Statistics & Probability Letters 17 (1993) 287-291

North-Holland

13 July 1993

An EDG/3 scheme having 5 associate classes

Sudhir Gupta Division of Statistics, Northern Illinois University, DeKalb, USA

Kishore Sinha Birsa Agricultural University, Ranchi, Bihar, India

Received September 1992

Abstract: A 5 associate class EGD/3 scheme is introduced for three factor experiments. Some methods of constructing designs

based on this scheme are given. The most important property of these designs is that they are balanced factorially.

Keywords: Balanced factorial experiment; block design; extended group divisible scheme.

1. Introduction

Consider a factorial experiment involving m factors, the ith factor being at pi levels denoted by

0, l,..., pi - 1. The u = FIy!lpi treatments will be labelled using m digit numbers 12,~~. . . a,, ai = 0, l,..., pi - 1. These m digit numbers will be arranged in increasing order of magnitude, and will be further coded as 1, 2,. . . , u respectively. Let x=(x,, x2,.. ., x,,,) where xi = 0 or 1. Hinkelmann and

Kempthorne (1963) defined the extended group divisible scheme (EGD/m) having 2” - 1 associate classes. Two treatments in the EGD/m scheme are x-associates where xi = 1 if the ith factor occurs at the same level in both the treatments and xi = 0 otherwise. Let h(x) denote the number of times two treatments which are x-associates occur together in the design. Note that A(X) depends only on x and is independent of the specific pair of x-associates chosen.

The EGD/m scheme was earlier considered by Nair and Rao (1948) and Shah (1958, 19601, and has been referred to as binary number association scheme by Paik and Federer (1973). A detailed study of EGD/m scheme is due to Hinkelmann (1964). The purpose of this paper is to consider a 5 associate class EGD/m scheme for m = 3 factors. The association scheme is defined in Section 2. Some methods of constructing designs based on this scheme are then presented in Section 3. Several authors have given construction procedures for EGD/m designs, a review of which can be found in Gupta and Mukerjee (1989, p. 21).

Correspondence to: Sudhir Gupta, Division of Statistics, Northern Illinois University, DeKalb, IL 601152854, USA.

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2. A 5 associate class EGD/3 scheme

We now define a 5 associate EGD/m scheme for the special case of m = 3 factors. Let v =p1p2p3 symbols 1, 2,. . . , u be arranged in p1 sets of p2p3 symbols each, the symbols in any set being arranged in pz rows of size p3 each as follows:

1+ (i - l)P,P,

1 !

2 + (i - ~)P,P, . . . {I+ (i - ~)P,)P,

Set i = l+(l+(i-l)p,}p, 2+{l+(i-l)p,}p, ... (2 + (i - ~)P,}P,

13

1 1 + (4, - 1)~~

i= 1,2 ,..., pl.

2+ (ip,- l)p, ... ip,p, 1

Any two symbols in the same set and the same row are first associates, in the same set and different rows are second associates, in the same position but different sets are third associates, in the same row but different sets and different positions are fourth associates, and in different sets and different rows are fifth associates. Let Ai, ni denote the number of blocks in which two ith associates occur together and the number of ith associates of any symbol respectively. Note that n, =p3 - 1, n2 =p3(p2 - l), n3 =p, - 1, n4 = (pl - 1X p3 - 0, n5 =p3(p1 - NP, - 1).

It may be verified that the association scheme defined above is EGD/3 with h(0, 0, 0) = h(0, 0, 1) = A,, A(0, 1, 0) = A,, A(0, 1, 1) = A,, A(1, 0, 0) = A(1, 0, 1) = A,, A(l, 1, 0) = A,.

It should be noted that Tharthare (1965) defined a four associate class scheme in a similar setting. However, a partially variance balanced design for a factorial experiment necessarily involves several associate classes. Therefore, reduction of associate classes is not always appropriate for factorial experiments. The 5 associate class scheme defined here is especially useful from this point of view.

It is well known that a design is balanced factorially if and only if it is a partially balanced incomplete block (PBIB) design with EGD/m scheme (Kshirsagar, 1966). The designs based upon the above association scheme are thus balanced factorially. Let F(x) denote a single d.f. normalized factorial contrast and let 0(x) denote its effective replication (Pearce, 1960) in an EGD/m design. If 0(x> is non-zero then the effect F(x) is estimated with a variance a2//3(x) where u2 is the random error variance. Otherwise F(x) is completely confounded in the design. Also, the efficiency factor of F(x) is given by 0(x)/r where r denotes the constant number of treatment replications. The e(x) can be obtained using the following equation which was proved by Shah (19601,

Here 13(0, 0,. ..,O) = 0, A(l, 1,. .., 1) = -r(k - l), the symbol @ denotes Kroenecker product, X de- notes the symbolic direct product (Shah, 1960) defined by

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and

3. Constructions

We now present some methods of constructing designs having EGD/3 scheme of Section 2.

Theorem 3.1. The existence of a balanced incomplete block (BIB) design D, with parameters p,, b,, r,, k,, A and a group divisible (GD) design D, with parameters v2, b,, r2, k,, A12, Az2, m =p2, n =p3 implies the existence of a 5 associate EGD/3 design with parameters

v =p1p2p3, b = b,b,, r=r1r2+(bl-rl)(b2-r2),

k = v - k,( v2 - k2) - kz( p1 - k,),

4 = (b, - rl)(b2 - 2r2) + blh12,

A2 = (4 - rl)(b2 - 2r2) + blhz2,

A3=(bl-2rl)(b2-r2) +Ab,,

A, = (b, - 2r,)( b, - 2r,) + (b, - 4r,)A12 + (b, - 4r,)A + 4AA,, + 2r,r2,

A, = (b, - 2r,)( b, - 2r,) + (6, - 4r,)A,, + A( b, - 4r, + 4A,,) + 2r,r,. •I

Let Ni denote the incidence matrix of Di, i = 1, 2. The incidence matrix N of a design of Theorem 3.1 is obtained from N, by replacing its 1 and 0 elements by IV2 and flz respectively. The p2 denotes the incidence matrix of the complimentary design obtained from D,.

The theorem can be proved by an explicit evaluation of NN’ which involves quite cumbersome expressions. The details are omitted here for the sake of brevity. A corollary follows.

Corollary 3.1. The existence of a BIB design D, withparametersp, = 2k,, b,, rl, k,, A and a GD design D, with parameters v2 = 2k,, b,, r2, k,, A12, A,,, m =p2, n =p3 implies the existence of a 5 associate EGD/3 design with parameters

v = 4k,k,, b = b,b,, r = 2r,r, , k = 2k,k,,

A, = b,A,,, A, = b,A,,, A, = Ab,,

A, = 2I(r1 - A)(r2 -42) + A4219 A,=2{(r,-A)(r,-A22+AA22). 0

The following corollaries are obtained from Theorem 3.1 by taking D, as given by the incidence matrix

NJ 0 1 [ 1 0 1’ (3.1)

Corollary 3.2. The existence of a GD design D, with parameters v2 = 2k,, b,, r2, k,, A,,, A22, m =p2, n =p3 implies the existence of a 5 associate EGD/3 design with parameters

v = 4k,, b = 2b,, r = 2r,, k = 2k,,

A, = 2A,,,‘ A, = 2A,, > A, = 0, A,=2(r2-A,,), A,=2(r,-A,,). 0

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Corollary 3.3. The existence of a BIB design D, with parameters v2 = 2k,, b,, r2, k,, A* implies the existence of a 5 associate EGD/3 design with parameters

v=4k,, b = 2b,, r=2r2, k = 2k,,

A,=A,=2A*, A, = 0, h,=A,=2(r,-A”). 0

When the design D, is self-complementary then in Corollaries 3.2 and 3.3, D, is taken as given by N, = [l 01’. The following example serves as an illustration.

Example 3.1. Take D, as the GD design [l 31, [l 41, [2 31, [2 41 which is self-complementary. Then Corollary 3.2 with Nr = [l 01’ yields the design [l 3 6 81, [l 4 6 71, [2 3 5 81, [2 4 6 71. The design has parameters

v = 8, b = 4, r=2, k = 4, A, = A, = 0, A,=A,=l, A, = 2, p, =p2 =p3 = 2.

The following theorem gives a method of construction using a balanced bipartite design. For notations and definition of balanced bipartite designs we refer to Kageyama and Sinha (1988).

Theorem 3.2. The existence of a balanced bipartite design D, with parameters v2 = vzl + vz2, vzl = vz2 = k,,

b, = r21 + r22, r21, r22, k,, A,,, A22, A,, implies the existence of a 5 associate EGD/3 design with parameters

v=4k,, b = 2b,, r=b2, k = 2k,,

A, = A11 + A227 A, = 242, A, = 0, A4=2(rl -A,,), A, = b, - 2A,,,

P1 ‘P2 = 2, p3=k2. 0

A design of the above theorem is also obtained as indicated after Theorem 3.1. The Nr is as given by (3.1).

Example 3.2. Take D, as the balanced bipartite design [l 21, [3 41, [l 41, [2 31, [l 31. Theorem 3.2 then yields the following design with parameters

v = 8, b = 10, r=5, k = 4, A, = 1, A, = 2, A, = 0, A, = 4, A, = 3:

[1278], [3456], [1467], [2358], [1368],

[3 4 5 61, [l 2 7 81, [2 3 5 81, [l 4 6 71, [2 4 5 71.

Finally, we remark that the latin square L, association scheme can also be used in the construction of 5 associate EGD/3 designs. However, in that case the definition of association scheme gets modified appropriately. Also, then D, is taken to be an L, design in Theorem 3.1 and Corollaries 3.1-3.3. The parameters of resulting designs are obtained in a similar way.

References

Gupta, S. and R. Mukerjee (1989), A Calculus for Factorial Arrangements. Lecture Notes in Statist. No. 59 (Springer, New York).

Hinkelmann, K. (1964), Extended group divisible partially

balanced incomplete block designs, Ann. Math. Statist. 35,

681-695.

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Hinkelmann, K. and 0. Kempthorne (1963), Two classes of

group divisible partial diallel crosses, Biometrika 50, 281-

291.

Kageyama, S. and K. Sinha (1988), Some constructions of

balanced bipartite block designs, Utilitas Math. 33, 137-

162.

Kshirsagar, A.M. (1966), Balanced factorial designs, J. Roy.

Statist. Sot. Ser. B 28, 559-569.

Nair, K.R. and C.R. Rao (1948), Confounding in asymmetric

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Paik, U.B. and W.T. Federer (19731, Partially balanced de-

signs and properties A and B, Comm. Statist.--Theory

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Pearce, S.C. (1976), Concurrence and quasi-replication: an

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Tharthare, S.K. (19631, Generalized right angular designs,

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