Construction of Three Associate Class PBIB Designs using Method of Juxtapositions

Two Series of three associate class Partially Balanced Incomplete Block (PBIB) Designs using a method of juxtaposing some association matrices are constructed in this paper. The designs used as base designs are four associate class PBIB designs with fixed replications, but in new series there is no restriction on the parameters of PBIB designs. New series are constructed by juxtaposing some association matrices of a series of four associate class PBIB designs of partial geometry (r,k,t) with finite graphs. Association schemes of both the series, Pmatrices of the association scheme, efficiencies factors along with overall efficiency factor and illustrations of both the series have also been given in this paper.


Introduction
In the construction of PBIB designs with higher associate classes, our emphasis will always be that the number of associate classes needed to be reduced so that we can estimate various treatment affects with more accuracies. Keeping this mind the existing four associate class PBIB designs have been reduced to new three associate class PBIB designs by using the method of juxtaposing some association matrices. In such a way, the new patterned matrix raises three associate class PBIB designs with higher efficiencies. Clatworthy [7] lists complete description on two associate class PBIB designs. Sinha.et.al [6], Kageyama.et.al [3,4,5] introduced some patterned constructions of PBIB designs. Recently,Garg,et.al [2] have also introduced some patterned constructions of PBIB designs by using juxtaposing some association matrices along with their association schemes.
Here, we have introduced two series of three associate class PBIB designs by reducing an already existing series of four associate class PBIB designs used by partial geometry (r,k,t) with finite graphs introduced by Garg,et.al [1]. Association scheme of series-Ι is based on occurrence of treatment pairs and association scheme of series-ΙΙ is based on occurrence of binary results (present/absent) of a treatments combination under consideration in the incidence matrix.

Proof:
Since N* denotes the incidence matrix of a series of three associate class PBIB designs, so it contains zeros and ones only. The order of N* matrix is 2s 2 x s 2 , where (s≥3) The product N*N* / can be written as Clearly, N*N* / provides the following observations 1) Each row sum in N*N* / is same.
2) Each column sum in N*N* / is same.
3) Each row sum is also equal to each column sum 4) Each diagonal position of N*N* / is also unique and equal to 's'.
Noting these observations, we say that N*N* / is a treatment structure matrix of a PBIB design. Since each row sum of patterned matrix N* is 's' taken as replication of each treatment as well as each column sum of patterned matrix N* is '2s' taken as block size. Also, every diagonal element of N*N* / is equal to 's' shows a replication size of a treatment of PBIB design.

Association Scheme -3.1
Association scheme is based on occurrence of treatment pairs. Let us we take a particular treatment say 'θ'. Treatment pairs of type (θ , θ i ) occurs 's' times in these blocks, where i =1,2,…,s-1 are 1 st associate of 'θ' and n 1 = s-1. Treatment pairs of type (θ , θ j ) occurs only once in these blocks , where j= 1,2,…,s 2 are 2 nd associate of 'θ' and n 2 = s 2 .
Remaining treatments which do not occur with 'θ' in any block or in other words, treatments pair of type (θ , θ k ) does not occur in these blocks. All these φ k 's where k = 1,2,…,s(s-1) are 3 rd associates of θ and n 3 =s(s-1).
Here, let B 0 , B 1 , B 2 , B 3 and B 4 denotes the association matrices of four associate PBIB design. Consider a patterned matrix The order of N* matrix is18 x 9 and N* N* / satisfies all the properties of a matrix to be treatment structure matrix. Each row sum of N* is '3' and each column sum is '6' as well as every diagonal element is '3', which shows that replication of each treatment is '3' and each block consist of exactly '6' treatments. Collection of numbers against ones from columnwise gives treatments of a particular block and collection of numbers against ones from row -wise gives replication of a particular treatment.

Proof:
Since matrix N* denotes the incidence matrix of a series of three associate class PBIB designs so it will contains zeros and ones only. Here, N* is a square matrix of order s 2 . Clearly, the product matrix N*N* / shows that each row sum and column sum of N*N* / is unique and same for every row and each column. The diagonal elements of N*N* / are also unique and represent replications of a PBIB design. In this way, N*N* / represent a treatment structure matrix of a new design in which rows correspond to treatment and columns correspond to blocks. Row sum and column sum of matrix N* is 2s-1, so each treatment is replicated '2s-1' times and each block contain '2s-1' treatments. Hence, patterned matrix N* denotes the incidence matrix of a series of three associate class PBIB designs with parameters v= s 2 = b, r =2s-1 = k, λ 1 = s, λ 2 = 2 = λ 3 , n 1 = 2(s-1), n 2 = s-1 , n 3 = (s-1)(s-2) follows an association scheme -3.2 defined below.

Association scheme -3.2
Each row/ column contains zeros and ones only. This association is based on result of binary number (one/zero) in rowwise. Let us consider a treatment 'θ' (say) in a particular row.
Treatments having '1's in the same row are first associates to each other and n 1 = 2(s-1). Treatments in a row under consideration showing '0' 's due to the structure of matrix B 3 C in patterned matrix Bs 2 are second associates of 'θ' and n 2 =s -1.
Remaining treatments in a row under consideration showing '0' 's due to the structure of matrix B 4 in Bs 2 are third associates of 'θ' and n 3 =(s -1)(s-2).

Illustration
In equation (3), if s=3, we have a three associate class PBIB designs with parameters.
The method of construction blocks, association scheme along with P-matrices are discussed as below.
Let us consider a treatment 'θ=1' (say) in a first row of incidence matrix. Treatments 2,3,4,7 having '1' 's in first row. It means 2,3,4,7 are present in first row, so these are 1 st associates of '1' and n 1 = 4. Treatments 5 and 9 in this row shows '0' 's means absent in this row due to the structure matrix B 3 c in the patterned matrix B 9 are 2 nd associates of '1' and n 2 =2. Treatments 6 and 8 in this also shows '0' 's in first row due to the structure matrix B 4 in the patterned matrix B 9 are 3 rd associates of '1' and n 3 = 2. The P -matrices of this design are

Conclusion
Here, we have constructed two series of three associate class PBIB designs by reducing already existing four associate class of PBIB designs. The base design has fixed replication (r = 3), but in newly constructed series, there is no restriction on any parameter of PBIB design. Seven PBIB designs with r ≤ 9 , v ≤ 162 are listed in table 1 and six PBIB designs for r ≤ 15, v ≤ 64 listed in table-2 . In both series, number of treatments increase very rapidly due to square power of 's'. In Table-Ι, efficiencies of treatments which are first associates to each other is exactly 100% even for less number of replications. Table-ΙΙ represents symmetric PBIB designs having higher replications. Therefore, efficiencies of newly constructed three associate class PBIB designs are much better than four associate class PBIB designs of base series and quite useful for comparing the efficiencies with existing three associate class PBIB designs. We observe that new series of designs are adjustable for particular parametric combinations.