Metodološki zvezki, Vol. 5, No. 1, 2008, 1-8
Some New Construction Methods of Variance Balanced Block Designs with Repeated Blocks
Bronislaw Ceranka and Malgorzata Graczyk1
Abstract
Some new construction methods of the variance balanced block designs with repeated blocks are given. They are based on the specialized product of incidence matrices of the balanced incomplete block designs.
1 Introduction
In the paper we present some types of block designs, which are used in practice as well as in the general theory of block designs. For a variety of reasons, it is desirable to have the balanced incomplete block design with the block repetitions, because it might be less expensive and easier to implement. In many applications, the experimenter may not wish to run certain treatment combinations. For example, it is physically impossible to run three or more treatments combinations in one block. However, this combination may produce observations which no longer conform to the homoscedastic linear model. Foody and Hedayat (1977) present some potential applications of the balanced incomplete block designs with repeated blocks to experimental designs and controlled sampling. Designs with repeated blocks with the equireplications and with equal size of each block are discussed in the literature: Hedayat and Li (1979), Hedayat and Hwang (1984). However from a practical point of view, it may not be possible to construct a design with equiblock sizes accommodating the equireplication of each treatment in all the blocks. Here we consider a class of block designs called variance balanced block designs which can be made available in unequal block sizes and for equal replications. In the variance balanced block design each elementary contrast is estimated with the same variance.
Let us consider v treatments arranged in b blocks in a block design with incidence matrix N = [nj], i = 1, 2,..., v, j = 1, 2,...,b, where nj denotes the number of experimental units in the jth block getting the ith treatment, n = YV=i i n%j ■ When nj = 1 or 0 for all i and j, the design is said to be binary. Otherwise, it is said to be nonbinary. In this paper we consider binary block designs, only. The following notation is used: r = [ri,r2, rv ]' is the vector of treatment replications, k = [ki, k2,kb]' is the vector of block sizes. Hence, N1b = r and Nlv = k, where 1a is the a x 1 vector of
1 Department of Mathematical and Statistical Methods, University of Life Sciences in Poznan, Wojska Polskiego 28, 60637 Poznan, Poland; bronicer@up.poznan.pl; magra@up.poznan.pl
ones.
The information matrix C for treatment effects is defined as
C = R - NK-1N',	(1.1)
where R = diag [r1, r2,.., rv], K = diag [k1, k2,..., kb]. The information matrix C is very suitable in determining properties of a block designs.
For several reasons, in particular from a practical point of view, it is desirable to have repeated blocks in the design. For example, some treatment combinations may be preferable over others, and also the design implementation may cost differently according to the design structure contains or not repeated blocks. The set of all distinct blocks in a block design is called the support of the design and the cardinality of the support is denoted by b* and is referred to as the support size of the design.
In the literature, see Calinski (1977), Puri and Nigam (1977), there are considered the balanced designs in various senses. In present paper we consider a balanced design of the following type, given in Rao (1958). A block design is said to be balanced if every elementary contrast of treatment effects is estimated with the same variance. In this sense the design is also called a variance balanced (VB) block design. It is well known that a block design is a VB if and only if it has
C
I__il
vv
(12)
where n is the unique nonzero eigenvalue of the C— matrix with multiplicity v — 1, Iv is the v X v identity matrix. For a binary block design
= Eti ^ - b
v — 1
(see Kageyama and Tsuji (1979)).
In the particular case, when the block design is equireplicated, then n
vr—b v-l •
n
v
2 Construction of the design matrices
Now, we consider balanced incomplete block design (BIBD) (See Raghavarao (1971)) as an arrangement of v treatments into b blocks each of k (< v) treatments, satisfying conditions: every treatment occurs at most once in each block and occurs in r blocks, every pair of treatments occurs together in A blocks. The parameters of the BIBD are v, b, r, k, A and they satisfy
vr = bk, A(v — 1) = r(k — 1).
Let N be an incidence matrix of the BIBD. We have, NN' = (r — A)Iv + A1v lV. It is known from literature, that any BIBD is VB.
Theorem 1 If Ni is BIBD with parameters v, bi, ri, ki, Ài and Ci is C— matrix for i = 1, 2then
N = [Ni N2 ... Nt]
is the incidence matrix of the VB block design.
C = rlv- NK-'N' = rlv - YLi	= Hi r& " Hi
EL (r^-^N; =EliCi.
The design Ni is VB as BIBD. Therefore from (1.2), we have
c = EI=1 Vi (I« - èMl) = V (I« - J Ml) , where rj = ELi V-.
Hence, the claim of the Theorem.
(2.1)
Proof. For the design N in (2.1), we have t = ei=1 ri, k
[kilb1 k2lb2 ... ktlbt] . Thus
We use the following specialized product of two matrices presented in Pal and Dutta
(1979). If A = (aast)
mxp
and B
(bzi )
mxqi
then the specialized product of the
matrices A and B is defined as
D = A * B = (dsi)
mx pq
(2.2)
where dsl = ast x biz, l being equal to (t — 1)q + z for s = 1, 2,..., m, t = 1, 2, ...,p, z = 1, 2,..., q.
Let Ni, i =1, 2, be an incidence matrix of the BIBD with parameters v, bi, ri,
ki, Ai. Let Ci be the C—matrix of this design defined by Ni. Now, we form the matrix N
as
N = N1 * N2.	(2.3)
Theorem 2 If N1 is an incidence matrix of the BIBD with parameters v, b1 = v(v — 1)/2, t1 = v — 1, k1 = 2, A1 = 1 and N2 is an incidence matrix of the BIBD with parameters v = b2, r2 = k2 = v — 1, À2 = v — 2, then N in the form (2.3) is an incidence matrix of the VB block design with repeated blocks and with parameters
b = v2 (v — 1)/2, t = (v — 1)2
k
2 ■ 1
v(v-1)(v-2)/2
l
v(v-1)
b* = v(v + 1)/2.
Proof. For the product (2.3) to hold, we have N = [N1 ® lV-2 Iv ® lV-J . Hence, the information matrix C = (v — 2)C1. Therefore, taking into consideration Theorem 1, N is an incidence matrix of the VB block design with repeated blocks. So, the Theorem is proven.
v
Let us consider the class of BIBD's for k = 3 and A =1 usually known as Steiner's triple system. There are only two series of Steiner's triple systems (See Raghavarao (1971)) with respective parameters for t = 1, 2,...
v = 6t + 1, b = t(6t +1), r = 31, k = 3, A = 1	(2.4)
v = 3(21 +1), b = (2t +1)(3t + 1), r = 31 + 1, k = 3, A = 1. (2.5)
Steiner (1853) posed the problem whether the two series of BIBD's with parameters given in (2.4) and (2.5) exist for every t and later on Moore (1893) and Hanani (1961) recursive methods of constructing Steiner's triple systems for all t are given. For a detailed account of showing the existence of such designs see Hall, Jr. (1967).
Theorem 3 If Ni is an incidence matrix of the BIBD with parameters given in (2.4) and N2 is an incidence matrix of the BIBD with parameters v = b2 = 6t +1, r2 = k2 = 6t, A2 = 6t — 1, then N in the form (2.3) is an incidence matrix of the VB block design with repeated blocks and with parameters
v, b = t(6t + 1)2, r = 18t2
k
3 ■ 1 2 ■ 1
2t(3t-1)(6t+1) 3t(6t+1)
b* = 4t(6t + 1).
Proof. For the product (2.3), we have N = N1 ® 12(3t-1) N3 , where N3 is an incidence matrix of BIBD with parameters v = 6t + 1, b3 = 3t(6t +1), r3 = 6t, k3 = 2, A3 = 1 and with information matrix C3. Thus C = 2(3t — 1)C1 + C3. Therefore, taking into consideration Theorem 1, N is an incidence matrix of the VB block design with repeated blocks. Hence, the result.
Theorem 4 If N1 is an incidence matrix of the BIBD with parameters given in (2.5) and N2 is an incidence matrix of the BIBD with parameters v = b2 = 3(2t + 1), r2 = k2 = 2(3t +1), A2 = 6t + 1, then N in the form (2.3) is an incidence matrix of the VB block design with repeated blocks and with parameters
3 ■ 16t(2t+1)(3t+1)
v, b = 3(2t + 1)2(3t + 1) b* = 4(2t + 1)(3t + 1).
r = 2(3t + 1)2
k
2 ■ 1
3(2t+1)(3t+1)
Proof. For the product (2.3), we have N = [N1 ® 1'6t N4] , where N4 is an incidence matrix of BIBD with parameters v = 3(2t + 1), b4 = 3(2t + 1)(3t + 1), r4 = 2(3t + 1), k4 = 2, A4 = 1 and with the information matrix C4. Hence the information matrix C is given as C = 6tC1 + C4. Owing to Theorem 1 it implies, that N is an incidence matrix of the VB block design with repeated blocks. So, the Theorem is proven.
Theorem 5 If N1 is an incidence matrix of the symmetrical BIBD with parameters v = b1, r1 = k1, A1, then N in the form
N = N1 * N1	(2.6)
is an incidence matrix of the VB block design with repeated blocks and with parameters
v, b = v2, r = r2, k =
kilv
Allv(v-l)
, b* = v(v + 1)/2.
where N5 is an incidence
r5 = ri(ri - 1)/2, k5 =
Proof. For the product (2.6), we have N = [N1 N5 N5] matrix of BIBD with parameters v, b5 = v(v — 1)/2. À1; À5 = (r1(r1 — 1)(A1 — 1)) / (2(v — 1)) and with information matrix C5. Thus the information matrix C = C1 + 2C5. That means N is an incidence matrix of the VB block design with repeated blocks, because of Theorem 1. Hence, the claim of the Theorem holds.
N1 is an incidence matrix of the symmetrical BIBD with parameters r1 = k1 = 2(t +1), A1 = t +1, 4t + 3 is a prime or a prime power,
Corollary 1 If
v = b1 = 4t+3
then N in the form (2.6) is an incidence matrix of the VB block design with repeated blocks and with parameters v, b = (4t + 3)2, r
" 2(t +1) ■ l4t+3
k
(t + 1) ■ 12(2t+1)(4t+3)
b*
(4t + 3)2, r = 4(t +1)2, 2(t + 1)(4t + 3).
Corollary 2 If N1 is an incidence matrix of the symmetrical BIBD with parameters v = b1 = 4t2, r1 = k1 = t(2t + 1), A1 = t(t + 1), then N in the form (2.6) is an incidence matrix of the VB block design with repeated blokks and with parameters
t(2t + 1) ■ l4t2
t(t + 1) ' 14t2(4t2-1)
b = 16t4, r = t2(2t + 1)2
k
b* = 2t2(4t2 + 1).
Corollary 3 If N1 is an incidence matrix of the symmetrical BIBD with parameters v = b1 = 4t2 — 1, r1 = k1 = 2t2, A1 = t2, then N in the form (2.6) is an incidence matrix of the VB block dfesign with repeated bklocks and with parameters
2t2 ■ 14t2 -1
t2 ■ 12(2t2 —1)(4t2-1)
b = (4t2 — 1)2
4t4
k
b* = 2t2 (4t2 — 1).
Corollary 4 If N1 is an incidence matrix of the symmetrical BIBD with parameters v = b1 = 8t + 7, r1 = k1 = 4(t + 1), A1 = 2(t + 1), then N in the form (2.6) is an incidence matrix of the VB block design with repeated blocks and with parameters
" 4(t + 1) ■ 18t+7
b*
b = (8t + 7)2 : 4(t + 1)(8t + 7).
r = 16(t + 1)2
k
2(t + 1) ' 12(4t+3)(8t+7)
Corollary 5 If N1 is an incidence matrix of the symmetrical BIBD with parameters v = b1 = t2 + t +1, r1 = k1 = t2, A1 = t(t — 1), where t is a prime or a prime power, then N in the form (2.6) is an incidence matrix of the VB block design with repefted blocks and with parameters v, b =(t2 + t + 1)2, r = t4, t2 ■ 1t2 +t+1 t(t — 1) ■ 1t(t+1)(t2 +t+1)
k
b* = (t2 + t + 1)(t2 + t + 2)/2.
Corollary 6 If N1 is an incidence matrix of the symmetrical BIBD with parameters v = b1 = (t + 1 )(t2 + 1), r1 = k1 = t3, A1 = t2(t — 1), where t is a prime or a prime power, then N in the form (2.6) is an incidence matrix of the VB block design with repeated blocks and with parameters v, b = (t + 1)2(t2 + 1)2,
13 ■ 1(t+1)(t2+1) t2(t — 1) ■ 1t(t+1)(t2+1)(t2 +t+1)
t6
k
b* = (t+1 )(t2 + 1)(t3+12+1+2)/2.
Corollary 7 If N1 is an incidence matrix of the symmetrical BIBD with parame-
ters v
b1 , r1 = k1
v — 1, A1
v — 2, then N in the form (2.6) is
an incidence matrix of the VB blocr design with repeated blocks and with parameters
b
(v — 1)2
k
(v — 1) ■ 1v (v — 2) ■ 1v(v-1)
b* = v(v + 1)/2.
2
v
v
r
3 Conclusions and examples
The importance of block repetition in a design is very well known, so many authors pay special attention to the construction rules and a practical properties of designs having repeated blocks. So, we present appropriate examples of constructions of the design matrices.
Example 1 Let us consider the BIBD (See Theorem 2) with parameters v = 4, b1 = 6, r1 = 3, k1 = 2, A1 = 1 with incidence matrix N1 and the BIBD with parameters v = 4, b2 = 4, r2 = 3, k2 = 3, À2 = 2 with incidence matrix N2, where
N1
10 10 10 10 0 10 1 0 110 0 1 0 10 110
N2
1110 1101 1011 0111
Based on the matrices N1 and N2, we form the incidence matrix N in the form (2.3) of the VB block design with repeated blocks and with parameters
2 ■ 112
v = 4, b = 24, r = 9, k
N
1
12
b* = 10,
1	1	1	0	0	0	0	0	1	1	1	0	0	0	0	0	1	1	1	0	0	0	0	0
1	1	0	1	0	0	0	0	0	0	0	0	1	1	0	1	0	0	0	0	1	1	0	1
0	0	0	0	1	0	1	1	1	0	1	1	0	0	0	0	0	0	0	0	1	0	1	1
0	0	0	0	0	1	1	1	0	0	0	0	0	1	1	1	0	1	1	1	0	0	0	0
Hence, after permutation of columns, we have N = [N1 N1 I4 I4 I4].
Example 2 Let us consider the BIBD (See Theorem 3) with parameters v = b1 = 7, r1 = k1 = 3, À1 = 1 with incidence matrix N1 and the BIBD with parameters v = b2 = 7, r2 = k2 = 6, À2 = 5 with incidence matrix N2, where
Ni
1	1	1	0	0	0	0		01	1	1	1	1
1	0	0	1	1	0	0		10	1	1	1	1
0	1	0	1	0	1	0		11	0	1	1	1
1	0	0	0	0	1	1	, N2 =	11	1	0	1	1
0	0	1	1	0	0	1		11	1	1	0	1
0	0	1	0	1	1	0		11	1	1	1	0
0	1	0	0	1	0	1		11	1	1	1	1
Based on the matrices Ni and N2, we form the incidence matrix N in the form (2.3) of the VB block design with repeated blocks and with parameters
3 ■ I28 2 ■ l2i
N = [Ni Ni N i Ni N3] , where N3 is an incidence matrix of BIBD with parameters v = 7, b3 = 21, r3 = 6, k3 = 2, À3 = 1,
7, b = 49, r = 18, k
b*
and, after permutation of columns,
N3
1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0
0	1	0	0	0	0	1	0	0	0	0	1	1	1	1	0	0	0	0	0	0
0	0	1	0	0	0	0	1	0	0	0	1	0	0	0	1	1	1	0	0	0
0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	1	0	0	1	1	0
0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	1	0	1	0	1
0	0	0	0	0	1	0	0	0	0	1	0	0	0	1	0	0	1	0	1	1
Example 3 Let us consider the BIBD (See Theorem 4) with parameters v = 9,
bi = 12, ri = 4, ki = 3, Ài = 1 with incidence matrix Ni and the BIBD with
parameters v = b2 = 9, r2 = k2 = 8, À2 = 7 with incidence matrix N2, where
	1	1	1	1	0	0	0	0	0	0	0	0		01	111	111
	1	0	0	0	1	1	1	0	0	0	0	0		10	111	111
	0	1	0	0	1	0	0	1	1	0	0	0		11	111	111
	0	0	1	0	0	1	0	1	0	1	0	0		11	011	111
Ni =	0	0	0	1	0	0	1	1	0	0	1	0	, N2 =	11	101	111
	1	0	0	0	0	0	0	0	1	1	1	0		11	1 1	111
	0	1	0	0	0	0	1	0	0	1	0	1		11	111	011
	0	0	1	0	1	0	0	0	0	0	1	1		11	111	101
	0	0	0	1	0	1	0	0	1	0	0	1		11	111	1 1 0
Based on the matrices Ni and N2, we form the incidence matrix N in the form (2.3) of the VB block design with repeated blocks and with parameters v = 9, b =
3 ■ I72
108, r = 32, k =
[Ni Ni Ni Ni parameters v = 9, b4
2 ■ I36 Ni Ni
36, r4 =
, b* = 48 and, after permutation of columns, N =
N4] , where N4 is an incidence matrix of BIBD with
8, k4 = 2, À4 = 1,
N4
1 1 1 1 1 1 1 1 0000000000000000000000000000 1 0000000 1 1 1 1 1 1 1 000000000000000000000 010000001000000111111000000000000000 001000000100000100000111110000000000 000100000010000010000100001111000000 000010000001000001000010001000111000 000001000000100000100001000100100110 000000100000010000010000100010010101 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1
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