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On multi-level supersaturated designs
Journal of Statistical Planning and Inference 136 (2006) 2805 – 2819 www.elsevier.com/locate/jspi
On multi-level supersaturated designs S. Georgiou, C. Koukouvinos∗ , P. Mantas Department of Mathematics, National Technical University of Athens, Zografou, 15773 Athens, Greece Received 19 November 2003; accepted 8 November 2004 Available online 24 December 2004
Abstract In this paper we present a method of construction E(fNOD )-optimal multi-level supersaturated design with n rows, m columns and the equal occurrence property, from a resolvable balanced incomplete block design. A connection between orthogonal arrays and resolvable balanced incomplete block designs is discussed and some E(fNOD )-optimal multi-level supersaturated designs are provided. © 2004 Elsevier B.V. All rights reserved. MSC: primary 62K10; 62K15; secondary 05B20 Keywords: Supersaturated designs; Factorial designs; Resolvable balanced incomplete block designs; Orthogonal arrays; Dependency; Efficiency
1. Introduction 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 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 ∗ Corresponding author.
E-mail address: [email protected] (C. Koukouvinos). 0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2004.11.002
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systematically and suggested the popular max |s| and E(s 2 )-optimality criteria. Since then, many researchers have considered the construction and the properties of two-level supersaturated designs. Important works include (Lin, 1993; Wu, 1993; Lin, 1995; Nguyen, 1996; Tang and Wu, 1997; Yamada and Lin, 1997; Li and Wu, 1997; Cheng, 1997; Lu and Meng, 2000; Liu and Zhang, 2000; Butler et al., 2001). Multi-level, especially three-level designs, are also useful in industrial and scientific experimentation and are also considered frequently.Yamada and Lin (1999) define a new class of three-level supersaturated designs. A measure for dependency between two columns of a design is defined by 2 statistic which is applied to the hypothesis test in a two-way contingency table. They also define criteria of dependency of a whole design and construct three-level supersaturated designs from two-level supersaturated designs. For supersaturated designs with any number of levels and the property of equal occurrence, Yamada and Matsui (2002) derived a lower bound on 2 dependency. Yamada et al. (1999) construct designs combining isomorphic saturated orthogonal designs to obtain a three-level supersaturated design. The isomorphic saturated orthogonal designs are derived by permutations of the rows of an initial saturated orthogonal design. Fang et al. (2000) discuss multilevel supersaturated designs. They propose some new criteria of comparing multi-level supersaturated designs and they obtain such designs by embedding a saturated orthogonal design into a uniform design. Lu and Sun (2001) propose the E(d 2 ) and max(d 2 ) criteria in evaluating multi-level supersaturated designs. E(d 2 ) and max(d 2 ) criteria are respectively equivalent to avex 2 and maxx 2 criteria proposed by Yamada and Lin (1999). They also provide a lower bound for E(d 2 ) and a necessary condition for E(d 2 ) reaching this lower bound. In the construction of multi-level supersaturated designs they use the same method thatYamada et al. (1999) have used. The designs constructed in this way achieve the lower bound for E(d 2 ). Lu et al. (2003) discuss multi-level supersaturated designs. They introduce a systematic procedure in the construction of multi-level supersaturated designs under the E(d 2 ) and max(d 2 ) criteria proposed by Lu and Sun (2001). They construct supersaturated designs from a factorial design induced from a resolvable balanced incomplete block design. Fang et al. (2003) propose a new criterion, called the E(fNOD ) criterion, for comparing supersaturated designs and obtain a lower bound of E(fNOD ). Aggarwal and Gupta (2004) construct multi-level supersaturated designs based on Galois field theory. This method is a generalization of the method given by Lu and Meng (2000). Yamada and Lin (2002) propose a construction method of mixed-level supersaturated designs consisting of two-level and three-level columns. They show that these newly constructed designs have low dependencies under the 2 statistic. Fang et al. (2002) propose a discrete discrepancy as a measure of uniformity for supersaturated designs, obtain a lower bound of this discrepancy and construct uniform supersaturated designs via resolvable balanced incomplete block designs. The paper is organized as follows. In Section 2 design criteria for comparing supersaturated designs are discussed. Some multi-level supersaturated designs from resolvable balanced incomplete block designs with specific parameters are provided in Section 3. In Section 4, a connection between orthogonal arrays and some resolvable balanced incomplete block designs is discussed, and a procedure of construction multi-level supersaturated designs using orthogonal arrays is proposed. Final, the results are tabulated in Section 5.
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2. Design criteria Let D be a n × m matrix of a factorial design with elements from a set of factor levels {1, 2, . . . , q}. We say that the matrix D has the equal occurrence property if each column contains the same number of {1, 2, . . . , q}. In this case the number n of runs is a multiple of q. A design with the equal occurrence property is called an orthogonal array of strength 2, denoted by Ln (q m ), if in any two columns all their level combinations appear the same number of times. In this case n is a multiple of q p and m(q − 1) (n − 1). When m(q − 1) = (n − 1) the design is called saturated and when m(q − 1) > (n − 1) the design is called supersaturated. A supersaturated design cannot be an orthogonal design, so it is necessary to define measures of departure from orthogonality. For multi-level supersaturated designs with any number q of levels, a measure for dependency between two columns di , dj of the matrix D = [d1 , d2 , . . . , dm ] is given by q (nab (i, j ) − n/q 2 )2 , (di , dj ) = n/q 2 2
(1)
a,b=1
where nab (i, j ) is the number of rows whose values are a, b in the n×2 matrix [di , dj ]. Two columns di , dj are said to be completely dependent if 2 (di , dj )=(q −1)n and independent if 2 (di , dj ) = 0. Criteria of dependency of a whole design D are given by m 2 2 (di , dj ) , (2) ave = 2 1 i
max 2 =
max
1 i
2 (di , dj ).
(3)
For supersaturated designs with any number q of levels and the property of equal occurrence, Yamada and Matsui (2002) derived a lower bound on ave 2 , which is given by ave 2
(q − 1)n[(q − 1)m − n + 1] . (n − 1)(m − 1)
(4)
Using the lower bound L2 , ave 2 -efficiency is defined as L2 ave 2
(5)
.
A design is ave 2 -optimal when ave 2 -efficiency is equal to 1.00. Yamada and Lin (1999) proved that 2 (di , dj ) is essentially the same measure as the squared inner product for two level factors. For multi-level supersaturated designs a measure for dependency between two columns di , dj of the design matrix D = [d1 , d2 , . . . , dm ] with n rows, m columns each having q levels, is given by q n f (di , dj ) = (6) nab (i, j ) − q 2 , a,b=1
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where nab (i, j ) is the number of (i, j )-pairs in (di , dj ). Criteria of dependency of a whole design D = [d1 , d2 , . . . , dm ] with n = pq are given by m f (di , dj ) , (7) ave |f | = 2 1 i
ave (f ) = 2
2
f (di , dj )
1 i
fmax =
max
1 i
m , 2
(8) (9)
f (di , dj ).
Fang et al. (2003) introduced the E(fNOD ) criterion for multi-level supersaturated designs with design matrix D = [d1 , d2 , . . . , dm ], and the property of equal occurrence. For any two columns di , dj of the design matrix D they define i,j
fNOD =
q
nab (i, j ) −
a,b=1
n q2
2 .
The new criterion E(fNOD ) is defined as m i,j . fNOD E(fNOD ) = 2
(10)
(11)
1 i
Fang et al. (2003) proved that n E(fNOD ) =
2 k,l=1,k=l kl
m(m − 1)
+
n n n2 m− − 2, m−1 q q
2 mn n n n n2 E(fNOD ) −1 + m− − 2, (m − 1)(n − 1) q m−1 q q
(12)
(13)
where kl is the number of coincidences between the kth and lth rows. The lower bound of E(fNOD ) can be achieved if and only if = m(n/q − 1)/(n − 1) is a positive integer and all the kl ’s for k = l are equal to . Using the lower bound LfNOD , E(fNOD )-efficiency is defined as LfNOD . E(fNOD )
(14)
It is easy to prove that for any design matrix D with the equal occurrence property the criteria E(fNOD ) and ave 2 satisfy the following relation: E(fNOD ) =
n ave 2 , q2
when q 3.
(15)
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3. E(fNOD )-optimal supersaturated designs from resolvable balanced incomplete block designs In this section we present the method we use to obtain multi-level supersaturated designs with the equal occurrence property using some resolvable balanced incomplete block designs. A balanced incomplete block design (BIBD) with parameters (v, b, r, k, ) denoted by BIBD (v, b, r, k, ) is an arrangement of v treatments belonging to a set = {1, 2, . . . , v}, into b blocks of size k such that each block contains k(< v) distinct treatments, each treatment appears in r blocks, and every pair of treatments appears in exactly blocks. The integers v, b, r, k, are called the parameters of the BIB design and must satisfy the following relations vr = bk, and (v − 1) = r(k − 1). For a BIBD (v, b, r, k, ), we can define an v × b matrix Z = (zij ), where zij =
1, when treatment i occurs in block j, i = 1, . . . , v; j = 1, . . . , b. 0, otherwise,
The matrix Z is called the incidence matrix of the design. The incidence matrix Z satisfy the following equations: ZZ T = (r − )Iv + Jv×v ,
Z Jb = rJ v×b ,
Jv×v Z = k Jv×b ,
(16)
where In is the identity matrix of order n, and J×b is the × b matrix whose elements are 1. A block design BIBD (v, b, r, k, ) is said to be resolvable, denoted by RBIBD (v, b, r, k, ), if its blocks can be grouped into r sets, called parallel classes, each set containing b/r blocks such that every treatment appears in each set precisely once. We can construct a q-level supersaturated design X with n = q p runs and m columns and the equal occurrence property using a RBIBD (q p , mq, m, q p−1 , m(q p−1 − 1)/(q p − 1)). The following theorem is the basis for the construction of the designs that follow, and the proof that the supersaturated designs constructed in this way are E(fNOD )-optimal. Theorem 1. Let Z be the incidence matrix of a RBIBD (q p , mq, m, q p−1 , m(q p−1 − 1)/(q p − 1)). Then there exists a q-level E(fNOD )-optimal supersaturated design X with n = q p runs, m columns and the equal occurrence property. Proof. Each of m parallel classes in the RBIBD, denoted by P1 , P2 , . . . , Pm consists of q disjoint blocks, say 1, 2, . . . , q of size n/q. Each parallel class is transformed into a column of the supersaturated design. In each parallel class each block is transformed into a level 1, 2, . . . , q and the numbers in each block are transformed into runs. In that way, each level appears in any column of the supersaturated design exactly n/q times and the design has the equal occurrence property. Since Z is the incidence matrix of a RBIBD, it is obvious from the definition of a resolvable BIBD that the number of coincidences between any two rows of the constructed supersaturated design is a constant = m(q p−1 − 1)/q p − 1. Substituting the parameters of this theorem in Eq. (12), it results that E(fNOD ) achieves the lower bound in (13) and the supersaturated design X is an E(fNOD )-optimal design.
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Example 1. Take the RBIBD(16, 40, 10, 4, 2) in Table 1. We can construct a supersaturated design in 4-levels with n = 16 rows and m = 10 columns as follows. There are 10 parallel classes in the RBIBD, denoted by P1 , P2 , . . . , P10 . Each parallel class is transformed into a column in the design. In each parallel class each of the four blocks is transformed into a level 1,2,3,4 and the numbers in each block are transformed into runs. For instance, in the parallel class P1 , the block {1, 2, 8, 11} is the first one, so we put 1 in the rows 1,2,8,11 of the first column in the design, the block {6, 10, 13, 14} is the second, so we put 2 in the rows 6,10,13,14 of the first column in the design, and so on. Continuing, we obtain a four-level column, the column 1 in Table 2. In this way, one can construct 10 columns which form a four-level supersaturated design with n = 16 rows and m = 10 columns and the equal occurrence property as shown in Table 2.
Table 1 RBIBD(16, 40, 10, 4, 2) P1
P2
P3
P4
P5
{1, 2, 8, 11} {6, 10, 13, 14} {3, 7, 9, 15} {4, 5, 12, 16}
{6, 8, 15, 16} {5, 9, 10, 11} {2, 3, 4, 13} {1, 7, 12, 14}
{4, 11, 14, 15} {8, 9, 12, 13} {2, 5, 6, 7} {1, 3, 10, 16}
{2, 10, 12, 15} {3, 5, 8, 14} {1, 4, 6, 9} {7, 11, 13, 16}
{1, 5, 13, 15} {4, 7, 8, 10} {2, 9, 14, 16} {3, 6, 11, 12}
P6
P7
P8
P9
P10
{2, 3, 9, 12} {7, 11, 14, 15} {4, 8, 10, 16} {1, 5, 6, 13}
{1, 7, 9, 16} {6, 10, 11, 12} {3, 4, 5, 14} {2, 8, 13, 15}
{5, 12, 15, 16} {9, 10, 13, 14} {3, 6, 7, 8} {1, 2, 4, 11}
{3, 11, 13, 16} {4, 6, 9, 15} {2, 5, 7, 10} {1, 8, 12, 14}
{2, 6, 14, 16} {5, 8, 9, 11} {1, 3, 10, 15} {4, 7, 12, 13}
Table 2 Four-level supersaturated design with n = 16 rows and m = 10 columns derived from RBIBD(16, 40, 10, 4, 2) Row
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1 3 4 4 2 3 1 3 2 1 4 2 2 3 4
4 3 3 3 2 1 4 1 2 2 2 4 3 4 1 1
4 3 4 1 3 3 3 2 2 4 1 2 2 1 1 4
3 1 2 3 2 3 4 2 3 1 4 1 4 2 1 4
1 3 4 2 1 4 2 2 3 2 4 4 1 3 1 3
4 1 1 3 4 4 2 3 1 3 2 1 4 2 2 3
1 4 3 3 3 2 1 4 1 2 2 2 4 3 4 1
4 4 3 4 1 3 3 3 2 2 4 1 2 2 1 1
4 3 1 2 3 2 3 4 2 3 1 4 1 4 2 1
3 1 3 4 2 1 4 2 2 3 2 4 4 1 3 1
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Remark 1. In the case that a RBIBD has repeated parallel classes which lead to completely dependent columns in the supersaturated design, this RBIBD is omitted. 4. E(fNOD )-optimal supersaturated designs from orthogonal arrays Lemma 1. The existence of an orthogonal array A with q p runs, m columns and strength 2, is equivalent to the existence of a resolvable balanced incomplete block design with parameters (q p , mq, m, q p−1 , m(q p−1 − 1)/(q p − 1)). Proof. Let B = [b11 , b12 , . . . , b1q , b21 , b22 , . . . , b2q , . . . , bm1 , bm2 , . . . , bmq ] be the dummy variable representation matrix of A, then ⎡ ⎤ q q q A=⎣ j b1j , j b2j , . . . , j bmj ⎦ j =1
j =1
(17)
(18)
j =1
and B is the incidence matrix of the RBIBD.
Theorem 2. If there exists an orthogonal array A with n runs, m columns in q level, and strength 2, then there exists a q-level E(fNOD )-optimal supersaturated design with n runs and ( + 1) · m columns, for all = 1, 2, . . . , n − 1. Proof. Let Qn be the circulant matrix of order n with first row (0, 1, 0, . . . , 0) and set Sn =QTn , where QTn is the transpose of Qn . Observe that Snn =In . It is easy to see that the n×m matrix defined by Sni A is an orthogonal array of n runs and m columns for any integer i. Select 1n − 1 and set G = [Sn0 A, Sn1 A, . . . , Sn A].
(19)
Matrix G is an n × m( + 1) matrix. Using Lemma 1 and Theorem 1 we have that G is the desirable q-level E(fNOD )-optimal supersaturated design with n runs and ( + 1) · m columns. Examples using Theorem 2 are given in Section 5. Remark 2. According to Theorem 2, one can construct multi-level supersaturated designs by permuting the rows of an orthogonal array, and the SSDs constructed in this way are optimal under the E(fNOD ) and ave 2 criteria. However, these criteria are not enough to prevent the existence of completely dependent columns, or columns with large values of fmax or max 2 . We need suitable permutations of the rows and columns of an orthogonal array, with appropriate parameters, to obtain SSDs without completely dependent columns, and with small values of fmax and ave 2 . The orthogonal arrays which are presented in our paper have been obtained by permuting rows and columns of some already known orthogonal arrays. The original arrays we have used can be found in Hedayat et al. (1999).
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5. The results In this section we present the results we have found using the previously mentioned method. 5.1. Three-level supersaturated designs Here, we present and discuss the results we have found on three-level supersaturated designs. Consider the orthogonal array ⎡
3 ⎢3 ⎢ ⎢1 ⎢ ⎢3 ⎢ A = ⎢1 ⎢ ⎢2 ⎢ ⎢1 ⎣ 2 2
3 2 2 1 3 3 1 1 2
3 2 3 1 1 2 2 3 1
⎤ 3 1⎥ ⎥ 2⎥ ⎥ 2⎥ ⎥ 1⎥ , ⎥ 2⎥ ⎥ 3⎥ ⎦ 1 3
with parameters n = 9, m = 4 with q = 3 levels and strength 2 and let ⎡
0 ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ B = ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎣ 0 0
0 0 0 0 0 1 0 1 1
1 1 0 1 0 0 0 0 0
0 0 0 1 0 0 1 1 0
0 1 1 0 0 0 0 0 1
1 0 0 0 1 1 0 0 0
0 0 0 1 1 0 0 0 1
⎤ 0 0 1 1 0 0⎥ ⎥ 0 1 0⎥ ⎥ 0 1 0⎥ ⎥ 1 0 0⎥ ⎥ 0 1 0⎥ ⎥ 0 0 1⎥ ⎦ 1 0 0 0 0 1
0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0
is the dummy variable representation matrix of A, then B is the incidence matrix of a RBIBD (9, 12, 4, 3, 1). Using Theorem 2 we obtain the three-level E(fNOD )-optimal supersaturated designs with n = 9 runs, m columns, max 2 = 10 and fmax = 8 given in Table 3. In this case, Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. Table 3 Three-level E(fNOD )-optimal supersaturated designs with n = 9 runs, m columns, max 2 = 10, and fmax = 8 m
ave |f |
ave(f 2 )
E(fNOD )
m
ave |f |
ave(f 2 )
E(fNOD )
8 12 16 20
2.357 3.000 3.317 3.516
13.286 16.909 18.500 19.453
2.571 3.273 3.600 3.789
24 28 32 36
3.652 3.746 3.810 3.857
20.087 20.508 20.831 21.086
3.913 4.000 4.065 4.114
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Let A be the three-level orthogonal array with n=27 runs, m=13 columns, and strength 2. To save space we give its transpose AT . ⎡
2 ⎢2 ⎢ ⎢3 ⎢ ⎢3 ⎢ ⎢2 ⎢ ⎢1 ⎢ T A =⎢ ⎢3 ⎢1 ⎢ ⎢3 ⎢ ⎢2 ⎢ ⎢2 ⎢ ⎣2 1
2 2 1 2 2 3 3 2 1 3 1 3 2
1 3 1 3 2 2 2 1 2 2 3 3 2
3 3 2 1 3 1 3 2 2 2 1 2 2
1 3 2 2 2 1 2 2 3 3 2 1 3
3 1 1 1 2 1 1 3 3 1 2 3 2
3 2 2 2 1 2 2 3 3 2 1 3 1
3 3 1 2 3 2 3 1 1 1 2 1 1
2 1 2 2 3 3 2 1 3 1 3 2 2
2 3 1 1 1 2 1 1 3 3 1 2 3
3 3 3 3 3 3 3 3 3 3 3 3 3
2 3 2 3 1 1 1 2 1 1 3 3 1
3 2 3 1 1 1 2 1 1 3 3 1 2
1 1 2 1 1 3 3 1 2 3 2 3 1
3 1 3 2 2 2 1 2 2 3 3 2 1
1 2 3 2 3 1 1 1 2 1 1 3 3
⎤ 2 1 2 1 2 3 2 3 1 1 1 2 1 3 3 1 1 1 2 2 2 1⎥ ⎥ 2 1 3 3 1 2 3 1 1 2 3⎥ ⎥ 1 2 2 1 3 3 1 3 1 3 3⎥ ⎥ 2 1 1 2 3 2 3 1 3 3 1⎥ ⎥ 2 1 3 3 1 3 2 3 3 2 2⎥ ⎥ 3 3 1 2 2 1 2 2 1 1 3⎥ ⎥. 3 3 3 3 3 1 2 2 2 3 2⎥ ⎥ 2 1 2 1 2 1 1 2 3 1 3⎥ ⎥ 1 2 2 1 3 2 2 1 2 3 1⎥ ⎥ 3 3 2 1 1 1 2 2 3 2 1⎥ ⎥ 1 2 1 2 1 1 3 2 1 2 1⎦ 3 3 2 1 1 3 3 3 1 2 2
Using Theorem 2 we obtain the three-level E(fNOD )-optimal supersaturated designs with n = 27 runs and m columns, given in Table 4. Table 4 Three-level E(fNOD )-optimal supersaturated designs with n = 27 runs and m columns m
max 2
ave 2
ave |f |
ave(f 2 )
fmax
E(fNOD )
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
13.333 13.333 16.000 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333
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 3.994 4.000 4.006 4.011
4.203 5.487 6.115 6.490 6.742 6.923 7.057 7.156 7.236 7.302 7.357 7.403 7.444 7.480 7.511 7.538 7.562 7.584 7.603 7.619 7.635 7.650 7.663 7.675 7.686 7.697
38.412 50.348 56.278 59.896 62.315 64.058 65.334 66.322 67.111 67.762 68.318 68.770 69.158 69.495 69.779 70.038 70.271 70.479 70.665 70.830 70.983 71.124 71.256 71.375 71.482 71.580
16 16 18 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
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
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5.2. Four-level supersaturated designs In this section we present and discuss the results we have found on four-level supersaturated designs. Consider the four-level orthogonal array with n = 16 runs, m = 5 columns, and strength 2. To save space we give its transpose AT . ⎡1
⎢4 ⎢ AT = ⎢ 4 ⎣ 3 1
1 3 4 4 2 3 3 3 2 1 3 4 1 3 3 1 2 3 2 3 3 4 2 1 4
3 4 3 4 2
1 1 2 2 2
3 2 2 3 3
2 2 4 1 2
1 2 1 4 4
4 4 2 1 4
2 2 3 4⎤ 3 4 1 1⎥ ⎥ 2 1 1 4⎥ . ⎦ 4 2 1 4 1 3 1 3
Using Theorem 2 we obtain the four-level E(fNOD )-optimal supersaturated designs with n = 16 runs, m columns, max 2 = 20, and fmax = 14 given in Table 5. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. Consider the four-level orthogonal array with n=64 runs, m=21 columns, and strength 2. To save space we give its transpose AT . ⎤ 1412324123412312344133413414124212123413232342312341241234143234 ⎢ 4213442134231213431231434343424244212121214324311212343114333221 ⎥ ⎢ ⎥ ⎢ 4412312341112343223214234413214234341142413232321441114333214322 ⎥ ⎢ ⎥ ⎢ 3132424231223141331331132244234111424424423134211331431432414222 ⎥ ⎥ ⎢ ⎢ 2222224444344433312234221334444313322442114411113311111133322213 ⎥ ⎥ ⎢ ⎢ 3241331324423123144233312312414421442314314232412413424121311232 ⎥ ⎥ ⎢ ⎢ 1333323333344433312221443111114112244221112232234422344414412142 ⎥ ⎥ ⎢ ⎢ 4234121432412312344111324322344114314322231433223214123441413312 ⎥ ⎥ ⎢ ⎢ 1241322413223141331342423313144412331132422422443113142324234411 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 4241314231323114212421244314234434113423243112434213213223123143 ⎥ ⎥ ⎢ AT = ⎢ 3124332134331242112121342213424321321122434342134334121341244144 ⎥ . ⎥ ⎢ ⎢ 1314244231123132423141313134234242342422133121342424124141332431 ⎥ ⎥ ⎢ ⎢ 1423111324323114212443133422414332324312244223141342131412433421 ⎥ ⎥ ⎢ ⎢ 2132412413323114212412311243144133242131242444222431324131343334 ⎥ ⎥ ⎢ ⎢ 1143242341312321412334323233214442114143323241413232223411441214 ⎥ ⎥ ⎢ ⎢ 1431243421231213431213343431244142421214213411224343421321223343 ⎥ ⎢ ⎥ ⎢ 1444432222144411123323223443334422222111221134433344311143313112 ⎥ ⎥ ⎢ ⎢ 3321422341412312344144142143214311232141233214144123332122323443 ⎥ ⎥ ⎢ ⎢ 2234132341212334131424411323214123423144143223232314441244132131 ⎥ ⎥ ⎢ ⎣ 4333311111244422231142224112224134422334333342243322411113341414 ⎦ 3111143333444444444411222221114241122223442223323333211123334311 ⎡
Table 5 Four-level E(fNOD )-optimal supersaturated designs with n = 16 runs, m columns, max 2 = 20, and fmax = 14 m
ave |f |
ave(f 2 )
E(fNOD )
m
ave |f |
ave(f 2 )
E(fNOD )
10 15 20 25 30 35 40 45
4.667 6.038 6.705 7.153 7.453 7.664 7.805 7.903
42.756 55.352 61.663 66.307 69.526 71.704 73.159 74.190
5.333 6.857 7.579 8.000 8.276 8.471 8.615 8.727
50 55 60 65 70 75 80
7.976 8.039 8.096 8.151 8.196 8.234 8.263
74.949 75.585 76.188 76.794 77.279 77.653 77.934
8.816 8.889 8.949 9.000 9.043 9.081 9.114
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Using Theorem 2 we obtain the four-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, max 2 = 62, and fmax = 52, given in Table 6. 5.3. Five-level supersaturated designs In this section we present and discuss the results we have found on five-level supersaturated designs. Consider the five-level orthogonal array with n = 25 runs, m = 6 columns, and strength 2. To save space we give its transpose AT . ⎡5
⎢5 ⎢ ⎢5 T A =⎢ ⎢5 ⎣ 5 5
4 1 5 4 3 2
5 4 4 4 4 4
1 3 4 5 1 2
1 5 1 2 3 4
2 3 5 2 4 1
1 1 2 3 4 5
5 1 1 1 1 1
3 2 5 3 1 4
2 5 2 4 1 3
3 3 1 4 2 5
1 4 5 1 2 3
4 2 1 5 4 3
5 2 2 2 2 2
3 5 3 1 4 2
2 4 1 3 5 2
4 5 4 3 2 1
1 2 3 4 5 1
5 3 3 3 3 3
2 1 3 5 2 4
3 4 2 5 3 1
2 2 4 1 3 5
4 3 2 1 5 4
3 1 4 2 5 3
4⎤ 4⎥ ⎥ 3⎥ ⎥. 2⎥ ⎦ 1 5
Table 6 Four-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, max 2 = 62, and fmax = 52 m
ave 2
ave |f |
ave(f 2 )
E(fNOD )
42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525 546 567 588 609 630 651 672 693
4.683 6.194 6.940 7.385 7.680 7.890 8.048 8.170 8.268 8.348 8.414 8.471 8.519 8.561 8.597 8.629 8.658 8.683 8.706 8.727 8.746 8.763 8.779 8.794 8.807 8.820 8.831 8.842 8.852 8.862 8.870 8.879
9.368 12.474 14.024 14.953 15.575 16.017 16.348 16.602 16.806 16.973 17.112 17.228 17.328 17.414 17.489 17.556 17.615 17.667 17.715 17.759 17.799 17.836 17.869 17.901 17.930 17.956 17.981 18.005 18.026 18.047 18.066 18.084
192.544 254.425 284.801 302.706 314.724 323.223 329.527 334.309 338.106 341.265 343.892 346.096 347.991 349.613 351.017 352.273 353.369 354.325 355.215 356.049 356.795 357.481 358.120 358.716 359.265 359.768 360.239 360.683 361.092 361.482 361.850 362.204
18.732 24.774 27.759 29.538 30.720 31.562 32.192 32.681 33.072 33.391 33.657 33.882 34.075 34.242 34.388 34.517 34.631 34.734 34.826 34.909 34.985 35.054 35.117 35.176 35.229 35.279 35.325 35.368 35.409 35.446 35.481 35.514
m
ave 2
ave |f |
ave(f 2 )
E(fNOD )
714 735 756 777 798 819 840 861 882 903 924 945 966 987 1008 1029 1050 1071 1092 1113 1134 1155 1176 1197 1218 1239 1260 1281 1302 1323 1344
8.886 8.893 8.901 8.907 8.913 8.919 8.925 8.930 8.935 8.940 8.945 8.949 8.953 8.957 8.961 8.965 8.969 8.972 8.975 8.978 8.981 8.984 8.987 8.990 8.993 8.995 8.998 9.000 9.002 9.004 9.007
18.101 18.117 18.132 18.146 18.160 18.172 18.184 18.196 18.207 18.218 18.228 18.238 18.247 18.256 18.265 18.273 18.281 18.288 18.295 18.302 18.308 18.315 18.321 18.326 18.332 18.337 18.342 18.347 18.352 18.357 18.361
362.539 362.859 363.156 363.441 363.710 363.963 364.202 364.432 364.652 364.863 365.060 365.253 365.441 365.615 365.778 365.935 366.084 366.224 366.359 366.488 366.611 366.731 366.846 366.953 367.057 367.157 367.255 367.349 367.441 367.531 367.619
35.546 35.575 35.603 35.629 35.654 35.677 35.700 35.721 35.741 35.761 35.779 35.797 35.813 35.830 35.845 35.860 35.874 35.888 35.901 35.914 35.926 35.938 35.949 35.960 35.970 35.981 35.990 36.000 36.009 36.018 36.027
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Using Theorem 2 we obtain the five-level E(fNOD )-optimal supersaturated designs with n = 25 runs and m columns, given in Table 7. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. 5.4. Seven-level supersaturated designs In this section we present and discuss the results we have found on seven-level supersaturated designs. Consider the seven-level orthogonal array with n = 49 runs, m = 8 columns, and strength 2. To save space we give its transpose AT . ⎡
⎤ 2213553521337657151366325174466371742445624427671 ⎢ 3447732253361744765145415172516556237711326466232 ⎥ ⎢ ⎥ ⎢ 5653515774621624146434733276275157272456243116133 ⎥ ⎢ ⎥ ⎢ 7166361525251574227723351373664451244124167536734 ⎥ AT = ⎢ ⎥. ⎢ 2372144346511454371312676477353752216562714256635 ⎥ ⎢ ⎥ ⎢ 4515627167141334452671224574742353251237631676536 ⎥ ⎣ ⎦ 6721473611471214533267542671431654223675555326437 1234256432731164614556167775127255265343472746331
Table 7 Five-level E(fNOD )-optimal supersaturated designs with n = 25 runs and m columns m
max 2
ave |f |
ave(f 2 )
fmax
E(fNOD )
12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150
30 30 30 30 32 32 32 32 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
7.818 10.261 11.391 12.069 12.527 12.836 13.059 13.235 13.366 13.479 13.569 13.643 13.705 13.757 13.806 13.846 13.884 13.916 13.945 13.973 14.000 14.023 14.047 14.067
117.697 156.209 173.536 184.193 191.835 196.767 200.188 202.980 205.037 206.911 208.336 209.461 210.366 211.095 211.840 212.454 213.043 213.540 213.961 214.408 214.836 215.207 215.578 215.893
20 22 22 22 22 22 22 22 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24
9.091 11.765 13.043 13.793 14.286 14.634 14.894 15.094 15.254 15.385 15.493 15.584 15.663 15.730 15.789 15.842 15.888 15.929 15.966 16.000 16.031 16.058 16.084 16.107
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Using Theorem 2 we obtain the seven-level E(fNOD )-optimal supersaturated designs with n = 49 runs, m columns, and max 2 = 70 given in Table 8. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. 5.5. Eight-level supersaturated designs In this section we present and discuss the results we have found on eight-level supersaturated designs. Consider the eight-level orthogonal array with n = 64 runs, m = 9 columns, and strength 2. To save space we give its transpose AT . ⎡
⎤ 7181338785456465257537814358341262122324876677581413834665621742 ⎢ 5842878221578643786263711241147871425333572581536734652636145486 ⎥ ⎢ ⎥ ⎢ 4144318626746332652341782571783224283856587173835576621841516547 ⎥ ⎢ ⎥ ⎢ 2157378133366517458215141882578258657618634876143222746234735446 ⎥ ⎢ ⎥ T ⎢ A = ⎢ 8162358542536288853774276613834275516237763271458164118516473244 ⎥ ⎥. ⎢ 3176388457226121556852328244657241334473111575664651277783368848 ⎥ ⎢ ⎥ ⎢ 6115368268176654151188467465222233878165258774377837353422144345 ⎥ ⎢ ⎥ ⎣ 1123328871686876355463553136416217465742342472212785485357887641 ⎦ 5138348314816743754626635727165286741581425378726348562178252143
Table 8 Seven-level E(fNOD )-optimal supersaturated designs with n = 49 runs, m columns, and max 2 = 70 m
ave |f |
ave(f 2 )
fmax
E(fNOD )
m
ave |f |
ave(f 2 )
fmax
E(fNOD )
16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200
16.100 21.138 23.597 25.072 26.016 26.690 27.200 27.594 27.907 28.170 28.390 28.580 28.744 28.884 29.005 29.112 29.206 29.291 29.366 29.433 29.493 29.548 29.598 29.644
503.600 662.855 740.645 787.826 816.918 837.940 853.917 866.074 875.801 884.246 891.306 897.549 902.909 907.418 911.235 914.659 917.643 920.326 922.711 924.724 926.579 928.286 929.828 931.195
46 46 46 46 46 46 46 46 46 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
19.600 25.565 28.452 30.154 31.277 32.073 32.667 33.127 33.494 33.793 34.042 34.252 34.432 34.588 34.724 34.844 34.951 35.046 35.132 35.210 35.280 35.344 35.403 35.457
208 216 224 232 240 248 256 264 272 280 288 296 304 312 320 328 336 344 352 360 368 376 384 392
29.686 29.724 29.759 29.793 29.823 29.851 29.878 29.903 29.927 29.950 29.971 29.991 30.011 30.030 30.048 30.066 30.082 30.098 30.113 30.127 30.141 30.154 30.166 30.178
932.413 932.532 934.587 935.588 936.462 937.281 938.066 938.799 939.513 940.173 940.787 941.385 941.990 942.573 943.161 943.717 944.235 944.734 945.212 945.639 946.063 946.470 946.864 947.225
48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
35.507 35.553 35.596 35.636 35.674 35.709 35.741 35.772 35.801 35.828 35.854 35.878 35.901 35.923 35.944 35.963 35.982 36.000 36.017 36.033 36.049 36.064 36.078 36.092
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Table 9 Eight-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, and max 2 = 84 m
ave |f |
ave(f 2 )
fmax
E(fNOD )
m
ave |f |
ave(f 2 )
fmax
E(fNOD )
18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 234 243 252 261 270 279 288 297
21.908 28.507 31.695 33.584 34.841 35.748 36.438 36.970 37.394 37.738 38.021 38.259 38.462 38.638 38.791 38.928 39.048 39.154 39.250 39.338 39.419 39.494 39.562 39.622 39.677 39.728 39.776 39.821 39.865 39.905 39.942 39.977
920.366 1191.601 1322.679 1400.331 1452.648 1490.585 1519.890 1542.525 1560.599 1575.190 1587.070 1597.042 1605.606 1612.985 1619.364 1625.106 1630.145 1634.483 1638.461 1642.159 1645.610 1648.733 1651.584 1654.040 1656.248 1658.254 1660.177 1662.034 1663.860 1665.546 1667.130 1668.598
54 54 54 54 54 54 54 54 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 62 62 62 62 62
26.353 34.462 38.400 40.727 42.264 43.355 44.169 44.800 45.303 45.714 46.056 46.345 46.592 46.806 46.993 47.158 47.304 47.435 47.553 47.660 47.756 47.845 47.926 48.000 48.069 48.132 48.191 48.246 48.297 48.345 48.390 48.432
306 315 324 333 342 351 360 369 378 387 396 405 414 423 432 441 450 459 468 477 486 495 504 513 522 531 540 549 558 567 576
40.010 40.041 40.071 40.100 40.127 40.153 40.177 40.200 40.221 40.241 40.260 40.279 40.298 40.315 40.331 40.347 40.362 40.376 40.390 40.404 40.416 40.428 40.440 40.452 40.463 40.474 40.484 40.494 40.504 40.513 40.523
1669.982 1671.273 1672.554 1673.799 1675.000 1676.111 1677.134 1678.046 1678.913 1679.737 1680.547 1681.341 1682.111 1682.819 1683.490 1684.148 1684.772 1685.368 1685.945 1686.496 1687.017 1687.517 1688.005 1688.477 1688.951 1689.416 1689.868 1690.298 1690.705 1691.085 1691.464
62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62
48.472 48.510 48.545 48.578 48.610 48.640 48.669 48.696 48.721 48.746 48.770 48.792 48.814 48.834 48.854 48.873 48.891 48.908 48.925 48.941 48.957 48.972 48.986 49.000 49.013 49.026 49.039 49.051 49.063 49.074 49.085
Using Theorem 2 we obtain the eight-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, and max 2 = 84 given in Table 9. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. References 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 E(s 2 )-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632. Cheng, C.S., 1997. E(s 2 )-optimal supersaturated designs. Statist. Sinica 7, 929–939.
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Fang, K.T., Lin, D.K.J., Ma, C.X., 2000. On the construction of multi-level supersaturated designs. J. Statist. Plann. Inference 86, 239–252. Fang, K.T., Ge, G., Liu, M.Q., 2002. Uniform supersaturated design and its construction. Sci. China 45, 1080–1088. 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, M., Zhang, R., 2000. Construction of E(s 2 ) 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. 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. Nguyen, N.K., 1996. An algorithmic approach to constructing supersaturated designs. Technometrics 38, 69–73. 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., Lin, D.K.J., 2002. Construction of mixed-level supersaturated design. Metrika 56, 205–214. 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.
On multi-level supersaturated designs S. Georgiou, C. Koukouvinos∗ , P. Mantas Department of Mathematics, National Technical University of Athens, Zografou, 15773 Athens, Greece Received 19 November 2003; accepted 8 November 2004 Available online 24 December 2004
Abstract In this paper we present a method of construction E(fNOD )-optimal multi-level supersaturated design with n rows, m columns and the equal occurrence property, from a resolvable balanced incomplete block design. A connection between orthogonal arrays and resolvable balanced incomplete block designs is discussed and some E(fNOD )-optimal multi-level supersaturated designs are provided. © 2004 Elsevier B.V. All rights reserved. MSC: primary 62K10; 62K15; secondary 05B20 Keywords: Supersaturated designs; Factorial designs; Resolvable balanced incomplete block designs; Orthogonal arrays; Dependency; Efficiency
1. Introduction 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 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 ∗ Corresponding author.
E-mail address: [email protected] (C. Koukouvinos). 0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2004.11.002
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systematically and suggested the popular max |s| and E(s 2 )-optimality criteria. Since then, many researchers have considered the construction and the properties of two-level supersaturated designs. Important works include (Lin, 1993; Wu, 1993; Lin, 1995; Nguyen, 1996; Tang and Wu, 1997; Yamada and Lin, 1997; Li and Wu, 1997; Cheng, 1997; Lu and Meng, 2000; Liu and Zhang, 2000; Butler et al., 2001). Multi-level, especially three-level designs, are also useful in industrial and scientific experimentation and are also considered frequently.Yamada and Lin (1999) define a new class of three-level supersaturated designs. A measure for dependency between two columns of a design is defined by 2 statistic which is applied to the hypothesis test in a two-way contingency table. They also define criteria of dependency of a whole design and construct three-level supersaturated designs from two-level supersaturated designs. For supersaturated designs with any number of levels and the property of equal occurrence, Yamada and Matsui (2002) derived a lower bound on 2 dependency. Yamada et al. (1999) construct designs combining isomorphic saturated orthogonal designs to obtain a three-level supersaturated design. The isomorphic saturated orthogonal designs are derived by permutations of the rows of an initial saturated orthogonal design. Fang et al. (2000) discuss multilevel supersaturated designs. They propose some new criteria of comparing multi-level supersaturated designs and they obtain such designs by embedding a saturated orthogonal design into a uniform design. Lu and Sun (2001) propose the E(d 2 ) and max(d 2 ) criteria in evaluating multi-level supersaturated designs. E(d 2 ) and max(d 2 ) criteria are respectively equivalent to avex 2 and maxx 2 criteria proposed by Yamada and Lin (1999). They also provide a lower bound for E(d 2 ) and a necessary condition for E(d 2 ) reaching this lower bound. In the construction of multi-level supersaturated designs they use the same method thatYamada et al. (1999) have used. The designs constructed in this way achieve the lower bound for E(d 2 ). Lu et al. (2003) discuss multi-level supersaturated designs. They introduce a systematic procedure in the construction of multi-level supersaturated designs under the E(d 2 ) and max(d 2 ) criteria proposed by Lu and Sun (2001). They construct supersaturated designs from a factorial design induced from a resolvable balanced incomplete block design. Fang et al. (2003) propose a new criterion, called the E(fNOD ) criterion, for comparing supersaturated designs and obtain a lower bound of E(fNOD ). Aggarwal and Gupta (2004) construct multi-level supersaturated designs based on Galois field theory. This method is a generalization of the method given by Lu and Meng (2000). Yamada and Lin (2002) propose a construction method of mixed-level supersaturated designs consisting of two-level and three-level columns. They show that these newly constructed designs have low dependencies under the 2 statistic. Fang et al. (2002) propose a discrete discrepancy as a measure of uniformity for supersaturated designs, obtain a lower bound of this discrepancy and construct uniform supersaturated designs via resolvable balanced incomplete block designs. The paper is organized as follows. In Section 2 design criteria for comparing supersaturated designs are discussed. Some multi-level supersaturated designs from resolvable balanced incomplete block designs with specific parameters are provided in Section 3. In Section 4, a connection between orthogonal arrays and some resolvable balanced incomplete block designs is discussed, and a procedure of construction multi-level supersaturated designs using orthogonal arrays is proposed. Final, the results are tabulated in Section 5.
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2. Design criteria Let D be a n × m matrix of a factorial design with elements from a set of factor levels {1, 2, . . . , q}. We say that the matrix D has the equal occurrence property if each column contains the same number of {1, 2, . . . , q}. In this case the number n of runs is a multiple of q. A design with the equal occurrence property is called an orthogonal array of strength 2, denoted by Ln (q m ), if in any two columns all their level combinations appear the same number of times. In this case n is a multiple of q p and m(q − 1) (n − 1). When m(q − 1) = (n − 1) the design is called saturated and when m(q − 1) > (n − 1) the design is called supersaturated. A supersaturated design cannot be an orthogonal design, so it is necessary to define measures of departure from orthogonality. For multi-level supersaturated designs with any number q of levels, a measure for dependency between two columns di , dj of the matrix D = [d1 , d2 , . . . , dm ] is given by q (nab (i, j ) − n/q 2 )2 , (di , dj ) = n/q 2 2
(1)
a,b=1
where nab (i, j ) is the number of rows whose values are a, b in the n×2 matrix [di , dj ]. Two columns di , dj are said to be completely dependent if 2 (di , dj )=(q −1)n and independent if 2 (di , dj ) = 0. Criteria of dependency of a whole design D are given by m 2 2 (di , dj ) , (2) ave = 2 1 i
max 2 =
max
1 i
2 (di , dj ).
(3)
For supersaturated designs with any number q of levels and the property of equal occurrence, Yamada and Matsui (2002) derived a lower bound on ave 2 , which is given by ave 2
(q − 1)n[(q − 1)m − n + 1] . (n − 1)(m − 1)
(4)
Using the lower bound L2 , ave 2 -efficiency is defined as L2 ave 2
(5)
.
A design is ave 2 -optimal when ave 2 -efficiency is equal to 1.00. Yamada and Lin (1999) proved that 2 (di , dj ) is essentially the same measure as the squared inner product for two level factors. For multi-level supersaturated designs a measure for dependency between two columns di , dj of the design matrix D = [d1 , d2 , . . . , dm ] with n rows, m columns each having q levels, is given by q n f (di , dj ) = (6) nab (i, j ) − q 2 , a,b=1
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where nab (i, j ) is the number of (i, j )-pairs in (di , dj ). Criteria of dependency of a whole design D = [d1 , d2 , . . . , dm ] with n = pq are given by m f (di , dj ) , (7) ave |f | = 2 1 i
ave (f ) = 2
2
f (di , dj )
1 i
fmax =
max
1 i
m , 2
(8) (9)
f (di , dj ).
Fang et al. (2003) introduced the E(fNOD ) criterion for multi-level supersaturated designs with design matrix D = [d1 , d2 , . . . , dm ], and the property of equal occurrence. For any two columns di , dj of the design matrix D they define i,j
fNOD =
q
nab (i, j ) −
a,b=1
n q2
2 .
The new criterion E(fNOD ) is defined as m i,j . fNOD E(fNOD ) = 2
(10)
(11)
1 i
Fang et al. (2003) proved that n E(fNOD ) =
2 k,l=1,k=l kl
m(m − 1)
+
n n n2 m− − 2, m−1 q q
2 mn n n n n2 E(fNOD ) −1 + m− − 2, (m − 1)(n − 1) q m−1 q q
(12)
(13)
where kl is the number of coincidences between the kth and lth rows. The lower bound of E(fNOD ) can be achieved if and only if = m(n/q − 1)/(n − 1) is a positive integer and all the kl ’s for k = l are equal to . Using the lower bound LfNOD , E(fNOD )-efficiency is defined as LfNOD . E(fNOD )
(14)
It is easy to prove that for any design matrix D with the equal occurrence property the criteria E(fNOD ) and ave 2 satisfy the following relation: E(fNOD ) =
n ave 2 , q2
when q 3.
(15)
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3. E(fNOD )-optimal supersaturated designs from resolvable balanced incomplete block designs In this section we present the method we use to obtain multi-level supersaturated designs with the equal occurrence property using some resolvable balanced incomplete block designs. A balanced incomplete block design (BIBD) with parameters (v, b, r, k, ) denoted by BIBD (v, b, r, k, ) is an arrangement of v treatments belonging to a set = {1, 2, . . . , v}, into b blocks of size k such that each block contains k(< v) distinct treatments, each treatment appears in r blocks, and every pair of treatments appears in exactly blocks. The integers v, b, r, k, are called the parameters of the BIB design and must satisfy the following relations vr = bk, and (v − 1) = r(k − 1). For a BIBD (v, b, r, k, ), we can define an v × b matrix Z = (zij ), where zij =
1, when treatment i occurs in block j, i = 1, . . . , v; j = 1, . . . , b. 0, otherwise,
The matrix Z is called the incidence matrix of the design. The incidence matrix Z satisfy the following equations: ZZ T = (r − )Iv + Jv×v ,
Z Jb = rJ v×b ,
Jv×v Z = k Jv×b ,
(16)
where In is the identity matrix of order n, and J×b is the × b matrix whose elements are 1. A block design BIBD (v, b, r, k, ) is said to be resolvable, denoted by RBIBD (v, b, r, k, ), if its blocks can be grouped into r sets, called parallel classes, each set containing b/r blocks such that every treatment appears in each set precisely once. We can construct a q-level supersaturated design X with n = q p runs and m columns and the equal occurrence property using a RBIBD (q p , mq, m, q p−1 , m(q p−1 − 1)/(q p − 1)). The following theorem is the basis for the construction of the designs that follow, and the proof that the supersaturated designs constructed in this way are E(fNOD )-optimal. Theorem 1. Let Z be the incidence matrix of a RBIBD (q p , mq, m, q p−1 , m(q p−1 − 1)/(q p − 1)). Then there exists a q-level E(fNOD )-optimal supersaturated design X with n = q p runs, m columns and the equal occurrence property. Proof. Each of m parallel classes in the RBIBD, denoted by P1 , P2 , . . . , Pm consists of q disjoint blocks, say 1, 2, . . . , q of size n/q. Each parallel class is transformed into a column of the supersaturated design. In each parallel class each block is transformed into a level 1, 2, . . . , q and the numbers in each block are transformed into runs. In that way, each level appears in any column of the supersaturated design exactly n/q times and the design has the equal occurrence property. Since Z is the incidence matrix of a RBIBD, it is obvious from the definition of a resolvable BIBD that the number of coincidences between any two rows of the constructed supersaturated design is a constant = m(q p−1 − 1)/q p − 1. Substituting the parameters of this theorem in Eq. (12), it results that E(fNOD ) achieves the lower bound in (13) and the supersaturated design X is an E(fNOD )-optimal design.
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Example 1. Take the RBIBD(16, 40, 10, 4, 2) in Table 1. We can construct a supersaturated design in 4-levels with n = 16 rows and m = 10 columns as follows. There are 10 parallel classes in the RBIBD, denoted by P1 , P2 , . . . , P10 . Each parallel class is transformed into a column in the design. In each parallel class each of the four blocks is transformed into a level 1,2,3,4 and the numbers in each block are transformed into runs. For instance, in the parallel class P1 , the block {1, 2, 8, 11} is the first one, so we put 1 in the rows 1,2,8,11 of the first column in the design, the block {6, 10, 13, 14} is the second, so we put 2 in the rows 6,10,13,14 of the first column in the design, and so on. Continuing, we obtain a four-level column, the column 1 in Table 2. In this way, one can construct 10 columns which form a four-level supersaturated design with n = 16 rows and m = 10 columns and the equal occurrence property as shown in Table 2.
Table 1 RBIBD(16, 40, 10, 4, 2) P1
P2
P3
P4
P5
{1, 2, 8, 11} {6, 10, 13, 14} {3, 7, 9, 15} {4, 5, 12, 16}
{6, 8, 15, 16} {5, 9, 10, 11} {2, 3, 4, 13} {1, 7, 12, 14}
{4, 11, 14, 15} {8, 9, 12, 13} {2, 5, 6, 7} {1, 3, 10, 16}
{2, 10, 12, 15} {3, 5, 8, 14} {1, 4, 6, 9} {7, 11, 13, 16}
{1, 5, 13, 15} {4, 7, 8, 10} {2, 9, 14, 16} {3, 6, 11, 12}
P6
P7
P8
P9
P10
{2, 3, 9, 12} {7, 11, 14, 15} {4, 8, 10, 16} {1, 5, 6, 13}
{1, 7, 9, 16} {6, 10, 11, 12} {3, 4, 5, 14} {2, 8, 13, 15}
{5, 12, 15, 16} {9, 10, 13, 14} {3, 6, 7, 8} {1, 2, 4, 11}
{3, 11, 13, 16} {4, 6, 9, 15} {2, 5, 7, 10} {1, 8, 12, 14}
{2, 6, 14, 16} {5, 8, 9, 11} {1, 3, 10, 15} {4, 7, 12, 13}
Table 2 Four-level supersaturated design with n = 16 rows and m = 10 columns derived from RBIBD(16, 40, 10, 4, 2) Row
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1 3 4 4 2 3 1 3 2 1 4 2 2 3 4
4 3 3 3 2 1 4 1 2 2 2 4 3 4 1 1
4 3 4 1 3 3 3 2 2 4 1 2 2 1 1 4
3 1 2 3 2 3 4 2 3 1 4 1 4 2 1 4
1 3 4 2 1 4 2 2 3 2 4 4 1 3 1 3
4 1 1 3 4 4 2 3 1 3 2 1 4 2 2 3
1 4 3 3 3 2 1 4 1 2 2 2 4 3 4 1
4 4 3 4 1 3 3 3 2 2 4 1 2 2 1 1
4 3 1 2 3 2 3 4 2 3 1 4 1 4 2 1
3 1 3 4 2 1 4 2 2 3 2 4 4 1 3 1
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Remark 1. In the case that a RBIBD has repeated parallel classes which lead to completely dependent columns in the supersaturated design, this RBIBD is omitted. 4. E(fNOD )-optimal supersaturated designs from orthogonal arrays Lemma 1. The existence of an orthogonal array A with q p runs, m columns and strength 2, is equivalent to the existence of a resolvable balanced incomplete block design with parameters (q p , mq, m, q p−1 , m(q p−1 − 1)/(q p − 1)). Proof. Let B = [b11 , b12 , . . . , b1q , b21 , b22 , . . . , b2q , . . . , bm1 , bm2 , . . . , bmq ] be the dummy variable representation matrix of A, then ⎡ ⎤ q q q A=⎣ j b1j , j b2j , . . . , j bmj ⎦ j =1
j =1
(17)
(18)
j =1
and B is the incidence matrix of the RBIBD.
Theorem 2. If there exists an orthogonal array A with n runs, m columns in q level, and strength 2, then there exists a q-level E(fNOD )-optimal supersaturated design with n runs and ( + 1) · m columns, for all = 1, 2, . . . , n − 1. Proof. Let Qn be the circulant matrix of order n with first row (0, 1, 0, . . . , 0) and set Sn =QTn , where QTn is the transpose of Qn . Observe that Snn =In . It is easy to see that the n×m matrix defined by Sni A is an orthogonal array of n runs and m columns for any integer i. Select 1n − 1 and set G = [Sn0 A, Sn1 A, . . . , Sn A].
(19)
Matrix G is an n × m( + 1) matrix. Using Lemma 1 and Theorem 1 we have that G is the desirable q-level E(fNOD )-optimal supersaturated design with n runs and ( + 1) · m columns. Examples using Theorem 2 are given in Section 5. Remark 2. According to Theorem 2, one can construct multi-level supersaturated designs by permuting the rows of an orthogonal array, and the SSDs constructed in this way are optimal under the E(fNOD ) and ave 2 criteria. However, these criteria are not enough to prevent the existence of completely dependent columns, or columns with large values of fmax or max 2 . We need suitable permutations of the rows and columns of an orthogonal array, with appropriate parameters, to obtain SSDs without completely dependent columns, and with small values of fmax and ave 2 . The orthogonal arrays which are presented in our paper have been obtained by permuting rows and columns of some already known orthogonal arrays. The original arrays we have used can be found in Hedayat et al. (1999).
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5. The results In this section we present the results we have found using the previously mentioned method. 5.1. Three-level supersaturated designs Here, we present and discuss the results we have found on three-level supersaturated designs. Consider the orthogonal array ⎡
3 ⎢3 ⎢ ⎢1 ⎢ ⎢3 ⎢ A = ⎢1 ⎢ ⎢2 ⎢ ⎢1 ⎣ 2 2
3 2 2 1 3 3 1 1 2
3 2 3 1 1 2 2 3 1
⎤ 3 1⎥ ⎥ 2⎥ ⎥ 2⎥ ⎥ 1⎥ , ⎥ 2⎥ ⎥ 3⎥ ⎦ 1 3
with parameters n = 9, m = 4 with q = 3 levels and strength 2 and let ⎡
0 ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ B = ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎣ 0 0
0 0 0 0 0 1 0 1 1
1 1 0 1 0 0 0 0 0
0 0 0 1 0 0 1 1 0
0 1 1 0 0 0 0 0 1
1 0 0 0 1 1 0 0 0
0 0 0 1 1 0 0 0 1
⎤ 0 0 1 1 0 0⎥ ⎥ 0 1 0⎥ ⎥ 0 1 0⎥ ⎥ 1 0 0⎥ ⎥ 0 1 0⎥ ⎥ 0 0 1⎥ ⎦ 1 0 0 0 0 1
0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0
is the dummy variable representation matrix of A, then B is the incidence matrix of a RBIBD (9, 12, 4, 3, 1). Using Theorem 2 we obtain the three-level E(fNOD )-optimal supersaturated designs with n = 9 runs, m columns, max 2 = 10 and fmax = 8 given in Table 3. In this case, Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. Table 3 Three-level E(fNOD )-optimal supersaturated designs with n = 9 runs, m columns, max 2 = 10, and fmax = 8 m
ave |f |
ave(f 2 )
E(fNOD )
m
ave |f |
ave(f 2 )
E(fNOD )
8 12 16 20
2.357 3.000 3.317 3.516
13.286 16.909 18.500 19.453
2.571 3.273 3.600 3.789
24 28 32 36
3.652 3.746 3.810 3.857
20.087 20.508 20.831 21.086
3.913 4.000 4.065 4.114
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Let A be the three-level orthogonal array with n=27 runs, m=13 columns, and strength 2. To save space we give its transpose AT . ⎡
2 ⎢2 ⎢ ⎢3 ⎢ ⎢3 ⎢ ⎢2 ⎢ ⎢1 ⎢ T A =⎢ ⎢3 ⎢1 ⎢ ⎢3 ⎢ ⎢2 ⎢ ⎢2 ⎢ ⎣2 1
2 2 1 2 2 3 3 2 1 3 1 3 2
1 3 1 3 2 2 2 1 2 2 3 3 2
3 3 2 1 3 1 3 2 2 2 1 2 2
1 3 2 2 2 1 2 2 3 3 2 1 3
3 1 1 1 2 1 1 3 3 1 2 3 2
3 2 2 2 1 2 2 3 3 2 1 3 1
3 3 1 2 3 2 3 1 1 1 2 1 1
2 1 2 2 3 3 2 1 3 1 3 2 2
2 3 1 1 1 2 1 1 3 3 1 2 3
3 3 3 3 3 3 3 3 3 3 3 3 3
2 3 2 3 1 1 1 2 1 1 3 3 1
3 2 3 1 1 1 2 1 1 3 3 1 2
1 1 2 1 1 3 3 1 2 3 2 3 1
3 1 3 2 2 2 1 2 2 3 3 2 1
1 2 3 2 3 1 1 1 2 1 1 3 3
⎤ 2 1 2 1 2 3 2 3 1 1 1 2 1 3 3 1 1 1 2 2 2 1⎥ ⎥ 2 1 3 3 1 2 3 1 1 2 3⎥ ⎥ 1 2 2 1 3 3 1 3 1 3 3⎥ ⎥ 2 1 1 2 3 2 3 1 3 3 1⎥ ⎥ 2 1 3 3 1 3 2 3 3 2 2⎥ ⎥ 3 3 1 2 2 1 2 2 1 1 3⎥ ⎥. 3 3 3 3 3 1 2 2 2 3 2⎥ ⎥ 2 1 2 1 2 1 1 2 3 1 3⎥ ⎥ 1 2 2 1 3 2 2 1 2 3 1⎥ ⎥ 3 3 2 1 1 1 2 2 3 2 1⎥ ⎥ 1 2 1 2 1 1 3 2 1 2 1⎦ 3 3 2 1 1 3 3 3 1 2 2
Using Theorem 2 we obtain the three-level E(fNOD )-optimal supersaturated designs with n = 27 runs and m columns, given in Table 4. Table 4 Three-level E(fNOD )-optimal supersaturated designs with n = 27 runs and m columns m
max 2
ave 2
ave |f |
ave(f 2 )
fmax
E(fNOD )
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
13.333 13.333 16.000 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333 17.333
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 3.994 4.000 4.006 4.011
4.203 5.487 6.115 6.490 6.742 6.923 7.057 7.156 7.236 7.302 7.357 7.403 7.444 7.480 7.511 7.538 7.562 7.584 7.603 7.619 7.635 7.650 7.663 7.675 7.686 7.697
38.412 50.348 56.278 59.896 62.315 64.058 65.334 66.322 67.111 67.762 68.318 68.770 69.158 69.495 69.779 70.038 70.271 70.479 70.665 70.830 70.983 71.124 71.256 71.375 71.482 71.580
16 16 18 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
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
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5.2. Four-level supersaturated designs In this section we present and discuss the results we have found on four-level supersaturated designs. Consider the four-level orthogonal array with n = 16 runs, m = 5 columns, and strength 2. To save space we give its transpose AT . ⎡1
⎢4 ⎢ AT = ⎢ 4 ⎣ 3 1
1 3 4 4 2 3 3 3 2 1 3 4 1 3 3 1 2 3 2 3 3 4 2 1 4
3 4 3 4 2
1 1 2 2 2
3 2 2 3 3
2 2 4 1 2
1 2 1 4 4
4 4 2 1 4
2 2 3 4⎤ 3 4 1 1⎥ ⎥ 2 1 1 4⎥ . ⎦ 4 2 1 4 1 3 1 3
Using Theorem 2 we obtain the four-level E(fNOD )-optimal supersaturated designs with n = 16 runs, m columns, max 2 = 20, and fmax = 14 given in Table 5. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. Consider the four-level orthogonal array with n=64 runs, m=21 columns, and strength 2. To save space we give its transpose AT . ⎤ 1412324123412312344133413414124212123413232342312341241234143234 ⎢ 4213442134231213431231434343424244212121214324311212343114333221 ⎥ ⎢ ⎥ ⎢ 4412312341112343223214234413214234341142413232321441114333214322 ⎥ ⎢ ⎥ ⎢ 3132424231223141331331132244234111424424423134211331431432414222 ⎥ ⎥ ⎢ ⎢ 2222224444344433312234221334444313322442114411113311111133322213 ⎥ ⎥ ⎢ ⎢ 3241331324423123144233312312414421442314314232412413424121311232 ⎥ ⎥ ⎢ ⎢ 1333323333344433312221443111114112244221112232234422344414412142 ⎥ ⎥ ⎢ ⎢ 4234121432412312344111324322344114314322231433223214123441413312 ⎥ ⎥ ⎢ ⎢ 1241322413223141331342423313144412331132422422443113142324234411 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 4241314231323114212421244314234434113423243112434213213223123143 ⎥ ⎥ ⎢ AT = ⎢ 3124332134331242112121342213424321321122434342134334121341244144 ⎥ . ⎥ ⎢ ⎢ 1314244231123132423141313134234242342422133121342424124141332431 ⎥ ⎥ ⎢ ⎢ 1423111324323114212443133422414332324312244223141342131412433421 ⎥ ⎥ ⎢ ⎢ 2132412413323114212412311243144133242131242444222431324131343334 ⎥ ⎥ ⎢ ⎢ 1143242341312321412334323233214442114143323241413232223411441214 ⎥ ⎥ ⎢ ⎢ 1431243421231213431213343431244142421214213411224343421321223343 ⎥ ⎢ ⎥ ⎢ 1444432222144411123323223443334422222111221134433344311143313112 ⎥ ⎥ ⎢ ⎢ 3321422341412312344144142143214311232141233214144123332122323443 ⎥ ⎥ ⎢ ⎢ 2234132341212334131424411323214123423144143223232314441244132131 ⎥ ⎥ ⎢ ⎣ 4333311111244422231142224112224134422334333342243322411113341414 ⎦ 3111143333444444444411222221114241122223442223323333211123334311 ⎡
Table 5 Four-level E(fNOD )-optimal supersaturated designs with n = 16 runs, m columns, max 2 = 20, and fmax = 14 m
ave |f |
ave(f 2 )
E(fNOD )
m
ave |f |
ave(f 2 )
E(fNOD )
10 15 20 25 30 35 40 45
4.667 6.038 6.705 7.153 7.453 7.664 7.805 7.903
42.756 55.352 61.663 66.307 69.526 71.704 73.159 74.190
5.333 6.857 7.579 8.000 8.276 8.471 8.615 8.727
50 55 60 65 70 75 80
7.976 8.039 8.096 8.151 8.196 8.234 8.263
74.949 75.585 76.188 76.794 77.279 77.653 77.934
8.816 8.889 8.949 9.000 9.043 9.081 9.114
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Using Theorem 2 we obtain the four-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, max 2 = 62, and fmax = 52, given in Table 6. 5.3. Five-level supersaturated designs In this section we present and discuss the results we have found on five-level supersaturated designs. Consider the five-level orthogonal array with n = 25 runs, m = 6 columns, and strength 2. To save space we give its transpose AT . ⎡5
⎢5 ⎢ ⎢5 T A =⎢ ⎢5 ⎣ 5 5
4 1 5 4 3 2
5 4 4 4 4 4
1 3 4 5 1 2
1 5 1 2 3 4
2 3 5 2 4 1
1 1 2 3 4 5
5 1 1 1 1 1
3 2 5 3 1 4
2 5 2 4 1 3
3 3 1 4 2 5
1 4 5 1 2 3
4 2 1 5 4 3
5 2 2 2 2 2
3 5 3 1 4 2
2 4 1 3 5 2
4 5 4 3 2 1
1 2 3 4 5 1
5 3 3 3 3 3
2 1 3 5 2 4
3 4 2 5 3 1
2 2 4 1 3 5
4 3 2 1 5 4
3 1 4 2 5 3
4⎤ 4⎥ ⎥ 3⎥ ⎥. 2⎥ ⎦ 1 5
Table 6 Four-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, max 2 = 62, and fmax = 52 m
ave 2
ave |f |
ave(f 2 )
E(fNOD )
42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525 546 567 588 609 630 651 672 693
4.683 6.194 6.940 7.385 7.680 7.890 8.048 8.170 8.268 8.348 8.414 8.471 8.519 8.561 8.597 8.629 8.658 8.683 8.706 8.727 8.746 8.763 8.779 8.794 8.807 8.820 8.831 8.842 8.852 8.862 8.870 8.879
9.368 12.474 14.024 14.953 15.575 16.017 16.348 16.602 16.806 16.973 17.112 17.228 17.328 17.414 17.489 17.556 17.615 17.667 17.715 17.759 17.799 17.836 17.869 17.901 17.930 17.956 17.981 18.005 18.026 18.047 18.066 18.084
192.544 254.425 284.801 302.706 314.724 323.223 329.527 334.309 338.106 341.265 343.892 346.096 347.991 349.613 351.017 352.273 353.369 354.325 355.215 356.049 356.795 357.481 358.120 358.716 359.265 359.768 360.239 360.683 361.092 361.482 361.850 362.204
18.732 24.774 27.759 29.538 30.720 31.562 32.192 32.681 33.072 33.391 33.657 33.882 34.075 34.242 34.388 34.517 34.631 34.734 34.826 34.909 34.985 35.054 35.117 35.176 35.229 35.279 35.325 35.368 35.409 35.446 35.481 35.514
m
ave 2
ave |f |
ave(f 2 )
E(fNOD )
714 735 756 777 798 819 840 861 882 903 924 945 966 987 1008 1029 1050 1071 1092 1113 1134 1155 1176 1197 1218 1239 1260 1281 1302 1323 1344
8.886 8.893 8.901 8.907 8.913 8.919 8.925 8.930 8.935 8.940 8.945 8.949 8.953 8.957 8.961 8.965 8.969 8.972 8.975 8.978 8.981 8.984 8.987 8.990 8.993 8.995 8.998 9.000 9.002 9.004 9.007
18.101 18.117 18.132 18.146 18.160 18.172 18.184 18.196 18.207 18.218 18.228 18.238 18.247 18.256 18.265 18.273 18.281 18.288 18.295 18.302 18.308 18.315 18.321 18.326 18.332 18.337 18.342 18.347 18.352 18.357 18.361
362.539 362.859 363.156 363.441 363.710 363.963 364.202 364.432 364.652 364.863 365.060 365.253 365.441 365.615 365.778 365.935 366.084 366.224 366.359 366.488 366.611 366.731 366.846 366.953 367.057 367.157 367.255 367.349 367.441 367.531 367.619
35.546 35.575 35.603 35.629 35.654 35.677 35.700 35.721 35.741 35.761 35.779 35.797 35.813 35.830 35.845 35.860 35.874 35.888 35.901 35.914 35.926 35.938 35.949 35.960 35.970 35.981 35.990 36.000 36.009 36.018 36.027
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Using Theorem 2 we obtain the five-level E(fNOD )-optimal supersaturated designs with n = 25 runs and m columns, given in Table 7. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. 5.4. Seven-level supersaturated designs In this section we present and discuss the results we have found on seven-level supersaturated designs. Consider the seven-level orthogonal array with n = 49 runs, m = 8 columns, and strength 2. To save space we give its transpose AT . ⎡
⎤ 2213553521337657151366325174466371742445624427671 ⎢ 3447732253361744765145415172516556237711326466232 ⎥ ⎢ ⎥ ⎢ 5653515774621624146434733276275157272456243116133 ⎥ ⎢ ⎥ ⎢ 7166361525251574227723351373664451244124167536734 ⎥ AT = ⎢ ⎥. ⎢ 2372144346511454371312676477353752216562714256635 ⎥ ⎢ ⎥ ⎢ 4515627167141334452671224574742353251237631676536 ⎥ ⎣ ⎦ 6721473611471214533267542671431654223675555326437 1234256432731164614556167775127255265343472746331
Table 7 Five-level E(fNOD )-optimal supersaturated designs with n = 25 runs and m columns m
max 2
ave |f |
ave(f 2 )
fmax
E(fNOD )
12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150
30 30 30 30 32 32 32 32 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
7.818 10.261 11.391 12.069 12.527 12.836 13.059 13.235 13.366 13.479 13.569 13.643 13.705 13.757 13.806 13.846 13.884 13.916 13.945 13.973 14.000 14.023 14.047 14.067
117.697 156.209 173.536 184.193 191.835 196.767 200.188 202.980 205.037 206.911 208.336 209.461 210.366 211.095 211.840 212.454 213.043 213.540 213.961 214.408 214.836 215.207 215.578 215.893
20 22 22 22 22 22 22 22 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24
9.091 11.765 13.043 13.793 14.286 14.634 14.894 15.094 15.254 15.385 15.493 15.584 15.663 15.730 15.789 15.842 15.888 15.929 15.966 16.000 16.031 16.058 16.084 16.107
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Using Theorem 2 we obtain the seven-level E(fNOD )-optimal supersaturated designs with n = 49 runs, m columns, and max 2 = 70 given in Table 8. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. 5.5. Eight-level supersaturated designs In this section we present and discuss the results we have found on eight-level supersaturated designs. Consider the eight-level orthogonal array with n = 64 runs, m = 9 columns, and strength 2. To save space we give its transpose AT . ⎡
⎤ 7181338785456465257537814358341262122324876677581413834665621742 ⎢ 5842878221578643786263711241147871425333572581536734652636145486 ⎥ ⎢ ⎥ ⎢ 4144318626746332652341782571783224283856587173835576621841516547 ⎥ ⎢ ⎥ ⎢ 2157378133366517458215141882578258657618634876143222746234735446 ⎥ ⎢ ⎥ T ⎢ A = ⎢ 8162358542536288853774276613834275516237763271458164118516473244 ⎥ ⎥. ⎢ 3176388457226121556852328244657241334473111575664651277783368848 ⎥ ⎢ ⎥ ⎢ 6115368268176654151188467465222233878165258774377837353422144345 ⎥ ⎢ ⎥ ⎣ 1123328871686876355463553136416217465742342472212785485357887641 ⎦ 5138348314816743754626635727165286741581425378726348562178252143
Table 8 Seven-level E(fNOD )-optimal supersaturated designs with n = 49 runs, m columns, and max 2 = 70 m
ave |f |
ave(f 2 )
fmax
E(fNOD )
m
ave |f |
ave(f 2 )
fmax
E(fNOD )
16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200
16.100 21.138 23.597 25.072 26.016 26.690 27.200 27.594 27.907 28.170 28.390 28.580 28.744 28.884 29.005 29.112 29.206 29.291 29.366 29.433 29.493 29.548 29.598 29.644
503.600 662.855 740.645 787.826 816.918 837.940 853.917 866.074 875.801 884.246 891.306 897.549 902.909 907.418 911.235 914.659 917.643 920.326 922.711 924.724 926.579 928.286 929.828 931.195
46 46 46 46 46 46 46 46 46 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
19.600 25.565 28.452 30.154 31.277 32.073 32.667 33.127 33.494 33.793 34.042 34.252 34.432 34.588 34.724 34.844 34.951 35.046 35.132 35.210 35.280 35.344 35.403 35.457
208 216 224 232 240 248 256 264 272 280 288 296 304 312 320 328 336 344 352 360 368 376 384 392
29.686 29.724 29.759 29.793 29.823 29.851 29.878 29.903 29.927 29.950 29.971 29.991 30.011 30.030 30.048 30.066 30.082 30.098 30.113 30.127 30.141 30.154 30.166 30.178
932.413 932.532 934.587 935.588 936.462 937.281 938.066 938.799 939.513 940.173 940.787 941.385 941.990 942.573 943.161 943.717 944.235 944.734 945.212 945.639 946.063 946.470 946.864 947.225
48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
35.507 35.553 35.596 35.636 35.674 35.709 35.741 35.772 35.801 35.828 35.854 35.878 35.901 35.923 35.944 35.963 35.982 36.000 36.017 36.033 36.049 36.064 36.078 36.092
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Table 9 Eight-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, and max 2 = 84 m
ave |f |
ave(f 2 )
fmax
E(fNOD )
m
ave |f |
ave(f 2 )
fmax
E(fNOD )
18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 234 243 252 261 270 279 288 297
21.908 28.507 31.695 33.584 34.841 35.748 36.438 36.970 37.394 37.738 38.021 38.259 38.462 38.638 38.791 38.928 39.048 39.154 39.250 39.338 39.419 39.494 39.562 39.622 39.677 39.728 39.776 39.821 39.865 39.905 39.942 39.977
920.366 1191.601 1322.679 1400.331 1452.648 1490.585 1519.890 1542.525 1560.599 1575.190 1587.070 1597.042 1605.606 1612.985 1619.364 1625.106 1630.145 1634.483 1638.461 1642.159 1645.610 1648.733 1651.584 1654.040 1656.248 1658.254 1660.177 1662.034 1663.860 1665.546 1667.130 1668.598
54 54 54 54 54 54 54 54 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 62 62 62 62 62
26.353 34.462 38.400 40.727 42.264 43.355 44.169 44.800 45.303 45.714 46.056 46.345 46.592 46.806 46.993 47.158 47.304 47.435 47.553 47.660 47.756 47.845 47.926 48.000 48.069 48.132 48.191 48.246 48.297 48.345 48.390 48.432
306 315 324 333 342 351 360 369 378 387 396 405 414 423 432 441 450 459 468 477 486 495 504 513 522 531 540 549 558 567 576
40.010 40.041 40.071 40.100 40.127 40.153 40.177 40.200 40.221 40.241 40.260 40.279 40.298 40.315 40.331 40.347 40.362 40.376 40.390 40.404 40.416 40.428 40.440 40.452 40.463 40.474 40.484 40.494 40.504 40.513 40.523
1669.982 1671.273 1672.554 1673.799 1675.000 1676.111 1677.134 1678.046 1678.913 1679.737 1680.547 1681.341 1682.111 1682.819 1683.490 1684.148 1684.772 1685.368 1685.945 1686.496 1687.017 1687.517 1688.005 1688.477 1688.951 1689.416 1689.868 1690.298 1690.705 1691.085 1691.464
62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62
48.472 48.510 48.545 48.578 48.610 48.640 48.669 48.696 48.721 48.746 48.770 48.792 48.814 48.834 48.854 48.873 48.891 48.908 48.925 48.941 48.957 48.972 48.986 49.000 49.013 49.026 49.039 49.051 49.063 49.074 49.085
Using Theorem 2 we obtain the eight-level E(fNOD )-optimal supersaturated designs with n = 64 runs, m columns, and max 2 = 84 given in Table 9. Eq. (15) shows that ave 2 = E(fNOD ), so we tabulate the E(fNOD ) values only. References 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 E(s 2 )-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632. Cheng, C.S., 1997. E(s 2 )-optimal supersaturated designs. Statist. Sinica 7, 929–939.
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