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treatments being the ith associates occur together in exactly λi blocks. ... signs based on a rectangular association scheme of v = mn treatments arranged in an m ...
J. Japan Statist. Soc. Vol. 33 No. 1 2003 137–144

SOME PATTERNED CONSTRUCTIONS OF RECTANGULAR DESIGNS Sanpei Kageyama* and Kishore Sinha** This paper describes some new patterned methods of constructing rectangular designs from balanced incomplete block (BIB) designs and nested BIB designs, and gives a table of rectangular designs in the range of r, k ≤ 10. Key words and phrases:

BIB design, nestedness, rectangular design.

1. Introduction A balanced incomplete block (BIB) design with parameters v, b, r, k, λ is a block design, BIBD(v, b, r, k, λ), with v treatments and b blocks of size k each such that every treatment occurs in exactly r blocks and that any two distinct treatments occur together in exactly λ blocks. This is a standard design used for constructing other designs (see Raghavarao, 1988). A nested BIB design with parameters v, b1 , r, k1 , b2 , k2 , λ1 , λ2 is a block design in which both the nesting blocks (superblocks) and the sub-blocks form BIBD(v, b1 , r, k1 , λ1 ) and BIBD(v, b2 , r, k2 , λ2 ), respectively (see Preece, 1967; Morgan, 1996). Thus, there are b2 sub-blocks of size k2 nested in each of b1 superblocks of size k1 . Note that parameters λ1 and λ2 in the nested BIB design have different meaning from those in partially balanced incomplete block (PBIB) designs. A PBIB design, based on an s-associate association scheme, with parameters v, b, r, k, λi , i = 1, 2, . . . , s, is a block design with v treatments and b blocks of size k each such that every treatment occurs in r blocks and any two distinct treatments being the ith associates occur together in exactly λi blocks. A group divisible (GD) design is a 2-associate PBIB design based on a group divisible association scheme, i.e., a set of the mn treatments can be divided into m groups of n treatments each such that any two treatments occur together in λ1 blocks if they belong to the same group, and in λ2 blocks if they belong to different groups. Rectangular designs, introduced by Vartak (1955), are 3-associate PBIB designs based on a rectangular association scheme of v = mn treatments arranged in an m × n rectangle such that, with respect to each treatment, the first associates are the other n − 1 (= n1 , say) treatments of the same row, the second associates are the other m − 1 (= n2 , say) treatments of the same column and the remaining (m − 1)(n − 1) (= n3 , say) treatments are the third associates. Received June 6, 2002. Revised August 26, 2002. Accepted February 11, 2003. *Hiroshima University, Higashi-Hiroshima 739-8524, Japan. **Birsa Agricultural University, Ranchi-834006, India.

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SANPEI KAGEYAMA AND KISHORE SINHA

That is, a rectangular design is an arrangement of v = mn treatments in b blocks such that (i) each block contains k distinct treatments, k < v, (ii) each treatment occurs in exactly r blocks, (iii) the mn treatments are arranged in a rectangle of m rows and n columns such that any two treatments in the same row (column) occur together in λ1 (λ2 ) blocks, respectively, and in λ3 blocks otherwise. These designs have been studied by Bhagwandas and Kageyama (1985), Suen (1989), Sinha (1991), Sinha et al. (1993, 1996, 1999), Kageyama and Miao (1995), Sinha et al. (2002b), and so on. The rectangular designs are useful as factorial experiments, having balance as well as orthogonality (Gupta and Mukerjee, 1989). In addition, if λ3 is bigger than λ1 and λ2 , the loss of information on the main effects becomes small (Suen, 1989), when these designs are used as an m×n complete confounded factorial experiments. A review of constructional procedures for these designs is given by Gupta and Mukerjee (1989). Recently rectangular designs have also been used in the construction of balanced arrays and orthogonal arrays in Sinha et al. (2002a). In this paper, some patterned constructions of rectangular designs are given along with a table of new designs in the range of r, k ≤ 10 and not found in the tables of Suen (1989), Sinha et al. (1993, 1996), and Sinha et al. (2002b). For convenience, I s denotes the identity matrix of order s, J s×t denotes the s × t matrix all of whose elements are unity, in particular, J s = J s×s and A ⊗ B denotes the Kronecker product of two matrices A and B. 2. Constructions It is known (Vartak, 1955) that the Kronecker product of incidence matrices of two BIB designs produces a rectangular design. Here other variations will be considered. The rectangular association is here arranged as 1 n+1 .. .

2 n+2 .. .

··· n · · · 2n , .. .

(m − 1)n + 1 (m − 1)n + 2 · · · mn where n1 = n − 1, n2 = m − 1 and n3 = (m − 1)(n − 1). Theorem 2.1. The existence of a BIB design with parameters (2.1)

v  = m, b , r , k  , λ

implies the existence of a rectangular design with parameters

SOME PATTERNED CONSTRUCTIONS OF RECTANGULAR DESIGNS

  n  b, 2   n−1   b , k = 2k  + (n − 2)v  , r = (n − 1)r + 2   n−2  b, λ1 = (2n − 3)r + 2   n−1   b, λ2 = (n − 1)λ + 2   n−2    b. λ3 = λ + 2(n − 2)r + 2 v = mn,

(2.2)

139

b=

Proof. Let N be the m × b incidence matrix of a BIB design with parameters (2.1). Further let Q be the v  ×b incidence matrix of an unreduced BIB design with parameters v  = n, b = n2 , r = n − 1, k  = 2, λ = 1. Then it follows that N ∗ = Q ⊗ N + (J n×b − Q) ⊗ J m×b n  is the mn × 2 b incidence matrix of the required rectangular design with parameters (2.2). Remark 1. In Theorem 2.1, as usual, if we take only Q ⊗ N as a design N ∗ , then a rectangular design with parameters v = mn, b = n2 b , r = (n − 1)r , k = 2k  , λ1 = r , λ2 = (n − 1)λ , λ3 = λ can be obtained. This simple structure can produce many designs with relatively small values of parameters. When n = 2, we get N ∗ = J 2×1 ⊗ N , which coincides with a group divisible PBIB design. Hence we only consider the case n ≥ 3, especially for the preparation of the table in this paper. Theorem 2.2. The existence of a BIB design with parameters (2.1) implies the existence of a rectangular design with parameters v = mn, (2.3)

b = nb ,

r = (n − 1)r ,

k = (n − 1)k  ,

λ1 = (n − 2)r ,

λ2 = (n − 1)λ ,

λ3 = (n − 2)λ .

Proof. Let N be the m × b incidence matrix of a BIB design with parameters (2.1). Then it follows that N ∗ = (J n − I n ) ⊗ N is the mn × nb incidence matrix of the required rectangular design with parameters (2.3). Note that in Theorem 2.2, when λ1 = λ2 , i.e., (n − 2)r = (n − 1)λ , and m = n, the design becomes a Latin square PBIB design (cf. Clatworthy, 1973).

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SANPEI KAGEYAMA AND KISHORE SINHA

Theorem 2.3. The existence of a BIB design with parameters (2.1) implies the existence of a rectangular design with parameters v = mn, (2.4)

b = nb ,

r = (n − 1)r + b , λ1 = nr ,

k = (n − 1)k  + v  ,

λ2 = (n − 1)λ + b ,

λ3 = (n − 2)λ + 2r .

Proof. Let N be the m × b incidence matrix of a BIB design with parameters (2.1). Then it follows that N ∗ = (J n − I n ) ⊗ N + I n ⊗ J m×b is the mn × nb incidence matrix of the required rectangular design with parameters (2.4). Remark 2. Theorem 2.3 is a generalization of Theorem 3.2 in Bhagwandas and Kageyama (1985). In Theorem 2.3, when λ1 = λ2 , i.e., b = nr − (n − 1)λ , and m = n, we get a Latin square PBIB design, while when λ2 = λ3 , i.e., b = 2r − λ , we get a GD design (cf. Clatworthy, 1973). Theorem 2.4. The existence of a BIB design with parameters (2.1) implies the existence of a rectangular design with parameters v = mn, (2.5)

b = nb ,

r = (n − 2)r + b , λ1 = (n − 2)r ,

k = (n − 2)k  + v  ,

λ2 = b − 2r + nλ ,

λ3 = (n − 4)λ + 2r .

Proof. Let N be the m × b incidence matrix of a BIB design with parameters (2.1). Then it follows that N ∗ = (J n − I n ) ⊗ N + I n ⊗ (J v ×b − N ) is the mn × nb incidence matrix of the required rectangular design with parameters (2.5). Remark 3. Theorem 2.4 is a generalization of Theorem 3.3 in Bhagwandas and Kageyama (1985). In Theorem 2.4, when λ1 = λ2 , i.e., b = n(r − λ ), and m = n, we get a Latin square PBIB design, while when λ2 = λ3 , i.e., b = 4(r − λ ), we get a GD design (cf. Clatworthy, 1973). A nested BIB design can also be used to construct rectangular designs. Theorem 2.5. The existence of a nested BIB design with parameters (2.6)

v  = m, r , k1 = 2k2 , k2 , b1 , b2 , λ1 , λ2

SOME PATTERNED CONSTRUCTIONS OF RECTANGULAR DESIGNS

141

implies the existence of a rectangular design with parameters v = 3m, (2.7)

b = 3b2 ,

r = 2r ,

λ1 = 0,

λ2 = 2λ2 ,

n1 = 2,

n2 = m − 1,

k = 2k2 ,

λ3 = λ1 − λ2 ,

n3 = 2(m − 1).

Proof. Let N 1 and N 2 be the v  × b1 incidence matrix of the first and second halves of the whole blocks of the nested BIB design with parameters (2.6), and O be the v  ×b1 matrix of zero’s, where N 1 +N 2 and [N 1 : N 2 ] are incidence matrices of whole blocks (superblocks) and sub-blocks of the nested BIB design, respectively. Then it follows that   N1 N2 N2 N1 O O   N ∗ = N 2 N 1 O O N 1 N 2  O O N1 N2 N2 N1 yields the required rectangular design with parameters (2.7). When 4t+1 is a prime or a prime power, an initial block (x0 , x2 , . . . , x4t−2 , x, mod (4t + 1) can yield a nested BIB design with parameters v  = 4t + 1, r = 4t, k1 = 4t, k2 = 2t, b1 = 4t + 1, b2 = 2(4t + 1), λ1 = 4t − 1, λ2 = 2t − 1, where x is a primitive element of GF(4t + 1) (see Sinha et al., 1993). This observation with Theorem 2.5 produces the following. x3 , . . . , x4t−1 )

Corollary 2.5.1. When 4t + 1 is a prime or a prime power, there exists a rectangular design with parameters v = 3(4t + 1), b = 6(4t + 1), r = 8t, k = 4t, λ1 = 0, λ2 = 2(2t−1), λ3 = 2t, n1 = 2, n2 = 4t, n3 = 8t, m = 4t+1, n = 3. By the application of Theorem 2.2 of Sinha et al. (2002a) to Corollary 2.5.1, we can get a balanced array BA(6(4t + 1), 4t + 1, 4, 2) {µii = 2(2t − 1), i = 0, 1, 2; µij = 2t = µji , i, j(i = j) = 0, 1, 2; µi3 = 2 = µ3i , µ33 = 0, i = 0, 1, 2}. For the definition of a balanced array, see Sinha et al. (2002a). Similarly, when 4t−1 is a prime or a prime power, an initial block (x0 , x2 , . . . , x4(t−1) , x, x3 , . . . , x4t−3 ) mod (4t − 1) can yield a nested BIB design with parameters v  = 4t − 1, r = 2(2t − 1), k1 = 2(2t − 1), k2 = 2t − 1, b1 = 4t − 1, b2 = 2(4t − 1), λ1 = 4t − 3, λ2 = 2(t − 1), where x is a primitive element of GF(4t − 1) (see Sinha et al., 1993). This observation with Theorem 2.5 produces the following. Corollary 2.5.2. When 4t − 1 is a prime or a prime power, there exists a rectangular design with parameters v = 3(4t−1), b = 6(4t−1), r = 4(2t−1), k = 2(2t − 1), λ1 = 0, λ2 = 4(t − 1), λ3 = 2t − 1, n1 = 2, n2 = 4t − 2, n3 = 4(2t − 1), m = 4t − 1, n = 3. By the application of Theorem 2.2 of Sinha et al. (2002a) to Corollary 2.5.2, we can get a balanced array BA(6(4t − 1), 4t − 1, 4, 2) {µii = 4(t − 1), i = 0, 1, 2; µij = 2t − 1 = µji , i, j(i = j) = 0, 1, 2; µi3 = 2 = µ3i , µ33 = 0, i = 0, 1, 2}.

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SANPEI KAGEYAMA AND KISHORE SINHA

Theorem 2.6. The existence of a nested BIB design with parameters (2.6) implies the existence of a rectangular design with parameters v = 3m, λ1 = 2r , n1 = 2,

b = 6b1 ,

r = 2r + 2b1 ,

λ2 = 2λ2 + 2b1 ,

n2 = m − 1,

k = 2k2 + m,

λ3 = 2r + λ1 − λ2 ,

n3 = 2(m − 1).

Proof. Let N 1 and N 2 be defined as in the proof it follows that  J m×b1 J m×b1 N 1 N2 N1  ∗ N =  N1 N 2 J m×b1 J m×b1 N 2 N2 N1 N2 N 1 J m×b1

of Theorem 2.5. Then  N2  N1  J m×b1

yields the required rectangular design. Corollary 2.6.1. When 4t + 1 is a prime or a prime power, there exists a rectangular design with parameters v = 3(4t+1), b = 6(4t+1), r = 2(8t+1), k = 8t+1, λ1 = 8t, λ2 = 12t, λ3 = 10t, n1 = 2, n2 = 4t, n3 = 8t, m = 4t+1, n = 3. Corollary 2.6.2. When 4t − 1 is a prime or a prime power, there exists a rectangular design with parameters v = 3(4t−1), b = 6(4t−1), r = 2(8t−3), k = 8t − 3, λ1 = 4(2t − 1), λ2 = 6(2t − 1), λ3 = 5(2t − 1), n1 = 2, n2 = 4t − 2, n3 = 4(2t − 1), m = 4t − 1, n = 3. The results given in this paper produce the following new rectangular designs within the range of r, k ≤ 10, that are not found in tables by Suen (1989), Sinha et al. (1993, 1996), and Sinha et al. (2002b). In the table, the designs of Nos. 1 and 19 are LS26 and LS43, respectively, after combining 1st and 2nd associate classes and then renaming the new 1st and 2nd associates. The designs of Nos. 2, 5 and 7 are R172, R176 and R203, respectively, after combining 2nd and 3rd associate classes. The design of No. 16 is LS116, after combining 1st and 2nd associate classes. Here LS stands for Latin square PBIB designs, while R stands for regular GD designs in tables given by Clatworthy (1973). Table. Rectangular designs with r, k ≤ 10. No.

v

m

n

b

r

k

λ1

λ2

λ3

1

9

3

3

9

4

4

2

2

1

2

9

3

3

9

7

7

6

5

5

3

12

4

3

12

6

6

3

4

2

Source Remark 1 or Theorem 2.2, BIBD(3,3,2,2,1) Theorem 2.1 or 2.3, BIBD(3,3,2,2,1) Remark 1 or Theorem 2.2, BIBD(4,4,3,3,2)

4

12

3

4

12

6

6

4

3

2

Theorem 2.2, BIBD(3,3,2,2,1)

SOME PATTERNED CONSTRUCTIONS OF RECTANGULAR DESIGNS

Table. (continued). No.

v

m

n

b

r

k

λ1

λ2

λ3

5

12

4

3

12

7

7

3

4

4

Theorem 2.4, BIBD(4,4,3,3,2)

Source

6

12

3

4

12

7

7

4

3

4

Theorem 2.4, BIBD(3,3,2,2,1)

7

12

4

3

12

10

10

9

8

8

Theorem 2.1 or 2.3,

8

12

4

3

18

6

4

3

2

1

9

12

3

4

18

6

4

2

3

1

Remark 1, BIBD(3,3,2,2,1)

10

12

4

3

18

9

6

3

3

5

Theorem 2.4, BIBD(4,6,3,2,1)

11

15

5

3

15

8

8

4

6

3

Remark 1 or Theorem 2.2,

BIBD(4,4,3,3,2) Remark 1 or Theorem 2.2, BIBD(4,6,3,2,1)

BIBD(5,5,4,4,3) 12

15

3

5

15

8

8

6

4

3

Theorem 2.2, BIBD(3,3,2,2,1)

13

15

5

3

15

9

9

4

6

5

Theorem 2.4, BIBD(5,5,4,4,3)

14

15

3

5

30

8

4

2

4

1

Remark 1, BIBD(3,3,2,2,1)

15

15

5

3

30

8

4

4

2

1

Remark 1 or Theorem 2.2,

16

16

4

4

16

9

9

6

6

4

Theorem 2.2, BIBD(4,4,3,3,2)

17

16

4

4

24

9

6

3

6

2

Remark 1, BIBD(4,4,3,3,2)

18

16

4

4

24

9

6

6

3

2

Theorem 2.2, BIBD(4,6,3,2,1)

19

16

4

4

36

9

4

3

3

1

Remark 1, BIBD(4,6,3,2,1)

20

18

6

3

18

10

10

5

8

4

Remark 1 or Theorem 2.2,

21

18

3

6

18

10

10

8

5

4

Theorem 2.2, BIBD(3,3,2,2,1)

22

18

6

3

30

10

6

5

4

2

Remark 1 or Theorem 2.2,

23

18

3

6

45

10

4

2

5

1

Remark 1, BIBD(3,3,2,2,1)

24

18

6

3

45

10

4

5

2

1

Remark 1 or Theorem 2.2,

25

21

7

3

21

6

6

3

2

1

26

21

7

3

21

8

8

4

4

2

27

21

7

3

21

10

10

3

4

5

Theorem 2.4, BIBD(7,7,3,3,1)

28

27

9

3

36

8

6

4

2

1

Remark 1 or Theorem 2.2,

29

28

7

4

28

9

9

6

3

2

Theorem 2.2, BIBD(7,7,3,3,1)

30

28

7

4

42

9

6

3

3

1

Remark 1, BIBD(7,7,3,3,1)

31

33

11

3

33

10

10

5

4

2

Remark 1 or Theorem 2.2,

32

39

13

3

39

8

8

4

2

1

33

48

16

3

60

10

8

5

2

1

34

63

21

3

63

10

10

5

2

1

BIBD(5,10,4,2,1)

BIBD(6,6,5,5,4)

BIBD(6,10,5,3,2)

BIBD(6,15,5,2,1) Remark 1 or Theorem 2.2, BIBD(7,7,3,3,1) Remark 1 or Theorem 2.2, BIBD(7,7,4,4,2)

BIBD(9,12,4,3,1)

BIBD(11,11,5,5,2) Remark 1 or Theorem 2.2, BIBD(13,13,4,4,1) Remark 1 or Theorem 2.2, BIBD(16,20,5,4,1) Remark 1 or Theorem 2.2, BIBD(21,21,5,5,1)

143

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References Bhagwandas and Kageyama, S. (1985). Patterned constructions of partially balanced incomplete block designs, Commun. Statist. -Theor. Meth., 14, 1259–1267. Clatworthy, W. H. (1973). Tables of Two-Associate-Classes Partially Balanced Designs, National Bureau of Standards, Applied Mathematics Series 63, Washington, D.C. Gupta, S. and Mukerjee, R. (1989). A Calculus for Factorial Arrangements, Lecture Notes in Statistics, Vol. 59, Springer-Verlag, New York. Kageyama, S. and Miao, Y. (1995). Some methods of constructions of rectangular PBIB designs, Combinatorics Theory, 1, 186–198. Morgan, J. P. (1996). Nested designs, Handbook of Statistics, Vol. 13, 939–976, Elsevier, NorthHolland. Preece, D. A. (1967). Nested balanced incomplete block designs, Biometrika, 43, 479–486. Raghavarao, D. (1988). Constructions and Combinatorial Problems in Design of Experiments, Dover, New York. Sinha, K. (1991). A construction of rectangular designs, J. Comb. Math. Comb. Computing, 9, 199–200. Sinha, K. and Mitra, R. K. (1999). Constructions of nested balanced incomplete block designs, rectanguar designs and q-ary codes, Ann. Combin., 3, 71–80. Sinha, K., Kageyama, S. and Singh, M. K. (1993). Construction of rectangular designs, Statistics, 25, 63–70. Sinha, K., Mitra, R. K. and Saha, G. M. (1996). Nested BIB designs, balanced bipartite weighing designs and rectangular designs, Utilitas Math., 49, 216–222. Sinha, K., Dhar, V., Saha, G. M. and Kageyama, S. (2002a). Balanced arrays of strength two from block designs, J. Combin. Designs, 10, 303–312. Sinha, K., Singh, M. K., Kageyama, S. and Singh, R. S. (2002b). Some series of rectangular designs, J. Statist. Plann. Inference, 106, 39–46. Suen, C. (1989). Some rectangular designs constructed by the method of differences, J. Statist. Plann. Inference, 21, 273–276. Vartak, M. N. (1955). On an application of Kronecker product of matrices to statistical designs, Ann. Math. Statist., 26, 420–438.