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Optimal multi-level supersaturated designs constructed from linear and quadratic functions
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Statistics & Probability Letters 69 (2004) 199–211 www.elsevier.com/locate/stapro
Optimal multi-level supersaturated designs constructed from linear and quadratic functions C. Koukouvinos, S. Stylianou Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece Received 1 October 2003; received in revised form 4 April 2004 Available Online 08 July 2004
Abstract In this paper, we present a method of construction Eðf NOD Þ-optimal multi-level supersaturated design with n rows, m columns and the equal occurrence property, using linear and quadratic functions. Using this method, as well as permutations and juxtapositions of the columns, many new Eðf NOD Þ-optimal multi-level supersaturated designs are provided. r 2004 Elsevier B.V. All rights reserved. MSC: primary 62K10; 62K15; secondary 05B20 Keywords: Supersaturated designs; Factorial designs; Linear functions; Quadratic functions; Orthogonal arrays; Dependency; Efficiency
1. Introduction A q-level supersaturated design is a factorial design having n ¼ qp experimental runs, while the number of factors m is more than ðn 1Þ=ðq 1Þ. Supersaturated design is used in the initial stage of an industrial or scientific experiment for screening the active factors, and is useful when there are a large number of factors under investigation while only a very limited number of experimental runs is available. The analysis of supersaturated designs rely on the assumption of Corresponding author.
E-mail address: [email protected] (C. Koukouvinos). 0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.06.030
ARTICLE IN PRESS C. Koukouvinos, S. Stylianou / Statistics & Probability Letters 69 (2004) 199–211
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effect sparsity (Box and Meyer (1986)). This assumes that only a few dominant factors actually affect the response. Satterthwaite (1959) proposed the idea of supersaturated design as a random balance design. Booth and Cox (1962) first examined two-level supersaturated designs systematically. Since then, many researchers have studied the construction and the properties of two-level supersaturated designs (see Butler et al., 2001; Cheng, 1997; Li and Wu, 1997; Lin, 1993; Lin, 1995; Liu and Dean, 2004; Liu and Zhang, 2000; Lu and Meng, 2000; Nguyen, 1996; Plackett and Burman, 1946; Tang and Wu, 1997; Wu, 1993; Yamada and Lin, 1997). However, all these authors considered only two-level designs. Designs with multi-levels are often requested in industrial and scientific experimentation. Especially, designs with three levels have been widely used (see for example Yamada and Lin, 1999; Yamada and Matsui, 2002). Fang et al. (2000), Lu and Sun (2001) and Lu et al. (2003) have discussed multi-level supersaturated designs. Aggarwal and Gupta (2004) have constructed multi-level supersaturated designs based on Galois field theory. This method is a generalization of the method given by Lu and Meng (2000). A supersaturated design cannot be an orthogonal design, so it is necessary to define measures of departure from orthogonality. Therefore, we present some design criteria for comparing supersaturated designs. Let D be an n m matrix of a factorial design with elements from a set of factor levels f0; 1; . . . ; q 1g. We say that the matrix D has the equal occurrence property if each column contains the same number of f0; 1; . . . ; q 1g. In this case, the number n of runs is a multiple of q. A column which satisfies the equal occurrence property is said to be balanced. A design with the equal occurrence property is called an orthogonal array of strength 2, denoted by OAðn; t; q; 2Þ, if in any two columns all their q2 level combinations appear the same number of times (see Hedayat et al., 1999). In this case, n is a multiple of q2 and tðq 1Þpðn 1Þ. When tðq 1Þ ¼ ðn 1Þ the design is called saturated and when tðq 1Þ4ðn 1Þ the design is called supersaturated. For q-level supersaturated designs, Yamada and Lin (1999) defined a measure for dependency between two columns d i ; d j of the matrix D ¼ ½d 1 ; d 2 ; . . . ; d m by q1 X ðnab ði; jÞ n=q2 Þ2 w ðd i ; d j Þ ¼ ; n=q2 a;b¼0 2
where nab ði; jÞ is the number of rows whose values are a; b in the n 2 matrix ½d i ; d j . Two columns d i ; d j are said to be completely dependent if w2 ðd i ; d j Þ ¼ nðq 1Þ and independent if w2 ðd i ; d j Þ ¼ 0. We can say that every two columns d i ; d j are independent if all the q2 possible pairs ði; jÞ occur with the same frequency. Yamada and Lin (1999) also defined criteria of dependency of a whole design, by , X m w2 ðd i ; d j Þ ; max w2 ¼ max w2 ðd i ; d j Þ: ave w2 ¼ 1piojpm 2 1piojpm Yamada and Matsui (2002) derived a lower bound on ave w2 , which is given by ave w2 X
ðq 1Þn½ðq 1Þm n þ 1 ¼ Lw2 : ðn 1Þðm 1Þ
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Using this lower bound, ave w2 -efficiency is defined as Lw2 =ave w2 . A design is ave w2 -optimal when ave w2 -efficiency is equal to 1. Using the criterion q1 X n f ðd i ; d j Þ ¼ nab ði; jÞ q2 a;b¼0 for measuring the orthogonality of two columns we can obtain the next criteria for the whole design , X m avejf j ¼ f ðd i ; d j Þ ; 2 1piojpm 2
aveðf Þ ¼
X
f ðd i ; d j Þ
1piojpm
2
,
m
2
and f max ¼
max
1piojpm
f ðd i ; d j Þ:
Fang et al. (2003) introduced the Eðf NOD Þ criterion for multi-level supersaturated designs with the property of equal occurrence. For any two columns d i ; d j of the design matrix D they defined q1 X n 2 i;j f NOD ¼ nab ði; jÞ 2 : q a;b¼0 The new criterion Eðf NOD Þ is given by , X m i;j : f NOD Eðf NOD Þ ¼ 2 1piojpm In the same article, they proved that Pn 2 n n n2 k;l¼1;kal lkl Eðf NOD Þ ¼ m 2 þ m1 q mðm 1Þ q 2 mn n n n n2 X 1 þ m 2; ðm 1Þðn 1Þ q m1 q q where lkl is the number of coincidences between the kth and lth rows. The lower bound of Eðf NOD Þ can be achieved if and only if l ¼ mðn=q 1Þ=ðn 1Þ is a positive integer and all the lkl s for kal are equal to l. So, the Eðf NOD Þ-efficiency is defined as Lf NOD =Eðf NOD Þ. It is easy to prove that for any design matrix D with the equal occurrence property the criteria Eðf NOD Þ and ave w2 satisfy the following relation: Eðf NOD Þ ¼ n=q2 ave w2 , when qX3. Mukerjee and Wu (1995) proved the following lemma. Lemma 1. Suppose that A is a saturated OAðn; t; q; 2Þ with t ¼ ðn 1Þ=ðq 1Þ. Then, the number of coincidences of rows i and j of A is li;j ¼ ðn qÞ=ðqðq 1ÞÞ for any iaj.
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Many multi-level supersaturated designs can be constructed combining together orthogonal arrays and Lemma 1 guarantees their optimality. This result is stated in next Theorem. Theorem 1. Let Ai , i ¼ 1; 2; . . . ; k be saturated OAðn; t; q; 2Þ, n ¼ qp , t ¼ ðqp 1Þ=ðq 1Þ. The matrix defined by juxtaposing all orthogonal arrays D ¼ ½A1 ; A2 ; . . . ; Ak is an Eðf NOD Þoptimal q-level supersaturated design with n rows, m ¼ kt columns and the property of equal occurrence. Proof. The equal occurrence property is satisfied since all columns are columns of different or the same OAðn; t; q; 2Þ. From Lemma 1, we have that for any pair ði; jÞ of rows of matrix D the number of coincidence li;j is a constant (actually li;j ¼ kðn qÞ=qðq 1ÞÞ. Thus, design D achieves Eðf NOD Þ-efficiency equal to 1 and so this is an Eðf NOD Þ-optimal q-level supersaturated design with n rows, m ¼ kt columns and the property of equal occurrence. & Example 1. Suppose we have the following three equivalent OAð9; 4; 3; 2Þ: 3 3 3 2 2 2 2 2 2 2 2 1 1 0 0 1 2 0 62 1 1 07 62 2 2 27 60 0 1 27 7 7 7 6 6 6 7 7 7 6 6 6 60 1 2 17 60 1 2 17 62 1 1 17 7 7 7 6 6 6 7 7 7 6 6 6 62 0 0 17 60 2 0 07 60 2 0 17 7 7 7 6 6 6 7 7 7 6 6 A1 ¼ 6 6 0 2 0 0 7 A2 ¼ 6 1 2 1 1 7 A3 ¼ 6 2 0 0 0 7: 7 7 7 6 6 6 61 2 1 17 60 0 1 27 61 0 2 17 7 7 7 6 6 6 60 0 1 27 62 0 0 17 62 2 2 27 7 7 7 6 6 6 7 7 7 6 6 6 41 0 2 05 41 0 2 05 41 2 1 05 1
1
0 2
1
1
0
2
1
1
0 2
Let D ¼ ½A1 ; A2 ; A3 be the array which contains all columns of A1 ; A2 ; A3 . Then, D is an Eðf NOD Þoptimal 3-level supersaturated design with 9 rows and m ¼ 4 3 ¼ 12 columns with the property of equal occurrence. Observe that the first column of A1 is fully aliased with the first column of A3 , but this does not effect the optimality of the design. & Remark 1. The optimal supersaturated designs obtained by Theorem 1 may contain fully aliased columns. Thus, Eðf NOD Þ criterion is not enough to prevent the existence of fully aliased columns in the designs. For this reason we also use the max value of the design criteria in all the supersaturated designs given. We study construction methods that produce optimal designs without fully aliased columns in the next sections. This paper studies the construction of multi-level supersaturated designs. In Section 2, a method for constructing optimal multi-level supersaturated designs is proposed and a procedure using this method is presented and explained with several examples. Finally, the results found with this method are tabulated and presented in Section 3.
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2. The construction method and the procedure In this section, we present the method we use to obtain multi-level supersaturated designs with the equal occurrence property using linear and quadratic functions.
2.1. Preliminaries Addelman and Kempthorne (1961) first used linear and quadratic functions to obtain orthogonal arrays. Let F q ¼ GFðqÞ be the finite field of q elements where q is a prime power. By F nq we denote the n-dimensional vector space over GFðqÞ. The columns of an array are labeled with linear or quadratic polynomials in p variables X 1 ; X 2 ; . . . ; X p and the rows are labeled with points from F nq . Let f 1 ðX 1 ; X 2 ; . . . ; X p Þ and f 2 ðX 1 ; X 2 ; . . . ; X p Þ be two functions, linear or nonlinear. They correspond to two columns of length n ¼ qp when evaluated at F nq . The two functions (or columns) are fully aliased (give the maximum possible value using any of the previously mentioned criteria) if the pair has only q level combinations, each combination occurring qp1 times; and orthogonal (give zero using any of the previously mentioned criteria) if the pair has q2 distinct level combinations, each combination occurring qp2 times. Let LðX 1 ; X 2 ; . . . ; X p Þ be the set of all nonzero linear functions of X 1 ; X 2 ; . . . ; X p , i.e., LðX 1 ; X 2 ; . . . ; X p Þ ¼ fc1 X 1 þ c2 X 2 þ þ cn X p : ci 2 F q ; not all ci are zerog: Every function in LðX 1 ; X 2 ; . . . ; X p Þ corresponds to a balanced column. Two functions f 1 and f 2 in LðX 1 ; X 2 ; . . . ; X p Þ are dependent if there is a nonzero constant c 2 F q such that f 1 ¼ cf 2 ; otherwise, they are independent. Clearly, dependent linear functions correspond to the same column up to level permutation and thus they arep fully aliased while independent linear functions 1Þ independent linear functions generates an correspond to orthogonal columns. A set of ðqðq1Þ p ðqp 1Þ OAðq ; ðq1Þ ; q; 2Þ. The traditional convention is to assume the first nonzero element being 1 for each column. Here, for convenience we assume the last nonzero element being 1 for each column. In particular, let HðX 1 ; X 2 ; . . . ; X p Þ be the set of all nonzero linear functions of X 1 ; X 2 ; . . . ; X p such thatp the last nonzero coefficient is 1. When evaluated at F nq , HðX 1 ; X 2 ; . . . ; X p Þ is a saturated 1Þ ; q; 2Þ. This is indeed the regular fractional factorial design and the construction is OAðqp ; ðqðq1Þ called the Rao–Hamming construction, see Hedayat et al. (Section 3.4). The key idea of the Addelman and Kempthorne (1961) construction is to use quadratic functions in addition to linear functions. Let Q1 ðX 1 ; X 2 ; . . . ; X p Þ ¼ fX 21 þ aX 1 þ h : a 2 F q ; h 2 HðX 2 ; . . . ; X p Þg S and Q1 ðX 1 ; X 2 ; . . . ; X p Þ ¼ fX 1 g Q1 ðX 1 ; X 2 ; . . . ; X p Þ. p 1Þ HðX 1 ; X 2 ; . . . ; X p Þ and Q1 ðX 1 ; X 2 ; . . . ; X p Þ have ðqðq1Þ columns each. The column juxtaposition of HðX 1 ; X 2 ; . . . ; X p Þ and Q1 ðX 1 ; X 2 ; . . . ; X p Þ forms a supersaturated design on q levels, n ¼ qp p 1Þ rows and m columns where m ¼ 2ðq ðq1Þ , which is a half of an Addelman and Kempthorne orthogonal array.
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Example 2. Consider q ¼ 3 and p ¼ 2. The functions are HðX 1 ; X 2 Þ ¼ fX 1 ; X 2 ; X 1 þ X 2 ; 2X 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ 2X 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 1 ; X 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ 2X 1 þ X 2 g: HðX 1 ; X 2 Þ is an OAð9; 4; 3; 2Þ when the functions are evaluated at F 93 ; so does Q1 ðX 1 ; X 2 Þ. The column juxtaposition of HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ forms a supersaturated design on 3 levels, n ¼ 3p rows and m columns where m ¼ 2ð32 1Þ=ð3 1Þ ¼ 8. However, this design has fully aliased columns since both HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ have column X 1 . This problem is solved using the method described in the following. 2.2. The new method As we have previously shown, the main problem we have to solve is to avoid having fully aliased columns in the design. Moreover, we need a method to systematically generate new columns, with the property of equal occurrence and the Eðf NOD Þ-optimality for the design. We select p orthogonal q-level columns X 1 ; X 2 ; . . . ; X p . Then, we find all functions of HðX 1 ; X 2 ; . . . ; X p Þ, where HðX 1 ; X 2 ; . . . ; X p Þ defined as before. If we calculate Q1 using columns X 1 ; X 2 ; . . . ; X p then we will have fully aliased columns since both, H and Q1 , include column X 1 . To avoid this we have to use a different set of p orthogonal columns Y 1 ; Y 2 ; . . . ; Y p for Q1 . These columns should have the property that no function in Q1 ðY 1 ; Y 2 ; . . . ; Y p Þ is not fully aliased with any functions in HðX 1 ; X 2 ; . . . ; X p Þ or Q1 ðY 1 ; Y 2 ; . . . ; Y p Þ except from itself. That means f acg; 8c 2 F q , for all f ag; f ; g 2 HðX 1 ; X 2 ; . . . ; X p Þ [ Q1 ðY 1 ; Y 2 ; . . . ; Y p Þ. A systematic method for producing new columns from a set of orthogonal columns is to apply cyclic permutations to each of the columns. Corollary 1. Let Ai , i ¼ 1; 2; . . . ; k be saturated OAðn; t; q; 2Þ, n ¼ qp , t ¼ ðqp 1Þ=ðq 1Þ. Suppose that these arrays are constructed by linear Hð Þ and quadratic Q1 ð Þ functions with a suitable selection of p orthogonal variables. The matrix defined by juxtaposing all orthogonal arrays D ¼ ½A1 ; A2 ; . . . ; Ak is an Eðf NOD Þ-optimal q-level supersaturated design with n rows and m ¼ kt columns with the property of equal occurrence and with no fully aliased columns. Proof. The result follows from Theorem 1 and the construction using linear and quadratic functions with a suitable selection of orthogonal variables. & Example 3. Set X T1 ¼ ð0; 1; 2; 0; 1; 0; 1; 2; 2Þ, X T2 ¼ ð2; 0; 0; 1; 1; 0; 2; 2; 1Þ, Y T1 ¼ ð1; 1; 2; 0; 0; 0; 1; 2; 2Þ, Y T2 ¼ ð0; 1; 0; 2; 0; 1; 2; 2; 1Þ, ZT1 ¼ ð1; 2; 0; 1; 0; 0; 1; 2; 2Þ and ZT2 ¼ ð0; 0; 2; 1; 0; 1; 2; 2; 1Þ. Define A1 ¼ HðX 1 ; X 2 Þ, A2 ¼ Q1 ðY 1 ; Y 2 Þ and A3 ¼ Q1 ðZ 1 ; Z 2 Þ. These are k ¼ 3 equivalent orthogonal arrays OA(9,4,3,2) with no fully aliased columns. From Corollary 1 we have that D ¼ ½A1 ; A2 ; A3 , the juxtaposition of all three orthogonal arrays, is an Eðf NOD Þ-optimal 3-level supersaturated design with n ¼ 9 rows and m ¼ kt ¼ 12 columns with the property of equal occurrence and with no fully aliased columns.
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In the next Theorem, we apply permutations of columns to obtain optimal supersaturated designs with more columns. Theorem 2. Let D ¼ ½A1 ; A2 ; . . . ; Ak be a q-level supersaturated design with n runs and m ¼ kt columns, where Ai are saturated orthogonal arrays with n runs, t columns and strength 2 and are constructed by linear Hð Þ and quadratic Q1 ð Þ functions. Then there exists a q-level Eðf NOD Þ-optimal supersaturated design with n runs and ð‘ þ 1Þt columns, for all ‘ ¼ 1; . . . ; nk 1. Proof. Let Pn be the circulant matrix of order n with first row ð0; 1; 0; . . . ; 0Þ and set S n ¼ PTn , where PTn is the transpose of Pn . Observe that Snn ¼ I n . It is easy to see that the n t matrix defined by Sin Aj is an orthogonal array of n runs and t columns for any integers i and j, where jpk. Select 1p‘pnk 1. From the Euclidean division algorithm we know that ‘ þ 1 ¼ dn þ r, where 0pdpk and 0pron. Now define 0 n1 0 r1 G ¼ ½S 0n A1 ; . . . ; S n1 n A1 ; . . . ; S n Ad ; . . . ; S n Ad ; S n Adþ1 ; . . . ; S n Adþ1 :
ð1Þ
Obviously if ‘ þ 1 ¼ dn (i.e. r ¼ 0) then 0 n1 G ¼ ½S 0n A1 ; . . . ; S n1 n A1 ; . . . ; S n Ad ; . . . ; S n Ad :
Clearly, G is an n tð‘ þ 1Þ matrix. Using Lemma 1 and Theorem 1 we have that G is the desirable q-level Eðf NOD Þ-optimal supersaturated design with n runs and ð‘ þ 1Þt columns. & Corollary 2. Suppose that ‘ þ 1 ¼ dn þ r and C¼
n1 d [ [ i¼0 j¼1
fS in Aj g
r1 [
fS in Adþ1 g
i¼0
a set of q-level columns. If jCj ¼ tð‘ þ 1Þ and no fully aliased columns exist in C then the q-level Eðf NOD Þ-optimal supersaturated design with n runs and tð‘ þ 1Þ columns derived from Theorem 2 has no fully aliased columns. Proof. From the fact that the cardinality of C is tð‘ þ 1Þ we have that there are no similar columns in G (as it is defined in the proof of Theorem 2). Then the property that no fully aliased columns exist in C gives that no fully aliased columns exist in G. & Example 4. In Example 3, we constructed three saturated orthogonal arrays with no fully aliased columns and used them to obtain an optimal supersaturated design with n ¼ 9 runs and m ¼ 12 columns. Generally for a chosen integer 1p‘p26 with ‘ þ 1 ¼ 9d þ r we define C to be the set of all columns constructed by the ith cyclic permutations of columns of the first d saturated orthogonal arrays and the j th cyclic permutations of the columns of the d þ 1 saturated orthogonal array, where i ¼ 0; 1; . . . ; 8, j ¼ 0; 1; . . . ; r 1 and by the 0th permutation of a column we mean the column itself. C is as defined in Corollary 2. The cardinality jCj of C is 4ð‘ þ 1Þ and a computer program reveals that no fully aliased columns exit in C. Thus, the conditions of Corollary 2 hold and from Theorem 2 we have that G ¼ S 09 A1 ; . . . ; S89 A1 ; . . . ; S 09 Ad ; . . . ; S 89 Ad ; S09 Adþ1 ; . . . ; S r1 9 Adþ1 is a 3-level Eðf NOD Þ-optimal supersaturated design with 9 runs and 4ð‘ þ 1Þ columns.
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Table 1 Three-level Eðf NOD Þ-optimal supersaturated designs with n ¼ 9 runs, m columns, max w2 ¼ 10, f max ¼ 8 and ave w2 =Eðf NOD Þ ‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
1 2 3 4 5 6 7 8 9 10 11 12 13
8 12 16 20 24 28 32 36 40 44 48 52 56
2.286 2.970 3.300 3.505 3.645 3.741 3.810 3.857 3.908 3.951 3.986 4.017 4.040
13.714 17.091 18.600 19.516 20.130 20.540 20.831 21.086 21.169 21.222 21.277 21.330 21.390
2.571 3.273 3.600 3.789 3.913 4.000 4.065 4.114 4.154 4.186 4.213 4.235 4.255
14 15 16 17 18 19 20 21 22 23 24 25 26
60 64 68 72 76 80 84 88 92 96 100 104 108
4.059 4.076 4.091 4.106 4.120 4.132 4.141 4.149 4.157 4.164 4.171 4.176 4.181
21.451 21.506 21.556 21.592 21.655 21.714 21.769 21.818 21.864 21.908 21.947 21.984 22.019
4.271 4.286 4.299 4.310 4.320 4.329 4.337 4.345 4.352 4.358 4.364 4.369 4.374
For ‘ ¼ 26 we have that ‘ þ 1 ¼ 27 ¼ 9 3 þ 0. C is generated by all cyclic permutations of all
columns of the three saturated orthogonal arrays. jCj ¼ 108 ¼ 27 4. It has been checked, by a computer, that no fully aliased columns occur in C. Thus, these columns form matrix G which is the desirable 3-level supersaturated design with 9 runs and ð‘ þ 1Þt ¼ 27 4 ¼ 108 columns. The maximum dependency max w2 is 10, ave w2 and Lw2 is 4:374. So this design is ave w2 -optimal. Moreover, avejf j ¼ 4:181, aveðf 2 Þ ¼ 22:019, f max ¼ 8, Eðf NOD Þ ¼ Lf NOD ¼ 4:374 and thus this design is also Eðf NOD Þ-optimal. For ‘ ¼ 20 we have that ‘ þ 1 ¼ 21 ¼ 9 2 þ 3. C is generated by all cyclic permutations of all columns of the orthogonal arrays A1 ; A2 and the three first permutations of all columns of the A3 array. jCj ¼ 84 ¼ 21 4. It has been checked, by a computer, that no fully aliased columns occur in C. Thus, these columns form matrix G which is the desirable 3-level supersaturated design with 9 runs and ð‘ þ 1Þt ¼ 21 4 ¼ 84 columns. The maximum dependency max w2 is 10, ave w2 and Lw2 is 4:337. So, this design is ave w2 -optimal. Moreover, ave jf j ¼ 4:141, ave ðf 2 Þ ¼ 21:769, f max ¼ 8, Eðf NOD Þ ¼ Lf NOD ¼ 4:337 and thus this design is also Eðf NOD Þ-optimal.
The results obtained for ‘ ¼ 1; 2; . . . ; 26 are given in Table 1.
3. The results In this section, we applied the method described previously and we come up with many new Eðf NOD Þ-optimal supersaturated designs with three and four levels. 3.1. Three-level supersaturated designs Here, we present the results we have found using the method described above and evaluating functions over F n3 .
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Table 2 Orthogonal columns X T1 X T2 X T3 Y T1 Y T2 Y T3 Y T4 Y T5 Y T6 X T4 X T5 X T6
0 1 1 0 0 0 0 0 0 2 2 2
0 0 0 0 2 0 0 0 2 0 1 1
0 1 0 2 1 1 1 0 1 1 1 1
0 1 2 2 1 2 0 1 2 0 0 2
0 2 2 0 2 1 1 1 0 1 0 2
1 0 0 0 2 2 2 2 0 2 0 0
1 0 1 1 0 0 0 0 1 1 1 2
0 0 1 0 0 2 2 2 1 0 2 2
0 0 2 1 0 1 2 1 2 2 1 2
1 0 2 1 0 2 1 0 0 0 0 1
1 1 1 1 1 1 2 1 0 1 0 0
2 1 2 1 1 2 1 2 0 2 2 0
2 2 0 1 2 2 0 2 0 1 2 1
1 1 2 2 0 0 2 1 1 1 2 2
1 2 1 2 0 1 1 1 1 1 1 0
1 2 2 2 0 2 1 1 2 0 0 0
1 1 0 0 1 0 1 2 2 0 1 2
2 0 0 0 1 1 1 0 2 1 0 1
2 1 0 1 2 0 2 0 0 2 1 0
0 2 0 1 2 1 2 0 1 2 2 1
1 2 0 0 1 2 0 1 1 0 2 1
0 2 1 1 1 0 2 0 2 0 2 0
2 1 1 2 1 0 0 1 0 1 2 0
2 2 1 2 2 0 1 2 1 2 1 1
2 0 1 0 0 1 0 2 1 2 0 1
2 0 2 2 2 1 0 2 2 0 1 0
2 2 2 2 2 2 2 2 2 2 0 2
When p ¼ 2 we obtain supersaturated designs with n ¼ 9 runs using the functions given in Example 3 and Theorem 2. The parameters of these three-level supersaturated designs are presented in Table 1, while the designs can be obtained from linear and quadratic functions using the ‘ value given in the same table. When p ¼ 3 we obtain supersaturated designs with n ¼ 27 runs. Using the orthogonal columns X 1 , X 2 , X 3 , Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 , X 4 , X 5 , X 6 , we obtain the four orthogonal arrays HðX 1 ; X 2 ; X 3 Þ, Q1 ðY 1 ; Y 2 ; Y 3 Þ, Q1 ðY 4 ; Y 5 ; Y 6 Þ, and HðX 4 ; X 5 ; X 6 Þ (Table 2). By the method given in a previous section and Theorem 2, we obtain many three-level supersaturated designs with the parameters presented in Table 3. The designs can be constructed from linear and quadratic functions using the ‘ value given in the same table. For 1p‘p8 we obtain designs with ðmax w2 ; f max Þ ¼ ð14; 18Þ, for 9p‘p26 we obtain designs with ðmax w2 ; f max Þ ¼ ð17:333; 20Þ, for 10p‘p80 we obtain designs with ðmax w2 ; f max Þ ¼ ð21:333; 20Þ, and for 81p‘p107 we obtain designs with ðmax w2 ; f max Þ ¼ ð26; 24Þ. 3.2. Four-level supersaturated designs When q ¼ 4, the elements of GFð4Þ are polynomials with coefficients over GFð2Þ and the desirable supersaturated designs are constructed using linear and quadratic functions over F n4 . Example 5. Consider q ¼ 4 and p ¼ 2. In this case GFð4Þ ¼ F 2 ½z =ðz2 þ z þ 1Þ. The functions are HðX 1 ; X 2 Þ ¼ fX 1 ; X 2 ; X 1 þ X 2 ; zX 1 þ X 2 ; ð1 þ zÞX 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ zX 1 þ X 2 ; X 21 þ ð1 þ zÞX 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 1 ; X 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ zX 1 þ X 2 ; X 21 þ ð1 þ zÞX 1 þ X 2 g: HðX 1 ; X 2 Þ is an OAð16; 5; 4; 2Þ when the functions are evaluated at F 16 4 ; so does Q1 ðX 1 ; X 2 Þ. The column juxtaposition of HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ forms a supersaturated design on 4 levels,
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Table 3 Three-level Eðf NOD Þ-optimal supersaturated designs with n ¼ 27 runs and m columns ‘
m
ave w2
ave jf j
ave ðf 2 Þ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 71 72 73 74 75 76 77 78 79 80 81 82
26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325 338 351 364 377 390 403 416 429 442 455 468 936 949 962 975 988 1001 1014 1027 1040 1053 1066 1079
4.105 5.406 6.060 6.447 6.707 6.891 7.029 7.138 7.226 7.296 7.353 7.402 7.446 7.485 7.519 7.549 7.574 7.597 7.618 7.637 7.654 7.669 7.683 7.696 7.708 7.719 7.726 7.733 7.740 7.747 7.753 7.759 7.764 7.770 7.775 7.872 7.874 7.876 7.877 7.879 7.880 7.882 7.883 7.885 7.886 7.887 7.888
38.511 50.575 56.730 60.390 62.791 64.458 65.707 66.673 67.450 68.076 68.617 69.080 69.490 69.854 70.173 70.461 70.708 70.928 71.123 71.297 71.451 71.590 71.718 71.839 71.947 72.046 72.125 72.201 72.275 72.344 72.408 72.469 72.529 72.583 72.633 73.569 73.583 73.597 73.610 73.622 73.634 73.646 73.658 73.670 73.681 73.685 73.688
6.480 8.526 9.529 10.125 10.519 10.800 11.010 11.172 11.302 11.408 11.497 11.571 11.635 11.691 11.739 11.782 11.820 11.854 11.884 11.912 11.937 11.960 11.981 12.000 12.018 12.034 12.050 12.064 12.077 12.090 12.101 12.112 12.122 12.132 12.141 12.302 12.304 12.306 12.308 12.310 12.312 12.314 12.316 12.318 12.319 12.321 12.323
Eðf NOD Þ 2.160 2.842 3.176 3.375 3.506 3.600 3.670 3.724 3.767 3.803 3.832 3.857 3.878 3.897 3.913 3.927 3.940 3.951 3.961 3.970 3.979 3.986 3.993 4.000 4.006 4.011 4.016 4.021 4.025 4.030 4.034 4.037 4.041 4.044 4.047 4.100 4.101 4.102 4.102 4.103 4.104 4.104 4.105 4.106 4.106 4.107 4.107
‘
m
ave w2
ave jf j
ave ðf 2 Þ
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 90 91 92 93 94 95 96 97 98 99 100 101
481 494 507 520 533 546 559 572 585 598 611 624 637 650 663 676 689 702 715 728 741 754 767 780 793 806 819 832 845 858 871 884 897 910 923 1183 1196 1209 1222 1235 1248 1261 1274 1287 1300 1313 1326
7.780 7.784 7.789 7.793 7.797 7.800 7.804 7.807 7.810 7.813 7.816 7.819 7.822 7.825 7.827 7.830 7.832 7.835 7.837 7.840 7.842 7.845 7.847 7.849 7.851 7.853 7.856 7.858 7.860 7.862 7.863 7.865 7.867 7.869 7.871 7.896 7.897 7.898 7.899 7.899 7.900 7.901 7.902 7.902 7.903 7.904 7.904
72.680 72.726 72.771 72.815 72.857 72.896 72.935 72.973 73.010 73.045 73.078 73.108 73.138 73.166 73.194 73.220 73.246 73.272 73.292 73.310 73.329 73.349 73.368 73.386 73.403 73.418 73.434 73.450 73.466 73.481 73.496 73.511 73.526 73.540 73.555 73.711 73.714 73.717 73.721 73.724 73.727 73.731 73.734 73.738 73.742 73.745 73.749
12.150 12.158 12.166 12.173 12.180 12.187 12.194 12.200 12.205 12.211 12.216 12.222 12.226 12.231 12.236 12.240 12.244 12.248 12.252 12.256 12.259 12.263 12.266 12.270 12.273 12.276 12.279 12.282 12.284 12.287 12.290 12.292 12.295 12.297 12.299 12.335 12.336 12.338 12.339 12.340 12.342 12.343 12.344 12.345 12.346 12.348 12.349
Eðf NOD Þ 4.050 4.053 4.055 4.058 4.060 4.062 4.064 4.066 4.068 4.070 4.072 4.074 4.075 4.077 4.078 4.080 4.081 4.083 4.084 4.085 4.086 4.087 4.089 4.090 4.091 4.092 4.093 4.094 4.095 4.095 4.096 4.097 4.098 4.099 4.100 4.111 4.112 4.112 4.113 4.113 4.114 4.114 4.114 4.115 4.115 4.116 4.116
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Table 3 (continued ) ‘
m
ave w2
ave jf j
ave ðf 2 Þ
83 84 85 86 87 88 89
1092 1105 1118 1131 1144 1157 1170
7.889 7.890 7.891 7.892 7.893 7.894 7.895
73.690 73.693 73.696 73.699 73.702 73.705 73.708
12.324 12.326 12.328 12.329 12.331 12.332 12.334
Eðf NOD Þ 4.108 4.108 4.109 4.110 4.110 4.111 4.111
‘
m
ave w2
ave jf j
ave ðf 2 Þ
102 103 104 105 106 107
1339 1352 1365 1378 1391 1404
7.905 7.906 7.906 7.907 7.907 7.908
73.752 73.755 73.759 73.762 73.765 73.768
12.350 12.351 12.352 12.353 12.354 12.355
Eðf NOD Þ 4.116 4.117 4.117 4.117 4.118 4.118
Table 4 Four-level Eðf NOD Þ-optimal supersaturated designs with n ¼ 16 runs, m columns and ave w2 =Eðf NOD Þ ‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165
4.711 6.095 6.705 7.356 7.356 7.539 7.685 7.808 7.909 7.987 8.051 8.106 8.147 8.183 8.213 8.246 8.276 8.303 8.328 8.351 8.370 8.386 8.401 8.415 8.429 8.441 8.451 8.461 8.471 8.480 8.488 8.494
45.689 58.590 64.400 68.240 70.713 72.477 73.841 75.131 76.140 76.929 77.548 78.092 78.506 78.844 79.129 79.287 79.442 79.591 79.742 79.900 8.370 80.112 80.210 80.311 80.405 80.494 80.556 80.626 80.698 80.767 80.835 80.917
5.333 6.857 7.579 8.000 8.276 8.471 8.615 8.727 8.816 8.889 8.949 9.000 9.043 9.081 9.114 9.143 9.169 9.191 9.212 9.231 9.248 9.263 9.277 9.290 9.302 9.313 9.324 9.333 9.342 9.351 9.358 9.366
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320
8.500 8.506 8.512 8.517 8.521 8.525 8.529 8.533 8.538 8.541 8.545 8.548 8.551 8.554 8.557 8.562 8.567 8.571 8.576 8.579 8.583 8.586 8.590 8.593 8.596 8.600 8.603 8.605 8.608 8.610 8.613
80.989 81.055 81.133 81.210 81.273 81.329 81.382 81.452 81.517 81.576 81.625 81.674 81.730 81.781 81.828 81.875 81.92 81.960 81.996 82.026 82.052 82.074 82.098 82.127 82.156 82.182 82.205 82.225 82.241 82.255 82.270
9.373 9.379 9.385 9.391 9.397 9.402 9.407 9.412 9.416 9.421 9.425 9.429 9.432 9.436 9.439 9.443 9.446 9.449 9.452 9.455 9.457 9.460 9.462 9.465 9.467 9.469 9.472 9.474 9.476 9.478 9.480
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2
1Þ n ¼ 4p rows and m columns where m ¼ 2ð4 ð41Þ ¼ 10. However, this design has fully aliased columns since both HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ have column X 1 . This problem is solved using the four orthogonal arrays Q1 ðY 1 ; Y 2 Þ, HðX 1 ; X 2 Þ, Q1 ðY 3 ; Y 4 Þ, and HðX 3 ; X 4 Þ, where columns Y 1 ; Y 2 ; X 1 ; X 2 ; Y 3 ; Y 4 ; X 3 ; X 4 are as follows:
Y1T Y2T X 1T X 2T Y3T Y4T X 3T X 4T
0, 0, 1 + z, 0, 1 + z, 1, z , 1, z , z , 1 + z, 0, 1, 1, z , 1 + z 0, z , 1 + z, 1 + z, 0, 1, 1, 0, 0, 1 + z, 1, 1, z , 1 + z, z, z 0, 1, 1 + z, 0, 1, 1, 0, z , 0, z , 1 + z, z, 1, z , 1 + z, 1 + z 0, 1 + z, 1 + z, 1, 0, 1, z , z , 1 + z, 1 + z, 0, 1, z , 0, 1, z 0, 0, 1 + z, 1 + z, 1, z , 1, z , 1 + z, 1, z , 0, 0, 1, z , 1 + z 0, z , 1 + z, 0, 1, 1, z , 1 + z, 1, 0, 0, 1, 1 + z, 1 + z, z, z 0, 1, 1, 0, 1, z , z , z , 1 + z, z, 1 + z, 1 + z, 0, 1, 0, 1 + z 0, 1 + z, 0, z , z , 0, z , 1 + z, 0, 1, 1, 1 + z, 1 + z, 1, 1, z
Using these four pairs of orthogonal columns, the orthogonal arrays derived from these and the permutation Theorem, we obtain many supersaturated designs with parameters given in Table 4. The designs, corresponding to ‘ given in this Table, can be constructed from the above functions and the orthogonal pairs given. For ‘ ¼ 1 we obtain a design with ðmax w2 ; f max Þ ¼ ð18; 14Þ and for ‘ ¼ 2 we obtain a design with ðmax w2 ; f max Þ ¼ ð20; 14Þ. When ‘42 we obtain designs with ðmax w2 ; f max Þ ¼ ð26; 16Þ.
References Addelman, S., Kempthorne, O., 1961. Some main-effect plans and orthogonal arrays of strength two. Ann. Math. Statist. 32, 1167–1176. Aggarwal, M.L., Gupta, S., 2004. A new method of construction of multi-level supersaturated designs. J. Statist. Plann. Inference 121, 127–134. Booth, K.H.V., Cox, D.R., 1962. Some systematic supersaturated designs. Technometrics 4, 489–495. Box, G.E.P., Meyer, R.D., 1986. An analysis for unreplicated fractional factorials. Technometrics 28, 11–18. Butler, N., Mead, R., Eskridge, K.M., Gilmour, S.G., 2001. A general method of constructing Eðs2 Þ-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632. Cheng, C.S., 1997. Eðs2 Þ-optimal supersaturated designs. Statist. Sin. 7, 929–939. Fang, K.T., Lin, D.K.J., Ma, C.X., 2000. On the construction of multi-ilevel supersaturated designs. J. Statist. Plann. Inference 86, 239–252. Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003. Optimal mixed-level supersaturated design. Metrika 58, 279–291. Hedayat, A.S., Sloane, N.J.A., Stufken, J., 1999. Orthogonal Arrays: Theory and Applications. Springer, New York. Li, W.W., Wu, C.F.J., 1997. Columnwise-pairwise algorithms with applications to the construction of supersaturated designs. Technometrics 39, 171–179. Lin, D.K.J., 1993. A new class of supersaturated designs. Technometrics 35, 28–31. Lin, D.K.J., 1995. Generating systematic supersaturated designs. Technometrics 37, 213–225. Liu, Y., Dean, A., 2004. k-circulant supersaturated designs. Technometrics 46, 32–43. Liu, M., Zhang, R., 2000. Construction of Eðs2 Þ optimal supersaturated designs using cyclic BIBDs. J. Statist. Plann. Inference 91, 139–150. Lu, X., Meng, Y., 2000. A new method in the construction of two-level supersaturated designs. J. Statist. Plann. Inference 86, 229–238.
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Lu, X., Sun, Y., 2001. Supersaturated design with more than two levels. Chinese Ann. Math. 22B (2), 183–194. Lu, X., Hu, W., Zheng, Y., 2003. A systematic procedure in the construction of multi-level supersaturated design. J. Statist. Plann. Inference 115, 287–310. Mukerjee, R., Wu, C.F.J., 1995. On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist. 23, 2102–2115. Nguyen, N.K., 1996. An algorithmic approach to constructing supersaturated designs. Technometrics 38, 69–73. Plackett, R.L., Burman, J.P., 1946. The design of optimum multifactorial experiments. Biometrika 33, 303–325. Satterthwaite, F.E., 1959. Random balance experimentation (with discussions). Technometrics 1, 111–137. Tang, B., Wu, C.F.J., 1997. A method for constructing supersaturated designs and its Es2 -optimality. Canad. J. Statist. 25, 191–201. Wu, C.F.J., 1993. Construction of supersaturated designs through partially aliased interactions. Biometrika 80, 661–669. Yamada, S., Lin, D.K.J., 1997. Supersaturated designs including an orthogonal base. Canad. J. Statist. 25, 203–213. Yamada, S., Lin, D.K.J., 1999. Three-level supersaturated designs. Statist. Probab. Lett. 45, 31–39. Yamada, S., Matsui, T., 2002. Optimality of mixed-level supersaturated designs. J. Statist. Plann. Inference 104, 459–468. Yamada, S., Ikebe, Y.T., Hashiguchi, H., Niki, N., 1999. Construction of three-level supersaturated design. J. Statist. Plann. Inference 81, 183–193.
Statistics & Probability Letters 69 (2004) 199–211 www.elsevier.com/locate/stapro
Optimal multi-level supersaturated designs constructed from linear and quadratic functions C. Koukouvinos, S. Stylianou Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece Received 1 October 2003; received in revised form 4 April 2004 Available Online 08 July 2004
Abstract In this paper, we present a method of construction Eðf NOD Þ-optimal multi-level supersaturated design with n rows, m columns and the equal occurrence property, using linear and quadratic functions. Using this method, as well as permutations and juxtapositions of the columns, many new Eðf NOD Þ-optimal multi-level supersaturated designs are provided. r 2004 Elsevier B.V. All rights reserved. MSC: primary 62K10; 62K15; secondary 05B20 Keywords: Supersaturated designs; Factorial designs; Linear functions; Quadratic functions; Orthogonal arrays; Dependency; Efficiency
1. Introduction A q-level supersaturated design is a factorial design having n ¼ qp experimental runs, while the number of factors m is more than ðn 1Þ=ðq 1Þ. Supersaturated design is used in the initial stage of an industrial or scientific experiment for screening the active factors, and is useful when there are a large number of factors under investigation while only a very limited number of experimental runs is available. The analysis of supersaturated designs rely on the assumption of Corresponding author.
E-mail address: [email protected] (C. Koukouvinos). 0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.06.030
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effect sparsity (Box and Meyer (1986)). This assumes that only a few dominant factors actually affect the response. Satterthwaite (1959) proposed the idea of supersaturated design as a random balance design. Booth and Cox (1962) first examined two-level supersaturated designs systematically. Since then, many researchers have studied the construction and the properties of two-level supersaturated designs (see Butler et al., 2001; Cheng, 1997; Li and Wu, 1997; Lin, 1993; Lin, 1995; Liu and Dean, 2004; Liu and Zhang, 2000; Lu and Meng, 2000; Nguyen, 1996; Plackett and Burman, 1946; Tang and Wu, 1997; Wu, 1993; Yamada and Lin, 1997). However, all these authors considered only two-level designs. Designs with multi-levels are often requested in industrial and scientific experimentation. Especially, designs with three levels have been widely used (see for example Yamada and Lin, 1999; Yamada and Matsui, 2002). Fang et al. (2000), Lu and Sun (2001) and Lu et al. (2003) have discussed multi-level supersaturated designs. Aggarwal and Gupta (2004) have constructed multi-level supersaturated designs based on Galois field theory. This method is a generalization of the method given by Lu and Meng (2000). A supersaturated design cannot be an orthogonal design, so it is necessary to define measures of departure from orthogonality. Therefore, we present some design criteria for comparing supersaturated designs. Let D be an n m matrix of a factorial design with elements from a set of factor levels f0; 1; . . . ; q 1g. We say that the matrix D has the equal occurrence property if each column contains the same number of f0; 1; . . . ; q 1g. In this case, the number n of runs is a multiple of q. A column which satisfies the equal occurrence property is said to be balanced. A design with the equal occurrence property is called an orthogonal array of strength 2, denoted by OAðn; t; q; 2Þ, if in any two columns all their q2 level combinations appear the same number of times (see Hedayat et al., 1999). In this case, n is a multiple of q2 and tðq 1Þpðn 1Þ. When tðq 1Þ ¼ ðn 1Þ the design is called saturated and when tðq 1Þ4ðn 1Þ the design is called supersaturated. For q-level supersaturated designs, Yamada and Lin (1999) defined a measure for dependency between two columns d i ; d j of the matrix D ¼ ½d 1 ; d 2 ; . . . ; d m by q1 X ðnab ði; jÞ n=q2 Þ2 w ðd i ; d j Þ ¼ ; n=q2 a;b¼0 2
where nab ði; jÞ is the number of rows whose values are a; b in the n 2 matrix ½d i ; d j . Two columns d i ; d j are said to be completely dependent if w2 ðd i ; d j Þ ¼ nðq 1Þ and independent if w2 ðd i ; d j Þ ¼ 0. We can say that every two columns d i ; d j are independent if all the q2 possible pairs ði; jÞ occur with the same frequency. Yamada and Lin (1999) also defined criteria of dependency of a whole design, by , X m w2 ðd i ; d j Þ ; max w2 ¼ max w2 ðd i ; d j Þ: ave w2 ¼ 1piojpm 2 1piojpm Yamada and Matsui (2002) derived a lower bound on ave w2 , which is given by ave w2 X
ðq 1Þn½ðq 1Þm n þ 1 ¼ Lw2 : ðn 1Þðm 1Þ
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Using this lower bound, ave w2 -efficiency is defined as Lw2 =ave w2 . A design is ave w2 -optimal when ave w2 -efficiency is equal to 1. Using the criterion q1 X n f ðd i ; d j Þ ¼ nab ði; jÞ q2 a;b¼0 for measuring the orthogonality of two columns we can obtain the next criteria for the whole design , X m avejf j ¼ f ðd i ; d j Þ ; 2 1piojpm 2
aveðf Þ ¼
X
f ðd i ; d j Þ
1piojpm
2
,
m
2
and f max ¼
max
1piojpm
f ðd i ; d j Þ:
Fang et al. (2003) introduced the Eðf NOD Þ criterion for multi-level supersaturated designs with the property of equal occurrence. For any two columns d i ; d j of the design matrix D they defined q1 X n 2 i;j f NOD ¼ nab ði; jÞ 2 : q a;b¼0 The new criterion Eðf NOD Þ is given by , X m i;j : f NOD Eðf NOD Þ ¼ 2 1piojpm In the same article, they proved that Pn 2 n n n2 k;l¼1;kal lkl Eðf NOD Þ ¼ m 2 þ m1 q mðm 1Þ q 2 mn n n n n2 X 1 þ m 2; ðm 1Þðn 1Þ q m1 q q where lkl is the number of coincidences between the kth and lth rows. The lower bound of Eðf NOD Þ can be achieved if and only if l ¼ mðn=q 1Þ=ðn 1Þ is a positive integer and all the lkl s for kal are equal to l. So, the Eðf NOD Þ-efficiency is defined as Lf NOD =Eðf NOD Þ. It is easy to prove that for any design matrix D with the equal occurrence property the criteria Eðf NOD Þ and ave w2 satisfy the following relation: Eðf NOD Þ ¼ n=q2 ave w2 , when qX3. Mukerjee and Wu (1995) proved the following lemma. Lemma 1. Suppose that A is a saturated OAðn; t; q; 2Þ with t ¼ ðn 1Þ=ðq 1Þ. Then, the number of coincidences of rows i and j of A is li;j ¼ ðn qÞ=ðqðq 1ÞÞ for any iaj.
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Many multi-level supersaturated designs can be constructed combining together orthogonal arrays and Lemma 1 guarantees their optimality. This result is stated in next Theorem. Theorem 1. Let Ai , i ¼ 1; 2; . . . ; k be saturated OAðn; t; q; 2Þ, n ¼ qp , t ¼ ðqp 1Þ=ðq 1Þ. The matrix defined by juxtaposing all orthogonal arrays D ¼ ½A1 ; A2 ; . . . ; Ak is an Eðf NOD Þoptimal q-level supersaturated design with n rows, m ¼ kt columns and the property of equal occurrence. Proof. The equal occurrence property is satisfied since all columns are columns of different or the same OAðn; t; q; 2Þ. From Lemma 1, we have that for any pair ði; jÞ of rows of matrix D the number of coincidence li;j is a constant (actually li;j ¼ kðn qÞ=qðq 1ÞÞ. Thus, design D achieves Eðf NOD Þ-efficiency equal to 1 and so this is an Eðf NOD Þ-optimal q-level supersaturated design with n rows, m ¼ kt columns and the property of equal occurrence. & Example 1. Suppose we have the following three equivalent OAð9; 4; 3; 2Þ: 3 3 3 2 2 2 2 2 2 2 2 1 1 0 0 1 2 0 62 1 1 07 62 2 2 27 60 0 1 27 7 7 7 6 6 6 7 7 7 6 6 6 60 1 2 17 60 1 2 17 62 1 1 17 7 7 7 6 6 6 7 7 7 6 6 6 62 0 0 17 60 2 0 07 60 2 0 17 7 7 7 6 6 6 7 7 7 6 6 A1 ¼ 6 6 0 2 0 0 7 A2 ¼ 6 1 2 1 1 7 A3 ¼ 6 2 0 0 0 7: 7 7 7 6 6 6 61 2 1 17 60 0 1 27 61 0 2 17 7 7 7 6 6 6 60 0 1 27 62 0 0 17 62 2 2 27 7 7 7 6 6 6 7 7 7 6 6 6 41 0 2 05 41 0 2 05 41 2 1 05 1
1
0 2
1
1
0
2
1
1
0 2
Let D ¼ ½A1 ; A2 ; A3 be the array which contains all columns of A1 ; A2 ; A3 . Then, D is an Eðf NOD Þoptimal 3-level supersaturated design with 9 rows and m ¼ 4 3 ¼ 12 columns with the property of equal occurrence. Observe that the first column of A1 is fully aliased with the first column of A3 , but this does not effect the optimality of the design. & Remark 1. The optimal supersaturated designs obtained by Theorem 1 may contain fully aliased columns. Thus, Eðf NOD Þ criterion is not enough to prevent the existence of fully aliased columns in the designs. For this reason we also use the max value of the design criteria in all the supersaturated designs given. We study construction methods that produce optimal designs without fully aliased columns in the next sections. This paper studies the construction of multi-level supersaturated designs. In Section 2, a method for constructing optimal multi-level supersaturated designs is proposed and a procedure using this method is presented and explained with several examples. Finally, the results found with this method are tabulated and presented in Section 3.
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2. The construction method and the procedure In this section, we present the method we use to obtain multi-level supersaturated designs with the equal occurrence property using linear and quadratic functions.
2.1. Preliminaries Addelman and Kempthorne (1961) first used linear and quadratic functions to obtain orthogonal arrays. Let F q ¼ GFðqÞ be the finite field of q elements where q is a prime power. By F nq we denote the n-dimensional vector space over GFðqÞ. The columns of an array are labeled with linear or quadratic polynomials in p variables X 1 ; X 2 ; . . . ; X p and the rows are labeled with points from F nq . Let f 1 ðX 1 ; X 2 ; . . . ; X p Þ and f 2 ðX 1 ; X 2 ; . . . ; X p Þ be two functions, linear or nonlinear. They correspond to two columns of length n ¼ qp when evaluated at F nq . The two functions (or columns) are fully aliased (give the maximum possible value using any of the previously mentioned criteria) if the pair has only q level combinations, each combination occurring qp1 times; and orthogonal (give zero using any of the previously mentioned criteria) if the pair has q2 distinct level combinations, each combination occurring qp2 times. Let LðX 1 ; X 2 ; . . . ; X p Þ be the set of all nonzero linear functions of X 1 ; X 2 ; . . . ; X p , i.e., LðX 1 ; X 2 ; . . . ; X p Þ ¼ fc1 X 1 þ c2 X 2 þ þ cn X p : ci 2 F q ; not all ci are zerog: Every function in LðX 1 ; X 2 ; . . . ; X p Þ corresponds to a balanced column. Two functions f 1 and f 2 in LðX 1 ; X 2 ; . . . ; X p Þ are dependent if there is a nonzero constant c 2 F q such that f 1 ¼ cf 2 ; otherwise, they are independent. Clearly, dependent linear functions correspond to the same column up to level permutation and thus they arep fully aliased while independent linear functions 1Þ independent linear functions generates an correspond to orthogonal columns. A set of ðqðq1Þ p ðqp 1Þ OAðq ; ðq1Þ ; q; 2Þ. The traditional convention is to assume the first nonzero element being 1 for each column. Here, for convenience we assume the last nonzero element being 1 for each column. In particular, let HðX 1 ; X 2 ; . . . ; X p Þ be the set of all nonzero linear functions of X 1 ; X 2 ; . . . ; X p such thatp the last nonzero coefficient is 1. When evaluated at F nq , HðX 1 ; X 2 ; . . . ; X p Þ is a saturated 1Þ ; q; 2Þ. This is indeed the regular fractional factorial design and the construction is OAðqp ; ðqðq1Þ called the Rao–Hamming construction, see Hedayat et al. (Section 3.4). The key idea of the Addelman and Kempthorne (1961) construction is to use quadratic functions in addition to linear functions. Let Q1 ðX 1 ; X 2 ; . . . ; X p Þ ¼ fX 21 þ aX 1 þ h : a 2 F q ; h 2 HðX 2 ; . . . ; X p Þg S and Q1 ðX 1 ; X 2 ; . . . ; X p Þ ¼ fX 1 g Q1 ðX 1 ; X 2 ; . . . ; X p Þ. p 1Þ HðX 1 ; X 2 ; . . . ; X p Þ and Q1 ðX 1 ; X 2 ; . . . ; X p Þ have ðqðq1Þ columns each. The column juxtaposition of HðX 1 ; X 2 ; . . . ; X p Þ and Q1 ðX 1 ; X 2 ; . . . ; X p Þ forms a supersaturated design on q levels, n ¼ qp p 1Þ rows and m columns where m ¼ 2ðq ðq1Þ , which is a half of an Addelman and Kempthorne orthogonal array.
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Example 2. Consider q ¼ 3 and p ¼ 2. The functions are HðX 1 ; X 2 Þ ¼ fX 1 ; X 2 ; X 1 þ X 2 ; 2X 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ 2X 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 1 ; X 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ 2X 1 þ X 2 g: HðX 1 ; X 2 Þ is an OAð9; 4; 3; 2Þ when the functions are evaluated at F 93 ; so does Q1 ðX 1 ; X 2 Þ. The column juxtaposition of HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ forms a supersaturated design on 3 levels, n ¼ 3p rows and m columns where m ¼ 2ð32 1Þ=ð3 1Þ ¼ 8. However, this design has fully aliased columns since both HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ have column X 1 . This problem is solved using the method described in the following. 2.2. The new method As we have previously shown, the main problem we have to solve is to avoid having fully aliased columns in the design. Moreover, we need a method to systematically generate new columns, with the property of equal occurrence and the Eðf NOD Þ-optimality for the design. We select p orthogonal q-level columns X 1 ; X 2 ; . . . ; X p . Then, we find all functions of HðX 1 ; X 2 ; . . . ; X p Þ, where HðX 1 ; X 2 ; . . . ; X p Þ defined as before. If we calculate Q1 using columns X 1 ; X 2 ; . . . ; X p then we will have fully aliased columns since both, H and Q1 , include column X 1 . To avoid this we have to use a different set of p orthogonal columns Y 1 ; Y 2 ; . . . ; Y p for Q1 . These columns should have the property that no function in Q1 ðY 1 ; Y 2 ; . . . ; Y p Þ is not fully aliased with any functions in HðX 1 ; X 2 ; . . . ; X p Þ or Q1 ðY 1 ; Y 2 ; . . . ; Y p Þ except from itself. That means f acg; 8c 2 F q , for all f ag; f ; g 2 HðX 1 ; X 2 ; . . . ; X p Þ [ Q1 ðY 1 ; Y 2 ; . . . ; Y p Þ. A systematic method for producing new columns from a set of orthogonal columns is to apply cyclic permutations to each of the columns. Corollary 1. Let Ai , i ¼ 1; 2; . . . ; k be saturated OAðn; t; q; 2Þ, n ¼ qp , t ¼ ðqp 1Þ=ðq 1Þ. Suppose that these arrays are constructed by linear Hð Þ and quadratic Q1 ð Þ functions with a suitable selection of p orthogonal variables. The matrix defined by juxtaposing all orthogonal arrays D ¼ ½A1 ; A2 ; . . . ; Ak is an Eðf NOD Þ-optimal q-level supersaturated design with n rows and m ¼ kt columns with the property of equal occurrence and with no fully aliased columns. Proof. The result follows from Theorem 1 and the construction using linear and quadratic functions with a suitable selection of orthogonal variables. & Example 3. Set X T1 ¼ ð0; 1; 2; 0; 1; 0; 1; 2; 2Þ, X T2 ¼ ð2; 0; 0; 1; 1; 0; 2; 2; 1Þ, Y T1 ¼ ð1; 1; 2; 0; 0; 0; 1; 2; 2Þ, Y T2 ¼ ð0; 1; 0; 2; 0; 1; 2; 2; 1Þ, ZT1 ¼ ð1; 2; 0; 1; 0; 0; 1; 2; 2Þ and ZT2 ¼ ð0; 0; 2; 1; 0; 1; 2; 2; 1Þ. Define A1 ¼ HðX 1 ; X 2 Þ, A2 ¼ Q1 ðY 1 ; Y 2 Þ and A3 ¼ Q1 ðZ 1 ; Z 2 Þ. These are k ¼ 3 equivalent orthogonal arrays OA(9,4,3,2) with no fully aliased columns. From Corollary 1 we have that D ¼ ½A1 ; A2 ; A3 , the juxtaposition of all three orthogonal arrays, is an Eðf NOD Þ-optimal 3-level supersaturated design with n ¼ 9 rows and m ¼ kt ¼ 12 columns with the property of equal occurrence and with no fully aliased columns.
ARTICLE IN PRESS C. Koukouvinos, S. Stylianou / Statistics & Probability Letters 69 (2004) 199–211
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In the next Theorem, we apply permutations of columns to obtain optimal supersaturated designs with more columns. Theorem 2. Let D ¼ ½A1 ; A2 ; . . . ; Ak be a q-level supersaturated design with n runs and m ¼ kt columns, where Ai are saturated orthogonal arrays with n runs, t columns and strength 2 and are constructed by linear Hð Þ and quadratic Q1 ð Þ functions. Then there exists a q-level Eðf NOD Þ-optimal supersaturated design with n runs and ð‘ þ 1Þt columns, for all ‘ ¼ 1; . . . ; nk 1. Proof. Let Pn be the circulant matrix of order n with first row ð0; 1; 0; . . . ; 0Þ and set S n ¼ PTn , where PTn is the transpose of Pn . Observe that Snn ¼ I n . It is easy to see that the n t matrix defined by Sin Aj is an orthogonal array of n runs and t columns for any integers i and j, where jpk. Select 1p‘pnk 1. From the Euclidean division algorithm we know that ‘ þ 1 ¼ dn þ r, where 0pdpk and 0pron. Now define 0 n1 0 r1 G ¼ ½S 0n A1 ; . . . ; S n1 n A1 ; . . . ; S n Ad ; . . . ; S n Ad ; S n Adþ1 ; . . . ; S n Adþ1 :
ð1Þ
Obviously if ‘ þ 1 ¼ dn (i.e. r ¼ 0) then 0 n1 G ¼ ½S 0n A1 ; . . . ; S n1 n A1 ; . . . ; S n Ad ; . . . ; S n Ad :
Clearly, G is an n tð‘ þ 1Þ matrix. Using Lemma 1 and Theorem 1 we have that G is the desirable q-level Eðf NOD Þ-optimal supersaturated design with n runs and ð‘ þ 1Þt columns. & Corollary 2. Suppose that ‘ þ 1 ¼ dn þ r and C¼
n1 d [ [ i¼0 j¼1
fS in Aj g
r1 [
fS in Adþ1 g
i¼0
a set of q-level columns. If jCj ¼ tð‘ þ 1Þ and no fully aliased columns exist in C then the q-level Eðf NOD Þ-optimal supersaturated design with n runs and tð‘ þ 1Þ columns derived from Theorem 2 has no fully aliased columns. Proof. From the fact that the cardinality of C is tð‘ þ 1Þ we have that there are no similar columns in G (as it is defined in the proof of Theorem 2). Then the property that no fully aliased columns exist in C gives that no fully aliased columns exist in G. & Example 4. In Example 3, we constructed three saturated orthogonal arrays with no fully aliased columns and used them to obtain an optimal supersaturated design with n ¼ 9 runs and m ¼ 12 columns. Generally for a chosen integer 1p‘p26 with ‘ þ 1 ¼ 9d þ r we define C to be the set of all columns constructed by the ith cyclic permutations of columns of the first d saturated orthogonal arrays and the j th cyclic permutations of the columns of the d þ 1 saturated orthogonal array, where i ¼ 0; 1; . . . ; 8, j ¼ 0; 1; . . . ; r 1 and by the 0th permutation of a column we mean the column itself. C is as defined in Corollary 2. The cardinality jCj of C is 4ð‘ þ 1Þ and a computer program reveals that no fully aliased columns exit in C. Thus, the conditions of Corollary 2 hold and from Theorem 2 we have that G ¼ S 09 A1 ; . . . ; S89 A1 ; . . . ; S 09 Ad ; . . . ; S 89 Ad ; S09 Adþ1 ; . . . ; S r1 9 Adþ1 is a 3-level Eðf NOD Þ-optimal supersaturated design with 9 runs and 4ð‘ þ 1Þ columns.
ARTICLE IN PRESS C. Koukouvinos, S. Stylianou / Statistics & Probability Letters 69 (2004) 199–211
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Table 1 Three-level Eðf NOD Þ-optimal supersaturated designs with n ¼ 9 runs, m columns, max w2 ¼ 10, f max ¼ 8 and ave w2 =Eðf NOD Þ ‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
1 2 3 4 5 6 7 8 9 10 11 12 13
8 12 16 20 24 28 32 36 40 44 48 52 56
2.286 2.970 3.300 3.505 3.645 3.741 3.810 3.857 3.908 3.951 3.986 4.017 4.040
13.714 17.091 18.600 19.516 20.130 20.540 20.831 21.086 21.169 21.222 21.277 21.330 21.390
2.571 3.273 3.600 3.789 3.913 4.000 4.065 4.114 4.154 4.186 4.213 4.235 4.255
14 15 16 17 18 19 20 21 22 23 24 25 26
60 64 68 72 76 80 84 88 92 96 100 104 108
4.059 4.076 4.091 4.106 4.120 4.132 4.141 4.149 4.157 4.164 4.171 4.176 4.181
21.451 21.506 21.556 21.592 21.655 21.714 21.769 21.818 21.864 21.908 21.947 21.984 22.019
4.271 4.286 4.299 4.310 4.320 4.329 4.337 4.345 4.352 4.358 4.364 4.369 4.374
For ‘ ¼ 26 we have that ‘ þ 1 ¼ 27 ¼ 9 3 þ 0. C is generated by all cyclic permutations of all
columns of the three saturated orthogonal arrays. jCj ¼ 108 ¼ 27 4. It has been checked, by a computer, that no fully aliased columns occur in C. Thus, these columns form matrix G which is the desirable 3-level supersaturated design with 9 runs and ð‘ þ 1Þt ¼ 27 4 ¼ 108 columns. The maximum dependency max w2 is 10, ave w2 and Lw2 is 4:374. So this design is ave w2 -optimal. Moreover, avejf j ¼ 4:181, aveðf 2 Þ ¼ 22:019, f max ¼ 8, Eðf NOD Þ ¼ Lf NOD ¼ 4:374 and thus this design is also Eðf NOD Þ-optimal. For ‘ ¼ 20 we have that ‘ þ 1 ¼ 21 ¼ 9 2 þ 3. C is generated by all cyclic permutations of all columns of the orthogonal arrays A1 ; A2 and the three first permutations of all columns of the A3 array. jCj ¼ 84 ¼ 21 4. It has been checked, by a computer, that no fully aliased columns occur in C. Thus, these columns form matrix G which is the desirable 3-level supersaturated design with 9 runs and ð‘ þ 1Þt ¼ 21 4 ¼ 84 columns. The maximum dependency max w2 is 10, ave w2 and Lw2 is 4:337. So, this design is ave w2 -optimal. Moreover, ave jf j ¼ 4:141, ave ðf 2 Þ ¼ 21:769, f max ¼ 8, Eðf NOD Þ ¼ Lf NOD ¼ 4:337 and thus this design is also Eðf NOD Þ-optimal.
The results obtained for ‘ ¼ 1; 2; . . . ; 26 are given in Table 1.
3. The results In this section, we applied the method described previously and we come up with many new Eðf NOD Þ-optimal supersaturated designs with three and four levels. 3.1. Three-level supersaturated designs Here, we present the results we have found using the method described above and evaluating functions over F n3 .
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Table 2 Orthogonal columns X T1 X T2 X T3 Y T1 Y T2 Y T3 Y T4 Y T5 Y T6 X T4 X T5 X T6
0 1 1 0 0 0 0 0 0 2 2 2
0 0 0 0 2 0 0 0 2 0 1 1
0 1 0 2 1 1 1 0 1 1 1 1
0 1 2 2 1 2 0 1 2 0 0 2
0 2 2 0 2 1 1 1 0 1 0 2
1 0 0 0 2 2 2 2 0 2 0 0
1 0 1 1 0 0 0 0 1 1 1 2
0 0 1 0 0 2 2 2 1 0 2 2
0 0 2 1 0 1 2 1 2 2 1 2
1 0 2 1 0 2 1 0 0 0 0 1
1 1 1 1 1 1 2 1 0 1 0 0
2 1 2 1 1 2 1 2 0 2 2 0
2 2 0 1 2 2 0 2 0 1 2 1
1 1 2 2 0 0 2 1 1 1 2 2
1 2 1 2 0 1 1 1 1 1 1 0
1 2 2 2 0 2 1 1 2 0 0 0
1 1 0 0 1 0 1 2 2 0 1 2
2 0 0 0 1 1 1 0 2 1 0 1
2 1 0 1 2 0 2 0 0 2 1 0
0 2 0 1 2 1 2 0 1 2 2 1
1 2 0 0 1 2 0 1 1 0 2 1
0 2 1 1 1 0 2 0 2 0 2 0
2 1 1 2 1 0 0 1 0 1 2 0
2 2 1 2 2 0 1 2 1 2 1 1
2 0 1 0 0 1 0 2 1 2 0 1
2 0 2 2 2 1 0 2 2 0 1 0
2 2 2 2 2 2 2 2 2 2 0 2
When p ¼ 2 we obtain supersaturated designs with n ¼ 9 runs using the functions given in Example 3 and Theorem 2. The parameters of these three-level supersaturated designs are presented in Table 1, while the designs can be obtained from linear and quadratic functions using the ‘ value given in the same table. When p ¼ 3 we obtain supersaturated designs with n ¼ 27 runs. Using the orthogonal columns X 1 , X 2 , X 3 , Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 , X 4 , X 5 , X 6 , we obtain the four orthogonal arrays HðX 1 ; X 2 ; X 3 Þ, Q1 ðY 1 ; Y 2 ; Y 3 Þ, Q1 ðY 4 ; Y 5 ; Y 6 Þ, and HðX 4 ; X 5 ; X 6 Þ (Table 2). By the method given in a previous section and Theorem 2, we obtain many three-level supersaturated designs with the parameters presented in Table 3. The designs can be constructed from linear and quadratic functions using the ‘ value given in the same table. For 1p‘p8 we obtain designs with ðmax w2 ; f max Þ ¼ ð14; 18Þ, for 9p‘p26 we obtain designs with ðmax w2 ; f max Þ ¼ ð17:333; 20Þ, for 10p‘p80 we obtain designs with ðmax w2 ; f max Þ ¼ ð21:333; 20Þ, and for 81p‘p107 we obtain designs with ðmax w2 ; f max Þ ¼ ð26; 24Þ. 3.2. Four-level supersaturated designs When q ¼ 4, the elements of GFð4Þ are polynomials with coefficients over GFð2Þ and the desirable supersaturated designs are constructed using linear and quadratic functions over F n4 . Example 5. Consider q ¼ 4 and p ¼ 2. In this case GFð4Þ ¼ F 2 ½z =ðz2 þ z þ 1Þ. The functions are HðX 1 ; X 2 Þ ¼ fX 1 ; X 2 ; X 1 þ X 2 ; zX 1 þ X 2 ; ð1 þ zÞX 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ zX 1 þ X 2 ; X 21 þ ð1 þ zÞX 1 þ X 2 g; Q1 ðX 1 ; X 2 Þ ¼ fX 1 ; X 21 þ X 2 ; X 21 þ X 1 þ X 2 ; X 21 þ zX 1 þ X 2 ; X 21 þ ð1 þ zÞX 1 þ X 2 g: HðX 1 ; X 2 Þ is an OAð16; 5; 4; 2Þ when the functions are evaluated at F 16 4 ; so does Q1 ðX 1 ; X 2 Þ. The column juxtaposition of HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ forms a supersaturated design on 4 levels,
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Table 3 Three-level Eðf NOD Þ-optimal supersaturated designs with n ¼ 27 runs and m columns ‘
m
ave w2
ave jf j
ave ðf 2 Þ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 71 72 73 74 75 76 77 78 79 80 81 82
26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325 338 351 364 377 390 403 416 429 442 455 468 936 949 962 975 988 1001 1014 1027 1040 1053 1066 1079
4.105 5.406 6.060 6.447 6.707 6.891 7.029 7.138 7.226 7.296 7.353 7.402 7.446 7.485 7.519 7.549 7.574 7.597 7.618 7.637 7.654 7.669 7.683 7.696 7.708 7.719 7.726 7.733 7.740 7.747 7.753 7.759 7.764 7.770 7.775 7.872 7.874 7.876 7.877 7.879 7.880 7.882 7.883 7.885 7.886 7.887 7.888
38.511 50.575 56.730 60.390 62.791 64.458 65.707 66.673 67.450 68.076 68.617 69.080 69.490 69.854 70.173 70.461 70.708 70.928 71.123 71.297 71.451 71.590 71.718 71.839 71.947 72.046 72.125 72.201 72.275 72.344 72.408 72.469 72.529 72.583 72.633 73.569 73.583 73.597 73.610 73.622 73.634 73.646 73.658 73.670 73.681 73.685 73.688
6.480 8.526 9.529 10.125 10.519 10.800 11.010 11.172 11.302 11.408 11.497 11.571 11.635 11.691 11.739 11.782 11.820 11.854 11.884 11.912 11.937 11.960 11.981 12.000 12.018 12.034 12.050 12.064 12.077 12.090 12.101 12.112 12.122 12.132 12.141 12.302 12.304 12.306 12.308 12.310 12.312 12.314 12.316 12.318 12.319 12.321 12.323
Eðf NOD Þ 2.160 2.842 3.176 3.375 3.506 3.600 3.670 3.724 3.767 3.803 3.832 3.857 3.878 3.897 3.913 3.927 3.940 3.951 3.961 3.970 3.979 3.986 3.993 4.000 4.006 4.011 4.016 4.021 4.025 4.030 4.034 4.037 4.041 4.044 4.047 4.100 4.101 4.102 4.102 4.103 4.104 4.104 4.105 4.106 4.106 4.107 4.107
‘
m
ave w2
ave jf j
ave ðf 2 Þ
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 90 91 92 93 94 95 96 97 98 99 100 101
481 494 507 520 533 546 559 572 585 598 611 624 637 650 663 676 689 702 715 728 741 754 767 780 793 806 819 832 845 858 871 884 897 910 923 1183 1196 1209 1222 1235 1248 1261 1274 1287 1300 1313 1326
7.780 7.784 7.789 7.793 7.797 7.800 7.804 7.807 7.810 7.813 7.816 7.819 7.822 7.825 7.827 7.830 7.832 7.835 7.837 7.840 7.842 7.845 7.847 7.849 7.851 7.853 7.856 7.858 7.860 7.862 7.863 7.865 7.867 7.869 7.871 7.896 7.897 7.898 7.899 7.899 7.900 7.901 7.902 7.902 7.903 7.904 7.904
72.680 72.726 72.771 72.815 72.857 72.896 72.935 72.973 73.010 73.045 73.078 73.108 73.138 73.166 73.194 73.220 73.246 73.272 73.292 73.310 73.329 73.349 73.368 73.386 73.403 73.418 73.434 73.450 73.466 73.481 73.496 73.511 73.526 73.540 73.555 73.711 73.714 73.717 73.721 73.724 73.727 73.731 73.734 73.738 73.742 73.745 73.749
12.150 12.158 12.166 12.173 12.180 12.187 12.194 12.200 12.205 12.211 12.216 12.222 12.226 12.231 12.236 12.240 12.244 12.248 12.252 12.256 12.259 12.263 12.266 12.270 12.273 12.276 12.279 12.282 12.284 12.287 12.290 12.292 12.295 12.297 12.299 12.335 12.336 12.338 12.339 12.340 12.342 12.343 12.344 12.345 12.346 12.348 12.349
Eðf NOD Þ 4.050 4.053 4.055 4.058 4.060 4.062 4.064 4.066 4.068 4.070 4.072 4.074 4.075 4.077 4.078 4.080 4.081 4.083 4.084 4.085 4.086 4.087 4.089 4.090 4.091 4.092 4.093 4.094 4.095 4.095 4.096 4.097 4.098 4.099 4.100 4.111 4.112 4.112 4.113 4.113 4.114 4.114 4.114 4.115 4.115 4.116 4.116
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Table 3 (continued ) ‘
m
ave w2
ave jf j
ave ðf 2 Þ
83 84 85 86 87 88 89
1092 1105 1118 1131 1144 1157 1170
7.889 7.890 7.891 7.892 7.893 7.894 7.895
73.690 73.693 73.696 73.699 73.702 73.705 73.708
12.324 12.326 12.328 12.329 12.331 12.332 12.334
Eðf NOD Þ 4.108 4.108 4.109 4.110 4.110 4.111 4.111
‘
m
ave w2
ave jf j
ave ðf 2 Þ
102 103 104 105 106 107
1339 1352 1365 1378 1391 1404
7.905 7.906 7.906 7.907 7.907 7.908
73.752 73.755 73.759 73.762 73.765 73.768
12.350 12.351 12.352 12.353 12.354 12.355
Eðf NOD Þ 4.116 4.117 4.117 4.117 4.118 4.118
Table 4 Four-level Eðf NOD Þ-optimal supersaturated designs with n ¼ 16 runs, m columns and ave w2 =Eðf NOD Þ ‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
‘
m
ave jf j
ave ðf 2 Þ
Eðf NOD Þ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165
4.711 6.095 6.705 7.356 7.356 7.539 7.685 7.808 7.909 7.987 8.051 8.106 8.147 8.183 8.213 8.246 8.276 8.303 8.328 8.351 8.370 8.386 8.401 8.415 8.429 8.441 8.451 8.461 8.471 8.480 8.488 8.494
45.689 58.590 64.400 68.240 70.713 72.477 73.841 75.131 76.140 76.929 77.548 78.092 78.506 78.844 79.129 79.287 79.442 79.591 79.742 79.900 8.370 80.112 80.210 80.311 80.405 80.494 80.556 80.626 80.698 80.767 80.835 80.917
5.333 6.857 7.579 8.000 8.276 8.471 8.615 8.727 8.816 8.889 8.949 9.000 9.043 9.081 9.114 9.143 9.169 9.191 9.212 9.231 9.248 9.263 9.277 9.290 9.302 9.313 9.324 9.333 9.342 9.351 9.358 9.366
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320
8.500 8.506 8.512 8.517 8.521 8.525 8.529 8.533 8.538 8.541 8.545 8.548 8.551 8.554 8.557 8.562 8.567 8.571 8.576 8.579 8.583 8.586 8.590 8.593 8.596 8.600 8.603 8.605 8.608 8.610 8.613
80.989 81.055 81.133 81.210 81.273 81.329 81.382 81.452 81.517 81.576 81.625 81.674 81.730 81.781 81.828 81.875 81.92 81.960 81.996 82.026 82.052 82.074 82.098 82.127 82.156 82.182 82.205 82.225 82.241 82.255 82.270
9.373 9.379 9.385 9.391 9.397 9.402 9.407 9.412 9.416 9.421 9.425 9.429 9.432 9.436 9.439 9.443 9.446 9.449 9.452 9.455 9.457 9.460 9.462 9.465 9.467 9.469 9.472 9.474 9.476 9.478 9.480
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2
1Þ n ¼ 4p rows and m columns where m ¼ 2ð4 ð41Þ ¼ 10. However, this design has fully aliased columns since both HðX 1 ; X 2 Þ and Q1 ðX 1 ; X 2 Þ have column X 1 . This problem is solved using the four orthogonal arrays Q1 ðY 1 ; Y 2 Þ, HðX 1 ; X 2 Þ, Q1 ðY 3 ; Y 4 Þ, and HðX 3 ; X 4 Þ, where columns Y 1 ; Y 2 ; X 1 ; X 2 ; Y 3 ; Y 4 ; X 3 ; X 4 are as follows:
Y1T Y2T X 1T X 2T Y3T Y4T X 3T X 4T
0, 0, 1 + z, 0, 1 + z, 1, z , 1, z , z , 1 + z, 0, 1, 1, z , 1 + z 0, z , 1 + z, 1 + z, 0, 1, 1, 0, 0, 1 + z, 1, 1, z , 1 + z, z, z 0, 1, 1 + z, 0, 1, 1, 0, z , 0, z , 1 + z, z, 1, z , 1 + z, 1 + z 0, 1 + z, 1 + z, 1, 0, 1, z , z , 1 + z, 1 + z, 0, 1, z , 0, 1, z 0, 0, 1 + z, 1 + z, 1, z , 1, z , 1 + z, 1, z , 0, 0, 1, z , 1 + z 0, z , 1 + z, 0, 1, 1, z , 1 + z, 1, 0, 0, 1, 1 + z, 1 + z, z, z 0, 1, 1, 0, 1, z , z , z , 1 + z, z, 1 + z, 1 + z, 0, 1, 0, 1 + z 0, 1 + z, 0, z , z , 0, z , 1 + z, 0, 1, 1, 1 + z, 1 + z, 1, 1, z
Using these four pairs of orthogonal columns, the orthogonal arrays derived from these and the permutation Theorem, we obtain many supersaturated designs with parameters given in Table 4. The designs, corresponding to ‘ given in this Table, can be constructed from the above functions and the orthogonal pairs given. For ‘ ¼ 1 we obtain a design with ðmax w2 ; f max Þ ¼ ð18; 14Þ and for ‘ ¼ 2 we obtain a design with ðmax w2 ; f max Þ ¼ ð20; 14Þ. When ‘42 we obtain designs with ðmax w2 ; f max Þ ¼ ð26; 16Þ.
References Addelman, S., Kempthorne, O., 1961. Some main-effect plans and orthogonal arrays of strength two. Ann. Math. Statist. 32, 1167–1176. Aggarwal, M.L., Gupta, S., 2004. A new method of construction of multi-level supersaturated designs. J. Statist. Plann. Inference 121, 127–134. Booth, K.H.V., Cox, D.R., 1962. Some systematic supersaturated designs. Technometrics 4, 489–495. Box, G.E.P., Meyer, R.D., 1986. An analysis for unreplicated fractional factorials. Technometrics 28, 11–18. Butler, N., Mead, R., Eskridge, K.M., Gilmour, S.G., 2001. A general method of constructing Eðs2 Þ-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632. Cheng, C.S., 1997. Eðs2 Þ-optimal supersaturated designs. Statist. Sin. 7, 929–939. Fang, K.T., Lin, D.K.J., Ma, C.X., 2000. On the construction of multi-ilevel supersaturated designs. J. Statist. Plann. Inference 86, 239–252. Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003. Optimal mixed-level supersaturated design. Metrika 58, 279–291. Hedayat, A.S., Sloane, N.J.A., Stufken, J., 1999. Orthogonal Arrays: Theory and Applications. Springer, New York. Li, W.W., Wu, C.F.J., 1997. Columnwise-pairwise algorithms with applications to the construction of supersaturated designs. Technometrics 39, 171–179. Lin, D.K.J., 1993. A new class of supersaturated designs. Technometrics 35, 28–31. Lin, D.K.J., 1995. Generating systematic supersaturated designs. Technometrics 37, 213–225. Liu, Y., Dean, A., 2004. k-circulant supersaturated designs. Technometrics 46, 32–43. Liu, M., Zhang, R., 2000. Construction of Eðs2 Þ optimal supersaturated designs using cyclic BIBDs. J. Statist. Plann. Inference 91, 139–150. Lu, X., Meng, Y., 2000. A new method in the construction of two-level supersaturated designs. J. Statist. Plann. Inference 86, 229–238.
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Lu, X., Sun, Y., 2001. Supersaturated design with more than two levels. Chinese Ann. Math. 22B (2), 183–194. Lu, X., Hu, W., Zheng, Y., 2003. A systematic procedure in the construction of multi-level supersaturated design. J. Statist. Plann. Inference 115, 287–310. Mukerjee, R., Wu, C.F.J., 1995. On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist. 23, 2102–2115. Nguyen, N.K., 1996. An algorithmic approach to constructing supersaturated designs. Technometrics 38, 69–73. Plackett, R.L., Burman, J.P., 1946. The design of optimum multifactorial experiments. Biometrika 33, 303–325. Satterthwaite, F.E., 1959. Random balance experimentation (with discussions). Technometrics 1, 111–137. Tang, B., Wu, C.F.J., 1997. A method for constructing supersaturated designs and its Es2 -optimality. Canad. J. Statist. 25, 191–201. Wu, C.F.J., 1993. Construction of supersaturated designs through partially aliased interactions. Biometrika 80, 661–669. Yamada, S., Lin, D.K.J., 1997. Supersaturated designs including an orthogonal base. Canad. J. Statist. 25, 203–213. Yamada, S., Lin, D.K.J., 1999. Three-level supersaturated designs. Statist. Probab. Lett. 45, 31–39. Yamada, S., Matsui, T., 2002. Optimality of mixed-level supersaturated designs. J. Statist. Plann. Inference 104, 459–468. Yamada, S., Ikebe, Y.T., Hashiguchi, H., Niki, N., 1999. Construction of three-level supersaturated design. J. Statist. Plann. Inference 81, 183–193.