Construction of Partially Balanced Incomplete Block Designs

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1 International Journal of Statistics and Systems ISS Volume, umber (06), pp Research India Publications Construction of Partially Balanced Incomplete Block Designs Jyoti Sharma, D. K. Ghosh and Jagdish Prasad 3 Centre for Mathematical Sciences, Banasthali University, Rajasthan, India. Department of Statistics, Saurashtra University, Rajkot, India. 3 School of Applied Sciences, Amity University Rajasthan, Jaipur, India. sharmajyoti09@gmail.com, 3 jprasad@jpr.amity.edu Abstract In this paper, some series of BIB and PBIB designs have been constructed using either Semi-Regular Group Divisible Designs or Regular Group Divisible Designs along with its corresponding group. Keywords: Association Scheme, Balanced incomplete block designs, Partially balanced incomplete block designs, group and Group Divisible designs.. Introduction Yates (936) introduced the concept of BIBD. BIBD is an arrangement of v treatments into b blocks each of k (<v) treatments, satisfying the following conditions:. Every treatment occurs at most once in each block.. Every treatment occurs in exactly r blocks. 3. Every pair of treatment occurs together in exactly blocks. A BIBD is said to be symmetrical if v=b and r=k. The terms v, b, r, k, are known as the parameters of BIBD. In this design we can estimate all possible treatment contrast with the same precision. The rich contribution of BIBD is mainly by Fisher (940, 94), Fisher and Yates (963) or Bose ( ). A BIBD is said to be resolvable if the b blocks are grouped into r classes of n blocks each, such that each class forms a complete replication of all the v treatments and each class however contains b / r blocks. BIBD are not available for every parametric combination. Also even if a BIBD exists for a given no. of treatments (v) and block size (k), it may require too many replications. To overcome this problem, Bose and

2 68 Jyoti Sharma et al air (939) introduced a class of binary, equi-replicate and proper designs, which we called PBIBD with m-associate classes. Bose and Shimamoto (95), air and Rao (94) have also contributed to the theory of PBIBD. A PBIBD with two associate classes is an arrangement of v treatments in b blocks such that:. Each of the v treatments is replicated r times in b blocks each of size k (k < v), and no treatments appears more than once in any block.. There exists a relationship of association between every pair of the v treatments satisfying the following conditions: a. Any two treatments are either first or second associates. b. Each treatment has exactly n i ith associates (i=, ). c. Given any two treatments which are i th associates, the number of treatments common to the j th associate of the first and k th associate of the second is p i jk and is independent of the pair of treatments. Also p i jk=p i kj, i, j, k=,. 3. Any pair of treatments which are i th associate occur together in exactly i blocks for i=,. air and Rao (94) modified the original definition of PBIB designs. For m=, Bose and Shimamoto (95) classified the known PBIB designs into () Group Divisible (GD), () Simple (S.I), (3) Triangular (T), (4) Latin Square Type (L i ) and (5) Cyclic Designs. A group divisible design is an arrangement of v=mn treatments into b blocks such that each block contains k(<v) distinct treatments which are partitioned into m( ) groups of n( ) treatments each, further any two distinct treatments occurring together in blocks if they belong to the same group, and in blocks if they belong to different groups. A group divisible design is classified into ) Singular Group Divisible Design ) Semi-Regular Group Divisible Design 3) Regular Group Divisible Design. A Group Divisible Design is said to be Singular Group Divisible Design if r- =0, a Group Divisible Design is said to be Semi-Regular Group Divisible Design if r- >0 and rk-v =0, and a Group Divisible Design is said to be Regular Group Divisible Design if r- >0 and rk-v >0. In this paper, we have discussed the method of construction of BIBD and PBIBD using GD design along with their corresponding group.. Method of Construction Let us consider a group (m,n) of a GD design, where v=mn treatments arranged in m groups of n treatments each. For an example: consider a Group (, ). ow this group

3 Construction of Partially Balanced Incomplete Block Designs 69 (, ) is expressed as an arrangement of in groups of treatments each. Let us call this group as design d, that is, d = Here the incidence matrix and concurrence matrix of d is denoted by, and which are as follows: = = Remark: If the same group is repeated p-times then is expressed as Im Im = p (.) I m... Im Where, an identity matrix I m is repeated n times in rows and m times in columns. Further consider a GD design with parameters v, b, r, k, m, n,,. ext add the corresponding group (m, n) to this GD design, which gives either resolvable BIBD or PBIBD according to its parameters. Case.: RGD designs from SRGD designs: Theorem.: If there exists a SRGD design with parameters v, b, r, k, m, n,, for which n=k, then a RGD design with parameters v=v, b=b +m, r=r +, k=k, = +, = always exists. Proof: Consider a SRGD design with parameters v, b, r, k, m, n,, whose incidence matrix is denoted by. ext add the corresponding group to this design and let the incidence matrix of the group is. ow we define the incidence matrix of resulting design which is given by, [ ] v b = ext the concurrence matrix of this design is as follows: = [ ] = [ + ] (.) Here we will obtain and separately. Since is the incidence matrix of SRGD and hence can be expressed as

4 70 Jyoti Sharma et al = r r r. r and = , ow using (.), is expressed as r + + r r = r (.3) Here all diagonal elements are same, that is, (r + ) and off diagonal elements are and ( + ). Hence we can say is the incidence matrix of a PBIB design with parameters v=v, b=b +m, r=r +, k=k, = +, =. Further we verified that r- >0 and (rk-v ) > 0 holds true and hence design is RGD. Example.: Consider SR with parameters v=4, r=4, k=, b=8, m=, n=, = 0, = with corresponding group (, ), whose blocks are as follows: ow by adding the group to this design we obtain another design whose blocks are, Which is a RGD design with parameters v=4, r=5, k=, b=0, n =, n =, m=, n=, = 0, =. This design is listed as R-3 in the Clatworthy (973).

5 Construction of Partially Balanced Incomplete Block Designs 7 Case.: RGD designs from RGD design. Theorem.: If there exists a RGD design with parameters v, b, r, k, m, n,, for which n=k, then a RGD design with parameters v=v, b=b +m, r=r + p, k=k, = + p, = always exists, where, p is the number of times a group is repeated. Proof: On the similar lines of theorem.. Example.: Consider a RGD design (R-) with parameters v=4, r=4, k=, b=8, m=, n=, =, = along with its group (, ), whose blocks are as follows: ow with this design, add its group (, ). we have another design whose blocks are, The resulting design is a RGD with parameters v=4, r=5, k=, b=0, m=, n=, = 3, = which is reported as R- in Clatworthy(973). Case.3: RGD design from a resolvable BIBD. Theorem.3: If there exists a resolvable BIBD with parameters v =s, b =s(s+), r =s+, k =s, =, then by adding a group (m, n) for which n=k, we obtained a RGD design with parameters v=v, b=b + m, r=r +, k=k, m*=m, n*=n, =+, =. Proof: consider a resolvable BIBD with parameters v =s, b = s( s + ), r =s+, k =s, = and let the incidence matrix of this design is denoted by. ext add the group (m, n) to this resolvable BIBD. Let the incidence matrix of group is. ow we define the incidence matrix of resulting design which is given by [ ] v b = ext the concurrence matrix of this design is as follows: = [ ] = [ + ] (.4) r + + r r + Hence = (.5). +. r + Here all diagonal elements are same, that is, (r + ) and off diagonal elements are of two types which are + = and =. Hence we can say is the incidence matrix of

6 7 Jyoti Sharma et al a PBIB design with parameters v=v, b=b + m, r=r +, k=k, m*=m, n*=n, =+, =. Further we verified that r- >0 and (rk-v ) > 0 holds true and hence the resulting design is RGD. Example.3: If we substitute s=3 in the above theorem, then a resolvable BIBD with parameters v=9, b=, r=4, k=3, = is considered, which is as follows: ow by adding the design of group (3, 3) to this BIBD we obtained the following design: The resulting design is RGD with parameters v=9, b=5, r=5, k=3,, =, = reported as R-59 in Clatworthy(973). Corollary.: If there exists a GD design with parameters v, b, r, k, m, n,,, then a resolvable BIBD exists provided =. Proof: Consider either a SRGD or RGD design such that n=k and let the incidence matrix of this design is denoted by. ext add the corresponding group to this design and let the incidence matrix of the group is. ow we add to and then we have the incidence matrix of resulting design is given by = [ ] ext the concurrence matrix of this design is as follows: r + + r r = r Here all the diagonal elements are r +, that is, r=r +. Further off-diagonal elements are either = or = +. Since it is given that = =, and r=r + and hence becomes the concurrence matrix of a BIBD. Example.4: Consider a SRGD design(sr-) with parameters v=4, r=, k=, b=4, m=, n=,, = 0, =,whose blocks are as follows:

7 Construction of Partially Balanced Incomplete Block Designs ow by adding the group (,) to the above design we have another design whose blocks are, This gives the blocks of resolvable BIBD. Therefore the resulting design is a resolvable BIBD with parameters v=4, r=3, k=, b=6, =. Table. SRGD used to construct BIBD Parameters of SRGD Parameters of resulting BIBD S.. v r k b m n Group v r k b Design SR (,) BIBD SR (3,3) BIBD SR (,4) BIBD SR (4,4) BIBD SR (5,5) BIBD SR (,6) 6 5 BIBD SR (7,7) BIBD SR (8,8) BIBD SR (9,9) BIBD SRGD used to construct PBIBD Parameters of SRGD Group Parameters of resulting PBIBD Design S.. v r k b m n v r k b SR (,) R-3 SR (,) R-7 SR (,) R-3 SR (,) 4 5 RGD* SR (3,3) R-6 SR (3,3) R-68 SR (4,4) R- SR (5,5) RGD* RGDs by which BIBD is obtained Parameters of RGD Group Parameters of resulting BIBD v r k b m n Λ v r k b Design R (,) 4 6 BIBD R (,) BIBD

8 74 Jyoti Sharma et al R (3,) BIBD R (3,) BIBD R (4,) BIBD R (5,) BIBD R (6,) 66 BIBD R (,3) BIBD R (3,3) BIBD RGDs through which PBIBD is obtained Design Parameters of RGD Group Parameters of resulting PBIBD v r k b m n Λ v r k b R (,) R- R (,) R-4 R (,) R-5 R (,) R-8 R (,) R-9 R (,) R-0 R (,) R- R (,) R- R (,) R-4 R (,) R-5 R (,) R-7 R (,) 4 9 RGD* R (,) 4 7 RGD* R (,) RGD* R (,) RGD* R (3,) R- R (3,) R- R (3,) R-6 R (3,) R-7 R (3,) R-8 R (3,) RGD* R (4,) R-3 R (4,) R-33 R (5,) RGD* R (,3) R-45 R (,3) R-47 R (,3) R-49 R (,3) R-53 R (,3) RGD* R (3,3) R-60 R (3,3) R-6 R (3,3) R-63

9 Construction of Partially Balanced Incomplete Block Designs 75 R (3,3) R-64 R (3,3) R-66 R (3,3) R-67 R (3,3) RGD* R (3,3) RGD* R (3,3) RGD* R (4,3) R-78 R (5,3) R-84 R (5,3) RGD* R (,4) R-00 R (4,4) R-0 R (4,4) R- R (4,4) R-3 R (4,4) RGD* R (7,4) RGD* R (5,5) R-57 R (5,5) R-58 R (5,5) RGD* R (7,7) RGD* Indicates design is not listed as r,k>0 References [] Bose, R. C. (939) On the Construction of Balanced Incomplete Block Designs. Annals of Eugenics, Vol. 9 (939), pp [] R. C. Bose (95), Partially balanced incomplete block d esigns with two associate classes involving only two replications, Calcutta Statistical Association Bulletin 3, 0. [3] Bose, R.C. and Connor, W.S.(95). Combinatorial properties of group divisible incomplete block designs. Ann. Math. Statist., 3, [4] Bose, R.C. and air, K.R. (939) Partially Balanced Incomplete Block Designs. Sankhya. 4, [5] Bose, R.C. and Shimamoto, T. (95) Classification and Analysis of Partially Balanced Incomplete Block Designs with two Associate Classes. J. Amer. Statist. Assoc., 47, [6] Clatworthy W. H. (973) Tables of Two-Associates-Class Partially Balanced Designs. BS Applied Mathematics Series 63, Washington, Dc. [7] Dey, A. (986) Theory of Block Designs. Willey Eastern. [8] Das, M.. and Giri,.C. (986) Design and Analysis of Experiments. Wiley Eastern Limited, ew Delhi. [9] Fisher, R.A. (940) An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugen., Lond.,

10 76 Jyoti Sharma et al [0] Fisher R.A. (94). The theory of confounding in factorial experiments in relation to the theory of groups. Ann. Eugenics,, [] Fisher, R.A. and Yates, F.(963). Statistical Tables for Biological, Agricultural and Medical Resuarch, Sixth edition, Oliver and Boyd, Edinburgh [] air, K. R. and Rao, C. R. (94) A ote on Partially Balanced Incomplete Block Designs. Science and Culture, 7, [3] Raghavarao, D. (970). Constructions and Combinatorial Problems in Design of Experiments. Willey, ew York. [4] Yates, F. (936) Incomplete Randomized Blocks. Ann. Eugen. 7, -40.