ON SOME METHODS OF CONSTRUCTION OF BLOCK DESIGNS

Transcription

1 ON SOME METHODS OF CONSTRUCTION OF BLOCK DESIGNS NURNABI MEHERUL ALAM M.Sc. (Agricultural Statistics), Roll No. I.A.S.R.I, Library Avenue, New Delhi- Chairperson: Dr. P.K. Batra Abstract: Block designs are useful in experiments requiring elimination of heterogeneity in one direction. A block design is said to be incomplete if it contains at least one block that does not have full set of treatments. The most commonly used block designs are Balanced Incomplete Block (BIB) design and Partially Balanced Incomplete Block (PBIB) designs. Many research workers have contributed to the construction of these designs. Many series of BIB designs have been constructed using finite geometries, method of symmetrically repeated differences, using Hadamard matrices etc. BIB designs and their properties can be used to obtain many new series of BIB and PBIB designs. Some methods of construction of BIB and PBIB designs using existing series of BIB designs would be described. Key words: Block Design, BIB design, PBIB design, Dual of Block Design, Block Section, Block Intersection, Complementary Design.. Introduction Block designs are useful in experiments requiring elimination of heterogeneity in one direction. The most commonly used block design in agricultural experimentation is the Randomized Complete Block (RCB) design, where entire experimental material is divided into homogenous groups called blocks and the treatments are allotted randomly to experimental units within blocks. In this design, every treatment occurs exactly once in each block. If the experiment involves large number of treatments so that it becomes difficult to have large blocks and preserve homogenous condition within each block, then incomplete block designs are used. An incomplete block design is one in which all the treatments do not occur in at least one block. Many incomplete block designs have been obtained keeping in view the ease of analysis and also their constructions. Construction of incomplete block designs having patterns admitting simplified analysis and possessing desirable statistical property was priority of research workers in this field. Many research workers notably Bose and Shimamoto (), Yates (), Das (), Shrikhande (), Dey (), Nigam et al. (), Kageyama () etc have contributed to the construction of important classes of block designs like Balanced Incomplete Block (BIB) design, Partially Balanced Incomplete Block (PBIB) design etc. The main advantage of these designs is the ease in analysis and interpretation of result. Many methods of construction of such designs are available using standard mathematical theories like- Galois field, projective geometries etc. Some methods are also available in the literature, which can fruitfully be used to construct new designs from the existing designs. Commonly used such methods are - block section, block intersection, complementary design, dual of block designs etc. Here, some methods in the construction of block designs, particularly BIB design and PBIB design would be discussed.

2 . Block Designs Block design D (v, b, r, k) is an arrangement of v treatments in b bocks in such a way that th th the i treatment occurs n times in j block. The v x b matrix ij N ( ( n ij ) ) (i,..., v; j,..., b) is called the incidence matrix of the design. Further N b r and N v k where r r, r,...,r ) is v x vector of treatment replications ( v and k k, k,..., k ) is b x vector of block sizes. Further r k n, the total ( b number of experimental units. A design is said to be binary if k j s are equal and equi-replicate if all r s are equal. If n i ij v b i j i j n ij or, proper if all i, j then the block design is said to be a complete block design and if for at least one i, j (i,..., v; j,..., b), then the block design is said to be an incomplete block design.. Model for Block Design Following is the general, additive, fixed effects model for a block design D(v, b, r, k): Y μ+ Δ τ + D β + e where E(e) and dispersion matrix D ( e ) σ I, Δ observation vs. treatment n incidence matrix, D observation vs. block incidence matrix, μ general mean effect, τ the column vector of treatment effects and β column vector of block effects. Using the principle of ordinary least squares, the reduced normal equations for estimating the linear functions of treatment effects are given by n ij where Cτ Q, Q T NK B denotes the vector of adjusted treatment totals and C R NK N is the coefficient matrix and is also known as Fisher s information matrix for estimating the treatment effects. It is a non-negative definite v x v symmetric matrix with row sum and column sum zero. R diagonal matrix of replications, K diagonal matrix of block sizes, T column vector of treatment totals. τˆ C Q C is a generalized inverse of C. A block design is said to be connected iff Rank ( C) v or C has v- non-zero eigen values. The analysis of variance of general block design is as follows: Source of Variation Degrees of Freedom Treatment v- Replication r- Error (r-)(v-) Total rv-

3 BIB and PBIB designs defined below are an important class of incomplete block designs. Balanced Incomplete Block Design: A BIB design is an arrangement of v treatments into b blocks such that (i) each block contains k (<v) distinct treatments, (ii) each treatment appears in r blocks, (iii) every pair of treatments appears together in λ blocks where v, b, r, k, λ are the parameters of the BIB design. Partially Balanced Incomplete Block Design: Given an association scheme with m classes ( m ), a PBIB design based on the association scheme is defined as arrangement of v treatments in b blocks, such that (i) each block contains k (< v ) distinct treatments, (ii) each treatment appears in r blocks, (iii) if the treatments α and β are mutually i th associates in the association scheme, then α and β occur together in λ i blocks, where the integer λ i does not depend on the pair (α, β) so long as they are mutually i th associates, i,,,m. Further not all λ s are equal. i Construction of BIB and PBIB have drawn attention of many research workers. Many methods in literature include the one where use of existing block design is made. These methods would be discussed in sequence.. Methods of Construction of Block Designs BIB designs and PBIB designs are important class of block designs and many research workers have contributed in the construction of different series of designs belonging to these classes. BIB designs are useful in varietial trials wherein it is of interest to make all elementary comparisons with equal precisions. Some of the methods of construction of these classes of block designs are based on mathematical developments in finite geometries, symmetricaliy repeated differences, Hadamard matrices etc. While few are based on using existing series of designs and their properties. The methods based on use of existing designs are discussed here.. Residual Design or Block Section A residual design of a block design D(v, b, r, k, N) is obtained by deleting any one block of the design and from the remaining b blocks deleting all those treatments, which appear in the deleted block and renumbering the treatments from,..., v k. The design so obtained is called residual design and the procedure is called the block section. Result.: The residual design of a symmetric BIB design D(v b, r k, λ) is a BIB design with parameters v v k, b b, r r, k k λ, λ λ. Example.: Consider a symmetric BIB design as given below with parameters v b, r k, λ. The residual design is obtained by deleting say the third block i.e. (,, ) and the treatments occurring in the block from other bocks. (Rows indicate block)

4 / / / / / / / / / The design obtained is as follows: Renumbering the treatments like the final design is which is again a BIB design with parameters (v, b, r, k, λ ).. Block Intersection or Derived Design Derived design from a block design D(v, b, r, k, N) is obtained by deleting any one block of the design and from the remaining b blocks deleting all those treatments, which do not appear in the deleted block and renumbering the treatments from,..., k. Result.: The derived design of a symmetric BIB design with parameters v b, r k, λ is a BIB design with parameters v k, b b, r r, k λ, λ λ. Example.: Consider a symmetric BIB design with v b, r k, λ. Deleting the fourth block and retaining the treatments appearing in the deleted block in the remaining blocks, a derived design is obtained.

5 / / / / / / / / / / / / / / / / Renumbering the treatments the final design with v, b, r, k, λ is as follows:. Complementary Design Given a design D with parameters v, b, r, k, N, another design called the complementary design D is obtained in which corresponding to each block of design D, a block of design D is formed by taking only those treatments, which are not appearing in the blocks of D. Alternatively, in the incidence matrix N of the design D, replacing s by and s by, the incidence matrix N of complementary design D is obtained. Result.: Complementary design of a BIB design ( v, b, r, k, λ) is a BIB design with parameters v v, b b, r b r, k v k, λ b r + λ. Example.: BIB design with parameters v, b, r, k, λ and its complementary design with parameters v, b, r, k, λ is given below:

6 Original design Complementary design The incidence matrix of BIB design is N Now replacing s by and s by in the incidence matrix, the incidence matrix of the complementary design N is as follows: N

7 Result.: The complementary of a Group Divisible (GD) PBIB design with parameters v, b, r, k,, λ is also a GD design with same association scheme and with parameters v v, b b, r b r, k v k, λ b r + λ, λ b + λ. λ r Example.: Consider a PBIB design with v, b, r, k, λ, λ. The complementary is also a PBIB design with parameters as, b, r, k, λ, λ v Original design Complementary design The association scheme in both the cases is as follows: For treatment, treatment is first associate that appears λ times in the original design and λ times in the complementary design. Remaining treatments are second associates of treatment that appear λ time in the original design and λ times in the complementary design. Result.: Complementary design of any equireplicate proper block design is also an equireplicate proper block design.. Construction by Deleting Blocks Containing a Particular Treatment Consider a BIB design D with parameters v, b, r, k, λ, then by deleting all those blocks containing a particular treatment, a GD design with parameters v v, b b r, r r, k k, λ, λ, m r, n k is obtained. Example.: BIB design with parameters v, b, r, k, λ is as follows:

8 Selecting treatment and deleting the blocks containing treatment, the resultant design is obtained. The above design is a GD design with parameters v, b, r, k, λ, λ, m, n and the association scheme is obtained from the deleted blocks after deleting treatment. Result.: Consider a symmetric BIB design with v n(n ) / + b, r n k, λ. Omitting all the blocks containing a particular treatment from this design, a triangular design with v n(n ) /, b (n )(n ) /, r n, k n, λ, λ is obtained. Example.: Consider the following BIB design with v b, r k, λ :

9 Deleting blocks containing treatment, the following triangular design is obtained: ` The parameters of this design are v, b, r, k, λ, λ. The arrangements of the treatments of the association scheme is. Replacement Method Consider a BIB design with parameters v, b, r, k, λ. In this design if each treatment is replaced by a group of s distinct treatments (s is constant), the resultant design is PBIB design with GD association scheme. The parameters of the resulting design are vs, b b, r r, k ks, λ r, λ λ, m v, n s. v Example.: Consider the following BIB design with v, b, r, k, λ :

10 Replacing each treatment by a group of two treatments i.e. s as follows: the resultant design becomes which is a PBIB design with parameters v, b, r, k, λ, λ with the following GD association scheme Result.: Consider a BIB design D with parameters v s, b, r, k, λ. Let the i-th treatment be replaced by the treatments in the i-th row of an association scheme with v s. Similarly i-th treatment is replaced by the treatments in the i-th column of association scheme, then a PBIB design with L association scheme with parameters L v s, b b, r r, k sk, λ r + λ, λ λ L is obtained. Example.: Consider the BIB design with parameters v b, k r, λ.

11 Let s. The L scheme on v treatments can be displayed as The design so obtained with, b, r, k, λ, λ is v. Dual Designs If N is an incidence matrix of a design D, then the design D, which has N as its incidence matrix, is said to be the dual design of D. Given a block designs D, by inter changing blocks and treatments, a new design D called the dual of the original design D is obtained. If the original design has parameters v, b, r, k then the dual design will have parameters v b, b v, r k, k r. Dualization of known designs sometimes yields new designs and sometimes known designs. Corresponding to any incomplete block design there is a dual design. If any incomplete block design can be used for experimentation, then its dual design can also be used for the same purpose without any fresh difficulty for the analysis of the block design. Given two designs D and E, one may be interested to know whether the design E can be the dual of D. For this purpose, we must necessarily have the number of treatments of D to be equal to the number of blocks in E, and the number of replications of D to be equal to the block size in E. Further if N is the incidence matrix of D and N is the incidence matrix of E, then the non zero characteristic roots along with their multiplicities should be the same for N N and N N. Example.: A BIB design with parameters (,,,, ) is as follows:

12 Further, N According to the definition of the dual design of D, D has the incidence matrix as the transpose of N i.e. N is given by N So the dual design obtained is

13 which is a PBIB design with, b, r, k, λ, λ. v Result.: Dual of a symmetric BIB design is also a symmetric BIB design with same parameters. Example.: Consider a symmetric BIB design with parameters (,,,, ) with N As the matrix is symmetric its transpose is same as same. N. So the resultant dual design is the Result.: If D is BIB design with parameters v, b, r, k, λ, then its dual D is a two associate-class PBIB design with GD association scheme and parameters as v b, b v, r k, k r, λ, λ, m r, n b r, p r + (k, k p. ) Example.: Consider a BIB design with parameters v, b, r, k, λ given as follows: Its dual design is

14 which is a PBIB design with parameters v, b, r, k, λ, λ and association scheme as where two treatments are first associate if they belong to the same row, otherwise they are second associates. r r Result.: If D is a BIB design with parameters v, b, r, k r, λ, then its dual design D is a PBIB design with the parameters v b, b v, r k, k r, λ, λ, p r-,. Example.: Consider the following BIB design with v, b, r, k, λ : p Here

15 N N So the dual design becomes which is PBIB design with parameters,, k, r, b, v λ λ follow the triangular association scheme. The arrangement of treatments in the association scheme is:. Conclusions Many mathematical theories like projective geometries, Galois theory, method of finite differences, Hadamard matrices have been used to construct various series of BIB designs.

16 Symmetrical BIB designs have property that any two blocks have λ treatments in common and the same has been used for construction of series of BIB and PBIB designs. Duals of BIB designs have also enriched the availability of series of BIB and PBIB designs. Many classes of designs like Reinforced designs, efficiency balanced designs, linked block designs have also been constructed using existing designs. Thus it is concluded that use of existing designs has been helpful in enriching the availability of block designs for experimentation. References Bose, R.C. and Shimamoto, T. (). Classification and analysis of PBIB designs with two associate classes. Journal of the American Statistical Association,, -. Das, M.N. and Kulkarni, G.A. (). Incomplete block designs for bio-assays. Biometrics,, -. Dey, A. (). A note on balanced designs. Sankhya, B,, -. Dey, A. (). Theory of block design, Wiley Eastren Limited. Kageyama, S. (). On properties of efficiency balanced designs. Communication in Statistics, A (), -. Nigam, A.K, Puri, P.D, Gupta, V.K. (). Characterizations and analysis of block designs. Wiley Eastren Limited. Raghavarao, D. (). Duals of partially balanced incomplete block designs and some non-existence theorems. Annals of Mathematical Statistics,, -. Shrikhande, S.S. (). On duals of some balanced incomplete block designs. Biometrics,, -. Shrikhande, S.S. and Bhagwandas (). Duals of incomplete block designs. Journal of the Indian Statistical Association,, -. Yates, F. (). A new method of arranging variety trials involving a large number of varieties. J. Agric. Sci.,, -.