- Email: [email protected]
Some D-optimal saturated designs for 3×m2×m3 factorials
Journal of Statistical Planning and Inference 136 (2006) 2820 – 2830 www.elsevier.com/locate/jspi
Some D-optimal saturated designs for 3×m2×m3 factorials St.A. Chatzopoulos, F. Kolyva-Machera∗ Faculty of Sciences, Department of Mathematics, Section of Statistics and O.R., Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Received 10 July 2002; accepted 18 October 2004
Abstract This paper extends the results concerning D-optimal saturated main effect designs for 2 × m2 × m3 to 3 × m2 × m3 factorials, when 3 m2 6 and m3 m2 . © 2004 Elsevier B.V. All rights reserved. MSC: 62K05 Keywords: D-optimal designs; Saturated factorial designs
1. Introduction An experimental design is said to be saturated if all degrees of freedom are consumed by the estimation of parameters, leaving no degrees of freedom for error variance estimation. Saturated resolution III factorial designs are commonly used in screening experiments, to determine which of many factors affects the measure of pertinent quality characteristics. The purpose of this paper is to give saturated resolution III designs, minimizing the generalized variance of the main effects and the general mean, that is, D-optimal designs. In recent years, there has been a considerable interest in optimal saturated main effect designs. Mukerjee et al. (1986) and Kraft (1990) showed that in the two-factor case all such designs are equivalent with respect to D-optimality. Later Mukerjee and Sinha (1990) considered optimality results on almost saturated main effect designs in the two-factor case. The result was also ∗ Corresponding author. Tel.: +30 310 997954; fax: +30 310 997983.
E-mail address: [email protected] (F. Kolyva-Machera). 0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2004.10.025
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2821
obtained by Pesotan and Raktoe (1988) in the special case for m2 factorials, who also worked on a subclass of m3 factorials. Our result on 3 × 3 × 3 factorial is D-better than Pesotan and Raktoe’s. The first attempt to extend the two factor results to three factors was done by Chatterjee and Mukerjee (1993). They consider 2×m2 ×m3 factorial to derive D-optimal saturated main effect designs. Later Chatterjee and Narasimhan (2002) using techniques from Graph Theory and Combinatorics, claimed about the upper bound of the determinant of the saturated 3 × m2 × m3 factorials when m2 = 2k + 1. In this paper we study some aspects of Doptimality. More specifically we provide alternative proofs for the saturated m1 × m2 and 2 × m2 × m3 designs, and we study the saturated 3 × m2 × m3 designs. Moreover, when this is feasible, we compare our findings with their claim. In the appendix we give explicitly a D-optimal 3 × 4 × m3 factorial, m3 5.
2. Preliminaries Following Chatterjee and Mukerjee (1993) we consider the setup of an m1 × m2 × m3 saturated factorial experiment, involving three factors F1 , F2 and F3 appearing at m1 , m2 and m3 levels, respectively where m3 m2 m1 2, with N = 1 + m1 − 1 + m2 − 1 + m3 − 1 = m1 + m2 + m3 − 2 runs. For 1i 3 let the levels of Fi be denoted by i and coded as 0, 1, . . . , mi − 1. Our interest is to find D-optimal resolution III designs. There are altogether u = m1 m2 m3 treatment combinations denoted by 1 2 3 , that will hereafter be assumed to be lexicographically ordered. Let, for 1i 3, 1i be the mi × 1 vector with each element unity, Ii the identity matrix of order mi , ⊗ denotes the Kronecker product of matrices and Pi be an (mi − 1) × mi −1/2 matrix such that (mi 1i , Pi ) is orthogonal. (A denotes the transpose of matrix A). In a complete experiment, where each treatment combination appears once, the usual fixed effect model under the absence of interactions is Y = W + , where Y is the response vector of the experiment, is a u × 1 vector of uncorrelated random errors with zero mean and the same variance 2 and is the vector of unknown parameters. In our case =(, 1 , 2 , 3 ) , where is the unknown general mean and the elements of the (mi − 1) × 1 vectors i are unknown parameters representing a full set of mutually orthogonal contrasts belonging to the main effects Fi and W = [11 ⊗ 12 ⊗ 13 , W1 , W2 , W3 ], where W1 = P1 ⊗ 12 ⊗ 13 , W2 = 11 ⊗ P2 ⊗ 13 and W3 = 11 ⊗ 12 ⊗ P3 . It is easy to see that the D-optimal design does not depend on the choice of Pi , 1 i 3. Following Mukerjee and Sinha (1990) let X0 = [11 ⊗ 12 ⊗ 13 , X1 , X2 , X3 ], where X1 = I1 ⊗12 ⊗13 , X2 =11 ⊗I2 ⊗13 and X3 =11 ⊗12 ⊗I3 . Consider the u×(m1 +m2 +m3 −2) (1) (1) matrix U which is a submatrix of X0 given by U = [X1 , X2 , X3 ], where for 1 i 2 (1) the matrices Xi are obtained from Xi by deleting the first column of the latter. The u rows of matrix U like those of W , correspond to the lexicographically ordered treatment combinations. Note that matrix U has full column rank. Moreover the columns of U span those of X0 and hence those of W , which also has full column rank. Following Mukerjee and Sinha (1990) one may obtain W = U H , where matrix H is a nonsingular matrix of order m1 + m2 + m3 − 2.
2822 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
For any design d in the class D of the saturated resolution III designs with N = m1 + m2 + m3 − 2 runs, the design matrix is Wd = Ud H , where Ud is a square matrix of order m1 + m2 + m3 − 2 such that for 1j m1 + m2 + m3 − 2 if the jth run in d is given by the treatment combination 1 2 3 then the jth row of Ud is the row of U corresponding to the treatment combination 1 2 3 . A design d is said to be D-optimal in the class D, if it maximizes the quantity det(Wd Wd ). Since matrix H is nonsingular a design is D-optimal if it maximizes the quantity |detU d |, where (1)
(1)
Ud = [Z1 , Z2 , Z3 ].
(2.1)
(1)
(1)
The matrices Zj , 1 j 2 and Z3 are obtained from the matrices Xj and X3 in a similar way as Ud is obtained from U . Definition 2.1. For 1i 2, if the ith factor enters the experiment at level 0 then the (1) corresponding row of the matrix Zi is a row vector with mi − 1 elements zero. On the other hand if the ith factor enters the experiment at level s, 1 s mi − 1, then the (1) corresponding row of the matrix Zi equals the sth row of the identity matrix of order s mi − 1. Let ni , 0 s mi − 1, denote the number of these rows. Similarly, if the third factor enters the experiment at level s, 0 s m3 − 1 then the corresponding row of the matrix Z3 equals to the (s + 1)th row of the identity matrix I3 . It holds that N=
m i −1
nsi .
(2.2)
s=0
Definition 2.2. For 1 i, j 3, i = j and 0 s mi − 1, 0 t mj − 1 let nst ij denote the number of runs where the ith factor appears at level s and the jth factor appears at level t. It holds that mj −1
nsi =
t=0
nst ij ,
ntj =
m i −1 s=0
nst ij
and
N=
m j −1 i −1 m s=0
nst ij .
(2.3)
t=0
Remark 2.1. It holds that nsi 1, 1 i 3, 0 s mi − 1, since the design matrix of a saturated design has full column rank. Remark 2.2. By the choice of the labels for the levels one can always assume without loss of generality (w.l.o.g.) that n0i = max{nsi , 0 s mi − 1}. Lemma 2.1. Consider the saturated m1 × · · · × mi × · · · × mk design d. If mi 2 and nsi = 1 for some 0 s mi − 1 then |detU d | = |detU d |, where d is a saturated m1 × · · · × (mi − 1) × · · · × mk design. Proof. If n0i = 1 then nsi = 1 for all 0 s mi − 1, because n0i = max{nsi }. So, nsi = 1 for (1) some 1s mi − 1. Obviously the sth column, 1 s mi − 1, of Zi contains exactly one 1. Expanding detU d along this column the result follows.
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2823
Corollary 2.1. Consider the saturated m1 × m2 × m3 , m3 m2 m1 2 design d with N = m1 + m2 + m3 − 2 runs. One can easily verify that ns3 = 1 at least w = m3 − (m1 + m2 − 2) times, using the pigeonhole principle. Applying Lemma 2.1 w times for k = 3 we get |detU d | = |detU d |, where d is a saturated m1 × m2 × (m1 + m2 − 2) design. Lemma 2.2. Consider the saturated m1 ×m2 designs e, m2 m1 2 with N =m1 +m2 −1 runs. It holds that |detU e | = 1. Proof. We can verify that ns2 =1, 0 s m2 −1, at least m2 +1−m1 times. Thus, applying Lemma 2.1 for k = 2 at least m2 + 1 − m1 times we get |detU e | = |detU e |, where e is a saturated m1 × (m1 − 1) design, that is (m1 − 1) × m1 with 2(m1 − 1) runs. Wecan continue the reduction until we have a saturated 1 × 2 design, for which Ue = 01 01 . Obviously, |detU e | = 1. This result is the same as that of Mukerjee et al. (1986), Dey and Mukerjee (1999), (Theorem 6.1.1), but the proof is different. Lemma m1 ×m2 ×· · ·×mk design d, it holds that |detU d | m1 −1 s2.3. For any saturated |, d is a saturated m2 × · · · × mk design}. n · max{|detU d s=1 1 Proof. We can evaluate |detU d | expanding matrix Ud along the m1 − 1 columns of the (1) (1) matrix Z1 . Observing that each row of Z1 has at most one 1 and the number of 1s in the (1) sth column of Z1 is ns1 , 1 s m1 −1, the result follows. 1 −1 s m1 −1 0 1 Lemma 2.4. The maximum of the quantity Q= m = s=1 n1 subject to n1 +n1 +· · ·+n1 s 0 1 1 −1 N, and all n1 , 0 s m1 − 1, positive integers is achieved when n1 n1 · · · nm 1 n01 − 1. Proof. Consider any pair of levels 0 s, t m1 −1. Given ns1 +nt1 =k, we have (j/jns1 )ns1 nt1 = (j/jns1 )ns1 (k − ns1 ) = k − 2ns1 , which is positive for all ns1 < k/2, i.e. for all ns1 < nt1 . So in order to maximize Q we have to increase ns1 to the maximum value such that ns1 nt1 , which is either ns1 =nt1 or ns1 =nt1 −1. By applying this to all pairs, the lemma follows. Theorem d be a saturated m1 × m2 × m3 , m3 m2 m1 2 design. If |detU d | = m1 −1 s 2.1. Let 0 n1 · · · nm1 −1 n0 − 1 then d is D-optimal. n and n 1 s=1 1 1 1 1 1 −1 s Proof. Applying Lemma 2.3 for k = 3 and using Lemma 2.2 we get |detU d | m s=1 n1 , m1 −1 s 0 m1 −1 s 1 2 . The quantity where N = s=0 n1 , n1 = max{n1 } and w.l.o.g n1 n1 · · · n1 |detU d | is maximized when equality holds in the above inequality.According to Lemma 2.4, 1 −1 s m1 −1 0 1 for the D-optimal design it holds that |detU d | = m s=1 n1 , where n1 n1 · · · n1 0 n1 − 1. Corollary 2.2. Consider the saturated 2 × m2 × m3 design d, m3 m2 2. We can prove Theorem 3.1 of Chatterjee and Mukerjee (1993), applying Corollary 2.1 and Lemma 2.3.
2824 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
We have |detU d | = |detU d |n11 · max{|detU e |}, where d is a saturated 2 × m2 × m2 design, with N = n01 + n11 = 2m2 runs and e is a saturated m2 × m2 design. Moreover from Remark 2.2 we have n01 n11 , that is n01 m2 and n11 m2 . Finally |detU d | = |detU d | m2 , because |detU e | = 1, as proved by Lemma 2.2. In what follows the quantity |detU e | is omitted when we evaluate |detU d | expanding along the elements of the two first columns of Ud , because |detU e | = 1.
3. Main results Lemma 3.1. Consider the saturated 3 × m2 × m3 design d, m2 3, m3 m2 + 2. It holds that |detU d | = |detU d |, where d is a saturated 3 × m2 × (m2 + 1) design. Proof. Applying Corollary 2.1 for m1 = 3 the result follows.
Theorem 3.1. Let d ∗ be a saturated 3 × m2 × m3 design, m3 > m2 3. If ∗
∗
∗
(1) |detU d ∗ | = n11 · n21 − (m2 + 1 − n01 ), ∗ ∗ ∗ ∗ (2) n01 n11 n21 n01 − 1, (3) |detU d ∗ | max{|detU d |}, where d is a saturated 3 × m2 × m2 design, then d ∗ is D-optimal. Proof. According to Lemma 3.1, the optimal D-criterion among saturated 3 × m2 × m3 designs is the same as the optimal D-criterion among saturated 3 × m2 × (m2 + 1) designs. (1) (1) Let Ud =[Z1 , Z2 , Z3 ] be the 2(m2 +1)×2(m2 +1) matrix corresponding to an arbitrary (1) (1) saturated 3 × m2 × (m2 + 1) design d, where Z1 , Z2 and Z3 are 2(m2 + 1) × 2, 2(m2 + 1) × (m2 − 1) and 2(m2 + 1) × (m2 + 1) matrices, respectively. 3 −1 s s It holds that m s=0 n3 = N = 2(m2 + 1). If n3 = 1 for some 0 s m2 then applying Lemma 2.1 we have that |detU d | = |detU d ||detU d ∗ |,
(3.1)
where d is a saturated 3 × m2 × m2 design. On the other hand if ns3 2 (actually ns3 = 2) for all 0 s m2 then each column of Z3 contains exactly two 1s. Let Z1,1 , Z1,2 and Z1,3 be n01 × 2, n11 × 2, n21 × 2 matrices, which each row is [0 0], [1 0] and [0 1], respectively. From Remark 2.2 it holds that (1) n01 = max{ns1 }, 0 s 2. Assume, w.l.o.g., that n01 n11 n21 . One can partition matrix Z1 Z1,1 (1) as Z1 = Z1,2 . Matrix Z3 is a 2(m2 + 1) × (m2 + 1) matrix in which each row contains Z1,3
exactly one 1. So, matrix Z3 has at least (m2 + 1) − n01 = columns, with all elements zero at the first n01 rows and at the other N − n01 rows each column contains two 1s. Assume
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2825
that > 0. Thus, w.l.o.g., matrix Ud can be partitioned inthe following way:
where Z3,1 , Z3,22 , Z3,32 are 2(m2 +1)×(m2 +1−), n11 ×, n21 × matrices respectively, with elements 0 or 1. We consider two cases: Case 1: If at least one of the last columns of Ud has the two 1s at matrix Z3,22 (or Z3,32 ) then by subtracting this column from the first (or the second) column of Ud we have that the number of 1s at the first (or the second) column of Ud becomes n11 −2 (or n21 −2). Expanding Ud along the two first columns and using Lemma 2.2 we have |detU d | (n11 − 2) · n21 (or |detU d | n11 · (n21 − 2)). Case 2: If each of the last columns of Ud contains one 1 at matrix Z3,22 and one 1 at matrix Z3,32 then the column vector which arises by adding these columns, contains 1s at rows corresponding to Z3,22 and 1s at rows corresponding to Z3,32 . By subtracting this vector from the first (or the second) column of Ud and expanding along the two first columns of Ud we get (in both cases) |detU d | (n11 − ) · n21 + | − | · (n21 − 1) = n11 · n21 − = (n11 − 1) · (n21 − 1) + m2 . It holds that (n21 − 2) · n11 (n11 − 2) · n21 because n11 n21 as assumed. Moreover (n11 − 2) · n21 < n11 · n21 − (m2 + 1 − n01 ),
(3.2)
because 2(m2 + 1) = n01 + n11 + n21 and n01 − n11 + 3n21 > 0. On the other hand, function (n11 − 1) · (n21 − 1) + m2 is maximized with respect to all pairs of positive integers (n11 , n21 ) satisfying equality n01 + n11 + n21 = 2(m2 + 1) and inequality n01 n11 n21 ,
when n21 n01 − 1 (see Lemma 2.4).
(3.3)
From (3.1)–(3.3), if > 0, that is m2 + 1 > n01 we have |detU d | |detU d ∗ |, where ∗ ∗ ∗ ∗ ∗ ∗ ∗ |detU d ∗ | = n11 · n21 − (m2 + 1 − n01 ) with n01 n11 n21 n01 − 1. If =m2 +1−n01 0, then n11 +n21 m2 +1 because n01 +n11 +n21 =2(m2 +1). By Lemmas 2.2 and 2.3 we have |detU d |n11 ·n21 . So |detU d |((m2 +1)/2)2 . Consequently, it remains ∗ ∗ ∗ to be proved that ((m2 + 1)/2)2 n11 · n21 − (m2 + 1 − n01 ). Previously in the proof it has ∗ ∗ ∗ ∗ ∗ been noted that n11 · n21 − (m2 + 1 − n01 ) = (n11 − 1) · (n21 − 1) + m2 . So we want to show ∗ ∗ (n11 − 1) · (n21 − 1) ((m2 + 1)/2)2 − m2 = ((m2 − 1)/2)2 . Condition (2) on d ∗ implies ∗ ∗ ∗ ∗ that (n11 − 1) · (n21 − 1)(n01 − 2)2 . So it suffices to show (n01 − 2)2 ((m2 − 1)/2)2 , or ∗ ∗ ∗ n01 −2 (m2 −1)/2, or n01 (m2 +3)/2. Condition (2) also implies that n01 (2m2 +2)/3, so it suffices to show (2m2 + 2)/3 (m2 + 3)/2, which is true for all m2 5. For m2 = 3 ∗ ∗ ∗ and 4 it holds that (n01 , n11 , n21 ) = (3, 3, 2) and (4, 3, 3), respectively, and one can verify ∗ ∗ that (n11 − 1) · (n21 − 1) ((m2 − 1)/2)2 .
2826 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
Theorem 3.2. Consider the saturated 3 × m2 × m2 design d ∗ with m2 4. If ∗
∗
∗
(1) |detU d ∗ | = n11 · n21 − (m2 + 1 − n01 ), ∗ ∗ ∗ ∗ (2) n01 n11 n21 n01 − 1, (3) |detU d ∗ | max{|detU d |}, where d is a saturated 3 × (m2 − 1) × m2 design, then d ∗ is D-optimal. (1)
(1)
Proof. Let Ud = [Z1 , Z2 , Z3 ] be the (2m2 + 1) × (2m2 + 1) matrix which corresponds to an arbitrary saturated 3 × m2 × m2 design d with N = 2m2 + 1 runs. It holds that N = n01 + n11 + n21 = 2m2 + 1, where n01 = max{ns1 }, 0 s 2 and w.l.o.g., n11 n21 . We can partition Ud as
(3.4) (1)
where Z2 and Z3 are (2m2 + 1) × (m2 − 1) and (2m2 + 1) × m2 matrices, with elements 0 or 1, respectively. We can always permute the second and the third factor because both appear at m2 levels. If ns2 = 1 or ns3 = 1 for some 0 s m2 − 1, then applying Lemma 2.1 for k = 3 and i = 2 or 3 we get |detU d | = |detU d ||detU d ∗ |,
(3.5)
where d is a saturated 3 × (m2 − 1) × m2 design. On the other hand if ntj > 1, 2 j 3, 0 t m2 − 1, then n0j = 3 and ntj = 2 for t = 0, mj −1 t nj = N = 2m2 + 1 and n0j = max{ntj }. We consider two cases: because t=0 st Case 1: n12 2 or nst 13 2 for some s 1 and some 0 t m2 − 1. W.l.o.g., we assume that nst 2. It holds that n03 = 3 and nt3 = 2 for t = 0. Then 13 st Case 1A: n13 = 2 for some s 1, and some t 1. It holds that nt3 = 2. So, the (t + 1)th column of matrix Z3 contains two 1s and since nst 13 = 2 these two 1s are at the same rows (1) s with two of the n1 1s of the sth column of Z1 . Then by subtracting the (t + 1)th column (1) of Z3 from the sth column of Z1 and expanding Ud along the first two columns we have
|detU d | (ns1 − 2)ns1 ,
(3.6)
where s = 2 (or 1) if s = 1 (or 2). Case 1B: ns0 13 = 3 for some s 1. Subtracting the first column of Z3 from the sth column (1) of Z1 and expanding Ud along the first two columns we get
|detU d | (ns1 − 3)ns1 . 0 Case 1C: ns0 13 = 2 for some s 1. It holds that n3 = 3 =
(3.7) 2
s0 s=0 n13
by (2.3).
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2827
Case 1C(i): If n00 13 = 1, then following the same procedure as in Case 1B we have
|detU d |(ns1 − 2) · ns1 + | − 1| · ns1 = (ns1 − 1) · ns1 .
(3.8)
s0
Case 1C(ii): If n13 = 1, then in a similar manner we get
|detU d |(ns1 − 2) · ns1 + | − 1| · (ns1 − 1) = ns1 · ns1 − ns1 − 1. For the case 1, comparing inequalities (3.6)–(3.9) for
n11 n21 ,
(3.9)
we get
|detU d | n11 · n21 − n21 .
(3.10)
st Case 2: nst 12 1 and n13 1 for all s 1 and all t. 0t t 0 Case 2A: n0t 12 2 or n13 2 for some t, because nj = 3 and nj = 2 for 1 t m2 − 1, 0t 00 2 j 3. Let, w.l.o.g. n13 =2 (or n13 =3), for some 1 t m2 −1. Consider (3.4); recalling 00 0 definition (2.2) for n0t 13 = 2 (or n13 = 3) we have that two (three) of the first n1 rows of Z3 are
the (t +1)th (or the first) row of I2 (because m3 =m2 ). Matrix Z3 is a (2m2 +1)×m2 matrix in which each row contains exactly one 1. There exist m2 −(n01 −1)= (m2 −(n01 −2)=+1) columns of Z3 (say the last), which have all elements zero at the first n01 rows. At the remaining (2m2 + 1) − n01 rows these columns contain each exactly two 1s, because nt3 = 2 for all 1 t m2 − 1. Assume that > 0. If there exists at least one column of the ( + 1) last columns of Z3 which contains the two 1s at the n11 or at the n21 rows of Ud then obviously 2t n1t 13 = 2 or n13 = 2, 1 t m2 − 1, respectively (case 1A). On the other hand if each of the ( + 1) last columns of Z3 contains one 1 at the n11 rows and one 1 at the n21 rows of Ud then by adding these columns, the resulting column will contain ( + 1) 1s at the n11 rows and ( + 1) 1s at the n21 rows of Ud . Subtracting from the first column of Ud the resulting column of Z3 and expanding along the first two columns we get |detU d | (n11 − ) · n21 + | − | · (n21 − 1)
if n0t 13 = 2, 0 t m2 − 1,
(|detU d | (n11 − ( + 1)) · n21 + | − ( + 1)| · (n21 − 1) if n00 13 = 3).
(3.11) (3.12)
Recalling that = m2 − (n01 − 1), for the case 2A, inequalities (3.11) and (3.12) yield |detU d | n11 · n21 − (m2 − n21 + 1).
(3.13)
Case 2B: n0t n0t 12 1 and 13 1 for all t. 0 00 10 20 It holds that n2 = 3 = 2s=0 ns0 12 by (2.3). So, n12 = 1 and n12 = 1 and n12 = 1, otherwise s0 s0 n12 = 2 or n12 = 3 for some 0 s 2. Recalling Definitions 2.1 and 2.2 for n00 12 = 1 we have (1) that there exists one row (say the first) of the (2m2 + 1) × (m2 − 1) matrix Z2 which has (1) 0 0 all its elements zero at the first n1 rows of Ud . The rest n1 − 1 first rows of Z2 will be the rows of the matrix Im2 −1 each one time, otherwise n0t 12 > 1. Consequently there exists (1) (m2 − 1) − (n01 − 1) = m2 − n01 (say 1) columns of Z2 (say that last) which have all 0 elements zero at the first n1 rows and at the rest (2m2 + 1) − n01 rows each column contains two 1s because nt2 = 2 for 1 t m2 − 1. Matrix Z3 is a (2m2 + 1) × m2 matrix which each row contains exactly one 1. In a similar manner one can verify that there exist m2 − n01 = columns of Z3 (say the last), which has
2828 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
all elements zero at the first n01 rows and at the other (2m2 + 1) − n01 rows have exactly two 1s, because nt3 = 2 for 1t m2 − 1. Assume that > 0. W.l.o.g. let
where Z2,1 , Z3,1 , Zj,22 and Zj,32 , 2 j 3 are (2m2 +1)×(m2 −−1), (2m2 +1)×(m2 −), n11 × and n21 × matrices, respectively. (1) If the two 1s of some of the last columns of Z2 (Z3 ) are at the matrix Z2,22 (Z3,22 ) or 2t 1t 2t at the matrix Z2,32 (Z3,32 ), then n1t 12 = 2 or n12 = 2, (n13 = 2 or n13 = 2), 1 t m2 − 1, st which does not hold, because we assumed that n1j 1. Thus matrix Z2,22 (Z3,22 ) contains one 1 and matrix Z2,32 (Z3,32 ) too. (1) Suppose the last m2 − n01 columns of Z2 and Z3 are the columns t with n0t 12 = 0 and 0t n13 = 0, respectively. Let S2 and S3 be the respective sums of these last columns. Since the design matrix of a saturated design has nonzero determinant we have that S2 = S3 . So at least one of the last columns of Z3 , say that tth column, has a 1 in a row different from the rows in which S2 has 1s. Obviously this column has another 1. There are two cases, depending on whether or not the other 1 is in a row different from the rows in which S2 (1) has 1s. In both cases, subtracting this column and S2 from the first column of Z1 and expanding Ud along the first two columns, we obtain |detU d | n11 · n21 − ( + 1). Recalling that = m2 − n01 we get |detU d | n11 · n21 − (m2 + 1 − n01 ). This inequality is identical with (3.13). So the upper bound in case 2 is n11 · n21 − (m2 + 1 − n01 ) which can be expressed as − 1) · (n21 − 1) + (m2 − 1) because n01 = 2m2 + 1 − n11 − n21 , which is maximized with respect to all pairs of integers (n11 , n21 ) satisfying n01 + n11 + n21 = 2m2 + 1 and n01 n11 n21 , when n21 n01 − 1. Comparing the upper bounds of cases 1 and 2 we get n11 ·n21 −n21 n11 ·n21 −(m2 +1−n01 ), that is n11 m2 . We assume that m2 − n01 > 0 i.e. m2 > n01 , with n01 n11 . So m2 > n11 . Consequently from (3.5) and the above notes we get |detU d | |detU d ∗ |, where |detU d ∗ | ∗ ∗ ∗ ∗ ∗ ∗ ∗ = n11 · n21 − (m2 + 1 − n01 ), with n01 n11 n21 n01 − 1, if m2 − n01 > 0. 0 If m2 − n1 0, then in a similar manner to the last paragraph of the proof of the Theorem 3.1 we get |detU d |((m2 + 1)/2)2 , which must be shown to be less than or equal to ∗ ∗ ∗ ∗ |detU d ∗ | = (n11 − 1) · (n21 − 1) + m2 − 1. So we have to prove that (n11 − 1) · (n12 − ∗ 1)((m2 + 1)/2)2 − m2 + 1 = ((m2 − 1)/2)2 + 1. It suffices to show that (n11 − 1) · ∗ ∗ ∗ ∗ (n21 − 1)(m2 /2)2 . Condition (2) on d ∗ implies that (n11 − 1) · (n21 − 1) (n01 − 2)2 . So ∗ ∗ ∗ 2 2 we have to show (n01 − 2) (m2 /2) , or n01 − 2 m2 /2, or n01 (m2 + 4)/2. Condition ∗ (2) also implies that n01 (2m2 + 1)/3, so it suffices to show (2m2 + 1)/3 (m2 + 4)/2, ∗ ∗ ∗ which is true for all m2 10. Finally for m2 =4, 5, 6, 7, 8 and 9 it holds that(n01 , n11 , n12 )= (3, 3, 3), (4, 4, 3), (5, 4, 4), (5, 5, 5), (6, 6, 5) and (7, 6, 6), respectively, and one can verify ∗ ∗ that (n11 − 1) · (n21 − 1) ((m2 − 1)/2)2 + 1. (n11
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2829
Corollary 3.1. The saturated 3 × 3 × 3 design d ∗ = {002, 021, 010, 101, 120, 200, 212} is a D-optimal design. One can verify that the conditions of Theorem 2.1 are satisfied. Corollary 3.2. Consider the saturated 3×3×m3 , m3 5, design d ∗ ={001, 012, 023, 024, ∗ . . . , 02(m3 − 1), 103, 110, 122, 200, 211}. It holds that |detU d ∗ | = |detU d ∗ |, where d ∗ is the saturated 3 × 3 × 4 design d = {001, 012, 023, 103, 110, 122, 200, 211}. Moreover ∗ 0 1 for the design d we have (n1 , n1 , n21 ) = (3, 3, 2) and n11 · n21 − (m2 + 1 − n01 ) = 5. On the other hand |detU d |4, where d is a saturated 3 × 3 × 3 design, as proved in Corollary ∗ 3.1. So, Theorem 3.1 implies that d is a D-optimal design in the class of 3 × 3 × 4saturated ∗ designs, that is, d is a D-optimal in the class of 3 × 3 × m3 , m3 5, saturated designs. Corollary 3.3. Let d ∗ = {003, 032, 020, 102, 111, 130, 201, 210, 223} be a saturated 3 × 4 × 4 design. It holds that |detU d ∗ | = 7. Applying Theorem 3.2 for (n01 , n11 , n21 ) = (3, 3, 3) and using Corollary 3.2 we have that d ∗ is a D-optimal 3 × 4 × 4 design. Using the procedures described in Corollaries 3.2 and 3.3 we can verify Corollaries 3.4, 3.7 and 3.5, respectively. Corollary 3.4. A D-optimal saturated 3 × 4 × m3 , m3 6, design is d ∗ = {001, 012, 023, 034, 035, . . . , 03(m3 − 1), 104, 120, 133, 200, 211, 222}. Corollary 3.5. The design d ∗ ={001, 012, 020, 034, 103, 111, 122, 140, 204, 230, 243} is a D-optimal saturated 3 × 5 × 5 design. Remark 3.1. Chatterjee and Narasimhan (2002) claimed that for the 3 × m2 × m3 designs, when m2 = 2k + 1, the upper bound of the determinant is (k + 1)2 + 1. Thus for the 3 × 3 × m3 , m3 3, the upper bound is 5, but the determinant of the design they give is 2. Corollaries 3.1 and 3.2 of the present paper prove that their claim fails. On the other hand Corollary 3.5 validates their claim for the 3 × 5 × 5 design. Corollary 3.6. Consider a saturated 3 × 5 × m3 design d, where m3 7. It holds that |detU d | = |detU d |14, where d is a saturated 3 × 5 × 6 design. The proof arises using Theorem 3.2 for (n01 , n11 , n21 ) = (4, 4, 4) and Corollary 3.5. Corollary 3.7. A D-optimal 3×6×6 saturated design d ∗ is d ∗ ={001, 012, 020, 034, 045, 103, 111, 122, 150, 205, 230, 244, 253}. Remark 3.2. Let us consider the 3×5×6 and 3×7×7 saturated designs d1 ={001, 012, 023, 034, 105, 114, 131, 140, 203, 210, 222, 245} and d2 = {001, 012, 020, 045, 056, 103, 136, 164, 141, 155, 202, 234, 263, 210, 226}, respectively. It holds that |detU d1 | = 12 and |detU d2 | = 19. Although these designs do not attain the upper bounds of Theorems 3.1 and 3.2 respectively, they are D-better than the designs given by Chatterjee and Narasimhan (2002). Moreover, we conjecture that the above designs are D-optimal.
2830 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
Acknowledgements The authors are thankful to two anonymous referees for many highly constructive suggestions. Appendix Let d ∗ = {001, 012, 023, 034, 035, . . . , 03(m3 − 1), 104, 120, 133, 200, 211, 222} be a saturated 3 × 4 × m3 design, where m3 5. Following definition (2.1) the corresponding matrix Ud ∗ is given by the form:
. Expanding matrix Ud ∗ along the last m3 − 5 columns, which each has one 1, we get |detU d ∗ | = |detU d ∗ |, where Ud ∗ is the 10 × 10 submatrix of matrix Ud ∗ . It holds that |detU d ∗ | = 8. In a similar manner one can verify the bounds for the rest of the designs given in the corollaries. References Chatterjee, K., Mukerjee, R., 1993. D-optimal saturated main effect plans for 2 × s2 × s3 factorials. J. Combin. Inform. System Sci. 18, 116–122. Chatterjee, K., Narasimhan, G., 2002. Graph theoretic techniques in D-optimal design problems. J. Statist. Plann. Inference 102, 377–387. Dey, A., Mukerjee, R., 1999. Fractional Factorial Plans, Wiley, New York. Kraft, O., 1990. Some matrix representations occurring in two factor models. In: Bahadur, R.R. (Ed.), Probability, Statistics and Design of Experiments. Wiley Eastern, New Delhi, pp. 461–470. Mukerjee, R., Sinha, B.K., 1990. Almost saturated D-optimal main effect plans and allied results. Metrika 37, 301–307. Mukerjee, R., Chatterjee, K., Sen, M., 1986. D-optimality of a class of saturated main effect plans and allied results. Statistics 17 (3), 349–355. Pesotan, H., Raktoe, B.L., 1988. On invariance and randomization in factorial designs with applications to Doptimal main effect designs of the symmetrical factorial. J. Statist. Plann. Inference 19, 283–298.
Some D-optimal saturated designs for 3×m2×m3 factorials St.A. Chatzopoulos, F. Kolyva-Machera∗ Faculty of Sciences, Department of Mathematics, Section of Statistics and O.R., Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Received 10 July 2002; accepted 18 October 2004
Abstract This paper extends the results concerning D-optimal saturated main effect designs for 2 × m2 × m3 to 3 × m2 × m3 factorials, when 3 m2 6 and m3 m2 . © 2004 Elsevier B.V. All rights reserved. MSC: 62K05 Keywords: D-optimal designs; Saturated factorial designs
1. Introduction An experimental design is said to be saturated if all degrees of freedom are consumed by the estimation of parameters, leaving no degrees of freedom for error variance estimation. Saturated resolution III factorial designs are commonly used in screening experiments, to determine which of many factors affects the measure of pertinent quality characteristics. The purpose of this paper is to give saturated resolution III designs, minimizing the generalized variance of the main effects and the general mean, that is, D-optimal designs. In recent years, there has been a considerable interest in optimal saturated main effect designs. Mukerjee et al. (1986) and Kraft (1990) showed that in the two-factor case all such designs are equivalent with respect to D-optimality. Later Mukerjee and Sinha (1990) considered optimality results on almost saturated main effect designs in the two-factor case. The result was also ∗ Corresponding author. Tel.: +30 310 997954; fax: +30 310 997983.
E-mail address: [email protected] (F. Kolyva-Machera). 0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2004.10.025
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2821
obtained by Pesotan and Raktoe (1988) in the special case for m2 factorials, who also worked on a subclass of m3 factorials. Our result on 3 × 3 × 3 factorial is D-better than Pesotan and Raktoe’s. The first attempt to extend the two factor results to three factors was done by Chatterjee and Mukerjee (1993). They consider 2×m2 ×m3 factorial to derive D-optimal saturated main effect designs. Later Chatterjee and Narasimhan (2002) using techniques from Graph Theory and Combinatorics, claimed about the upper bound of the determinant of the saturated 3 × m2 × m3 factorials when m2 = 2k + 1. In this paper we study some aspects of Doptimality. More specifically we provide alternative proofs for the saturated m1 × m2 and 2 × m2 × m3 designs, and we study the saturated 3 × m2 × m3 designs. Moreover, when this is feasible, we compare our findings with their claim. In the appendix we give explicitly a D-optimal 3 × 4 × m3 factorial, m3 5.
2. Preliminaries Following Chatterjee and Mukerjee (1993) we consider the setup of an m1 × m2 × m3 saturated factorial experiment, involving three factors F1 , F2 and F3 appearing at m1 , m2 and m3 levels, respectively where m3 m2 m1 2, with N = 1 + m1 − 1 + m2 − 1 + m3 − 1 = m1 + m2 + m3 − 2 runs. For 1i 3 let the levels of Fi be denoted by i and coded as 0, 1, . . . , mi − 1. Our interest is to find D-optimal resolution III designs. There are altogether u = m1 m2 m3 treatment combinations denoted by 1 2 3 , that will hereafter be assumed to be lexicographically ordered. Let, for 1i 3, 1i be the mi × 1 vector with each element unity, Ii the identity matrix of order mi , ⊗ denotes the Kronecker product of matrices and Pi be an (mi − 1) × mi −1/2 matrix such that (mi 1i , Pi ) is orthogonal. (A denotes the transpose of matrix A). In a complete experiment, where each treatment combination appears once, the usual fixed effect model under the absence of interactions is Y = W + , where Y is the response vector of the experiment, is a u × 1 vector of uncorrelated random errors with zero mean and the same variance 2 and is the vector of unknown parameters. In our case =(, 1 , 2 , 3 ) , where is the unknown general mean and the elements of the (mi − 1) × 1 vectors i are unknown parameters representing a full set of mutually orthogonal contrasts belonging to the main effects Fi and W = [11 ⊗ 12 ⊗ 13 , W1 , W2 , W3 ], where W1 = P1 ⊗ 12 ⊗ 13 , W2 = 11 ⊗ P2 ⊗ 13 and W3 = 11 ⊗ 12 ⊗ P3 . It is easy to see that the D-optimal design does not depend on the choice of Pi , 1 i 3. Following Mukerjee and Sinha (1990) let X0 = [11 ⊗ 12 ⊗ 13 , X1 , X2 , X3 ], where X1 = I1 ⊗12 ⊗13 , X2 =11 ⊗I2 ⊗13 and X3 =11 ⊗12 ⊗I3 . Consider the u×(m1 +m2 +m3 −2) (1) (1) matrix U which is a submatrix of X0 given by U = [X1 , X2 , X3 ], where for 1 i 2 (1) the matrices Xi are obtained from Xi by deleting the first column of the latter. The u rows of matrix U like those of W , correspond to the lexicographically ordered treatment combinations. Note that matrix U has full column rank. Moreover the columns of U span those of X0 and hence those of W , which also has full column rank. Following Mukerjee and Sinha (1990) one may obtain W = U H , where matrix H is a nonsingular matrix of order m1 + m2 + m3 − 2.
2822 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
For any design d in the class D of the saturated resolution III designs with N = m1 + m2 + m3 − 2 runs, the design matrix is Wd = Ud H , where Ud is a square matrix of order m1 + m2 + m3 − 2 such that for 1j m1 + m2 + m3 − 2 if the jth run in d is given by the treatment combination 1 2 3 then the jth row of Ud is the row of U corresponding to the treatment combination 1 2 3 . A design d is said to be D-optimal in the class D, if it maximizes the quantity det(Wd Wd ). Since matrix H is nonsingular a design is D-optimal if it maximizes the quantity |detU d |, where (1)
(1)
Ud = [Z1 , Z2 , Z3 ].
(2.1)
(1)
(1)
The matrices Zj , 1 j 2 and Z3 are obtained from the matrices Xj and X3 in a similar way as Ud is obtained from U . Definition 2.1. For 1i 2, if the ith factor enters the experiment at level 0 then the (1) corresponding row of the matrix Zi is a row vector with mi − 1 elements zero. On the other hand if the ith factor enters the experiment at level s, 1 s mi − 1, then the (1) corresponding row of the matrix Zi equals the sth row of the identity matrix of order s mi − 1. Let ni , 0 s mi − 1, denote the number of these rows. Similarly, if the third factor enters the experiment at level s, 0 s m3 − 1 then the corresponding row of the matrix Z3 equals to the (s + 1)th row of the identity matrix I3 . It holds that N=
m i −1
nsi .
(2.2)
s=0
Definition 2.2. For 1 i, j 3, i = j and 0 s mi − 1, 0 t mj − 1 let nst ij denote the number of runs where the ith factor appears at level s and the jth factor appears at level t. It holds that mj −1
nsi =
t=0
nst ij ,
ntj =
m i −1 s=0
nst ij
and
N=
m j −1 i −1 m s=0
nst ij .
(2.3)
t=0
Remark 2.1. It holds that nsi 1, 1 i 3, 0 s mi − 1, since the design matrix of a saturated design has full column rank. Remark 2.2. By the choice of the labels for the levels one can always assume without loss of generality (w.l.o.g.) that n0i = max{nsi , 0 s mi − 1}. Lemma 2.1. Consider the saturated m1 × · · · × mi × · · · × mk design d. If mi 2 and nsi = 1 for some 0 s mi − 1 then |detU d | = |detU d |, where d is a saturated m1 × · · · × (mi − 1) × · · · × mk design. Proof. If n0i = 1 then nsi = 1 for all 0 s mi − 1, because n0i = max{nsi }. So, nsi = 1 for (1) some 1s mi − 1. Obviously the sth column, 1 s mi − 1, of Zi contains exactly one 1. Expanding detU d along this column the result follows.
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2823
Corollary 2.1. Consider the saturated m1 × m2 × m3 , m3 m2 m1 2 design d with N = m1 + m2 + m3 − 2 runs. One can easily verify that ns3 = 1 at least w = m3 − (m1 + m2 − 2) times, using the pigeonhole principle. Applying Lemma 2.1 w times for k = 3 we get |detU d | = |detU d |, where d is a saturated m1 × m2 × (m1 + m2 − 2) design. Lemma 2.2. Consider the saturated m1 ×m2 designs e, m2 m1 2 with N =m1 +m2 −1 runs. It holds that |detU e | = 1. Proof. We can verify that ns2 =1, 0 s m2 −1, at least m2 +1−m1 times. Thus, applying Lemma 2.1 for k = 2 at least m2 + 1 − m1 times we get |detU e | = |detU e |, where e is a saturated m1 × (m1 − 1) design, that is (m1 − 1) × m1 with 2(m1 − 1) runs. Wecan continue the reduction until we have a saturated 1 × 2 design, for which Ue = 01 01 . Obviously, |detU e | = 1. This result is the same as that of Mukerjee et al. (1986), Dey and Mukerjee (1999), (Theorem 6.1.1), but the proof is different. Lemma m1 ×m2 ×· · ·×mk design d, it holds that |detU d | m1 −1 s2.3. For any saturated |, d is a saturated m2 × · · · × mk design}. n · max{|detU d s=1 1 Proof. We can evaluate |detU d | expanding matrix Ud along the m1 − 1 columns of the (1) (1) matrix Z1 . Observing that each row of Z1 has at most one 1 and the number of 1s in the (1) sth column of Z1 is ns1 , 1 s m1 −1, the result follows. 1 −1 s m1 −1 0 1 Lemma 2.4. The maximum of the quantity Q= m = s=1 n1 subject to n1 +n1 +· · ·+n1 s 0 1 1 −1 N, and all n1 , 0 s m1 − 1, positive integers is achieved when n1 n1 · · · nm 1 n01 − 1. Proof. Consider any pair of levels 0 s, t m1 −1. Given ns1 +nt1 =k, we have (j/jns1 )ns1 nt1 = (j/jns1 )ns1 (k − ns1 ) = k − 2ns1 , which is positive for all ns1 < k/2, i.e. for all ns1 < nt1 . So in order to maximize Q we have to increase ns1 to the maximum value such that ns1 nt1 , which is either ns1 =nt1 or ns1 =nt1 −1. By applying this to all pairs, the lemma follows. Theorem d be a saturated m1 × m2 × m3 , m3 m2 m1 2 design. If |detU d | = m1 −1 s 2.1. Let 0 n1 · · · nm1 −1 n0 − 1 then d is D-optimal. n and n 1 s=1 1 1 1 1 1 −1 s Proof. Applying Lemma 2.3 for k = 3 and using Lemma 2.2 we get |detU d | m s=1 n1 , m1 −1 s 0 m1 −1 s 1 2 . The quantity where N = s=0 n1 , n1 = max{n1 } and w.l.o.g n1 n1 · · · n1 |detU d | is maximized when equality holds in the above inequality.According to Lemma 2.4, 1 −1 s m1 −1 0 1 for the D-optimal design it holds that |detU d | = m s=1 n1 , where n1 n1 · · · n1 0 n1 − 1. Corollary 2.2. Consider the saturated 2 × m2 × m3 design d, m3 m2 2. We can prove Theorem 3.1 of Chatterjee and Mukerjee (1993), applying Corollary 2.1 and Lemma 2.3.
2824 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
We have |detU d | = |detU d |n11 · max{|detU e |}, where d is a saturated 2 × m2 × m2 design, with N = n01 + n11 = 2m2 runs and e is a saturated m2 × m2 design. Moreover from Remark 2.2 we have n01 n11 , that is n01 m2 and n11 m2 . Finally |detU d | = |detU d | m2 , because |detU e | = 1, as proved by Lemma 2.2. In what follows the quantity |detU e | is omitted when we evaluate |detU d | expanding along the elements of the two first columns of Ud , because |detU e | = 1.
3. Main results Lemma 3.1. Consider the saturated 3 × m2 × m3 design d, m2 3, m3 m2 + 2. It holds that |detU d | = |detU d |, where d is a saturated 3 × m2 × (m2 + 1) design. Proof. Applying Corollary 2.1 for m1 = 3 the result follows.
Theorem 3.1. Let d ∗ be a saturated 3 × m2 × m3 design, m3 > m2 3. If ∗
∗
∗
(1) |detU d ∗ | = n11 · n21 − (m2 + 1 − n01 ), ∗ ∗ ∗ ∗ (2) n01 n11 n21 n01 − 1, (3) |detU d ∗ | max{|detU d |}, where d is a saturated 3 × m2 × m2 design, then d ∗ is D-optimal. Proof. According to Lemma 3.1, the optimal D-criterion among saturated 3 × m2 × m3 designs is the same as the optimal D-criterion among saturated 3 × m2 × (m2 + 1) designs. (1) (1) Let Ud =[Z1 , Z2 , Z3 ] be the 2(m2 +1)×2(m2 +1) matrix corresponding to an arbitrary (1) (1) saturated 3 × m2 × (m2 + 1) design d, where Z1 , Z2 and Z3 are 2(m2 + 1) × 2, 2(m2 + 1) × (m2 − 1) and 2(m2 + 1) × (m2 + 1) matrices, respectively. 3 −1 s s It holds that m s=0 n3 = N = 2(m2 + 1). If n3 = 1 for some 0 s m2 then applying Lemma 2.1 we have that |detU d | = |detU d ||detU d ∗ |,
(3.1)
where d is a saturated 3 × m2 × m2 design. On the other hand if ns3 2 (actually ns3 = 2) for all 0 s m2 then each column of Z3 contains exactly two 1s. Let Z1,1 , Z1,2 and Z1,3 be n01 × 2, n11 × 2, n21 × 2 matrices, which each row is [0 0], [1 0] and [0 1], respectively. From Remark 2.2 it holds that (1) n01 = max{ns1 }, 0 s 2. Assume, w.l.o.g., that n01 n11 n21 . One can partition matrix Z1 Z1,1 (1) as Z1 = Z1,2 . Matrix Z3 is a 2(m2 + 1) × (m2 + 1) matrix in which each row contains Z1,3
exactly one 1. So, matrix Z3 has at least (m2 + 1) − n01 = columns, with all elements zero at the first n01 rows and at the other N − n01 rows each column contains two 1s. Assume
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2825
that > 0. Thus, w.l.o.g., matrix Ud can be partitioned inthe following way:
where Z3,1 , Z3,22 , Z3,32 are 2(m2 +1)×(m2 +1−), n11 ×, n21 × matrices respectively, with elements 0 or 1. We consider two cases: Case 1: If at least one of the last columns of Ud has the two 1s at matrix Z3,22 (or Z3,32 ) then by subtracting this column from the first (or the second) column of Ud we have that the number of 1s at the first (or the second) column of Ud becomes n11 −2 (or n21 −2). Expanding Ud along the two first columns and using Lemma 2.2 we have |detU d | (n11 − 2) · n21 (or |detU d | n11 · (n21 − 2)). Case 2: If each of the last columns of Ud contains one 1 at matrix Z3,22 and one 1 at matrix Z3,32 then the column vector which arises by adding these columns, contains 1s at rows corresponding to Z3,22 and 1s at rows corresponding to Z3,32 . By subtracting this vector from the first (or the second) column of Ud and expanding along the two first columns of Ud we get (in both cases) |detU d | (n11 − ) · n21 + | − | · (n21 − 1) = n11 · n21 − = (n11 − 1) · (n21 − 1) + m2 . It holds that (n21 − 2) · n11 (n11 − 2) · n21 because n11 n21 as assumed. Moreover (n11 − 2) · n21 < n11 · n21 − (m2 + 1 − n01 ),
(3.2)
because 2(m2 + 1) = n01 + n11 + n21 and n01 − n11 + 3n21 > 0. On the other hand, function (n11 − 1) · (n21 − 1) + m2 is maximized with respect to all pairs of positive integers (n11 , n21 ) satisfying equality n01 + n11 + n21 = 2(m2 + 1) and inequality n01 n11 n21 ,
when n21 n01 − 1 (see Lemma 2.4).
(3.3)
From (3.1)–(3.3), if > 0, that is m2 + 1 > n01 we have |detU d | |detU d ∗ |, where ∗ ∗ ∗ ∗ ∗ ∗ ∗ |detU d ∗ | = n11 · n21 − (m2 + 1 − n01 ) with n01 n11 n21 n01 − 1. If =m2 +1−n01 0, then n11 +n21 m2 +1 because n01 +n11 +n21 =2(m2 +1). By Lemmas 2.2 and 2.3 we have |detU d |n11 ·n21 . So |detU d |((m2 +1)/2)2 . Consequently, it remains ∗ ∗ ∗ to be proved that ((m2 + 1)/2)2 n11 · n21 − (m2 + 1 − n01 ). Previously in the proof it has ∗ ∗ ∗ ∗ ∗ been noted that n11 · n21 − (m2 + 1 − n01 ) = (n11 − 1) · (n21 − 1) + m2 . So we want to show ∗ ∗ (n11 − 1) · (n21 − 1) ((m2 + 1)/2)2 − m2 = ((m2 − 1)/2)2 . Condition (2) on d ∗ implies ∗ ∗ ∗ ∗ that (n11 − 1) · (n21 − 1)(n01 − 2)2 . So it suffices to show (n01 − 2)2 ((m2 − 1)/2)2 , or ∗ ∗ ∗ n01 −2 (m2 −1)/2, or n01 (m2 +3)/2. Condition (2) also implies that n01 (2m2 +2)/3, so it suffices to show (2m2 + 2)/3 (m2 + 3)/2, which is true for all m2 5. For m2 = 3 ∗ ∗ ∗ and 4 it holds that (n01 , n11 , n21 ) = (3, 3, 2) and (4, 3, 3), respectively, and one can verify ∗ ∗ that (n11 − 1) · (n21 − 1) ((m2 − 1)/2)2 .
2826 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
Theorem 3.2. Consider the saturated 3 × m2 × m2 design d ∗ with m2 4. If ∗
∗
∗
(1) |detU d ∗ | = n11 · n21 − (m2 + 1 − n01 ), ∗ ∗ ∗ ∗ (2) n01 n11 n21 n01 − 1, (3) |detU d ∗ | max{|detU d |}, where d is a saturated 3 × (m2 − 1) × m2 design, then d ∗ is D-optimal. (1)
(1)
Proof. Let Ud = [Z1 , Z2 , Z3 ] be the (2m2 + 1) × (2m2 + 1) matrix which corresponds to an arbitrary saturated 3 × m2 × m2 design d with N = 2m2 + 1 runs. It holds that N = n01 + n11 + n21 = 2m2 + 1, where n01 = max{ns1 }, 0 s 2 and w.l.o.g., n11 n21 . We can partition Ud as
(3.4) (1)
where Z2 and Z3 are (2m2 + 1) × (m2 − 1) and (2m2 + 1) × m2 matrices, with elements 0 or 1, respectively. We can always permute the second and the third factor because both appear at m2 levels. If ns2 = 1 or ns3 = 1 for some 0 s m2 − 1, then applying Lemma 2.1 for k = 3 and i = 2 or 3 we get |detU d | = |detU d ||detU d ∗ |,
(3.5)
where d is a saturated 3 × (m2 − 1) × m2 design. On the other hand if ntj > 1, 2 j 3, 0 t m2 − 1, then n0j = 3 and ntj = 2 for t = 0, mj −1 t nj = N = 2m2 + 1 and n0j = max{ntj }. We consider two cases: because t=0 st Case 1: n12 2 or nst 13 2 for some s 1 and some 0 t m2 − 1. W.l.o.g., we assume that nst 2. It holds that n03 = 3 and nt3 = 2 for t = 0. Then 13 st Case 1A: n13 = 2 for some s 1, and some t 1. It holds that nt3 = 2. So, the (t + 1)th column of matrix Z3 contains two 1s and since nst 13 = 2 these two 1s are at the same rows (1) s with two of the n1 1s of the sth column of Z1 . Then by subtracting the (t + 1)th column (1) of Z3 from the sth column of Z1 and expanding Ud along the first two columns we have
|detU d | (ns1 − 2)ns1 ,
(3.6)
where s = 2 (or 1) if s = 1 (or 2). Case 1B: ns0 13 = 3 for some s 1. Subtracting the first column of Z3 from the sth column (1) of Z1 and expanding Ud along the first two columns we get
|detU d | (ns1 − 3)ns1 . 0 Case 1C: ns0 13 = 2 for some s 1. It holds that n3 = 3 =
(3.7) 2
s0 s=0 n13
by (2.3).
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2827
Case 1C(i): If n00 13 = 1, then following the same procedure as in Case 1B we have
|detU d |(ns1 − 2) · ns1 + | − 1| · ns1 = (ns1 − 1) · ns1 .
(3.8)
s0
Case 1C(ii): If n13 = 1, then in a similar manner we get
|detU d |(ns1 − 2) · ns1 + | − 1| · (ns1 − 1) = ns1 · ns1 − ns1 − 1. For the case 1, comparing inequalities (3.6)–(3.9) for
n11 n21 ,
(3.9)
we get
|detU d | n11 · n21 − n21 .
(3.10)
st Case 2: nst 12 1 and n13 1 for all s 1 and all t. 0t t 0 Case 2A: n0t 12 2 or n13 2 for some t, because nj = 3 and nj = 2 for 1 t m2 − 1, 0t 00 2 j 3. Let, w.l.o.g. n13 =2 (or n13 =3), for some 1 t m2 −1. Consider (3.4); recalling 00 0 definition (2.2) for n0t 13 = 2 (or n13 = 3) we have that two (three) of the first n1 rows of Z3 are
the (t +1)th (or the first) row of I2 (because m3 =m2 ). Matrix Z3 is a (2m2 +1)×m2 matrix in which each row contains exactly one 1. There exist m2 −(n01 −1)= (m2 −(n01 −2)=+1) columns of Z3 (say the last), which have all elements zero at the first n01 rows. At the remaining (2m2 + 1) − n01 rows these columns contain each exactly two 1s, because nt3 = 2 for all 1 t m2 − 1. Assume that > 0. If there exists at least one column of the ( + 1) last columns of Z3 which contains the two 1s at the n11 or at the n21 rows of Ud then obviously 2t n1t 13 = 2 or n13 = 2, 1 t m2 − 1, respectively (case 1A). On the other hand if each of the ( + 1) last columns of Z3 contains one 1 at the n11 rows and one 1 at the n21 rows of Ud then by adding these columns, the resulting column will contain ( + 1) 1s at the n11 rows and ( + 1) 1s at the n21 rows of Ud . Subtracting from the first column of Ud the resulting column of Z3 and expanding along the first two columns we get |detU d | (n11 − ) · n21 + | − | · (n21 − 1)
if n0t 13 = 2, 0 t m2 − 1,
(|detU d | (n11 − ( + 1)) · n21 + | − ( + 1)| · (n21 − 1) if n00 13 = 3).
(3.11) (3.12)
Recalling that = m2 − (n01 − 1), for the case 2A, inequalities (3.11) and (3.12) yield |detU d | n11 · n21 − (m2 − n21 + 1).
(3.13)
Case 2B: n0t n0t 12 1 and 13 1 for all t. 0 00 10 20 It holds that n2 = 3 = 2s=0 ns0 12 by (2.3). So, n12 = 1 and n12 = 1 and n12 = 1, otherwise s0 s0 n12 = 2 or n12 = 3 for some 0 s 2. Recalling Definitions 2.1 and 2.2 for n00 12 = 1 we have (1) that there exists one row (say the first) of the (2m2 + 1) × (m2 − 1) matrix Z2 which has (1) 0 0 all its elements zero at the first n1 rows of Ud . The rest n1 − 1 first rows of Z2 will be the rows of the matrix Im2 −1 each one time, otherwise n0t 12 > 1. Consequently there exists (1) (m2 − 1) − (n01 − 1) = m2 − n01 (say 1) columns of Z2 (say that last) which have all 0 elements zero at the first n1 rows and at the rest (2m2 + 1) − n01 rows each column contains two 1s because nt2 = 2 for 1 t m2 − 1. Matrix Z3 is a (2m2 + 1) × m2 matrix which each row contains exactly one 1. In a similar manner one can verify that there exist m2 − n01 = columns of Z3 (say the last), which has
2828 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
all elements zero at the first n01 rows and at the other (2m2 + 1) − n01 rows have exactly two 1s, because nt3 = 2 for 1t m2 − 1. Assume that > 0. W.l.o.g. let
where Z2,1 , Z3,1 , Zj,22 and Zj,32 , 2 j 3 are (2m2 +1)×(m2 −−1), (2m2 +1)×(m2 −), n11 × and n21 × matrices, respectively. (1) If the two 1s of some of the last columns of Z2 (Z3 ) are at the matrix Z2,22 (Z3,22 ) or 2t 1t 2t at the matrix Z2,32 (Z3,32 ), then n1t 12 = 2 or n12 = 2, (n13 = 2 or n13 = 2), 1 t m2 − 1, st which does not hold, because we assumed that n1j 1. Thus matrix Z2,22 (Z3,22 ) contains one 1 and matrix Z2,32 (Z3,32 ) too. (1) Suppose the last m2 − n01 columns of Z2 and Z3 are the columns t with n0t 12 = 0 and 0t n13 = 0, respectively. Let S2 and S3 be the respective sums of these last columns. Since the design matrix of a saturated design has nonzero determinant we have that S2 = S3 . So at least one of the last columns of Z3 , say that tth column, has a 1 in a row different from the rows in which S2 has 1s. Obviously this column has another 1. There are two cases, depending on whether or not the other 1 is in a row different from the rows in which S2 (1) has 1s. In both cases, subtracting this column and S2 from the first column of Z1 and expanding Ud along the first two columns, we obtain |detU d | n11 · n21 − ( + 1). Recalling that = m2 − n01 we get |detU d | n11 · n21 − (m2 + 1 − n01 ). This inequality is identical with (3.13). So the upper bound in case 2 is n11 · n21 − (m2 + 1 − n01 ) which can be expressed as − 1) · (n21 − 1) + (m2 − 1) because n01 = 2m2 + 1 − n11 − n21 , which is maximized with respect to all pairs of integers (n11 , n21 ) satisfying n01 + n11 + n21 = 2m2 + 1 and n01 n11 n21 , when n21 n01 − 1. Comparing the upper bounds of cases 1 and 2 we get n11 ·n21 −n21 n11 ·n21 −(m2 +1−n01 ), that is n11 m2 . We assume that m2 − n01 > 0 i.e. m2 > n01 , with n01 n11 . So m2 > n11 . Consequently from (3.5) and the above notes we get |detU d | |detU d ∗ |, where |detU d ∗ | ∗ ∗ ∗ ∗ ∗ ∗ ∗ = n11 · n21 − (m2 + 1 − n01 ), with n01 n11 n21 n01 − 1, if m2 − n01 > 0. 0 If m2 − n1 0, then in a similar manner to the last paragraph of the proof of the Theorem 3.1 we get |detU d |((m2 + 1)/2)2 , which must be shown to be less than or equal to ∗ ∗ ∗ ∗ |detU d ∗ | = (n11 − 1) · (n21 − 1) + m2 − 1. So we have to prove that (n11 − 1) · (n12 − ∗ 1)((m2 + 1)/2)2 − m2 + 1 = ((m2 − 1)/2)2 + 1. It suffices to show that (n11 − 1) · ∗ ∗ ∗ ∗ (n21 − 1)(m2 /2)2 . Condition (2) on d ∗ implies that (n11 − 1) · (n21 − 1) (n01 − 2)2 . So ∗ ∗ ∗ 2 2 we have to show (n01 − 2) (m2 /2) , or n01 − 2 m2 /2, or n01 (m2 + 4)/2. Condition ∗ (2) also implies that n01 (2m2 + 1)/3, so it suffices to show (2m2 + 1)/3 (m2 + 4)/2, ∗ ∗ ∗ which is true for all m2 10. Finally for m2 =4, 5, 6, 7, 8 and 9 it holds that(n01 , n11 , n12 )= (3, 3, 3), (4, 4, 3), (5, 4, 4), (5, 5, 5), (6, 6, 5) and (7, 6, 6), respectively, and one can verify ∗ ∗ that (n11 − 1) · (n21 − 1) ((m2 − 1)/2)2 + 1. (n11
St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
2829
Corollary 3.1. The saturated 3 × 3 × 3 design d ∗ = {002, 021, 010, 101, 120, 200, 212} is a D-optimal design. One can verify that the conditions of Theorem 2.1 are satisfied. Corollary 3.2. Consider the saturated 3×3×m3 , m3 5, design d ∗ ={001, 012, 023, 024, ∗ . . . , 02(m3 − 1), 103, 110, 122, 200, 211}. It holds that |detU d ∗ | = |detU d ∗ |, where d ∗ is the saturated 3 × 3 × 4 design d = {001, 012, 023, 103, 110, 122, 200, 211}. Moreover ∗ 0 1 for the design d we have (n1 , n1 , n21 ) = (3, 3, 2) and n11 · n21 − (m2 + 1 − n01 ) = 5. On the other hand |detU d |4, where d is a saturated 3 × 3 × 3 design, as proved in Corollary ∗ 3.1. So, Theorem 3.1 implies that d is a D-optimal design in the class of 3 × 3 × 4saturated ∗ designs, that is, d is a D-optimal in the class of 3 × 3 × m3 , m3 5, saturated designs. Corollary 3.3. Let d ∗ = {003, 032, 020, 102, 111, 130, 201, 210, 223} be a saturated 3 × 4 × 4 design. It holds that |detU d ∗ | = 7. Applying Theorem 3.2 for (n01 , n11 , n21 ) = (3, 3, 3) and using Corollary 3.2 we have that d ∗ is a D-optimal 3 × 4 × 4 design. Using the procedures described in Corollaries 3.2 and 3.3 we can verify Corollaries 3.4, 3.7 and 3.5, respectively. Corollary 3.4. A D-optimal saturated 3 × 4 × m3 , m3 6, design is d ∗ = {001, 012, 023, 034, 035, . . . , 03(m3 − 1), 104, 120, 133, 200, 211, 222}. Corollary 3.5. The design d ∗ ={001, 012, 020, 034, 103, 111, 122, 140, 204, 230, 243} is a D-optimal saturated 3 × 5 × 5 design. Remark 3.1. Chatterjee and Narasimhan (2002) claimed that for the 3 × m2 × m3 designs, when m2 = 2k + 1, the upper bound of the determinant is (k + 1)2 + 1. Thus for the 3 × 3 × m3 , m3 3, the upper bound is 5, but the determinant of the design they give is 2. Corollaries 3.1 and 3.2 of the present paper prove that their claim fails. On the other hand Corollary 3.5 validates their claim for the 3 × 5 × 5 design. Corollary 3.6. Consider a saturated 3 × 5 × m3 design d, where m3 7. It holds that |detU d | = |detU d |14, where d is a saturated 3 × 5 × 6 design. The proof arises using Theorem 3.2 for (n01 , n11 , n21 ) = (4, 4, 4) and Corollary 3.5. Corollary 3.7. A D-optimal 3×6×6 saturated design d ∗ is d ∗ ={001, 012, 020, 034, 045, 103, 111, 122, 150, 205, 230, 244, 253}. Remark 3.2. Let us consider the 3×5×6 and 3×7×7 saturated designs d1 ={001, 012, 023, 034, 105, 114, 131, 140, 203, 210, 222, 245} and d2 = {001, 012, 020, 045, 056, 103, 136, 164, 141, 155, 202, 234, 263, 210, 226}, respectively. It holds that |detU d1 | = 12 and |detU d2 | = 19. Although these designs do not attain the upper bounds of Theorems 3.1 and 3.2 respectively, they are D-better than the designs given by Chatterjee and Narasimhan (2002). Moreover, we conjecture that the above designs are D-optimal.
2830 St.A. Chatzopoulos, F. Kolyva-Machera / J. Statistical Planning and Inference 136 (2006) 2820 – 2830
Acknowledgements The authors are thankful to two anonymous referees for many highly constructive suggestions. Appendix Let d ∗ = {001, 012, 023, 034, 035, . . . , 03(m3 − 1), 104, 120, 133, 200, 211, 222} be a saturated 3 × 4 × m3 design, where m3 5. Following definition (2.1) the corresponding matrix Ud ∗ is given by the form:
. Expanding matrix Ud ∗ along the last m3 − 5 columns, which each has one 1, we get |detU d ∗ | = |detU d ∗ |, where Ud ∗ is the 10 × 10 submatrix of matrix Ud ∗ . It holds that |detU d ∗ | = 8. In a similar manner one can verify the bounds for the rest of the designs given in the corollaries. References Chatterjee, K., Mukerjee, R., 1993. D-optimal saturated main effect plans for 2 × s2 × s3 factorials. J. Combin. Inform. System Sci. 18, 116–122. Chatterjee, K., Narasimhan, G., 2002. Graph theoretic techniques in D-optimal design problems. J. Statist. Plann. Inference 102, 377–387. Dey, A., Mukerjee, R., 1999. Fractional Factorial Plans, Wiley, New York. Kraft, O., 1990. Some matrix representations occurring in two factor models. In: Bahadur, R.R. (Ed.), Probability, Statistics and Design of Experiments. Wiley Eastern, New Delhi, pp. 461–470. Mukerjee, R., Sinha, B.K., 1990. Almost saturated D-optimal main effect plans and allied results. Metrika 37, 301–307. Mukerjee, R., Chatterjee, K., Sen, M., 1986. D-optimality of a class of saturated main effect plans and allied results. Statistics 17 (3), 349–355. Pesotan, H., Raktoe, B.L., 1988. On invariance and randomization in factorial designs with applications to Doptimal main effect designs of the symmetrical factorial. J. Statist. Plann. Inference 19, 283–298.