Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 1(9), 18-22, September (2012) Res.J.Recent Sci. International Science Congress Association 18 Some Construction Methods of Variance and Efficiency Balanced Block Designs with Repeated BlocksAwad Rashmi1 and Banerjee Shakti Dept. of Statistics, Govt. Holkar Science College, Indore, MP, INDIA School of Statistics, D.A.V.V., Indore, MP, INDIAAvailable online at: www.isca.in Received 10th April 2012, revised 16th May 2012, accepted 21st June 2012Abstract Some construction methods of the variance and efficiency balanced block designs with repeated blocks are proposed which are based on the incidence matrices of the known balanced incomplete block designs with repeated blocks. Keywords: Balance incomplete block design, balance incomplete block design with repeated blocks, variance balance design, efficiency balance design. Introduction The concept of repeated blocks came into existence in Van Lint. He noticed that many of the BIB designs constructed by Hanani have repeated blocks. In the early years, on the concept of repeated blocks, different type of work has been done by many statisticians, like Parker, Seiden, Stanton5,6, Sprott 7,8, Ryser etc. In 1973, Van Lint systematically studied the problems of the construction of BIB designs with repeated blocks. In the year 1977, Foody and Hedayat10 presented some potential applications of the balanced incomplete block designs with repeated blocks. Designs with repeated blocks with the equireplications and with equal size of each block were discussed in the literature; Hedayat and Li11, Hedayat and Hwang12. However from the practical point of view, it may not be possible to construct the design with equi block size accommodating the equireplication of each treatment in all the blocks. Here we consider a class of block designs called variance and efficiency balanced block designs which can be made available in unequal block sizes and for varying replications. From the point of view of application there is no reason to exclude the possibility that a BIB design would contain repeated blocks. For a variety of reasons, it is desirable to have the balanced incomplete block designs 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 treatment combinations in one block. Thus we need BIB designs with 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. It’s not always necessary that every variance balanced design is also an efficiency balanced but in the present paper we proposed some construction methods of variance balanced block designs with repeated blocks which are also efficiency balanced with unequal block sizes and unequal replications. For this we take the reference of some construction methods of variance balanced and efficiency balanced block designs with repeated blocks given in research papers of Bronislaw Ceranka and Malgorzata Graczyk13-15. Using these construction schemes, which are based on the incidence matrices of the known BIB designs with repeated blocks; we proposed some new construction methods of variance and efficiency balanced designs with repeated blocks for treatments which are generalization of the schemes given in reference papers. Let us consider treatments arranged in b blocks, such that the th block contains k. experimental units and the ith treatment appears r times in the entire design, i = 1,2,………, ; j = 1,2,……….,b. For any block design there exist a incidence matrix N = [nij] of order x b, where nij denotes the number of experiment units in the jth block getting the ith treatment. When ij = 1 or 0 " 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 additional notations are used k = [k k…..k]’ is the column vector of block sizes, r = [r…..r]’ is the column vector of treatment replication, Kbxb = diag [k k…..k] , Rvxv = diag [r r…..r] , = =n is the total number of experimental units , with this N1 = and N’1= , Where 1 is the a x 1 vector of ones. The information matrix for treatment effects C defined below as C = R – NK-1N' (1) Where R= diag (r, r..........r), K = diag (k, k..........k) Though there have been balanced designs in various sense (see Puri and Nigam16, Cali´nski17, we will consider a balanced design of the following type. A block design is said to be Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(9), 18-22, September (2012) Res. J. Recent Sci. International Science Congress Association 19 balanced if every elementary contrast of treatment is estimated with the same variance (Rao18). In this sense this design is also called a variance balance design. It is well known that block design is a variance balanced if and only if it has C = ( I – ' ) (2) where is the unique nonzero eigenvalue of the matrix C with the multiplicity v , v is the v × v identity matrix. For binary block design 11--=  (3) (Kageyama and Tsuji19) In particular case when block design is a balanced incomplete block design then . A block design is called efficiency balanced if every contrast of treatment effects is estimated through the design with the same efficiency factor. Let us consider the matrix Mo given by Cali´nski20 = R-1NK-1N' – n r' (4) o S = S Where T = [T T.......T]' is the vector of treatment totals ; Ti is the total yield for the ith treatment. is the unique non zero eigen value of M with multiplicity (-1) and M is given as (4). Cali´nski20showed that for such designs every treatment contrast is estimated with the same efficiency (1-) and N is a EB block design if and only if o = (I – n 1r') (5) Kageyama21 proved that for the EB block design N, eq (5) is fulfilled if and only if C = (1- ) (R – n r r') (6) Construction for treatments Let N, i = 1, 2, ..., t, be the incidence matrix of the balanced incomplete block design with repeated blocks with the parameters , b, r, k, , b*. Let C be the C-matrix of this design defined by N, i = 1, 2,.....,t. Now, we form the matrix N as N = [N N ...... N] (7) and prove the following theorem. Theorem 1 : Block design with the incidence matrix N of the form (7) is the variance and efficiency balanced block design with repeated blocks with the parameters n , , , , and . Proof : The matrix C of the block design (7) is NN k () =-\n \r--=tiiiiiiikIrkrC1'111nnnll where h , h is the unique nonzero eigenvalue of the matrix C , ,.........2,1 = . So, the theorem is proved and the Variance and Efficiency is given as, Variance Efficiency Example 1 : Let us consider the balanced incomplete block design with the parameters = 7, b = 21, r = 6, k = 2, = 1, * = 21 with the incidence matrix N given through the blocks (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (3, 4), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7) and the balanced incomplete block design with the parameters = 7, b = 14, r = 6, k = 3, = 2, b*= 7 with the incidence matrix N given through the blocks (1, 2, 4), (2, 3, 5), (3, 4, 6), (4, 5, 7), (1, 5, 6), (2, 6, 7), (1, 3, 7), each block is repeated two times. Based on the matrices N and N for t = 2 we form the incidence matrix N in the form (7) of the variance and efficiency balanced block design with repeated blocks with the parameters = 7, b = 21+14=35, r = 6+6=12, =1+2= 3, b= 21+7=28 and [ ] 1421and the structure is variance and efficiency balanced. Hence the matrix C is given as 49 and 1667.849Variance 319444.0722312Efficiency Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(9), 18-22, September (2012) Res. J. Recent Sci. International Science Congress Association 20 Some New Construction methods of Variance and Efficiency Balanced Block Designs with repeated blocks Theorem 2: Let N be the incidence matrix of the balanced incomplete block design with repeated blocks with the parameters , b, r, k, , b*. Then the Block design with the incidence matrix N of the form N = [ N 1] (8) is the variance and efficiency balanced block design with repeated blocks with the parameters , b = b + 1, r = r + 1, = 1 + 1 , b* = b* + 1 , [ ] .1 b and the variance and efficiency is given as , Variance EfficiencyTheorem 3 : Let N and N2 be the incidence matrices of the balanced incomplete block design with repeated blocks with the parameters , b, r, k, , b* and , b, r, k, , b* respectively. Then the block design with the incidence matrix N of the form N = [ N N2 ] (9) is the variance and efficiency balanced block design with repeated blocks with the parameters , b = b + b + 1, r = r + + 1, = 1 + + 1 , b* = b* + b* + 1 , [ ] .2.1 and the variance and efficiency is given as, \n \r++= k k Variance l l n EfficiencyTheorem 4 : Let N, i = 1, 2, ..., t, be the incidence matrix of the balanced incomplete block design with repeated blocks with the parameters , b, r, k, , b*. Then block design with the incidence matrix N of the form N = [ N N2 ……. N 1] (10) is the variance and efficiency balanced block design with repeated blocks with the parameters n , , , , , .2.1 and the variance and efficiency is given as, =+=tiii k Variance l n EfficiencyTheorem 5 : Let N be the incidence matrix of the balanced incomplete block design with repeated blocks with the parameters , b, r, k, , b*. Then block design with the incidence matrix N of the form times............. (11) is the variance and efficiency balanced block design with repeated blocks with the parameters n , , r r , l l , times and b b , then the variance and efficiency is given as Variance EfficiencyTheorem 6 : Let N be the incidence matrix of the balanced incomplete block design with repeated blocks with the parameters , b, r, k, , b*. Then block design with the incidence matrix N of the form .............times (12) is the variance and efficiency balanced block design with repeated blocks with the parameters n , , r r , l l , b b and .............times and the variance and efficiency is given as, Variance ( ) EfficiencyTheorem 7: Let N, i = 1, 2, ..., t, be the incidence matrix of the balanced incomplete block design with repeated blocks with the Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(9), 18-22, September (2012) Res. J. Recent Sci. International Science Congress Association 21 parameters , b, r, k, , b*. Then block design with the incidence matrix N of the form times..................... (13) is the variance and efficiency balanced block design with repeated blocks with the parameters n , , , , .2.1.........times , and the variance and efficiency is given as, Variance EfficiencyConclusion The construction methods discussed above are flexible enough to incorporate number of incidence matrices of balanced incomplete block design with repeated blocks. In the pattern given in (10), Variance increases but efficiency doesn’t show any regular pattern. In the patterns given in (11) and (13), Variance increases and efficiency decreases as unit vector 1repeated number of times. In the pattern given in (12), Variance and efficiency both increases as incidence matrix Ni repeated number of times. References 1.Van Lint J.H. , Block Designs with repeated blocks and (b, r, ) = 1, Journal of Combi Theory Series, (A15), 288-309 (1973) 2.Hanani H., The existence and construction of balanced incomplete block designs, Ann. 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