Incomplete block designs with nested rows and columns are designs for v treatments in b blocks of size k =pq
Biometrika (1990), 77, 1, pp. 193-202
Printed in Great Britain
Some constructions for balanced incomplete block designs with nested rows and columns BY NIZAM UDDIN A N D JOHN P. MORGAN
Department of Mathematics and Statistics, Old Dominion University, Norfolk,
Virginia 23529, U. S.A.
SUMMARY A method of construction of balanced incomplete block designs with nested rows and columns is developed using difference techniques, from which many infinite series of designs are obtained. Some key words: Balanced incomplete block design; Nested row-column design; Supplementary difference set.
Incomplete block designs with nested rows and columns are designs for v treatments in b blocks of size k = pq < v, where each block is composed of p rows and q columns. Such a design is balanced if: (a) a treatment occurs at most once in each block, (b) each treatment occurs in r blocks, and (c) pqrIv - p N I N : -qN2N:+ N N 1 = aI, -AJv. Here I, is the v x v identity matrix, J, is the v x v matrix of ones, N , , N, and N are respectively the treatment-row, treatment-column and treatment-block incidence matrices, and a and A are integers. Nested row-column designs satisfying (a)-(c) are called balanced incomplete block designs with nested rows and columns, and will be denoted by BIBRC (v, b, r, p, q, A), or BIBRC for short. Such designs were introduced by Singh & Dey (1979) for the elimination of heterogeneity in two directions within each block. Constructions have been given by Singh & Dey (1979), Street (1981), Agrawal & Prasad (1982, 1983) and Cheng (1986). A few of these designs show up in Ipinyomi & John's (1985) listing of nested row-column designs. Condition (c) says that the C-matrix for estimation of treatment contrasts has the same form as that of an ordinary balanced incomplete block design, in the absence of nested rows and columns; hence the balance. In this paper we present a technique for construction of BIBRC designs, based on the method of differences, that takes advantage of the fact that if p = q a sufficient condition for (c) to hold is that botk ( N , , N,) and N are incidence matrices for balanced incomplete block designs. For vectors a and b of lengths n, and n, respectively, we shall use B(a, b) to denote a n, x n, array whose (i, j)th element is equal to the sum of the ith element of a and the jth element b. The symbol R(a, b) will be used to denote a vector whose elements are, in some order, those of B(a, b). Also 1 +yA will be used to denote (1 +yz 1 z E A).
The construction method may be summarized as follows. CONSTRUCTION. Let G be an abelian group of order v. Suppose that we canjnd two sets of vectors b,, . . . ,bm and b : , . . . ,bh on G which are m-supplementary difference sets of B I B designs, i.e. ( i ) each bj has p distinct elements of G and each bi has q distinct elements of G, and (ii) each nonzero element of G occurs mp(p - l ) / ( v- 1 ) times among the symmetric diflerences arising from the bj's and mq(q - l ) / ( v- 1 ) times among the symmetric diferences arising from the bj's. Suppose further that (iii) the m vectors R, = R ( d i ,d : ) are together composed of A = mpq(p - l ) ( q- l ) / ( v- 1 ) occurrences of each nonzero element of G, where di and d : are the vectors of symmetric diflerences corresponding to bi and b: respectively. Then there exists a B I B R C (v, b, r, p, q, A ) design with b = mu, r = mpq, p, q and A. ProoJ: Define m initial p x q blocks Bi = B(bi,b:). These m blocks, when developed, give our design. By ( i ) and (ii), N 1 and N, are incidence matrices o f B I B designs. The symmetric differencesarising from Bi are q copies o f di from columns, p copies o f d : from rows, and the elements o f R,, all diagonal differences;so, by (ii)and (iii), N is the incidence matrix o f a B I B design. Example. For v = 19 and G = write b1= (0,2,3, 14),b, = (0,4,6,9),b3= (0, 1,7,1 I ) , b: = (0,6,l o ) , b; = (0, 1 , 12) and bi = (O,2, 5 ) . The initial blocks are
A B I B R C (19, 57, 36, 4, 3, 12) is found by successively adding 0 , . . . , 18 (mod 19) to the initial blocks. I f p = q, it is sufficient for (ii) that the 2mp(p - 1 ) combined symmetric differences from the bi's and b:'s are balanced, a fact which is taken advantage o f in 9 3 below. Applying this technique we obtain several infinite series o f designs as presented in the following theorems and corollaries. In all cases G will be taken as the finite field G F , with primitive element x.
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THEOREM 1. Let v = 2tm 1 be a prime power and write xu(= 1 - xmi. ( a ) I f there exists a positive integer u (ui- u j ) (mod m ) for i,j = 1 , . .. , t, then there exists a B I B R C with b = mu, r = 4mt2,p = q = 2t and A = 2t(2t - I),. ( b ) If there exists a positive integer u $ ui, (ui- uj) (mod m ) for i,j = 1, . .. , t, then there exists a B I B R C with b = mv, r = m(2t I)', p = q = 2t 1 and A = 2t(2t I),. ( c ) If there exists a positive integer u $ u,, (ui- u j ) (mod m ) for i,j = 1, ... , t, then there exists a BIBRC with b = m v , r = 2 m t ( 2 t + l ) , p=2t, q = 2 t + l and A = 2t(2t + 1)(2t- 1 ) .
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Roo$ ( a ) Let bl = ( x Oxm, , .. . , X ( 2 t - l ) m )
bi = xi-lbl ,
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Balanced incomplete block designs with nested rows and columns Sprott (1954) has shown that bl, . .. , b,, and hence b:, difference sets for a BIBD for which
.. . , b;,
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are m-supplementary
and O means the Kronecker product. Using x'" = -1 we obtain = Xim(l - X(f-i)m)= Xim+u,-t 1+ SO
that dl = blOw where w = (xul,xul, ... , xu", xu!-',xu!). Now Ri = R(di, d:) = xi-'R(d1, d:) = x i - ' ~ ( d , ,x udl),so
Using R(xnlbl,xn2bl)= x n l b l O ( l+xn2-"lb,), which can be verified easily, we obtain R ( d l , xudl)= {wjbl O (1 + w,wjlxubl)I j, 1= 1, . .. , 2 t - I), wj being the jth component of w. Since (xO,x l , . . . ,xm-I)O bl = G - {0), condition (iii) is satisfied and the proof is completed. To prove (b), adjoin 0 to the bi's and b:'s in the proof of (a). To prove (c), adjoin 0 to the bl's in the proof of (a). The conditions on u are needed to give di fl d: = 0, ensuring the binary property, i.e. that a treatment occurs at most once in each block. Sufficient, but often not necessary, conditions for the existence of u, and hence the designs of Theorem 1, are for (a), (b) and (c) respectively. The following two corollaries illustrate the application of this theorem. Note that the sufficient conditions are not necessary in Corollaries l(b), 2(a) and 2(c). COROLLARY 1. Let v = 4m + 1 be a prime power. ( a ) If m 2 4, then there exists a BIBRC with b = v(v - 1)/4, r = 4(v - I), p = q = 4 and A = 36. (b) If m > 7, then there exists a BIBRC with b = V ( V- 1)/4, r = 25(v - 1)/4, p = q = 5 and A = 100. (c) If m 6, then there exists a BIBRC with b = v(v - 1)/4, r = 5(v - I), p = 4, q = 5 and A = 60. Agrawal & Prasad (1982, 1983) have constructed 4 x 4 designs with the same r and prime power v = 16m+ 1, and with r = 8(v - 1) for prime power v = 2m + 1. Similar comparisons for the 4 x 5 and 5 x 5 designs may be made to the results of Agrawal & Prasad (1982, 1983) and Street (1981): in each case our series are for smaller r or a less sparse series of v. COROLLARY 2. Let v = s n be an odd prime power. 2, then there exists a BIBRC with b = s n ( s n- l ) / ( s - I), r = ( s - l ) ( s n- I), ( a ) If n p = q = s - 1 and A = ( ~ - l ) ( s - 2 ) ~ . (b) n 3, then there exists a BIBRC with b = s n ( s n- l ) / ( s - I), r = s2(sn- l ) / ( s - I), p = q = s and A = (s - l)s2. (c) If n 2 2, then there exists a BIBRC with b = s n ( s n- l ) / ( s - I), r = s ( s n- I), p = s - 1, q = s and A = s ( s - l ) ( s -2). Corollary 2 follows upon taking m = (sn - l ) / ( s - I), in which case ui = 0 (mod m) for every i. The condition n 2 3 in (b) is required to give incomplete blocks.
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THEOREM 2. Let v = 2 t m 1 be a prime power where t > 1 is odd, and write xu$= 1 - x2"'. ( a ) If there exists a positive integer u ( u i- u,) (mod m ) for i, j = 1 , .. . ,i ( t - I ) , then there exists a BIBRC with b = mv, r = mt2, p = q = t and A = i t ( t - I ) ~ . ( b ) If there exists a positive integer u 8f u i , ( u i- u j ) (mod m ) for i, j = 1,. .. ,i ( t - I ) , then there exists a BIBRC with b = mv, r = m ( t I ) ~ p, = q = t + 1 and A = i t ( t + 1)'. ( c ) If there exists a positive integer u ui, ( u i - u j ) (mod m ) for i, j = 1, . . . , i ( t - I ) , then there exists a BIBRC with b = mu, r = m t ( t I ) , p = t, q = t 1 and A = i t ( t 2 - 1).
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ProoJ: Here we prove (b) first. Let b , = ( 0 , xO,x2", . . . ,x ( " - ~ ) "= ) ( 0 , bT). The symmetric differences arising from b1 are the nonzero elements of B ( b l ,- b,). Hence
see also Sprott (1954). Using x'"
= - 1,
and for odd i, 1 +xi" = xirn+"(l-i)/2, gives
say. Taking bi = xi-'b1 and bj = xu+'-'b1, for i = 1, .. . , m, it is easily seen that each of rows and columns will be BIBDS. The diagonal differences are Ri = x i - ' ~ ( d , x, u d l ) for i = 1,. .. ,m. Upon noting that ( b f ,xmbT) is the b , of Theorem 1, we obtain
SO
{ R 1 ,. . . , R,) = ( x O x, l , . . . ,x r n p 1 C )3 R ( d l , x u d l ) is a balanced list. To prove (a), delete 0 from the hi's and bj's in the proof of (b). To prove (c), delete 0 from the bi's in the proof of (b). Sufficient conditions for the existence of u in (a), (b) and (c) of Theorem 2 are
respectively. We present three corollaries as applications of Theorem 2. We then show in § 3 how, under certain conditions, the number of blocks in the first two theorems may be reduced.
3. Let v = 6 m + 1 be a prime power. COROLLARY ( a ) If m 2 2, then there exists BIBRC with b = V ( V - 1 ) / 6 , r = 3 ( v - 1 ) / 2 , p = q = 3 and A =6. ( b ) I f m 2 4, then there exists a BIBRC with b = v ( v - 1 ) / 6 , r = 8 ( v - 1 ) / 3 ,p = q = 4 and A = 24. ( c ) If m 2 3, then there exists a BIBRC with b = V(V - 1 ) / 6 , r = 2 ( v - I ) , p = 3, q = 4 and A = 12. The 4 x 4 designs constructed here may be compared to those of Corollary 1 and the previously mentioned designs of Agrawal & Prasad (1982, 1983). Designs with the same parameters as the 3 x 3 and 3 x 4 series are constructed by Street (1981), but for m 2 3 and m 2 4, respectively.
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4. Let v = 10m 1 be a prime power. COROLLARY ( a ) If m 2 4, then there exists a BIBRC with b = V ( V - 1)/10, r = 5 ( v - 1 ) / 2 , p and A = 40. ( b ) If m 2 7, then there exists a BIBRC with b = V ( V - 1 ) / 10, r = 18(v - 1 ) / 5 , p and A = 90.
=q =5 =q
=6
Balanced incomplete block designs with nested rows and columns
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( c ) I f m > 6, then there exists a B I B R C with b = V ( V - 1 ) / 10, r = 3(v - I ) , p = 5, q = 6 and h = 60. COROLLARY5. Let v = sn ( n 2 2 ) be an odd prime power, and let s = 4 a +3. ( a ) If a 2 1 , then there exists a BIBRC with b = s n ( s n- l ) / ( s- I ) , r = ( s n- l ) ( s- 1)/4, p = q = ( s - 1 ) / 2 and A = ( s-3)'(s - 1 ) / 16. ( b ) If a 2 0 , then there exists a B I B R C with b = s n ( s n- l ) / ( s- I ) , r = ( s I ) ~ ( s-"1 ) / { 4 ( s- I ) ) , p = q = i ( s 1 ) and A = ( s I ) ~ (-s1)/16. ( c ) If a 2 1 , then there exists a B I B R C with b = s n ( s n- 1 ) / ( s- I ) , r = ( S l ) ( s n- 1)/4, p = i ( s - I ) , q = i ( s + 1 ) and A = ( s 2- 1 ) ( s- 3)/16.
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Combining the constructions o f Theorems 1 and 2 gives the following.
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THEOREM 3. Let v = 2tm 1 be a prime power where t > 1 is odd and write xu(= 1 - xmi. ( a ) If there exists a positive integer u $ - u j ) (mod m ) for i = 1 , ... ,i ( t - 1 ) and j = 1 , . .. , t, then there exists a B I B R C with b = mv, r = 2mt2, p = t, q = 2t and A = t ( t - 1)(2t- 1 ) . ( b ) Ifthere exists a positive integer u uZi,( u2i- u j ) (mod m )for i = 1 , . .. ,i ( t - 1 ) and j = 1 , . .. , t, then there exists a B I B R C with b = mv, r = mt(2t 1 ) , p = t, q = 2t 1 and A = t ( t - 1 ) ( 2 t + l ) . ( c ) If there exists a positive integer u -uj, ( u Z i uj) (mod m ) for i = 1 , . . . ,i ( t - 1 ) and j = 1, .. . , t, then there exists a BIBRC with b = mv, r = 2mt(t I ) , p = t 1, q = 2t and A = t ( t l ) ( 2 t- 1 ) . ( d ) Ifthere exists apositive integer u $ u2,,- uj, ( u Z i uj)(mod m )for i = 1 , . . . ,i ( t - 1 ) a n d j = l , . .. , t, t h e n t h e r e e x i s t s a ~ ~ withb=mv, ~~c r = m ( t + 1 ) ( 2 t + l ) , p = t+1, q = 2 t + l and A = t ( t + l ) ( 2 t + l ) .
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Proof: In each case take the bi's from Theorem 2 and the b:'s from Theorem 1. Sufficient conditions for the existence o f u in ( a ) - ( d )o f Theorem 3 are
respectively. All the designs given in this section serve also as nested balanced incomplete block designs (Preece, 1967).
In this section we turn our attention to reducing the number o f blocks for sub-series o f the Theorems 1 and 2 designs.
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THEOREM 4. Let v = 4tm 1 be a prime power and write xul = 1 - x2"'. ( a ) If ui - uj $ m (mod 2 m ) for i,j = 1 , . . . , t, then there exists a B I B R C with b = mv, r = 4mt2,p = q = 2t and A = t(2t - 1)'. ( b ) If u,, ui - uj m (mod 2 m )for i,j = 1, .. . , t, then there exists a B I B R C with b = mv, r=m(2t+l)', p = q = 2 t + l andA=t(2t+l)'.
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Proof: ( a ) In Theorem l ( a ) write v =2tmo+l, where m o = 2 m ; it will be shown that with the conditions o f this theorem and proper choice o f u, it is sufficientto use just the first m initial blocks given there. With b, = ( x O x, 2", .. . , x ' ~ ' - ~)' "take
i.e. take u = m = $moin Theorem l ( a ) ,for i = 1 , .. . , m. Then dl = b,O ( x Y ,x Y,x u ! ) ,where xY = ( x u ' ,. . . , xu[-'),S O the bi's and bj's are together a 2m-supplementary difference set, which by the conditions on the uj's satisfy di fl dj = 0. The diagonal differences are
Now R ( d l ,x m d l ) can be written as four copies of R l l , two copies of each of R,, and R,, , and one copy of R,,, where
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Factor these as follows, using R(xnlbl, xn2bl)= xnlbl0( 1 xn2-nlbl)and xmbl= x-" b, = b: :
Since
(x', X I , . . . , x m - ' ) @ ( b l ,b : ) = G F , - {o), ( x Ox, l , .. . ,x m - ' )O R l 1is a balanced list, as is ( x Ox, l , . .. , x m - ' )O (R,,, R,,). The proof is complete if ( x O x, l , . . . , ~ " - ~ ) O b , C 3+( b1 : ) is a balanced list, which will be the case provided the elements of 1 + b{ = 1 + xmbl can be partitioned into t pairs of the form x91, i = 1 , . . ., t where ki is odd. One such partition is ~
~
g
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~
l
~
To prove (b), adjoin 0 to the b,'s and hi's in (a). As an example of Theorem 4, t = 2 gives 4 x 4 blocks with r = 2(v - I ) , and 5 x 5 blocks with r = 25(v - 1)/8, for v = 8 m 1 a prime power; compare Corollary 1. For v < 500 the conditions fail to hold for v = 81, and for v = 289 in the 5 x 5 case. Setting t = 3, the conditions for v = 12m 1 in 6 x 6 and 7 x 7 blocks also fail for two values of v < 500: v = 37 in 6 x 6 blocks, and v = 169. Corresponding to Corollary 2 we have the following.
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COROLLARY 6 . Let v = sn be an odd prime power where n is even. ( a ) I f n 2 2, then there exists a BIBRC with b = s n ( s n- 1)/{2(s- I ) ) , r = i ( s - l ) ( s n- I ) , p = q = s - 1 and A = ; ( ~ - 2 ) ~ ( s - l ) . ( b ) I f n 2 4 , then there exists a BIBRC with b = s n ( s n- 1 ) / { 2 ( s - I ) ) , r = s2(sn- 1)/{2(s- I ) ) , p = q = s and A = i s 2 ( s- 1 ) .
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THEOREM 5. Let v = 4tm 1 be a prime power, where t > 1 is odd and write xu[= 1 - x4"'. 1 ( a ) If ui- uj 8 m (mod 2 m ) for i, j = 1 , . . . ,,(t - I ) , then there exists a BIBRC with 2 b = m v , r = m t 2 , p = q = t a n d A = t ( t - 1 ) 14.
Balanced incomplete block designs with nested rows and columns
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1 (b) If u,, ui - uj m (mod 2m) for i, j = I, . . . ,z(t - I), then there exists a b=mv, r = m ( t + ~ ) p~=, q = t + l a n d ~ = t ( t + 1 ) ~ / 4 .
199 BIBRC
with
ProoJ: Theorem 5 stands in relation to Theorem 2 as Theorem 4 to Theorem 1, and the proof is similar. The initial blocks are B(bi, b:) where for (a),
and for (b), 0 is adjoined to the bi's and hi's of (a).
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COROLLARY 7. Let v = 12m 1 be a prime power. ( a ) There exists a BIBRC with b = V ( V- 1)/12, r = 3(v - 1)/4, p = q = 3 and A = 3. (b) If m 2 2, then there exists a BIBRC with b = v(v - 1)/12, r = 4(v - 1)/3, p = q = 4 and A = 12. ProoJ: The proof of (a) is immediate from Theorem 5 since there is only a single ui. For (b) the condition is u, m (mod 2m) where xul = 1-x4". If this fails take the bi's as given in the theorem, but bl = xb,. Then dl = (xO,x2", . . . , X ~ ~ " ) O (xXm~) ,showing , that the rows and columns will each be BIBDS. NOWRi = xi-' ~ ( d ]xdl) , simplifies to
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and the diagonal differences are balanced as well. COROLLARY 8. If v = 20m + 1 (m 2 2) is a prime power, then there exists a b = v(v - 1)/20, r = 5(v - 1)/4, p = q = 5 and A = 20.
BIBRC
with
ProoJ: The condition is S = u2- u, 8 m (mod 2m) where xS = 1 + x4". If this fails take the bi's as given in Theorem 5 but bl = xb,. The proof follows easily upon noting that x-Uldl= (XO,x ~x ,~ " ,~ ~bl. ~ ) 0 The conditions for v = 2 0 m + l in 6 x 6 blocks fail twice for v
