CONSTRUCTION OF OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS

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1 Statistica Sinica 19 (2009), CONSTRUCTION OF OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS Min-Qian Liu and Dennis K. J. Lin Nankai University and The Pennsyvania State University Abstract: Supersaturated designs (SSDs) offer a potentiay usefu way to investigate many factors with ony a few experiments during the preiminary stages of experimentation. Whie the construction and anaysis of symmetrica SSDs have been widey expored, asymmetrica (or mixed-eve) SSDs deserve further investigation. Mixed-eve SSDs can be judged by various criteria. But, justified by existing resuts, the χ 2 criterion proposed by Yamada and Lin (1999) is adopted here. Optimaity resuts for mixed-eve SSDs are provided. A new construction method for χ 2 -optima SSDs is proposed, and we discuss properties of the resuting designs. Many new designs are tabuated for practica use. Key words and phrases: Baanced design, coumn juxtaposition, Kronecker sum, orthogona array. 1. Introduction A supersaturated design (SSD) is essentiay a factoria design whose run size is insufficient for estimating a the main effects represented by the design matrix. In an industria or scientific experiment, if many factors are to be investigated (e.g. in a screening study) and the experiment is expensive to conduct, economic considerations may compe the adoption of an SSD. The data coected by SSDs are typicay anayzed under the assumption of effect sparsity, i.e., the response of interest depends mainy on a few dominant or active factors, and the interactions and the effects of the remaining factors are reativey negigibe. SSDs were introduced by Box (1959), but not studied further unti the appearance of the work by Lin (1991, 1993) and Wu (1993). Since then there has been a arge number of papers on this subject, for exampe, Xu and Wu (2005), Georgiou, Koukouvinos and Mantas (2006), Yamada, Matsui, Matsui, Lin and Tahashi (2006) Zhang, Zhang and Liu (2007) Liu, Liu and Zhang (2007), Chen and Liu (2008) and Nguyen and Cheng (2008). Various fieds of research may benefit from the use of SSDs, incuding computer and medica experiments (Lin (1995)), industria and engineering experiments (Wu (1993), Lin (1999, 2003) and Nguyen (1996)). To set the issues, consider a study conducted by Nguyen and Cheng (2008) to examine the factors affecting the therma performance of project homes. They

2 198 MIN-QIAN LIU AND DENNIS K. J. LIN needed an SSD with 16 runs and 18 two-eve factors. Motivated by their study, suppose the factors are as foows: (1) wa insuation (R1, R1.5 or R2); (2) roof insuation (R2.5, R3 or R3.5); (3) foor insuation (R0, R0.5 or R1); (4) foor type (timber, eather or tie); (5) wa type (brick veneer, cavity or concrete); (6) north gass (5% or 20%); (7) east gass (5% or 15%); (8) west gass (5% or 15%); (9) south gass (5% or 15%); (10) north binds (yes or no); (11) east binds (yes or no); (12) west binds (yes or no); (13) south binds (yes or no); (14) north eave overhang (20% or 100%); (15) east eave overhang (20% or 70%); (16) west eave overhang (20% or 70%); and (17) south eave overhang (20% or 100%). The number of homes that can be used for this study is 12. Then we need an SSD with 12 runs, 5 three-eve factors and 12 two-eve factors. For the purpose of screening the active factors and keeping the prices of these homes comparabe, it is further asked that (i) any two-eve factor and three-eve factor be orthogona to each other; (ii) each home have six factors at the ow eve and six at the high eve for a 12 two-eve factors; (iii) for any two homes, either they take the same eve on each of the three-eve factors and different eves on each of the two-eve factors, or they take the same eve on ony one of the three-eve factors and take the eve combinations on the two-eve factors equay often. These constraints make existing mixed-eve SSDs inappicabe. See, for exampe, Deng, Lin and Wang (1996), Liu and Zhang (2001), Yamada and Matsui (2002), Yamada and Lin (2002), Fang, Lin and Liu (2003b), Li, Liu and Zhang (2004), Fang, Ge, Liu and Qin (2004a), Koukouvinos and Mantas (2005), Yamada, Matsui, Matsui, Lin and Tahashi (2006) and Chen and Liu (2008). This paper attempts to provide further optimaity resuts for mixed-eve SSDs and to find a combinatoria soution to the probems exempified above. Section 2 reviews the χ 2 (D) criterion (Yamada and Lin (1999) and Yamada and Matsui (2002)) and other optimaity criteria for mixed-eve SSDs. In particuar, the χ 2 (D) is we justified by some existing resuts, and is adopted as the optimaity criterion for evauating mixed-eve SSDs in this paper. Section 3 presents some optimaity resuts for mixed-eve SSDs. Especiay, optima mixed-eve SSDs are shown to be periodic. These optimaity resuts indicate a feasibe way to construct (neary) χ 2 (D)-optima mixed-eve SSDs. And a new method for constructing them is proposed in Section 4. Many designs constructed from this new method are tabuated in the Appendix. 2. Optimaity Criteria Some definitions and notation are necessary in order to review the optimaity criteria. Thus, a mixed-eve (or asymmetrica) design of n runs and m factors with eves q 1,..., q m, denoted by D(n; q 1,..., q m ), is an n m matrix D = (d ij ) in which the jth coumn takes vaues from a set of q j symbos {0,..., q j 1}. A D(n; q 1,..., q m ) is caed an orthogona array (OA) of strength two, denoted

3 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 199 by L n (q 1,..., q m ), if a possibe eve combinations for any two factors appear equay often. When m j=1 (q j 1) = n 1, the design D(n; q 1,..., q m ) is caed a saturated design; when m j=1 (q j 1) > n 1, the design is caed a supersaturated design, denoted by S(n; q 1,..., q m ). When some q j s are equa, we use the notations D(n; q r 1 1 qr ), L n (q r 1 1 qr ), and S(n; q r 1 1 qr ), respectivey, where j=1 r j = m. Two coumns (or rows) are caed orthogona if they (or their transposes) form an OA of strength two, and caed fuy aiased if one can be obtained from the other by permuting eves. In a design, it is necessary that no coumns are fuy aiased. Throughout the paper, we ony consider baanced (with equa occurrence property) designs in which a eves appear equay often for any coumn χ 2 (D) and E(f NOD ) criteria Let c 1,..., c m be the coumns of an S(n; q 1,..., q m ) design D, and n (ij) uv be the number of (u, v)-pairs in (c i, c j ). Yamada and Lin (1999) defined an index between c i and c j, by anaogy with the χ 2 statistic, χ 2 (c i, c j ) = q i 1 q j 1 u=0 v=0 [n (ij) uv n/(q i q j )] 2, n/(q i q j ) to evauate the dependency of the two coumns. The χ 2 (D) criterion defined by Yamada and Matsui (2002) is to minimize χ 2 (D) = χ 2 (c i, c j ). 1 i<j m Fang, Lin and Liu (2003b) proposed the E(f NOD ) criterion for comparing mixed-eve SSDs from the viewpoint of orthogonaity and uniformity: minimize E(f NOD ) = 2 m(m 1) 1 i<j m χ 2 (c i, c j )n q i q j. Note that the χ 2 (D) considers different weights for factors with different eves, whie E(f NOD ) does not. It is obvious that the χ 2 (D) and E(f NOD ) criteria are equivaent in the symmetric case. It has been shown as we that they are extensions of existing criteria defined for symmetrica SSDs, see Fang, Lin and Liu (2003b) Xu (2003) and Li, Liu and Zhang (2004) for detais Other optimaity criteria and connections There are severa other optimaity criteria for evauating mixed-eve SSDs. One is the generaized minimum aberration criterion deveoped by Ma and Fang

4 200 MIN-QIAN LIU AND DENNIS K. J. LIN (2001) and Xu and Wu (2001). Based on the ANOVA decomposition mode, for a design D(n; q 1,..., q m ), et X j = (x j ik ) be the matrix consisting of a j-factor contrast coefficients for j = 0,..., m. If A j (D) = 1 n 2 the generaized minimum aberration criterion is to sequentiay minimize A j (D) for j = 1,..., m. For a design D = (d ij ), et δ ij (D) = k m n i=1 x j ik q k δ (k) ij, where δ (k) ij = 1 if d ik = d jk, and 0 otherwise; δ ij (D) is caed the natura weighted coincidence number between the ith and jth rows of D. Define the tth power moment to be M t (D) = 2 n(n 1) 1 i<j n 2, [δ ij (D)] t, where t is a positive integer. The minimum moment aberration criterion proposed by Xu (2003) is to sequentiay minimize M t (D) for t = 1,..., m. Hickerne and Liu (2002) deveoped the minimum projection uniformity criterion for a D(n; q 1,..., q m ) design D. Define the t-dimensiona projection discrepancy D (t) (D; K) as the non-negative square root of D 2 (t) (D; K) = 1 n 2 n i,j=1 1 1 < < t m g=1 t ( ) 1 + q g δ ( g) ij. The minimum projection uniformity criterion is to sequentiay minimize D (t) (D; K) for t = 1,..., m. Recenty, Liu, Fang and Hickerne (2006) generaized the χ 2 (D) criterion to the so-caed minimum χ 2 criterion, and investigated the connections among these four criteria. Especiay, their Coroary 1 impies the foowing. Lemma 1. For any S(n; q 1,..., q m ) design D, D 2 (1) (D; K) = A 1(D) = 0, M 1 (D) is minimized, and D 2 (2) (D; K) = A 2(D) = χ2 (D) n = n 1 2n [M 2(D) γ 1 ], where γ 1 is a constant depending on n, m, and the eves q 1,..., q m.

5 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 201 This resut impies that, though A 2 (D), M 2 (D), D(2) 2 (D; K) and χ2 (D) arise from distinct considerations, they are strongy connected: an S(n; q 1,..., q m ) design that minimizes one of these criteria wi minimize them a. This concusion provides a strong justification for using χ 2 (D) as an optimaity criterion for choosing mixed-eve SSDs, and we adopt it as the optimaity criterion for assessing mixed-eve SSDs. 3. Optimaity Properties of the χ 2 (D) Criterion This section provides some optimaity resuts on χ 2 (D) for mixed-eve SSDs χ 2 (D) for the design obtained by coumn juxtaposition For any D(n; q r 1 1 qr ) design D, it is obvious that n i=1,i j δ ij (D) = r k (n q k ). (3.1) Theorem 1 of Li, Liu and Zhang (2004) shows the foowing. Lemma 2. For any D(n; q r 1 1 qr ) design D with m = r k, n χ 2 i,j=1,i j (D)= [δ ij(d)] {[ ] 2 [ ]} r k q k n r k q k + m(m 1) (3.2) 2n 2 n [ ] 2 n(n 1) + 2mn r k q k 2(n 1) 2(n 1) mn(m + n 1) r k q k +. (3.3) 2(n 1) Equaity hods if and ony if δ ij (D) is a constant for a i j. Further, if D is a saturated L n (q s 1 1 qs ), then s k(q k 1) = n 1 and, from Mukerjee and Wu (1995), δ ij (D) = s k 1, for i j, (3.4) which impies that D is χ 2 (D) optima. Theorem 4 of Li, Liu and Zhang (2004) and Coroary 3 of Liu, Fang and Hickerne (2006) show the χ 2 (D) optimaity of mixed-eve SSDs obtained by coumn juxtaposition of two or more SSDs with constant natura weighted coincidence numbers. The theorem beow gives the change in χ 2 (D) vaues when two designs are coumn juxtaposed, in particuar when one of the two designs is a saturated OA.

6 202 MIN-QIAN LIU AND DENNIS K. J. LIN Theorem 1. Suppose D 0 is a D(n; q r 1 1 qr ) and D 1 is a D(n; q s 1 1 qs ). Let D 0 D 1 be the coumn juxtaposition of D 0 and D 1. Then n χ 2 (D 0 D 1 ) = χ 2 (D 0 ) + χ 2 i,j=1,i j (D 1 ) + δ ij(d 0 )δ ij (D 1 ) n [ ][ ] + r k q k s k q k n s k. (3.5) Further, if D 1 has constant δ ij (D 1 ) s for i j, then r k χ 2 (D 0 D 1 ) = χ 2 (D 0 ) + γ 2, (3.6) where γ 2 is a constant depending on n, q i, r i and s i for i = 1,...,. In particuar, if D 1 is a saturated L n (q s 1 1 qs ), then χ 2 (D 1 ) = 0, and (3.7) χ 2 (D 0 D 1 ) = χ 2 (D 0 ) + n r k (q k 1). (3.8) Proof. To derive (3.5), we first express χ 2 (D 0 D 1 ) in terms of δ ij (D 0 D 1 ) based on (3.2), then note that δ ij (D 0 D 1 ) = δ ij (D 0 ) + δ ij (D 1 ). Using the expressions for χ 2 (D 0 ) and χ 2 (D 1 ) in (3.2), (3.5) is obtained foowing engthy but straightforward agebra. Equation (3.6) foows from (3.5) by noting that χ 2 (D 1 ) attains its ower bound in (3.3), and that (3.1) hods for D 0. Equations (3.7) and (3.8) foow directy since (3.4) hods for D 1. Theorem 1 provides a method for constructing χ 2 (D)-optima or neary optima SSDs by coumn-juxtaposing a design D 0 to a saturated OA, or an SSD D 1 with constant δ ij (D 1 ) s for i j. From (3.6) and (3.8), if D 0 is χ 2 (D)-optima, then the resuting design D is χ 2 (D)-optima among those designs obtained by coumn-juxtaposing a design to D 1, which is aso an optima design. Of course, optimaity may not be achievabe among D(n; q (r 1+s 1 ) 1 q (r +s ) ) s; the resuting design does have a χ 2 (D) vaue very cose to the ower bound in Lemma 2, thus it is a neary χ 2 (D)-optima SSD. For exampe, the design D 0 can be seected to be a design with ony one baanced coumn, or with two orthogona (or two neary orthogona) coumns c 1 and c 2, or more generay a χ 2 (D)-optima design. The next subsection shows when the resuting design is optima among designs with the same parameters. Remark 1. Theorem 2 of Yamada and Matsui (2002) showed the χ 2 (D) optimaity of a design D obtained by coumn-juxtaposing severa symmetrica saturated

7 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 203 OA s D 1,..., D s. The χ 2 (D) vaue of this design can be easiy obtained by using (3.8) recursivey since χ 2 (D k ) = 0 for k = 1,..., s, and the optimaity of this design is ensured since δ ij (D) = s δ ij(d k ) and (3.4) hods for each D k. When the saturated OA s are asymmetrica, the resuting design is sti χ 2 (D)-optima based on Theorem 4 of Li, Liu and Zhang (2004), or Coroary 3 of Liu, Fang and Hickerne (2006). If not a the D k s are saturated OA s, the χ 2 (D) optimaity of the resuting design is uncear, but see the subsection beow Periodicity of minimum χ 2 (D) Given n and q 1,..., q, et f(r 1,..., r ) = min{χ 2 (D) : D is an S(n; q r 1 1 q r )}, where designs may have fuy aiased coumns. The foowing resut shows that for certain n, f(r 1,..., r ) is periodic when the number of factors is sufficienty arge. Theorem 2. Suppose a saturated design L n (q s 1 1 qs ) exists. Then for i = 1,...,, there exist positive integers R i such that for r i R i, we have f(r 1 + s 1,..., r + s ) = f(r 1,..., r ) + n r k (q k 1). (3.9) Proof. Denote the right-hand side of (3.3) by LB(n, q 1,..., q, r 1,..., r ). Let g(r 1,..., r ) = f(r 1,..., r ) LB(n, q 1,..., q, r 1,..., r ). Inequaity (3.3) impies that g(r 1,..., r ) 0. From (3.8) we have f(r 1 + s 1,..., r + s ) f(r 1,..., r ) + n r k (q k 1). Then we have 0 g(r 1 + s 1,..., r + s ) g(r 1,..., r ) after some straightforward agebra. Note that since 2n(n 1)f(r 1,..., r ) is an integer, so is 2n(n 1)g(r 1,..., r ). Therefore, for any (t 1,..., t ) satisfying 1 t j s j for j = 1,...,, 2n(n 1)g(ks 1 + t 1,..., ks + t ) is a decreasing integer sequence in k and has a ower bound. There must exist a positive integer k 0 = k 0 (t 1,..., t ) such that, for k k 0, 2n(n 1)g(ks 1 + t 1,..., ks + t ) = 2n(n 1)g(k 0 s 1 + t 1,..., k 0 s + t ). Let K = max{k 0 (t 1,..., t ) : 1 t j s j for j = 1,..., }, and R i = (K + 1)s i, for i = 1,...,. Then for any r i R i with i = 1,...,, g(r 1 + s 1,..., r + s ) = g(r 1,..., r ) or, equivaenty, (3.9) hods.

8 204 MIN-QIAN LIU AND DENNIS K. J. LIN Remark 2. The resut of this theorem can be generaized to the case where the saturated L n (q s 1 1 qs ) is repaced by a design D 1 with constant δ ij (D 1 ) s for i j. This periodicity property of minimum χ 2 (D) heps us understand mixedeve SSDs of arge size; it shows how arger χ 2 (D)-optima mixed-eve SSDs are connected with smaer ones. From (3.8) and (3.9), when the number of factors is sufficient arge, the coumn juxtaposition of a χ 2 (D)-optima design and a saturated OA (as we as a design D 1 with constant δ ij (D 1 ) s for i j) is sti a χ 2 (D)-optima design. When the number of factors is not so arge, the design obtained in this way wi aso be satisfactory according to the χ 2 (D) criterion. The optima SSDs obtained through coumn juxtaposition may contain fuy aiased coumns; the next section presents an expicit construction method that produces optima SSDs without them. 4. Construction of χ 2 -Optima Mixed-Leve SSDs Let G be an additive group of eements, say {0, 1,..., 1}. For a vector A = (a 1,..., a u ) and a matrix B of order v r, both with entries from G, define their Kronecker sum to be the uv r matrix A B = B + a 1. B + a u where B + k is obtained from adding k, over G, to the eements of B. Let 0 q denote a q 1 vector of 0 s and L q = (0, 1,..., q 1). Theorem 3. Suppose p, q, s, t, λ and m 0 are positive integers satisfying, m 0 (s 1) = λ(ps 1), (4.1) pm 0 = pλ + q 2 t. (4.2) Let n 0 = ps and m 1 = q 2 t. If there exist two designs D 0 and D 1 such that (i) D 0 is an S(n 0 ; p m 0 ) design with λ coincidence positions between any two distinct rows, and (ii) D 1 is the transpose of an L m1 (q n 0 ), then D = [0 q p D 0, L q q D 1 ] (4.3) is an S(qn 0 ; p m 0 q m 1 ) design with the natura weighted coincidence number pm 0 between any two distinct rows, hence it is χ 2 (D)-optima. For the symmetric case, D is a χ 2 (D)-optima S(qn 0 ; q m 0+m 1 ) design. Proof. We need ony prove that the resuting design has the natura weighted coincidence number pm 0 between any two distinct rows. For the ith and jth rows

9 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 205 Tabe 1. An S(6; 3 5 ). Tabe 2. Transpose of an L 12 (2 6 ). Run Run Tabe 3. S(12; ) constructed from the two designs in Tabes 1 and 2. Run of D, where 1 i < j qn 0, if j i = 0 mod n 0, they have m 0 coincidence positions at the p-eve factors, and no coincidence position at the q-eve factors, so the natura weighted coincidence number between the two rows is pm 0 ; otherwise, they have λ coincidence positions at the p-eve factors, and qt coincidence positions at the q-eve factors, and then the natura weighted coincidence number between them is pλ+q 2 t. Hence, from (4.2), the natura weighted coincidence number between any two distinct rows of D is pm 0. Exampe 1. Here is an exampe of the construction method using (4.3). It can be verified that p = 3, q = 2, s = 2, t = 3, λ = 1 and m 0 = 5 satisfy (4.1) and (4.2). There exist designs D 0 and D 1, as shown in Tabes 1 and 2 respectivey, where D 0 is a χ 2 (D)-optima S(6; 3 5 ) design obtained by Fang, Ge and Liu (2004b), and D 1 is the transpose of an L 12 (2 6 ) which is found at the website Designs.txt. From these two designs, an S(12; ) is constructed using(4.3); it is shown in Tabe 3. It has the natura weighted coincidence number 15 between any two distinct rows, and thus is a χ 2 (D)-optima SSD.

10 206 MIN-QIAN LIU AND DENNIS K. J. LIN Remark 3. Note that this optima S(12; ) design provides a soution for the motivating exampe in the Introduction, as a the constraints given in the exampe are satisfied. Here are some properties of the designs constructed from Theorem 3. Coroary 1. If D is an S(qn 0 ; p m 0 q m 1 ) constructed through (4.3), then any p-eve and q-eve coumns in D are orthogona to each other. Further, if there are no fuy aiased coumns in D 0 or D 1, then there are no fuy aiased coumns in D. Coroary 2. If D is an S(qn 0 ; p m 0 q m 1 ) constructed through (4.3), then (i) each run has m 1 /q q-eve factors at each of the q eves; (ii) for any two runs, either they take the same eve on each of the p-eve factors and different eves on each of the q-eve factors, or they take the same eve on each of some λ p-eve factors and the eve combinations on the q-eve factors equay often. Based on Theorem 3, we can construct χ 2 (D)-optima mixed-eve SSDs that have the properties described in Coroaries 1 and 2. There are very rich resuts in the iterature for muti-eve SSDs with a constant number of coincidence positions between any two distinct rows. As for OAs, there is a ibrary of over 200 OAs maintained by Dr. N.J.A. Soane ( njas/oadir/). This ibrary has been recenty updated by Dr. W.F. Kuhfed at his OA site ( This site contains a OAs isted in the Appendix of Kuhfed and Tobias (2005), as we as new ones contributed by other authors. Appendix A dispays optima muti-eve SSDs that can be constructed by the new method, whie Appendix B tabuates optima mixed-eve SSDs which can be constructed from existing muti-eve SSDs and OAs. Except for those designs marked with in Appendix A, which can aso be constructed by a method proposed by Georgiou, Koukouvinos and Mantas (2006), a other SSDs in these tabes are apparenty new. Note that there are no fuy aiased coumns in any of the initia SSDs used in the construction, thus if there are no fuy aiased rows in the OAs, the resuting SSDs have no fuy aiased coumns. Further, any p-eve and q-eve coumns are orthogona to each other in any of the resuting SSDs, and these designs possess the properties isted in Coroary 2. Remark 4. The construction method proposed in Theorem 3 can aso be modified to construct E(f NOD )-optima designs. For this case, we need ony change the condition (4.2) to m 0 = λ + qt. Then many E(f NOD )-optima SSDs can be generated through (4.3) from existing muti-eve SSDs with λ coincidence positions between any two distinct rows, and OAs at Dr. Kuhfed s OA website.

11 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 207 Acknowedgements This work is supported by the Program for New Century Exceent Taents in University (NCET ) of China and the NNSF of China Grant Dennis Lin is partiay supported by Research Grant from Smea Coege of Business Administration at Penn State. The authors thank the Co-Editors, an associate editor and the referees for their vauabe comments. Appendix A. Optima S(qn 0 ; q m 0+m 1 ) designs for q > 2. q n 0 m 0 m 1 initia SSD [Source] L m1 (q n 0 ) fina SSD S(6; 3 15 ) [GK2006 ] L 36 (3 6 ) S(18; 3 51 ) S(9; 3 12 ) [Fang, Ge and Liu (2004b)] L 27 (3 9 ) S(27; 3 39 ) S(9; 3 16 ) [Fang, Ge and Liu (2004b)] L 36 (3 9 ) S(27; 3 52 ) S(9; 3 20 ) [Fang, Ge and Liu (2004b)] L 45 (3 9 ) S(27; 3 65 ) S(9; 3 24 ) [Fang, Ge and Liu (2004b)] L 54 (3 9 ) S(27; 3 78 ) S(9; 3 28 ) [Fang, Ge and Liu (2004b)] L 63 (3 9 ) S(27; 3 91 ) S(9; 3 32 ) [GKM2006 ] L 72 (3 9 ) S(27; ) S(9; 3 36 ) [GKM2006] L 81 (3 9 ) S(27; ) S(9; 3 40 ) [GK2006] L 90 (3 9 ) S(27; ) S(9; 3 48 ) [GK2006] L 108 (3 9 ) S(27; ) S(12; 3 33 ) [GK2006] L 72 (3 12 ) S(36; ) S(18; 3 51 ) [New] L 108 (3 18 ) S(54; ) S(27; 3 39 ) [New] L 81 (3 27 ) S(81; ) S(27; 3 52 ) [New] L 108 (3 27 ) S(81; ) S(27; 3 65 ) [New] L 135 (3 27 ) S(81; ) S(8; 4 14 ) [Fang, Ge and Liu (2002a)] L 48 (4 8 ) S(32; 4 62 ) S(8; 4 28 ) [GK2006] L 96 (4 8 ) S(32; ) S(8; 4 42 ) [GK2006] L 144 (4 8 ) S(32; ) S(16; 4 20 ) [FGLQ2004c ] L 64 (4 16 ) S(64; 4 84 ) S(16; 4 30 ) [FGLQ2004c] L 96 (4 16 ) S(64; ) S(16; 4 40 ) [GKM2006] L 128 (4 16 ) S(64; ) S(16; 4 45 ) [GKM2006] L 144 (4 16 ) S(64; ) S(25; 5 30 ) [GKM2006] L 125 (5 25 ) S(125; ) GK2006: Georgiou and Koukouvinos (2006). GKM2006: Georgiou, Koukouvinos and Mantas (2006). Fang, Ge, Liu and Qin (2004c). * Designs can aso be constructed via the method in Georgiou, Koukouvinos and Mantas (2006)

12 208 MIN-QIAN LIU AND DENNIS K. J. LIN Appendix B. Optima S(qn 0 ; p m 0 q m 1 ) designs for p = 2, 3. p q n 0 m 0 m 1 initia SSD [Source] L m1 (q n 0 ) fina SSD S(12; 2 33 ) [Liu and Zhang (2000)] L 36 (3 12 ) S(36; ) S(12; 2 66 ) [Liu and Zhang (2000)] L 72 (3 12 ) S(36; ) S(12; 2 99 ) [Liu and Zhang (2000)] L 108 (3 12 ) S(36; ) S(12; ) [Liu and Zhang (2000)] L 144 (3 12 ) S(36; ) S(16; ) [EGMBT2004 ] L 144 (3 16 ) S(48; ) S(18; 2 68 ) [Liu and Zhang (2000)] L 72 (3 18 ) S(54; ) S(18; ) [Liu and Zhang (2000)] L 108 (3 18 ) S(54; ) S(18; ) [EGMBT2004] L 144 (3 18 ) S(54; ) S(24; 2 69 ) [Liu and Zhang (2000)] L 72 (3 24 ) S(72; ) S(24; ) [Liu and Zhang (2000)] L 144 (3 24 ) S(72; ) S(8; 2 28 ) [Liu and Zhang (2000)] L 32 (4 8 ) S(32; ) S(12; 2 44 ) [Liu and Zhang (2000)] L 48 (4 12 ) S(48; ) S(12; 2 88 ) [Liu and Zhang (2000)] L 96 (4 12 ) S(48; ) S(12; ) [Liu and Zhang (2000)] L 144 (4 12 ) S(48; ) S(16; 2 60 ) [Liu and Zhang (2000)] L 64 (4 16 ) S(64; ) S(16; 2 90 ) [Liu and Zhang (2000)] L 96 (4 16 ) S(64; ) S(16; ) [EGMBT2004] L 128 (4 16 ) S(64; ) S(16; ) [EGMBT2004] L 144 (4 16 ) S(64; ) S(18; ) [EGMBT2004] L 144 (4 18 ) S(72; ) S(24; ) [Liu and Zhang (2000)] L 144 (4 24 ) S(96; ) S(20; 2 95 ) [Liu and Zhang (2000)] L 100 (5 20 ) S(100; ) S(16; ) [EGMBT2004] L 128 (8 16 ) S(128; ) S(6; 3 5 ) [Fang, Ge and Liu (2004b)] L 12 (2 6 ) S(12; ) S(6; 3 10 ) [GK2006 ] L 24 (2 6 ) S(12; ) S(6; 3 15 ) [GK2006] L 36 (2 6 ) S(12; ) S(9; 3 16 ) [Fang, Ge and Liu (2004b)] L 36 (2 9 ) S(18; ) S(9; 3 32 ) [GK2006] L 72 (2 9 ) S(18; ) S(9; 3 48 ) [GK2006] L 108 (2 9 ) S(18; ) S(12; 3 11 ) [Lu, Hu and Zheng (2003)] L 24 (2 12 ) S(24; ) S(12; 3 22 ) [GK2006] L 48 (2 12 ) S(24; ) S(12; 3 33 ) [GK2006] L 72 (2 12 ) S(24; ) S(12; 3 44 ) [GK2006] L 96 (2 12 ) S(24; ) S(12; 3 55 ) [GK2006] L 120 (2 12 ) S(24; ) S(15; 3 28 ) [GK2006] L 60 (2 15 ) S(30; ) S(18; 3 51 ) [New in Appendix A] L 108 (2 18 ) S(36; ) S(27; 3 52 ) [Fang, Lin and Ma (2000)] L 108 (2 27 ) S(54; ) S(12; 3 22 ) [GK2006] L 48 (4 12 ) S(48; ) S(12; 3 44 ) [GK2006] L 96 (4 12 ) S(48; ) EGMBT2004: Eskridge, Gimour, Mead, Buter and Travnicek (2004). GK2006: Georgiou and Koukouvinos (2006)

13 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 209 Appendix B. Optima S(qn 0 ; p m 0 q m 1 ) designs for p > 3. p q n 0 m 0 m 1 initia SSD [Source] L m1 (q n 0) fina SSD S(8; 4 7 ) [Fang, Ge and Liu (2002a)] L 24 (2 8 ) S(16; ) S(8; 4 14 ) [Fang, Ge and Liu (2002a)] L 48 (2 8 ) S(16; ) S(8; 4 21 ) [GK2006 ] L 72 (2 8 ) S(16; ) S(8; 4 28 ) [GK2006] L 96 (2 8 ) S(16; ) S(8; 4 35 ) [GK2006] L 120 (2 8 ) S(16; ) S(8; 4 42 ) [GK2006] L 144 (2 8 ) S(16; ) S(12; 4 11 ) [FGLQ2003a ] L 36 (2 12 ) S(24; ) S(12; 4 22 ) [GK2006] L 72 (2 12 ) S(24; ) S(12; 4 33 ) [GK2006] L 108 (2 12 ) S(24; ) S(16; 4 10 ) [FGLQ2003a] L 32 (2 16 ) S(32; ) S(16; 4 15 ) [FGLQ2003a] L 48 (2 16 ) S(32; ) S(16; 4 20 ) [FGLQ2003a] L 64 (2 16 ) S(32; ) S(16; 4 25 ) [FGLQ2003a] L 80 (2 16 ) S(32; ) S(16; 4 30 ) [FGLQ2003a] L 96 (2 16 ) S(32; ) S(16; 4 35 ) [FGLQ2003a] L 112 (2 16 ) S(32; ) S(16; 4 40 ) [GKM2006] L 128 (2 16 ) S(32; ) S(16; 4 45 ) [GKM2006] L 144 (2 16 ) S(32; ) S(20; 4 19 ) [LFXY2002 ] L 60 (2 20 ) S(40; ) S(24; 4 23 ) [LFXY2002] L 72 (2 24 ) S(48; ) S(8; 4 21 ) [GK2006] L 72 (3 8 ) S(24; ) S(8; 4 42 ) [GK2006] L 144 (3 8 ) S(24; ) S(12; 4 11 ) [FGLQ2003a] L 36 (3 12 ) S(36; ) S(12; 4 22 ) [GK2006] L 72 (3 12 ) S(36; ) S(12; 4 33 ) [GK2006] L 108 (3 12 ) S(36; ) S(16; 4 45 ) [GKM2006] L 144 (3 16 ) S(48; ) S(24; 4 23 ) [LFXY2002] L 72 (3 24 ) S(72; ) S(16; 4 40 ) [GK2006] L 128 (8 16 ) S(128; ) S(10; 5 9 ) [Fang, Ge and Liu (2002b)] L 40 (2 10 ) S(20; ) S(10; 5 18 ) [GK2006] L 80 (2 10 ) S(20; ) S(10; 5 27 ) [GK2006] L 120 (2 10 ) S(20; ) S(15; 5 14 ) [Fang, Ge and Liu (2004b)] L 60 (2 15 ) S(30; ) S(15; 5 28 ) [Fang, Ge and Liu (2004b)] L 120 (2 15 ) S(30; ) S(20; 5 19 ) [LFXY2002] L 80 (2 20 ) S(40; ) S(25; 5 24 ) [Fang, Lin and Ma (2000)] L 100 (2 25 ) S(50; ) S(30; 5 29 ) [LFXY2002] L 120 (2 30 ) S(60; ) S(15; 5 21 ) [Fang, Ge and Liu (2004b)] L 90 (3 15 ) S(45; ) S(10; 5 18 ) [GK2006] L 80 (4 10 ) S(40; ) S(12; 6 11 ) [Lu, Hu and Zheng (2003)] L 60 (2 12 ) S(24; ) S(12; 6 22 ) [GK2006] L 120 (2 12 ) S(24; ) S(24; 6 23 ) [Lu, Hu and Zheng (2003)] L 120 (2 24 ) S(48; ) S(18; 6 17 ) [Lu, Hu and Zheng (2003)] L 90 (3 18 ) S(54; ) S(14; 7 13 ) [Fang, Ge and Liu (2002b)] L 84 (2 14 ) S(28; ) S(28; 7 9 ) [Fang, Ge and Liu (2002b)] L 56 (2 28 ) S(56; ) S(64; 8 18 ) [GKM2006] L 128 (2 64 ) S(128; ) GK2006: Georgiou and Koukouvinos (2006). FGLQ2003a: Fang, Ge, Liu and Qin (2003a). LFXY2002: Lu, Fang, Xu and Yin (2002).

14 210 MIN-QIAN LIU AND DENNIS K. J. LIN References Box, G. E. P. (1959). Discussion on Random baance experimentation by F. E. Satterthwaite. Technometrics 1, Chen, J. and Liu, M. Q. (2008). Optima mixed-eve k-circuant supersaturated designs. J. Statist. Pann. Inference 138, Eskridge, K. M., Gimour, S. G., Mead, R., Buter, N. A. and Travnicek, D. A. (2004). Large supersaturated designs. J. Stat. Comput. Simu. 74, Deng, L. Y., Lin, D. K. J. and Wang, J. N. (1996). Marginay over saturated designs. Comm. Statist. Theory Methods 25, Fang, K. T., Ge, G. N. and Liu, M. Q. (2002a). Construction of E(f NOD )-optima supersaturated designs via Room squares. Cacutta Statist. Assoc. Bu. 52, Fang, K. T., Ge, G. N. and Liu, M. Q. (2002b). Uniform supersaturated design and its construction. Sci. China Ser. A 45, Fang, K. T., Ge, G. N., Liu, M. Q. and Qin, H. (2003a). Construction of minimum generaized aberration designs. Metrika 57, Fang, K. T., Ge, G. N., Liu, M. Q. and Qin, H. (2004a). Combinatoria constructions for optima supersaturated designs. Discrete Math. 279, Fang, K. T., Ge, G. N. and Liu, M. Q. (2004b). Construction of optima supersaturated designs by the packing method. Sci. China Ser. A 47, Fang, K. T., Ge, G. N., Liu, M. Q. and Qin, H. (2004c). Construction of uniform designs via super-simpe resovabe t-designs. Uti. Math. 66, Fang, K. T., Lin, D. K. J. and Liu, M. Q. (2003b). Optima mixed-eve supersaturated design. Metrika 58, Fang, K. T., Lin, D. K. J. and Ma, C. X. (2000). On the construction of muti-eve supersaturated designs. J. Statist. Pann. Inference 86, Georgiou, S. and Koukouvinos, C. (2006). Muti-eve k-circuant supersaturated designs. Metrika 64, Georgiou, S., Koukouvinos, C. and Mantas, P. (2006). On muti-eve supersaturated designs. J. Statist. Pann. Inference 136, Hickerne, F. J. and Liu, M. Q. (2002). Uniform designs imit aiasing. Biometrika 89, Koukouvinos, C. and Mantas, P. (2005). Construction of some E(f NOD ) optima mixed-eve supersaturated designs. Statist. Probab. Lett. 74, Kuhfed, W. F. and Tobias, R. D. (2005). Large factoria designs for product engineering and market research appications. Technometrics 47, Li, P. F., Liu, M. Q. and Zhang, R. C. (2004). Some theory and the construction of mixed-eve supersaturated designs. Statist. Probab. Lett. 69, Lin, D. K. J. (1991). Systematic supersaturated designs. Working paper No. 264, Coege of Business Administration, University of Tennessee. Lin, D. K. J. (1993). A new cass of supersaturated designs. Technometrics 35, Lin, D. K. J. (1995). Generating systematic supersaturated designs. Technometrics 37, Lin, D. K. J. (1999). Supersaturated designs. Encycopedia of Statistica Science (ed. S. Kotz), John Wiey & Sons, New York. Lin, D. K. J. (2003). Industria experimentation for screening. Chapter 2 in Handbook of Statistics (Edited by R. Khattree and C. R. Rao). North Hoand, New York.

15 OPTIMAL MIXED-LEVEL SUPERSATURATED DESIGNS 211 Liu, M. Q., Fang, K. T. and Hickerne, F. J. (2006). Connections among different criteria for asymmetrica fractiona factoria designs. Statist. Sinica 16, Liu, M. Q. and Zhang, R. C. (2000). Construction of E(s 2 ) optima supersaturated designs using cycic BIBDs. J. Statist. Pann. Inference 91, Liu, M. Q. and Zhang, R. C. (2001). Construction of mixed-eve supersaturated designs. Chinese J. App. Probab. Statist. 17, Liu, Y. K., Liu, M. Q. and Zhang, R. C. (2007). Construction of muti-eve supersaturated design via Kronecker product. J. Statist. Pann. Inference 137, Lu, X., Fang, K. T., Xu, Q. and Yin, J. X. (2002). Baance pattern and BP-optima factoria designs. Technica Report MATH-324, Hong Kong Baptist University. Lu, X., Hu, W. and Zheng, Y. (2003). A systematica procedure in the construction of muti-eve supersaturated designs. J. Statist. Pann. Inference 115, Ma, C. X. and Fang, K. T. (2001). A note on generaized aberration factoria designs. Metrika 53, Mukerjee, R. and Wu, C. F. J. (1995). On the existence of saturated and neary saturated asymmetrica orthogona arrays. Ann. Statist. 23, Nguyen, N-K. (1996). An agorithmic approach to constructing supersaturated designs. Technometrics 38, Nguyen, N-K. and Cheng, C. S. (2008). New E(s 2 )-optima supersaturated designs constructed from incompete bock designs. Technometrics 50, Wu, C. F. J. (1993). Construction of supersaturated designs through partiay aiased interactions. Biometrika 80, Xu, H. (2003). Minimum moment aberration for nonreguar designs and supersaturated designs. Statist. Sinica 13, Xu, H. and Wu, C. F. J. (2001). Generaized minimum aberration for asymmetrica fractiona factoria designs. Ann. Statist. 29, Xu, H. and Wu, C. F. J. (2005). Construction of optima muti-eve supersaturated designs. Ann. Statist. 33, Yamada, S. and Lin, D. K. J. (1999). Three-eve supersaturated designs. Statist. Prob. Lett. 45, Yamada, S. and Lin, D. K. J. (2002). Construction of mixed-eve supersaturated design. Metrika 56, Yamada, S. and Matsui, T. (2002). Optimaity of mixed-eve supersaturated designs. J. Statist. Pann. Inference 104, Yamada, S., Matsui, M., Matsui, T., Lin, D. K. J. and Tahashi, T. (2006). A genera construction method for mixed-eve supersaturated design. Comput. Statist. Data Ana. 50, Zhang, Q. Z., Zhang, R. C. and Liu, M. Q. (2007). A method for screening active effects in supersaturated designs. J. Statist. Pann. Inference 137, Department of Statistics, Schoo of Mathematica Sciences and LPMC, Nankai University, Tianjin , China. E-mai: mqiu@nankai.edu.cn Department of Suppy Chain and Information Systems, The Pennsyvania State University, University Park, PA 16802, U.S.A. E-mai: DKL5@psu.edu (Received January 2007; accepted Juy 2007)