break the aliasing in two-level designs, a standard strategy is to add a foldover of the initial design by reversing the signs of one or more of its factors. A foldover plan refers to the collection of columns whose signs are reversed in the foldover design. Box et al. (1978) firstly discussed the foldover plan for reversing only one factor to de-alias the specific factor from all other factors. Montgomery and Runger (1996) studied foldover plan for resolution IV designs. Li and Mee Corresponding author.
E-mail address: [email protected] (H. Qin). 0378-3758/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.12.008
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(2002) and Li and Lin (2003), respectively, proposed the concept of optimal foldover plan for combined designs which consist of the initial design and the foldover design, and by an exhausted search, gave all optimal foldover plans for regular two-level fractional factorial designs, in terms of the aberration criterion of the combined design. Li et al. (2003) extended optimal foldover plans to nonregular two-level factorial designs in terms of the generalized minimum aberration criterion of combined designs. Ye and Li (2003) studied the theoretical properties of the foldover design and the resulting combined design under a general foldover plan. In order to control systematic noises, blocking is a commonly used technique in experiments. For a given blocked regular two-level design, Li and Jacroux (2007) provided the optimal treatment foldover plans by an algorithm under two proposed optimality criteria. Ai et al. (2010) considered the optimal plans for regular two-level designs when both blocking and foldover techniques are employed and provided some theoretical insight into the relationships between an initial design and the resulting combined blocked design under a general foldover plan. This paper is intended to compliment the work of Ai et al. (2010) by simultaneously considering optimal blocking and foldover plans for nonregular two-level designs via the tool of indicator function. Indicator function is an effective tool in studying two-level factorial designs. Fontana et al. (2000) firstly introduced indicator function to study fractional factorial designs with no replicates. Recently, many papers focus on the application of indicator function in factorial designs. Among these papers are Ye (2003), Balakrishnan and Yang (2006a,b), Balakrishnan and Yang (2009), Ou and Qin (2010). In particular, Cheng et al. (2004) discussed the optimal blocking criteria for nonregular two-level designs via the tool of indicator function. Our research focuses on the optimal plans for nonregular two-level designs when both blocking and foldover techniques are simultaneously employed. Some general theoretical results are obtained and described. This paper is organized as follows. In Section 2, some preliminary concepts of the indicator function of unblocked or blocked two-level factorial designs are introduced. Some related optimality criteria based on indicator function are also introduced in this section. In Section 3, the indictor function and related optimality criteria of combined blocked design are obtained under a general foldover plan. The relationships between the treatment and block generalized wordlength patterns of an initial design and its combined blocked design are obtained in Section 4. Based on these theoretical results, a catalog of optimal blocking and foldover plans in terms of generalized aberration is, respectively, tabulated and compared for 12-run, 16-run, 20-run nonregular initial factorial designs in Section 5. Some concluding remarks are given in Section 6. 2. Indicator function and related criteria for blocked designs Let D be a 2s full factorial design with levels being 1 and 1 throughout this paper. The design points of D are the solutions of the polynomial system fx21 1 ¼ 0,x22 1 ¼ 0, . . . ,x2s 1 ¼ 0g. A n runs fractional factorial design F is an any subset of D without any restriction on its run size. We firstly introduce indicator functions of an unblocked two-level design presented by Fontana et al. (2000) and Ye (2003) as follows. Definition 1. Let D be a 2s design. The indicator function f ðxÞ of its fraction F is a function defined on D such that ( rx if x 2 F , f ðxÞ ¼ 0 if x 2 DF ,
ð1Þ
where rx is the number of replicates of point x in design F . ¨ Using the theory of Grobner basis and algebraic geometry, Fontana et al. (2000) and Ye (2003) showed that the indicator function of F has a unique polynomial representation as follows: X bI XI ðxÞ, ð2Þ f ðxÞ ¼ I2P
Q where XI ðxÞ ¼ i2I xi is defined on D for I 2 P and P is the collection of all subsets of f1,2, . . . ,sg. The coefficient of f ðxÞ in (2), bI, can be calculated by the formula bI ¼
1X XI ðxÞ: 2s x2F
ð3Þ
In particular, bf ¼ n=2s , where n is the run size of F . This implies that bf is just the ratio between the number of points of F and the number of points of D. The coefficients of indicator functions satisfy jbI =bf jr 1 for any I 2 P such that bI a0. Fontana et al. (2000) showed that a two-level factorial design F is a regular design if and only if jbI =bf j ¼ 1 for any I 2 P such that bI a0, otherwise F is a nonregular design. Moreover, F is an orthogonal array of strength t if and only if all the coefficients of the indicator function up to the order t are zero, that is, bI = 0 for any I 2 P such that 1 rjIjr t, where bI is the coefficient of XI ðxÞ of the indicator function f ðxÞ. Those polynomial terms XI ðxÞ with nonzero coefficients are called as words. The wordlength of the word XI ðxÞ is defined as jIj þ 1jbI =bf j, where jIj is the cardinality of the set I. The integer part of wordlength is just the number of letters in the word, while the degree of aliasing is presented by the fractional part. The resolution R of F refers to the smallest wordlength of the words in f ðxÞ.
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For i= 1, y, s, define Ai ðF Þ ¼
X jIj ¼ i,I2P
bI bf
!2 ,
then the vector ðA1 ðF Þ, . . . ,As ðF ÞÞ is called generalized wordlength pattern by Ye (2003). The generalized minimum aberration (GMA) criterion due to Ye (2003) sequentially minimizes ðA1 ðF Þ, . . . ,As ðF ÞÞ. Now we introduce the definition of indicator function of a blocked two-level design. Arranging a n runs and s two-level factors design F into 2q blocks is equivalent to selecting q independent factors out of the s independent factors as the q blocking factors, referred to as a blocking plan. Considering that the block factors should be orthogonal to treatment factors, we assign a set of q columns which no words in f ðxÞ are solely associated with these columns as the blocking factors, following the line of Cheng et al. (2004). In particularly, any q columns in an orthogonal array of strength t, where qr t, are eligible for constructing 2q blocks. In general, consider a blocked design with p treatment factors and q block factors, where p+ q= s and 1 r qr p1. One can start with an n ðp þ qÞ two-level design F and assign eligible q columns fxB1 , . . . ,xBq g as block factors, and assign the reminding p columns fx1 , . . . ,xs gfxB1 , . . . ,xBq g as treatment factors, where 1r B1 oB2 o . . . o Bq r s. This blocked design is formally denoted by ðn,2p : 2q Þ design F b . Under the assumption that the interactions between the q block factors are as important as the main effects of block factors as usual, therefore we denote xB1 xB2 ,xB1 xB3 , . . . ,xB1 xB2 xBq as xBq þ 1 , . . . ,xB2q 1 . Following Cheng et al. (2004), the indicator function for nonregular two-level design with blocks can be obtained by replacing xB1 xB2 ,xB1 xB3 , . . . ,xB1 xB2 xBq by xBq þ 1 , . . . ,xB2q 1 as follows: X X bI XI ðxÞ þ bI XI ðxÞ, ð4Þ fb ðxÞ ¼ I2P t
I2P b
where P t is the collection of all subsets of f1,2, . . . ,sgfB1 , . . . ,Bq g and P b ¼ PP t ¼ fI1 [ I2 jI1 2 P t ,I2 fB1 , . . . ,Bq g and I2 afg. The resulting polynomial function is referred to as blocked indicator function. The values of xBq þ 1 , . . . ,xB2q 1 depend on the value of xB1 , . . . ,xBq . Based on the usual assumptions that the block-by-treatment interactions are negligible and that the interactions between block factors are as important as the main effects of block factors, every word XI ðxÞ,I 2 P b represents a treatment effect confounded with a block effect. In a blocked indicator function, we can see that there are two types of polynomial terms with nonzero coefficients (i.e., words), one involving treatment factors only for the words XI ðxÞ,I 2 P t , and the other involving both block and treatment factors for the words XI ðxÞ,I 2 P b . We call the former pure-type words and the latter mixed-type words following the language of Cheng et al. (2004). The wordlength of the two type words is defined as follows. Definition 2. Suppose fb ðxÞ is the blocked indicator function of blocked design F b . Then the wordlength L(XI) of the word XI ðxÞ is defined as 8 bI > > if I 2 P t , jIj þ 1 > b < f ð5Þ LðXI Þ ¼ b > > > jI1 jþ 2 I if Ið ¼ I1 [ I2 Þ 2 P b : : bf The treatment resolution Rt is defined as the smallest wordlength of pure-type words and the block resolution Rb as the smallest wordlength of mixed-type words. Definition 3. Suppose fb ðxÞ is the blocked indicator function of blocked design F b . For j =1, y, p, define !2 X bI Aj,0 ðF b Þ ¼ bf jIj ¼ j,I2P t
and Aj,1 ðF b Þ ¼
X jI1 j ¼ j,I2P b
bI bf
!2 ,
then the vectors W t ðF b Þ ¼ ðA1,0 ðF b Þ, . . . ,Ap,0 ðF b ÞÞ and W b ðF b Þ ¼ ðA1,1 ðF b Þ, . . . ,Ap,1 ðF b ÞÞ are called treatment generalized wordlength pattern and block generalized wordlength pattern of F b , respectively. For a given design F , a GMA blocking plan is sequentially to minimize the components in W b ðF b Þ. Since there are two wordlength patterns for blocked designs, one usually combines the treatment generalized wordlength pattern and block generalized wordlength pattern into a combined sequence based on some ordering scheme for comparing and selecting optimal blocked designs, and then sequentially minimizes this sequence among all blocked designs with the same parameters. Based on the two wordlength patterns defined above, Cheng et al. (2004) suggested two optimality criteria for nonregular two-level designs below, which follows the line of Cheng and Wu (2002) for regular two-level designs. The
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generalized wordlength patterns for the blocked nonregular designs F b are defined as W1 ðF b Þ ¼ ðA3,0 ðF b Þ,A4,0 ðF b Þ,A2,1 ðF b Þ,A5,0 ðF b Þ,A6,0 ðF b Þ,A3,1 ðF b Þ,A7,0 ðF b Þ, . . .Þ, b
b
ð6Þ
b
where Ai,1 ðF Þ is inserted between A2i,0 ðF Þ and A2i þ 1,0 ðF Þ, and W2 ðF b Þ ¼ ðA3,0 ðF b Þ,A2,1 ðF b Þ,A4,0 ðF b Þ,A5,0 ðF b Þ,A3,1 ðF b Þ,A6,0 ðF b Þ,A7,0 ðF b Þ, . . .Þ,
ð7Þ
where Ai,1 ðF b Þ is inserted between A2i1,0 ðF b Þ and A2i,0 ðF b Þ. The GMA blocked designs can be obtained by sequentially minimizing the wordlength patterns W1 or W2. A catalog of the GMA blocking plans for some nonregular two-level designs with small runs can be found in Cheng et al. (2004). Although the orderings are different, it is essential that the number {Aj,1} is always put behind the number {Aj,0} for j =1, y, p, based on the popular effect hierarchy principle (Wu and Hamada, 2000).
3. Foldover plans for blocked designs and related criteria Let B ¼ f1,2, . . . ,sgfB1 , . . . ,Bq g and O ¼ fðgt , gb Þjgt ¼ ðgi1 , . . . , gip Þ,ij 2 B,1 rj rp, gb ¼ ðgB1 , . . . , gBq Þg, where gt and gb are the p- and q-dimensional row vectors with elements 0 or 1, respectively. Then any g ¼ ðgt , gb Þ 2 O defines a pair of treatment and block foldover plan for design F b , where gt and gb , respectively, represent the foldover plans of the treatment factors and blocking factors. For convenience, the factors corresponding to 0 are called unfoldover factors with respect to the foldover plan, and those corresponding to 1 are referred to as foldover factors. For a foldover plan g ¼ ðgt , gb Þ, the combined blocked design, denoted by F b ðgÞ, consisting of the initial design F b and its foldover design F bg , that is, F b ðgÞ ¼
Fb F bg
! :
Noting that the combined blocked design F b ðgÞ is blocked into 2q blocks of block size 2n/2q by the same block factors used in the initial design. For any g ¼ ðgt , gb Þ 2 O, let m1 and m2 be the number of nonzero elements of gt and gb , respectively. In particular, when m1 +m2 = 0, g ¼ ðgt , gb Þ is called the null foldover plan, the combined blocked design reduces to two replicates of the blocked design F b , when m1 + m2 = s, g ¼ ðgt , gb Þ is called the full foldover plan. For any g ¼ ðgt , gb Þ 2 O, let Egt ¼ fij : gij ¼ 1g, Egb ¼ fBj :
gBj ¼ 1g and Eg ¼ Egt [ Egb . The following theorem gives the indicator function of the combined blocked design F b ðgÞ. Theorem 1. Suppose fb ðxÞ is the blocked indicator function of blocked design F b as defined in (4). Then for any g ¼ ðgt , gb Þ 2 O, the indicator function Fb ðxÞ of the combined blocked design F b ðgÞ can be expressed as follows: X X Fb ðxÞ ¼ 2 bI XI ðxÞ þ 2 bI XI ðxÞ, ð8Þ I2P 2t
I2P 2b
where P 2t ¼ fIjI 2 P t ,jI \ Egt j is0 or even} and P 2b ¼ fIjI 2 P b ,jI \ Eg j is0 or even}. Proof. Let P 1t ¼ fIjI 2 P t ,jI \ Egt j is oddg and P 1b ¼ fIjI 2 P b ,jI \ Eg j is oddg, fb, g ðxÞ be the indicator function of F bg , respectively. Then fb ðxÞ and fb, g ðxÞ can be, respectively, expressed as follows: X X X X fb ðxÞ ¼ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ ð9Þ I2P 2t
I2P 1t
I2P 2b
I2P 1b
and fb, g ðxÞ ¼
X
bI XI ðxÞ
I2P 2t
X
bI XI ðxÞ þ
I2P 1t
X
bI XI ðxÞ
I2P 2b
X
bI XI ðxÞ:
Combining (9) and (10), we have X X Fb ðxÞ ¼ fb ðxÞ þ fb, g ðxÞ ¼ 2 bI XI ðxÞ þ 2 bI XI ðxÞ, I2P 2t
which completes the proof.
&
For j =1, y, p, define Aj,0 ðF b ðgÞÞ ¼
X jIj ¼ j,I2P 2t
bI bf
!2
ð10Þ
I2P 1b
I2P 2b
ð11Þ
Z. Ou et al. / Journal of Statistical Planning and Inference 141 (2011) 1635–1645
1639
and X
Aj,1 ðF b ðgÞÞ ¼
jI1 j ¼ j,I2P 2b
bI bf
!2 ,
then the vectors ðA1,0 ðF b ðgÞÞ, . . . ,Ap,0 ðF b ðgÞÞÞ and ðA1,1 ðF b ðgÞÞ, . . . ,Ap,1 ðF b ðgÞÞÞ are the treatment generalized wordlength pattern and block generalized wordlength pattern of the combined blocked design F b ðgÞ, respectively. A foldover plan g is called W1 (or W2) optimal foldover plan, if it sequentially minimizes the generalized wordlength pattern of the combined blocked designs as defined in (6) (or (7)). 4. Characterization of optimal blocking and foldover plans From the point of blocking, the combined design can be considered to be clearly classified into the initial design and the foldover design since the sequential nature of a foldover plan, i.e., there exists an additional blocking factor which takes the value 1 for the first half and + 1 for the other half. This blocking factor is called the implicit blocking factor, denoted by xb . Therefore, the combined design is blocked into 2q + 1 blocks of block size n/2q. Without loss of generality, it is assumed that the implicit blocking factor xb is in the last column of the combined blocked designs. Its design matrix has the form ! F b 1n b F ðgÞ ¼ , F bg 1n where 1n is the n-dimensional column vector of ones. The following theorem gives the indicator function of the combined blocked design F b ðgÞ. Theorem 2. Suppose fb ðxÞ is the blocked indicator function of blocked design F b as defined in (4). Then for any g ¼ ðgt , gb Þ 2 O, the indicator function Fb ðx,xb Þ of the combined blocked design F b ðgÞ can be expressed as follows: X X X X Fb ðx,xb Þ ¼ bI XI ðxÞ þ bI XI ðxÞxb bI XI ðxÞxb bI XI ðxÞ, ð12Þ I2P 2t
I2P 2b
I2P 1t
I2P 1b
where P it and P ib are defined as in Theorem1 for i =1, 2. Proof. Since the implicit blocking factor xb takes the value 1 for the initial design F b , the indicator function of the first half of F b ðgÞ can be expressed as follows: 0 1 X X X X 1 1 bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞA fb ðx,xb Þ ¼ ð1xb Þfb ðxÞ ¼ @ 2 2 I2P 2t I2P 1t I2P 2b I2P 1b 0 1 X X X 1 @X ð13Þ xb bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞ þ bI XI ðxÞA: 2 2 1 2 1 I2P t
I2P t
I2P b
I2P b
Moreover, the implicit blocking factor xb takes the value + 1 for the foldover design F b ðgÞ, it implies that the implicit blocking factor xb always exchanges its sign in any foldover plan g ¼ ðgt , gb Þ 2 O, that is, F b ðgÞ is obtained by the foldover plan g ¼ ðgt , gb ,1Þ. Therefore the indicator function of the second half of F b ðgÞ can be expressed as follows: 0 1 0 1 X X X X X X X 1 @X 1 fb, g ðx,xb Þ ¼ bI XI ðxÞ bI XI ðxÞ þ bI XI ðxÞ bI XI ðxÞA þ xb @ bI XI ðxÞ bI XI ðxÞ þ bI XI ðxÞ bI XI ðxÞA: 2 2 2 1 2 1 2 1 2 1 I2P t
I2P t
I2P b
I2P b
I2P t
I2P t
I2P b
I2P b
ð14Þ Noting that Fb ðx,xb Þ ¼ fb ðx,xb Þ þ fb, g ðx,xb Þ, combing (13) and (14), (12) follows.
&
Theorem 3. Suppose fb ðxÞ is the blocked indicator function of blocked design F b as defined in (4). Then for 1 rj rp, we have the following relationships: (i) Aj,0 ðF b ðgÞÞ ¼ Aj,0 ðF b ðgÞÞ; (ii) Aj,1 ðF b ðgÞÞ ¼ Aj,1 ðF b Þ þAj,0 ðF b ÞAj,0 ðF b ðgÞÞ.
Proof. From Definition 3 and (12), we have !2 X bI Aj,0 ðF b ðgÞÞ ¼ ¼ Aj,0 ðF b ðgÞÞ, bf 2 jIj ¼ j,I2P t
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hence (i) follows. Similarly, Aj,1 ðF b ðgÞÞ ¼
X jI1 j ¼ j,I2P 2b
bI bf
!2 þ
X jIj ¼ j,I2P 1t
bI bf
!2 þ
X jI1 j ¼ j,I2P 1b
bI bf
!2 ¼
X jI1 j ¼ j,I2P b
bI bf
!2 þ
X jIj ¼ j,I2P t
bI bf
!2
X
jIj ¼ j,I2P 2t
bI bf
!2
¼ Aj,1 ðF b Þ þ Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ ¼ Aj,1 ðF b Þ þ Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ, which completes the proof.
&
Remark 1. From (i) of Theorem 3, one can find that the treatment generalized wordlength pattern of F b ðgÞ is the same as the one of F b ðgÞ. It implies that the treatment generalized wordlength pattern is independent of the consideration of the implicit blocking factor in the combined blocked design. Remark 2. Ai et al. (2010) provided a similar relationship in Theorem 3 for regular two-level designs. In fact, if F is a regular two-level design, then the results of Theorem 3 cover the corresponding results of Ai et al. (2010) as a special case. However, as pointed in Ai et al. (2010), the treatment and block wordlength pattern of F b ðgÞ are independent of the choice of the block foldover plan when F is a regular two-level design. This may be invalid when F is nonregular, because it is dependent of both the choice of the block factors and the foldover plan gb for the block factors. For a fixed blocked design F b , if there exists a foldover plan for which the corresponding combined blocked design has GMA under the criterion of W1 or W2, then it is called a generalized minimum aberration foldover plan for the combined blocked design. Although the orderings of the generalized treatment and block wordlength patterns vary with the optimality criteria W1 and W2, it is essential that the number Aj,1 is always put behind the number Aj,0 for j= 1, 2, y, p, therefore we have the following theorem. Theorem 4. For a given blocked design F b , a foldover plan g has generalized minimum aberration for the combined blocked design F b ðgÞ if and only if it sequentially minimizes Aj,0 ðF b ðgÞÞ for j= 1, 2, y, p. From Theorem 4, we know that for a fixed initial blocked design, the GMA foldover plan is just the GMA foldover plan for the unblocked case. Therefore, we need only consider the GMA foldover plans for the unblocked cases. Some examples can be found in Li et al. (2003). A blocking plan for a given two-level design F , is said to have GMA if it sequentially minimizes the generalized block wordlength pattern W b ðF b Þ of the blocked design F b . A blocking plan for the design F and a fixed foldover plan g have GMA if it sequentially minimizes the block wordlength pattern W b ðF b ðgÞÞ of the combined blocked design F b ðgÞ. Noting that the P Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ ¼ jIj ¼ j,I2P 1 ðbI =bf Þ2 are fixed in this case, therefore from (ii) of Theorem 3, we have the following t conclusion. Theorem 5. For a two-level design F with a fixed foldover plan, a blocking plan has GMA if and only if it has GMA for F without consideration of the foldover plans. A pair of blocking and foldover plans is said to be optimal if it together ensures that the combined blocked design of a twolevel design has GMA. By combining Theorems 4 and 5, we have the following result. Theorem 6. For a two-level design F , a pair of blocking and foldover plans has GMA for the combined blocked design F b ðgÞ if and only if the foldover plan has GMA for the design F without consideration of the blocking plans and the blocking plan has GMA for F without consideration of the foldover plans. Remark 3. Ai et al. (2010) also provided the similar results corresponding to Theorems 4–6 for two-level regular design in terms of minimum aberration criterion. Example 1. Consider a nonregular design F listed in Table 1, where n =16 and s =6. The indicator function of the unblocked design F is f ðx1 ,x2 ,x3 ,x4 ,x5 ,x6 Þ ¼ 14 ð1x1 x3 x4 12 x3 x5 x6 12 x4 x5 x6 þ 12 x1 x3 x5 x6 þ 12x1 x4 x5 x6 þ 12 x2 x3 x5 x6 12 x2 x4 x5 x6 þ 12 x1 x2 x3 x5 x6 12x1 x2 x4 x5 x6 Þ: If we assign the second column as blocking factor and the first, sixth columns as foldover plan, respectively, then F is blocked into two blocks of size eight, and the corresponding blocked indicator function of F b is fb ðx1 ,xB ,x3 ,x4 ,x5 ,x6 Þ ¼ 14 ð1x1 x3 x4 12 x3 x5 x6 12 x4 x5 x6 þ 12 x1 x3 x5 x6 þ 12x1 x4 x5 x6 þ 12 xB x3 x5 x6 12 xB x4 x5 x6 þ 12 x1 xB x3 x5 x6 12x1 xB x4 x5 x6 Þ,
Z. Ou et al. / Journal of Statistical Planning and Inference 141 (2011) 1635–1645
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Table 1 F. Run
x1
x2
x3
x4
x5
x6
Block II
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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
B1 B1 B1 B1 B2 B2 B2 B2 B1 B1 B1 B1 B2 B2 B2 B2
Table 2 Optimal blocking and foldover plans for (12,2p: 21) designs. p
Design
3
4.1
4
5.1
5
6.1
6
7.1
7
8.1
8
9.1
9
10.1
Rt
Rb III.67
(0)
(0.333, 0.222)
1
IV.67
III.67
(0, 0.111)
(0.667, 0.889, 0)
2/1
IV.67
III.67
(0, 0.556, 0)
(1.111, 2.222, 0, 0.444)
1/6
IV.67
III.67
(0, 1.667, 0, 0)
(1.667, 4.444, 0.889, 0.889, 0.111)
1
IV.67
III.67
(0, 3.889, 0, 0.444, 0)
(2.333, 7.778, 2.222, 2.667, 0.778)
1
IV.67
III.67
(0, 7.778, 0, 1.778, 0, 0.111)
(3.111, 12.444, 4.444, 7.111, 3.111)
1
IV.67
III.67
(0, 14, 0, 5.333, 0, 1)
(4, 18.667, 8, 16, 9.333)
Foldover plan
Blocking plan
gf gf gf gf gf gf gf
1
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
which is obtained by replacing x2 in f(x1, x2, x3, x4, x5, x6) with xB. From Theorem 2, we can obtain the indicator function of the combined blocked design F b ðgÞ as follows: Fb ðx1 ,xB ,x3 ,x4 ,x5 ,x6 ,xb Þ ¼ 14 ð1 þx1 x3 x4 xb þ 12 x1 x3 x5 x6 þ 12 x1 x4 x5 x6 þ 12 x3 x5 x6 xb þ 12x4 x5 x6 xb þ 12 x1 xB x3 x5 x6 12 x1 xB x4 x5 x6 12 xB x3 x5 x6 xb þ 12xB x4 x5 x6 xb Þ, where xb is the implicit blocking factor. From fb(x1, xB, x3, x4, x5, x6), we know that Rt = III and Rb = IV.5, Wt ðF b Þ ¼ ð0,0,1:5,0:5,0Þ and Wb ðF b Þ ¼ ð0,0,0:5,0:5,0Þ for the n blocked design. From Fb(x1, xB, x3, x4, x5, x6, xb), we have Rt = IV.5 and Rb =IV, Wt ðF b ðgÞÞ ¼ ð0,0,0,0:5,0Þ and Wb ðF b ðgÞÞ ¼ ð0,0,2,0:5,0Þ for the combined blocked design. Here the combined blocked design F b ðgÞ is a (16, 25: 22) blocked design. One can find that Aj,1 ðF b ðgÞÞ ¼ Aj,1 ðF b Þ þ Aj,0 ðF b ÞAj,0 ðF b ðgÞÞ for j =1, y, 5, which is indicated by part (ii) of Theorem 3. Moreover, the blocking plan and foldover plan considered above is the optimal GMA blocking plan and foldover plan for F , respectively. Therefore, from Theorem 6, the pair of blocking and foldover plans considered above is optimal for design F . 5. Optimal blocking and foldover plans for 12, 16 and 20-run designs In this section, we present some optimal blocking and foldover plans for two-level designs with 12-run, 16-run and 20-run based on the generalized minimum aberration criterion discussed in the previous section. We only focus on the blocked designs with Rt Z 3 and Rb Z 3. Specifically, the restriction Rt Z 3 ensures the orthogonality among treatment main effects, and the restriction Rb Z 3 guarantees that block effects are orthogonal to treatment main effects. A necessary condition for Rb Z 3 is that n/2q must be an even number (Cheng et al., 2004). Therefore, we only consider the case of one or two block factors for 16-run designs and one block factor for 12 and 20-run designs. 5.1. Optimal blocking and foldover plans for 16-run designs Tables 3 and 4 give the optimal blocking and foldover plans for two-level designs as given in Sun et al. (2002). The first column of each table gives the number of treatment factors p, and the second column of each table gives the design index which is corresponding to the index number used by Sun et al. (2002). The third and fourth columns show the optimal
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Table 3 Optimal blocking and foldover plans for (16,2p:21) designs. p
Design
2
Foldover plan
Blocking plan
Rt
Rb
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
Rank
3.1 3.2 3.3
3 0 3
1 1 1
III
(1)
III.5
(0.25)
3 1 2
4.1 4.3 4.4 4.5
3 0 4 4
4 1 1 1
IV IV.5 III.5
(0) (0) (0) (0)
(0, 1) (0, 0) (0, 0.5) (0.25, 0.25)
3 1 2 4
4
5.1 5.2 5.3 5.5 5.6 5.7 5.8 5.10 5.11
3, 5 3 4 5 1, 5 5 5 5 1, 2
2 4 5 1 4 1 1 1 1
III IV V IV.5 III.5 IV.5 III.5 III.5 III.5
(0, (0, (0, (0, (0, (0, (0, (0, (0,
0) 0) 0) 0) 0) 0) 0) 0) 0)
(1, 2, 0) (0, 1, 0) (0, 0, 1) (0, 0.5, 0.5) (0.5, 1.5, 0) (0, 1, 0) (0.25, 0.5, 0.25) (0.5, 0.5, 0) (0.5, 1, 0)
9 4 1 2 8 3 5 6 7
5
6.1 6.2 6.3 6.4 6.6 6.7 6.9 6.12 6.13 6.18 6.19
3, 3, 3, 3, 3, 1, 1, 6 6 5, 2,
5, 6 5 6 6 6 6 6
1 6 1 4 4 2 4 1 1 1 1
IV IV
IV.5
III IV III IV III.5 IV III.5 IV.5 IV.5 III.5 IV.5
(0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0,
1, 0) 1, 0) 0, 0) 0, 0) 0, 0) 0.5, 0) 0.5, 0) 1, 0) 0, 0) 0, 0) 0.5, 0)
(2, 4, 0, 0) (0, 2, 0, 0) (1, 1, 0, 1) (0, 2, 1, 0) (0.25, 1.75, 0.75, 0.25) (0, 2, 0.5, 0) (0.25, 1.75, 0.25, 0.25) (0, 2, 0, 0) (0, 2, 1, 0) (1, 1, 0, 1) (0, 2, 0.5, 0)
19 9 5 2 3 7 10 8 1 4 6
7.1 7.2 7.3 7.4 7.5 7.7 7.8 7.9 7.10
3, 3, 3, 3, 3, 1, 1, 1, 1,
5, 5, 5, 5, 5, 7 2, 7 2,
6 6 7 7 7
1 7 2 4 6 6 6 2 1
IV IV IV IV IV IV IV IV IV.5
III IV III III IV III.5 III.5 IV III
(0, (0, (0, (0, (0, (0, (0, (0, (0,
3, 3, 1, 1, 1, 1, 1, 2, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0,
0) 0) 0) 0) 0) 0) 0) 0) 0)
(3, 8, 0, 0, 1) (0, 4, 0, 0, 0) (1, 4, 0, 0, 1) (1, 3, 1, 1, 0) (0, 4, 2, 0, 0) (0.5, 3.5, 1.5, 0.5, 0) (0.75, 3.5, 1, 0.5, 0.25) (0, 4, 1, 0, 0) (1, 3, 1, 1, 0)
9 3 8 7 1 4 5 2 6
8.1 8.2 8.3 8.5 8.7 8.8 8.9 8.10
3, 3, 3, 3, 1, 1, 1, 1,
5, 5, 5, 5, 2, 8 2, 2,
6 6, 8 6 7, 8 3, 8 3, 8 8
8 7 7 3 7 2 3 7
IV IV IV IV IV IV IV.5 IV
IV III IV III III.5 III III III.5
(0, (0, (0, (0, (0, (0, (0, (0,
7, 3, 3, 2, 3, 2, 2, 3,
0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 1, 0, 0, 1, 0,
0) 0) 0) 0) 0) 0) 0) 0)
(0, 7, 0, 0, 0) (1, 6, 2, 2, 1) (0, 7, 4, 0, 0) (1, 6, 3, 2, 0) (0.75, 6.25, 2.5, 1.5, 0.75) (1, 6, 3, 1, 1) (1, 6, 3, 2, 0) (1, 6, 2, 2, 1)
2 8 1 4 6 5 3 7
8
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
3, 3, 3, 3, 3, 1, 1, 1,
5, 5, 5, 5, 5, 2, 2, 8,
6, 6, 6, 7, 7, 3, 3, 9
9 8, 9 8 8 8, 9 9 9
8 4 8 2 1 8 9 2
IV IV IV IV IV IV IV IV
III III III III III III.5 III.5 III
(0, (0, (0, (0, (0, (0, (0, (0,
7, 5, 5, 3, 5, 7, 5, 5,
0, 0, 0, 4, 0, 0, 0, 2,
0, 2, 2, 0, 2, 0, 2, 0,
0, 0, 0, 0, 0, 0, 0, 0,
(1, 10, 4, 4, 3) (2, 9, 4, 6, 2) (1, 10, 6, 4, 1) (1, 10, 8, 0, 3) (2, 8, 5, 8, 0) (1, 10, 4, 4, 3) (1.5, 9.5, 5, 5, 1.5) (1, 10, 6, 2, 3)
5 8 2 1 7 4 6 3
9
10.1 10.2 10.3 10.6
3, 3, 3, 1,
5, 5, 5, 2,
6, 6, 6, 3,
9, 10 8, 9, 10 8, 9 10
8 1 6 10
IV IV IV IV
III III III III.5
(0, (0, (0, (0,
10, 0, 4, 0, 1) 9, 0, 6, 0, 0) 9, 0, 6, 0, 0) 9, 0, 6, 0, 0)
(2, 14, 8, 12, 6) (3, 13, 6, 15, 7) (2, 14, 9, 12, 4) (2.25, 13.75, 8.25, 12.75, 4.75)
2 4 1 3
10
11.1 11.2
3, 5, 6, 9, 10 3, 5, 6, 9, 10, 11
4 8
IV IV
III III
(0, 16, 0, 12, 0, 3) (0, 16, 0, 12, 0, 3)
(3, 20, 12, 24, 16) (3, 19, 13, 27, 13)
2 1
11
12.1 12.5 12.8
3, 5, 6, 9, 10, 12 1, 2, 3, 10, 11, 12 1, 2, 10, 11, 12
12 1 1
IV IV IV
III III III
(0, 26, 0, 24, 0, 13) (0, 25.5, 0, 25.5, 0, 11.5) (0, 25, 0, 27, 0, 10)
(0, 4, 26, 0, 25) (4, 25.5, 19.5, 51, 29.5) (4, 26, 19, 50, 31)
3 2 1
12
13.1
3, 5, 6, 9, 10, 12
2
IV
III
(0, 38, 0, 52, 0, 33)
(5, 34, 28, 88, 62)
1
3
6
7
6 3
7 7
IV.5 IV.5 IV
0) 0) 0) 0) 0) 0) 0) 0)
Z. Ou et al. / Journal of Statistical Planning and Inference 141 (2011) 1635–1645
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Table 4 Optimal blocking and foldover plans for (16,2p:22) designs. p
Design
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
Rank
Blocking plan
4.1 4.2 4.3 4.4
3 4 0 4
1, 1, 1, 1,
4 2 2 2
III.5 III
3
5.1 5.2 5.5 5.6 5.7 5.10 5.11
3, 5 3 5 1, 5 5 5 1, 2
2, 4, 1, 1, 1, 1, 1,
4 5 2 4 2 2 2
III IV IV.5 III.5 III.5 III.5 III.5
(0) (0) (0) (0) (0) (0) (0)
4
6.1 6.2 6.3 6.6 6.7 6.10 6.17 6.18 6.22 6.23 6.25 6.26
3, 3, 3, 3, 1, 1, 5, 5, 1, 1, 1, 1,
5, 6 5 6 6 6 6 6 6 2, 3 2 2 2
1, 2, 1, 1, 2, 1, 5, 1, 1, 3, 1, 1,
6 6 4 4 5 6 6 2 2 4 2 2
IV
III III III III III.5 III III.5 III.5 III.5 III.5 III.5 III.5
(0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0, (0,
1) 0) 0) 0) 0) 1) 1) 0) 0) 0.25) 1) 0)
(6, 0, 0) (1, 2, 0) (2, 0, 1) (1.5, 1, 0.5) (1, 2, 0) (4, 0, 0) (2, 0, 0) (2, 0, 1) (1.5, 1.5, 0) (1.5, 1, 0.25) (3, 0, 0) (3, 0, 0)
5
7.2 7.3 7.4 7.8 7.9
3, 3, 3, 1, 1,
5, 5, 5, 2, 7
6 7 7 7
1, 2, 4, 1, 2,
7 4 6 6 7
IV
IV
III III III III III.5
(0, (0, (0, (0, (0,
1, 0, 0, 0, 1,
0) 0) 0) 0) 0)
(2, 4, 0, 0) (3, 3, 0, 1) (2, 4, 1, 0) (2.5, 3.5, 0.5, 0.5) (2, 4, 0, 0)
5 3 1 2 4
6
8.1 8.2 8.5 8.7 8.10
3, 3, 3, 1, 1,
5, 5, 5, 2, 2,
6 6, 8 7, 8 3, 8 8
1, 6, 3, 1, 1,
8 7 8 7 7
IV IV VI IV IV
III III III III III
(0, (0, (0, (0, (0,
3, 1, 0, 1, 1,
0, 0, 0, 0, 0,
0) 0) 1) 0) 0)
(3, 8, 0, 0, 1) (3, 8, 2, 0, 1) (3, 8, 3, 0, 0) (3.5, 7, 2, 1, 0.5) (4, 6, 2, 2, 0)
3 2 1 4 5
7
9.1 9.2 9.4 9.5 9.8
3, 3, 3, 3, 1,
5, 5, 5, 5, 8,
6, 6, 7, 7, 9
2, 4, 2, 1, 2,
8 8 4 6 4
IV IV IV IV IV
III III III III III
(0, (0, (0, (0, (0,
3, 2, 1, 3, 2,
0, 0, 2, 0, 1,
0, 1, 0, 0, 0,
(5, (5, (5, (6, (5,
4 2 1 5 3
8
10.1 10.2 10.6
3, 5, 6, 9, 10 3, 5, 6, 8, 9, 10 1, 2, 3, 10
4, 8 1, 6 1, 10
IV IV IV
III III III
(0, 5, 0, 2, 0, 0) (0, 5, 0, 2, 0, 0) (0, 5, 0, 2, 0, 0)
(7, 18, 10, 12, 7) (8, 16, 9, 16, 6) (7.5, 17, 9.5, 14, 6.5)
1 3 2
9
11.1 11.4 11.7
3, 5, 6, 9, 10 1, 2, 3, 11 1, 2, 3, 10, 11
4, 8 1, 10 2, 10
IV IV IV
III III III
(0, 9, 0, 6, 0, 0) (0, 10, 0, 4, 0, 1) (0, 9.5, 0, 5, 0, 0.5)
(9, 27, 18, 27, 21) (10, 24, 18, 32, 18) (9.5, 25.5, 18, 29.5, 19.5)
1 3 2
10
12.1 12.2 12.5
3, 5, 6, 9, 10, 12 3, 5, 6, 8, 9, 12 1, 2, 3, 10, 11, 12
4, 9 1, 6 1, 6
IV IV IV
III III III
(0, 16, 0, 12, 0, 3) (0, 18, 0, 8, 0, 5) (0, 17, 0, 10, 0, 4)
(12, 36, 30, 60, 48) (13, 32, 32, 64, 42) (12.5, 34, 31, 62, 45)
1 3 2
11
13.1
3, 5, 6, 9, 10, 12
2, 12
IV
III
(0, 26, 0, 24, 0, 13)
(15, 48, 48, 112, 102)
1
2
9 8, 9 8 8, 9
Rt
Rb
Foldover plan
III.5
IV IV
IV IV
0) 0) 0) 0) 0)
(1) (1) (0) (0.5)
3 4 1 2
(3, 0) (0, 1) (0, 1) (2, 0) (0.5, 0.5) (1, 0) (1.5, 0)
7 2 1 6 3 4 5
12, 4, 4, 3) 12, 5, 4, 2) 12, 6, 2, 3) 9, 6, 6, 0) 12, 5, 3, 3)
12 2 6 3 1 11 9 5 4 8 10 7
foldover plan and blocking plan. For ease of comparison and selection of optimal designs, the treatment generalized wordlength pattern (GTWP) and the block generalized wordlength pattern (GBWP) of the optimal combined blocked design are listed in the fifth and sixth columns, respectively. Note that only some initial components of GTWP and GBWP are given for the sake of brevity. In order to compare combined blocked designs in terms of their aberration, we adopt the combined wordlength pattern W2 to rank-order these designs which are suggested by Cheng et al. (2004). According to the combined wordlength pattern W2 defined in (7), the rank of each design is given in the last column. Certainly, other criteria based on different combined wordlength patterns can be adopted to rank-order designs. From Tables 3 and 4, one can easily find that the optimal foldover plan g of some designs depend on the block foldover plan gb , which is different from the ones as shown in Ai et al. (2010), the optimal foldover plan is independent of the choice of block foldover plan for regular designs. For example, the optimal blocking plan is 1 for design 5.11 in Table 3, the optimal foldover plan is 1, 2 which is dependent on the block foldover plan for this design.
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Table 5 Optimal blocking and foldover plans for (20,2p:21) designs. p
Design
2 3
3.3 4.3
4
5.11
5
6.74
6
7.469
7
8.1540
7
8.1601
8
9.2444
8
9.2477
9
10.2389
10
11.1913
11
12.1292
12
13.728
13
14.320
14
15.92
Foldover plan
Blocking plan
3
1 1
gf gf gf gf gf gf gf gf gf gf gf gf gf gf
Rt
Rb
GTWP (A3,0,y,A8,0)
GBWP (A2,1,y,A6,1)
III.8 III.8
(0)
(0.04) (0.12, 0.08) (0.24, 0.32, 0)
1
IV.8
III.8
(0, 0.04)
1/2
IV.8
III.8
(0, 0.2, 0)
(0.4, 0.8, 0.48, 0.32)
5/1
IV.4
III.8
(0, 0.92, 0, 0)
(0.6, 1.92, 1.28, 0.64, 0.04)
8/1
IV.4
III.8
(0, 2.68, 0, 0.16, 0)
(0.84, 3.44, 2.72, 1.6, 0.28)
5/2
IV.4
III.8
(0, 3, 0, 0.16, 0)
(0.84, 3.76, 1.76, 1.92, 0.28)
6/3
IV.4
III.8
(0, 4.4, 0, 0.96, 0, 0.04)
(1.12, 6.72, 3.84, 5.12, 1.44)
3/1
IV.4
III.8
(0, 6, 0, 0.64, 0, 0.04)
(1.12, 7.04, 3.2, 3.52, 2.08)
2/1
IV.4
III.8
(0, 10.8, 0, 1.92, 0, 0.36)
(1.44, 10.56, 6.08, 8.32, 5.92)
1/3
IV.4
III.4
(0, 14.16, 0, 7.36, 0, 3.08)
(2.12, 14.72, 10.24, 21.44, 11.6)
1/2
IV.4
III.4
(0, 20.24, 0, 21.44, 0, 7.88)
(2.84, 19.92, 16.96, 39.36, 23.6)
5/3
IV.4
III.4
(0, 30.04, 0, 41.28, 0, 27.16)
(3.6, 25.92, 24, 73.28, 47.84)
1/3
IV.4
III.4
(0, 42.36, 0, 79.68, 0, 67.8)
(4.4, 33.76, 34.24, 120.48, 88.16)
15/3
IV.4
III.4
(0, 59.24, 0, 139.84, 0, 157.24)
(5.24, 42.56, 48, 189.12, 152.44)
5.2. Optimal blocking and foldover plans for 12 and 20-run designs Tables 2 and 5 give the optimal blocking and foldover plans for two-level designs which also considered by Cheng et al. (2004). They only considered the optimal block plans based on the combined wordlength patterns W1 or W2. From Tables 2 and 5, we can find that the full foldover plan gf is the unique optimal foldover plan for these designs except design 3.3. This is indicated by Theorem 6 that the full foldover plan is the unique optimal foldover plan these designs as shown in Li et al. (2003). The fourth column of Tables 2 and 5 gives the optimal blocking plan for both this paper and Cheng et al. (2004). For example, the optimal blocking plan is 2 of this paper and 1 of Cheng et al. (2004) for design 6.1 as denoted by 2/1 in Table 2. If there is only one number in the fourth column, it implies that the optimal blocking plan of us is the same as the one of Cheng et al. (2004). 6. Concluding remarks In this paper, using the effective tool of indicator function and based on the assumption of the implicit block effect is significant, optimal blocking and foldover plan of nonregular two-level designs are studied. It is shown that, for a two-level design F , a pair of blocking and foldover plans has GMA for the combined blocked design F b ðgÞ if and only if the blocking plan has GMA for F without consideration of the foldover plans and the foldover plan has GMA for design F without consideration of the blocking plans. All of these GMA results hold true for any generalized aberration criterion for blocked designs which orders Aj,1 behind Aj,0. These results extend the ones of Ai et al. (2010) from regular designs to nonregular designs.
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