Construction of fractional factorial split-plot designs with weak minimum aberration

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Statistics & Probability Letters 77 (2007) 1567–1573 www.elsevier.com/locate/stapro

Construction of fractional factorial split-plot designs with weak minimum aberration Jianfeng Yang, Runchu Zhang, Minqian Liu Department of Statistics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China Received 20 April 2006; received in revised form 4 February 2007; accepted 20 March 2007 Available online 19 April 2007

Abstract Fractional factorial split-plot (FFSP) designs with minimum aberration have been applied in industrial experiments. But, they are not so easy to construct for cases having many whole plot (or subplot) factors and only few subplot (or whole plot) factors. Weak minimum aberration is a weak version of minimum aberration. Based on the theory of complementary designs, this paper provides some theoretical results which are useful for constructing FFSP designs with weak minimum aberration. From these results, many such FFSP designs are constructed and tabulated, and it is further shown that quite a few of them are also minimum aberration designs. r 2007 Elsevier B.V. All rights reserved. MSC: primary 62K15; secondary 62K05 Keywords: Factorial design; Resolution; Split-plot; Weak minimum aberration

1. Introduction A 2nk design usually denotes a regular two-level fractional factorial (FF) design with n 2-level factors and 2 runs. But if the levels of some of the factors are difficult to be changed or controlled, it may be impractical or even impossible to perform the experimental runs of FF designs in a completely random order. In situations of this kind, fractional factorial split-plot (FFSP) designs, which involve a two-phase randomization, can be conveniently used to reduce costs and hence represent a practical design option. Suppose we wish to run an experiment with n two-level factors in 2nk runs, and there are n1 ð1pn1 onÞ hard-to-change factors with 2n1 k1 distinct level combinations. To perform such a design, we often first randomly choose one of the level combinations of these n1 factors and then run all of the level combinations of the remaining n2 ð¼ n  n1 Þ factors in a random order with the n1 factors fixed. This is repeated for each level combination of the n1 factors. Then the design is said to be a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design, where the n1 and n2 factors are called whole plot (WP) factors and subplot (SP) factors, respectively, and there are k1 and k2 ð¼ k  k1 Þ WP and SP factorial defining words, respectively. nk

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E-mail address: [email protected] (R. Zhang). 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.03.043

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FFSP designs have received much attention in recent years, see for example, Huang et al. (1998), Bingham and Sitter (1999a, b), Bingham and Sitter (2001), Bisgaard (2000), Goos (2002), Mukerjee and Fang (2002), Bingham et al. (2004, 2005), Bingham and Mukerjee (2006), Yang et al. (2006) and Zi et al. (2006) for details. In particular, Huang et al. (1998) and Bingham and Sitter (1999a) gave several algorithms to search FFSP designs with minimum aberration (MA). But when there are many WP (or SP) factors and only few SP (or WP) factors, the construction of MA FFSP designs is not so easy to implement. Based on the theory of complementary designs, the present paper provides some theoretical results which are useful for constructing FFSP designs with weak minimum aberration (WMA), and hence many such FFSP designs are constructed. Section 2 introduces some notation and existing results. Section 3 contains the theoretical results. The constructed FFSP designs with WMA are tabulated in Section 4, most of which are also shown to have MA. The last section contains a remark.

2. Existing results A 2ðn1 þn2 Þðk1 þk2 Þ FFSP design can be represented by k1 þ k2 generators, in which the numbers 1; 2; . . . ; n1 þ n2 representing the factors are called letters, and a product (juxtaposition) of letters is called a word. Considering the structure of FFSP designs, only words involving none or at least two SP factors are permitted. The number of letters in a word is called the length of the word. Associated with every 2ðn1 þn2 Þðk1 þk2 Þ FFSP design is a set of k1 þ k2 words called generators. The set of distinct words formed by all possible products involving the k1 þ k2 generators gives the defining contrast subgroup of the design. Let Ai denote the number of words of length i in the defining contrast subgroup, then the vector W ¼ ðA3 ; A4 ; . . . ; An1 þn2 Þ is called the word length pattern of the FFSP design. The resolution of a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design is defined to be the smallest r such that Ar 40. Such a design is said to have maximum resolution if no other 2ðn1 þn2 Þðk1 þk2 Þ FFSP design has a larger resolution. Also a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design with maximum resolution R is said to have WMA if it has the smallest AR among all candidate 2ðn1 þn2 Þðk1 þk2 Þ FFSP designs and is said to have MA if it minimizes Ai sequentially for i ¼ 3; 4; . . . ; n1 þ n2 . The definitions of resolution, WMA and MA for FFSP designs are extensions of the corresponding ones for FF designs (see Box and Hunter, 1961; Chen and Hedayat, 1996; Fries and Hunter, 1980, respectively). Let P be a regular two-level saturated design with resolution III and n  k independent columns, e.g. for the case of n  k ¼ 3, we have P ¼ f1; 2; 12; 3; 13; 23; 123g, where 12; 13; 23 and 123 denote the columns generated by the element-by-element product of the corresponding independent columns 1, 2 and 3, respectively. Suppose D is an FF design chosen from P. Then the design formed by all the columns not involved in D, denoted by D, is called the complementary design of D (Tang and Wu, 1996). Hence, for a saturated design P, we have P ¼ D [ D. The following result due to Tang and Wu (1996) will be the main tool in the following for constructing WMA FFSP designs. Lemma 1. For a 2nk FF design D, we have A3 ðDÞ ¼ C  A3 ðDÞ,

(1)

where C is a constant not depending on the choice of D. 3. Main results Now let P ¼ P1 [ P2 be a saturated design with ðn1  k1 Þ þ ðn2  k2 Þ independent columns, where P1 involves n1  k1 independent columns and all the columns generated from these independent ones. Furthermore, P1 and P2 can be divided into two parts, respectively, i.e. P1 ¼ D1 [ D1 , P2 ¼ D2 [ D2 . We, therefore, have P ¼ P1 [ P2 ¼ D1 [ D1 [ D2 [ D2 . Following this way, a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design can be represented by D1 [ D2 with n1 ¼ jD1 j and n2 ¼ jD2 j, where jSj means the number of columns in S and Di contains ni  ki independent columns for i ¼ 1; 2 (Yang et al., 2006).

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3.1. Connection of A3 between D1 [ D2 and D1 [ D2 According to the definition of A3 and the structure of 2ðn1 þn2 Þðk1 þk2 Þ FFSP designs, we have A3 ðD1 [ D2 Þ ¼ A3 ðD1 Þ þ A3 ðD2 Þ þ M 2 ðD1 ; D2 Þ.

(2)

Similarly, A3 ðD1 [ D2 Þ ¼ A3 ðD1 Þ þ A3 ðD2 Þ þ M 2 ðD1 ; D2 Þ.

(3)

Here, M 2 ðQ1 ; Q2 Þ means the number of words of length three with one column from Q1 and the other two from Q2 . Then from (1), we have A3 ðD1 [ D2 Þ þ A3 ðD1 [ D2 Þ ¼ C þ 2A3 ðD2 Þ þ M 2 ðP1 ; D2 Þ, i.e. A3 ðD1 [ D2 Þ ¼ C þ 2A3 ðD2 Þ þ M 2 ðP1 ; D2 Þ  A3 ðD1 [ D2 Þ.

(4)

So we have the following theorem. Theorem 1. For a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design D1 [ D2 , minimizing A3 ðD1 [ D2 Þ is equivalent to minimizing 2A3 ðD2 Þ þ M 2 ðP1 ; D2 Þ  A3 ðD1 [ D2 Þ: This theorem is very useful in the construction of FFSP designs especially when D1 has much more columns than D1 and D2 involves only few ones. Corollary 1. In the case of n2  k2 ¼ 1, for given n1 and n2 , minimizing A3 ðD1 [ D2 Þ is equivalent to maximizing A3 ðD1 [ D2 Þ. Proof. For this special case, we have A3 ðD2 Þ ¼ 0 and M 2 ðP1 ; D2 Þ ¼ 0 from the structure of a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design. Then the assertion follows directly from Theorem 1. & 3.2. Connection of A3 between D1 [ D2 and D1 [ D2 With all the results in mind, the connection of A3 between D1 [ D2 and D1 [ D2 can be easily obtained, which is the following theorem. Theorem 2. For a 2ðn1 þn2 Þðk1 þk2 Þ FFSP design D1 [ D2 , minimizing A3 ðD1 [ D2 Þ is equivalent to maximizing 2A3 ðD2 Þ þ M 2 ðP1 ; D2 Þ  A3 ðD1 [ D2 Þ: Proof. According to (1), we have A3 ðD1 [ D2 Þ ¼ C 1  A3 ðD1 [ D2 Þ. Then from Theorem 1, A3 ðD1 [ D2 Þ ¼ C 2 þ 2A3 ðD2 Þ þ M 2 ðP1 ; D2 Þ  A3 ðD1 [ D2 Þ. Combining the above two equations, we have A3 ðD1 [ D2 Þ ¼ C  ð2A3 ðD2 Þ þ M 2 ðP1 ; D2 Þ  A3 ðD1 [ D2 ÞÞ, where C 1 ; C 2 ; C are constants that do not depend on the choice of D1 and D2 . Thus, the conclusion follows directly. & This theorem can be used to construct WMA FFSP designs when the SP section in an FFSP design has almost as many factors as P2 has but the WP section contains few ones. 4. Applications In this section, WMA designs will be constructed based on the above theoretical results. For a 2ðn1 þnþ2Þðk1 þk2 Þ FFSP design, Yang et al. (2006) pointed out that when n1 X2n1 k1 1 þ 1 or

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n1 þ n2 X2ðn1 þn2 Þðk1 þk2 Þ1 þ 1, the maximum resolution is III. Therefore, for the parameters satisfying these conditions, by minimizing A3 , we can obtain the corresponding FFSP designs with WMA. Based on Theorems 1, 2 and Corollary 1, we have constructed many such FFSP designs with WMA, which are tabulated in Tables 1–5. Note that we only list one WMA FFSP design and its corresponding word length pattern for each case. For simplicity, only the first five elements of each word length pattern are given in these tables except for the case of n1 ¼ 5; n2 ¼ 1 in Table 1. Comparing with the FF designs listed in Chen et al. Table 1 FFSP designs with WMA obtained through Corollary 1 (n1  k1 ¼ 3; n2  k2 ¼ 1) n1

n2

k1

k2

jD1 j

jD2 j

D1

D2

W

7 6 5

1 1 1

4 3 2

0 0 0

0 1 2

1 1 1

f f123g f123; 13g

f4g f4g f4g

ð7 7 0 0 1Þþ ð4 3 0 0 0Þþ ð2 1 0 0Þþ

7 6 5

2 2 2

4 3 2

1 1 1

0 1 2

2 2 2

f f12g f12; 13g

f4; 14g f4; 124g f4; 124g

ð8 10 4 4 4Þþ ð4 6 4 0 0Þ ð2 3 2 0 0Þ

7 6 5

3 3 3

4 3 2

2 2 2

0 1 2

3 3 3

f f12g f12; 13g

f4; 14; 24g f4; 14; 24g f4; 124; 134g

ð10 16 12 12 10Þ ð6 10 8 4 2Þ ð3 7 4 0 1Þþ

7 6 5

4 4 4

4 3 2

3 3 3

0 1 2

4 4 4

f f12g f12; 13g

f4; 14; 24; 34g f4; 124; 134; 234g f4; 124; 134; 234g

ð13 25 25 27 23Þþ ð8 18 16 8 8Þ ð4 14 8 0 4Þ

Table 2 FFSP designs with WMA obtained through Theorem 2 (n1  k1 ¼ 3; n2  k2 ¼ 1) n1

n2

k1

k2

jD1 j

jD2 j

D1

D2

W

3 3 3

8 7 6

0 0 0

7 6 5

3 3 3

0 1 2

f1; 2; 3g f1; 2; 3g f1; 2; 3g

f f1234g f124; 34g

ð12 26 28 24 20Þ ð9 16 15 12 7Þþ ð6 9 9 6 0Þþ

4 4 4 4

8 7 6 5

1 1 1 1

7 6 5 4

4 4 4 4

0 1 2 3

f1; 2; 3; 123g f1; 2; 3; 123g f1; 2; 3; 123g f1; 2; 3; 123g

f f124g f14; 24g f14; 24; 34g

ð16 39 48 48 48Þ ð12 26 28 24 20Þ ð8 18 16 8 8Þ ð4 14 8 0 4Þ

Table 3 FFSP designs with WMA obtained through Theorem 2 (n1  k1 ¼ 2; n2  k2 ¼ 2) n1

n2

k1

k2

jD1 j

jD2 j

D1

D2

W

2 2 2 2 2

12 11 10 9 8

0 0 0 0 0

10 9 8 7 6

2 2 2 2 2

0 1 2 3 4

f1; 2g f1; 2g f1; 2g f1; 2g f1; 2g

f f1234g f13; 23g f13; 23; 124g f13; 24; 1234; 34g or

ð28 77 112 168 232Þþ ð22 55 72 96 116Þþ ð16 39 48 48 48Þ ð12 26 28 24 20Þ ð8 18 16 8 8Þ

3 3 3 3 3

12 11 10 9 8

1 1 1 1 1

10 9 8 7 6

3 3 3 3 3

0 1 2 3 4

f1; 2; 12g f1; 2; 12g f1; 2; 12g f1; 2; 12g f1; 2; 12g

f f1234g f13; 23g f13; 24; 1234g f13; 24; 1234; 14g

ð35 ð28 ð22 ð16 ð12

105 168 280 435Þþ 77 112 168 232Þþ 55 72 96 116Þþ 39 48 48 48Þ 26 28 24 20Þ

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Table 4 FFSP designs with WMA obtained through Corollary 1 (n1  k1 ¼ 4; n2  k2 ¼ 1) n1

n2

k1

k2

jD1 j

jD2 j

D1

D2

W

15 15 15 15

1 2 3 4

11 11 11 11

0 1 2 3

0 0 0 0

1 2 3 4

f f f f

f5g f5; 15g f5; 15; 25g f5; 15; 25; 35g

ð35 ð36 ð38 ð41

105 112 126 147

14 14 14 14

1 2 3 4

10 10 10 10

0 1 2 3

1 1 1 1

1 2 3 4

f1234g f1234g f123g f12g

f5g f5; 12345g f5; 1235; 2345g f5; 125; 35; 1235g

ð28 ð28 ð30 ð32

77 112 168 232Þþ 84 140 224 344Þ 96 184 348 598Þ 116 256 528 992Þ

13 13 13 13

1 2 3 4

9 9 9 9

0 1 2 3

2 2 2 2

1 2 3 4

f12; 13g f12; 13g f12; 13g f12; 13g

f5g f5; 125g f5; 125; 135g f5; 125; 135; 145g

ð17 ð22 ð23 ð26

39 61 73 88

48 65 86Þþ 94 136 188Þ 132 216 347Þ 184 356 610Þ

12 12 12 12

1 2 3 4

8 8 8 8

0 1 2 3

3 3 3 3

1 2 3 4

f12; 13; 23g f12; 13; 23g f12; 13; 23g f12; 13; 23g

f5g f5; 125g f5; 125; 135g f5; 125; 135; 235g

ð16 ð16 ð16 ð16

39 45 57 76

48 48 48Þþ 64 72 96Þ 96 120 192Þ 144 192 352Þ

11 11 11 11

1 2 3 4

7 7 7 7

0 1 2 3

4 4 4 4

1 2 3 4

f12; 13; 23; 14g f12; 13; 23; 14g f12; 13; 23; 14g f12; 13; 23; 14g

f5g f5; 125g f5; 125; 135g f5; 125; 135; 235g

ð12 ð12 ð12 ð12

26 31 41 57

28 24 20Þþ 40 40 48Þ 64 72 104Þ 100 120 200Þ

10 10 10 10

1 2 3 4

6 6 6 6

0 1 2 3

5 5 5 5

1 2 3 4

f12; 13; 23; 14; 24g f12; 13; 23; 14; 24g f12; 13; 23; 14; 24g f12; 13; 23; 14; 24g

f5g f5; 125g f5; 125; 135g f5; 125; 135; 235g

ð8 ð8 ð8 ð8

18 23 31 45

16 24 40 64

9 9 9 9

1 2 3 4

5 5 5 5

0 1 2 3

6 6 6 6

1 2 3 4

f12; 13; 23; 124; 134; 234g f12; 13; 23; 124; 134; 234g f12; 13; 23; 124; 134; 234g f12; 13; 23; 124; 134; 234g

f5g f5; 125g f5; 125; 135g f5; 125; 135; 235g

ð4 ð4 ð4 ð4

14 18 26 39

8 0 4Þþ 12 8 12Þ 20 24 28Þ 32 48 56Þ

168 196 252 337

280 364 532 791

435Þþ 624Þ 1002Þ 1597Þ

8 8Þþ 16 24Þ 40 56Þ 72 112Þ

Table 5 FFSP designs with WMA obtained through Theorem 2 (n1  k1 ¼ 3; n2  k2 ¼ 2) n1

n2

k1

k2

jD1 j

jD2 j

D1

D2

W

7 6 5

2 2 2

4 3 2

0 0 0

0 1 2

2 2 2

f f123g f12; 13g

f4; 5g f4; 5g f4; 5g

ð7 7 0 0 1Þ ð4 3 0 0 0Þ ð2 1 0 0 0Þ

7 6 5

3 3 3

4 3 2

1 1 1

0 1 2

3 3 3

f f123g f12; 13g

f4; 5; 145g f4; 5; 1234g f4; 5; 124g

ð7 8 3 4 5Þ ð4 6 4 0 0Þ ð2 3 2 0 0Þ

7 6 5

4 4 4

4 3 2

2 2 2

0 1 2

4 4 4

f f123g f12; 13g

f4; 5; 145; 245g f4; 5; 1234; 1235g f4; 5; 145; 234g

ð8 12 10 12 12Þ ð4 10 8 0 4Þ ð3 5 2 2 3Þ

7 7

5 6

4 4

3 4

0 0

5 6

f f

f4; 5; 14; 245; 345g f4; 5; 14; 15; 245; 345g

ð9 17 21 27 27Þ ð10 25 38 50 58Þ

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(1993) and Bingham and Sitter (1999a), we can find that many designs in our tables are optimal in terms of MA. Example 1. Let us take the 2ð6þ4Þð3þ3Þ FFSP design as an example to illustrate how to obtain a WMA design. In this case, we have jD1 j ¼ 1; jD2 j ¼ 4. To minimize A3 ðD1 [ D2 Þ, from Corollary 1, we can maximize A3 ðD1 [ D2 Þ.So our purpose is to choose D2 to constitute as many length-three words with one factor from D1 and two from D2 as possible. One possible choice is to let D1 ¼ f12g and D2 ¼ f4; 124; 134; 234g, which is listed in Table 1. The choice says that only the column 12 is not in D1 . As P1 ¼ f1; 2; 12; 3; 13; 23; 123g, then D1 ¼ P1 nD1 ¼ f1; 2; 3; 13; 23; 123g. Suppose the WP and SP-generated factors are 5; 6; 7 and 8; 9; t, respectively, where t stands for 10. Then, the WP and SP defining relations are 5 ¼ 13; 6 ¼ 23; 7 ¼ 123, and 8 ¼ 124; 9 ¼ 134; t ¼ 234, respectively. That is to say, the WP and SP generators are 135; 236; 1237 and 1248; 1349; 234t, respectively. The resulting design D1 [ D2 has the same word length pattern as that of the MA 2106 FF design listed in Chen et al. (1993). Hence, the constructed FFSP design with WMA is also an FFSP design with MA. Note that for the constructed FFSP designs, especially for those not listed in Bingham and Sitter (1999a), many of the word length patterns are inferior to those of the corresponding FF designs given in Chen et al. (1993). One reason lies in the additional constraints imposed by the structure of FFSP designs. In other words, many MA FF designs have no such a structure required by FFSP designs. However, all the FFSP designs given in our tables have WMA, which are guaranteed by Theorems 1, 2 or Corollary 1. In the tables, each word length pattern marked with an asterisk ð Þ means that the corresponding FFSP design has MA by comparing with the word length pattern in Chen et al. (1993). And the one marked with superscript ðþ Þ also means that the corresponding design has MA. But this can be confirmed in the following two ways: one is that the design can be declared to have MA if its word length pattern is the best among those of the corresponding FF designs given in Chen et al. (1993) with the same A3 value; the other is for the case when the SP section has only one factor (i.e. n2 ¼ 1). In this case, the 2ðn1 þn2 Þðk1 þk2 Þ FFSP design is judged to have MA if its word length pattern is the same with that of the MA 2n1 k1 FF design. Note that for some of the parameters given in our tables, MA designs were also derived by Huang et al. (1998) and Bingham and Sitter (1999a). Comparing with their methods, we see that our approaches have the following two advantages. Firstly, for some specific parameters, the methods in Huang et al. (1998) cannot produce MA FFSP designs if the corresponding MA FF designs have no split structure. Ours, however, do not have this restriction. All the designs marked with superscript ðþ Þ in Tables 1–5 are of this kind. For example, for any 2ð7þ1Þð4þ0Þ design, split-plot structure indicates that the only SP factor should not be generated by the rest ones, i.e. WP factors. But all MA 284 FF designs do not have this kind of column because all factors appear in the defining relations (Fries and Hunter, 1980). Secondly, Bingham and Sitter (1999a) obtained MA FFSP designs by search algorithms. But here we construct the FFSP designs based on the theoretical results, which may be regarded as an initial work in this direction. 5. A remark MA criterion is proposed originally for ranking FF designs. It is often used directly for ranking FFSP designs. But recently Kulahci et al. (2006) criticized this criterion since it does not distinguish WP-by-WP interaction effects, WP-by-SP interaction effects and SP-by-SP interaction effects. Based on the nature of splitplot designs, Kulahci et al. (2006) provided a modified version of the maximum number of clear two-factor interactions’ criterion. Along this line, other criteria special for split-plot designs may be considered and the corresponding optimal designs need to be constructed. Acknowledgements The authors thank the Chief Editor and one referee for their valuable comments. This work was partially supported by the NNSF of China Grants 10571093 and 10671099, and the SRFDP of China Grant 20050055038. Zhang and Liu’s researches were also supported by the Visiting Scholar Program at Chern Institute of Mathematics.

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