A note on universally optimal row-column designs with empty nodes

STATII IC

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Statistics & Probability Letters 33 (1997) 145 149

ELSEVIER

A note on universally optimal row-column designs with empty nodes Chao-Ping Ting a'*, Bing-Ying L. Lin b, Feng-Shun Chai c aDepartment of Statistics, National Chengchi University, No• 64, Chi-Nan Road Sec• 2, Wenshan, Taiwan 11623, ROC bDepartment of Mathematical Sciences, National Chengchi University ~lnstitute of Statistical Science, Academia Sinica

Received April 1996; revised June 1996

Abstract

Identification of universally optimal row-column designs is investigated. This paper shows that Kunert's (1993) examples of universally optimal generalized non-binary designs are not special cases. One can construct an universally optimal generalized non-binary design by use of a binary one. Keywords." Balanced block design; Balanced incomplete block design; Generalized binary design; row-column design;

Universally optimal design

1. I n t r o d u c t i o n

The problem of identifying and constructing optimal row-column designs with empty nodes has been widely studied recently. The attempt has been seen in Saharay (1986), Jacroux and Saharay (1991), Stewart and Bradley (1991), Baksalary and Pukelsheim (1992), and Shah and Sinha (1993). Assume there are n experimental units to be arranged in bl rows of size kl each and b2 columns of size k2 each, where kl < b2, k2 < bl, and n = blk~ = b2k 2. Let ndijh denote the number of times treatment i (1 ~< i ~< v), where v is the b2 ndijh denote number of treatments, appears in row j (1 ~
* Corresponding author• Tel.: + 88 62 938 7468; fax: + 8862 937 9336; e-mail: [email protected]. 0167-7152/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PI1 S01 67-7 1 52(96)00122-8

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still have to be binary or generalized binary designs. Here generalized binary means that ndij. = [ n / v b l ] or [ n / v b a ] + 1, and ndi.h = [ n / v b 2 ] or, [ n / v b 2 ] + 1, Vi, j,h. Recently, K u n e r t (1993) has found a class of universally optimal designs with v, bl = v, k~ = m(v - 1) z, b 2 = mv(v - 1), k 2 = v - - 1, N d 3 = l'm(v- 11 @ (Jr - - Iv), and Hdij. = m(v - 1) or m(v - 2), rldi.h = 0 or 1, Vi, j , h , where m is a positive integer, Ts an s x 1 vector of ones, Js an s x s matrix of ones, Is the s x s identity matrix, ® denotes the K r o n e c k e r product, and A ' or ~i' the transpose of matrix A or vector ci, respectively. As one can see that when m becomes larger, the nd~. does not even come close to as equal as possible. In this paper, however, we show that K u n e r t ' s (1993) examples are not special cases. As a matter of fact, we can prove the optimality p r o p e r t y of m a n y generalized n o n - b i n a r y designs by use of the optimal binary ones.

2. Preliminaries The general additive linear model without interaction is assumed. Let Yijhl = fl + "Ci + flj + 7h + I~ijhl,

where Yiih~ denote t h e / t h m e a s u r e m e n t receiving treatment i on the (j, h)-node of the lattice, where 1 <~ i <~ v, 1 <~j <~ bl, 1 <~ h <~ b2, l = 0 or 1, and/~, z, fl, 7 denote the overall mean effect, the treatment effect, the row effect, and column effect, respectively. The eijh~ are independently and identically distributed n o r m a l r a n d o m variables with mean zero and variance a 2. Let D(v, b l , k l , b 2 , k2) be the class of all connected row-column designs with v treatments arranged in bl rows of size kx each and b2 columns of size k2 each. Then the reduced information matrix, the so-called C-matrix, of design d for estimating the treatment contrasts, zi - r j, Vi # j, is Cd = Cdl -- Ma2Kd2M'a2, or

C d = Cd2 - -

MdlKdlM'dl,

where Call = diag(rdl . . . . . rdv) -- ( 1 / k l ) N d l N ' d l , bl

b2

bl

b2

j=l

h=l

j=l

h=l

Md2 = N ~ : Kd2 = k2lb2 -

(1/kl)N~lNd3, (1/kl)N'azNd3

,

Cd2 = diag(rdl . . . . . r~v) -- (1/k2)Nd2N'd2 , Mdl = Ndl -- (1/kE)NdES'd3, Kdl = kllb~ -- (1/kE)N~aN'a3,

and A - denotes a generalized inverse of A. As one can observe, Cgl(Cd2) is the C-matrix of the c o m p o n e n t block design when rows (columns) are treated as blocks. Since Cd <<. Cdl (CUE), with equality holding if and only if M d 2 = O v x b2 ( M d l = O v x b l), where Os ×t is the s × t matrix of zeros. T h a t is, the C-matrix of a row-column design is the same as the C-matrix of the c o m p o n e n t

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block design when Md2 = Ovxb2 or Mall = Or×b1. Hence, finding optimal row-column designs can be simplified by finding optimal block designs with rows (columns) as blocks, and Md2 = Ov ×b2 (Mall = Oo ×~). There are many optimality criteria existing in the field of optimal designs; however, in this paper we adopt the concept of universal optimality from Kiefer (1975). A design d * ~ D ( v , ba, k l , b 2 , k 2 ) such that C* = x(I~ - J~/v), for some x > 0, and d* maximizes the trace of Cd (denoted as tr(Cd)) over all design in D(v, bl, k~, b2, k2); then d* is universally optimal in D(v, bl, kl, b2, k2). Therefore, if the component block design with rows (columns) as blocks of d* is a balanced incomplete block design (BIBD) or balanced block design (BBD), whose universal optimality properties have been verified, and has Md.2 = O~,×b2 (Mn*l = Or×h,), then d* is an universally optimal row-column design.

3. Main result In this section, we show that if an universally optimal row-column design d* is copied t times in the column direction as shown in Fig. 1, or in the row direction as shown in Fig. 2, the resulting design is still universally optimal, provided that d* satisfies some conditions.

d,I I Fig. 1.

d*!

d,! d* Fig. 2.

Proposition. Suppose d ~ D(v, bl, kl, b2, k2). L e t d ~° ~ D(v, bl, t × kl, t × b2, k2) be the t copies o l d in the column direction. T h e n Md~,q = t × Mdl, Md")2 ~- ltt ~ Md2.

Proof. For d t° as defined above, Nd~,,1 = t × Ndl, Nd~o2 ----T; ® Nd2, and Nd,.3 = 1; ® Nd3. Hence, Md,q = Nd.q -- (l /kE)Na.)2N'd~o3 = t x Ndl -- (t/kE)Nd2S'a2 =t×Mal, Ma"'2 = Na,,,e -- (t x k l ) - I Na,,,1Na,,,3 ---- 1; Q Na2 -- (1/k1)N~tl(1; (~ Na3) = T; ®

Md2.

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C.-P. Ting et al. / Statistics & Probability Letters 33 (1997) 145-149

Suppose d* E D(v, bl,kl,b2,k2) has Md.~ = Oo×bl and the columns of d* form a BIBD or BBD; then d* is universally optimal in D(v, bl, k~, b2, k2) as stated in Section 2. If d* is copied t times in the column direction resulting in d* tt), then by the proposition, Md.,)l = O~ ×b~ and the columns of d *~° still form a BIBD or a BBD; hence the following theorem is a direct consequence. Theorem. I f d* ~ D(v, bl, kl, b2, k2) is an universally optimal design having Md.1 = Ov ×bl, and the columns of d* form a BIBD or a BBD. Then d *tt) is universally optimal in D(v, bl, t x kl, t x bE, k2). Example. The design d* e D(6, 7, 6, 14, 3) given in the following, which is listed in Table 5 of Stewart and Bradely (1991), has Md.1 = 06 × 7, and is universally optimal in D(4, 7, 6, 14, 3): 2

5 4

3

6 5

7 6

5

2

6

1 2

1

4

6 5

2 2 1

5

. . . . .

4

4 1

7

1

7

7

7 4

7

6 3

. . . . .

3 3 2

6

4 3

5

If d* is copied t times in the column direction, the resulting design is with nd.i~. = 0 or t, nd*i.h = 0 or 1, V/,j, h, which is not a generalized binary design, is universally optimal in D(7, 7, 6t, 14t, 3) by the theorem. Suppose t = 2; the following design with nd.lj. = O, 2, nd*i.h = O, 1, Vi, j,h, is universally optimal in D(7,7, 12,28,3): 2

5 4

7 6

5

3

7 4

7

7 46

1 7

6

2

3

6 5

. . . . .

1 1 7

24 1

. . . . .

46

3 5

5 4

2 2 1

5 6

3

. . . .

2

4

5 6

2

7

7 46

1

1 7

1 1

. . . . .

46 1

724 7

4

7 3

6

3 5

6 5

3

3

2

5 2

2 1

5

3 3 2

6

4 3

5

The above theorem works for the rows too. Let d ttJ ~ D(v, t x bl, kl, b2, t × k2) be the t copies of d in the row direction. Then by the same argument as in the proposition, one can show that Mdt,J2 = t x Mn2 and Ma~,Jl = T; ® Md~. Thus, if d* is an universally optimal design having Md.2 = Or×b2, and the rows form a BIBD or a BBD, then d *ttj is universally optimal in D(v, t x ba, kl, b2, t x k2), and the columns o f d *ttl do not have the property of being generalized binary, either.

References Baksalary, J.K. and F. Pulelsheim (1992), Adjusted orthogonality properties in multiway block design, in: S. Schach and G. Trenkler, eds., Data Analysis and Statistical Inference (Festschrift in Honor of Freidhelm Eicker, Eul Verlag) pp. 413-420.

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Jacroux, M. and R. Saharay (1991), On the determination and construction of optimal row-column designs having unequal row and column sizes, Ann. Inst. Statist. Math. 43, 377-390. Kiefer, J. (1975), Construction and optimality of generalized Youden designs, in: J.N. Srivastava, ed., A Survey of Statistical Design and Linear Models (North-Holland, Amsterdam) pp. 333-353. Kunert, J. (1993a), A note on optimal designs with a non-orthogonal row-column-structure, J. Statist. Plann. Inference 37, 265-270. Kunert, J. (1993b), On designs with a non-orthogonal row-column structure, in: W.G. Miiller, H.P. Wynn, A.A. Zhigljavsky, eds., Model Oriented Data Analysis (Physica Verlag, Wurzburg) pp. 105-112. Saharay, R. (1986), Optimal designs under a certain class of non-orthogonal row-column structure, Sankhy~ B 48, 44-67. Shah, K.R. and B.K. Sinha (1993), Optimality aspects of row-column designs with non-orthogonal structure, J. Statist. Plann. Inference 36, 331-346. Stewart, F.P. and R.A. Bradley (1991), Some universally optimal row-column designs with empty nodes, Biometrika 78, 337-348.