![Construction of Balanced Incomplete…
www.ijmsi.org 11 | Page
EXAMPLE 2.2: Consider a PBIBD with parameters v = 9, b= 6, k=3, r=2, λ0= 0, λ1=1. The resulting BIBD
with parameters v=9, b=18, r=8, k=4, n1=4, n2=4, λ=3 is presented below. The efficiency of the BIBD is
PBIBD Incidence Matrix of PBIBD BIBD Incidence Matrix of BIBD
963
852
741
987
654
321
100100100
010010010
001001001
111000000
000111000
000000111
5421
6431
6532
8721
9531
9832
8754
9764
9865
6387
5297
9841
9354
8264
7165
9621
8531
7432
000011011
000101101
000110110
011000011
100010101
110000110
011011000
101101000
110110000
011100100
101010010
110001001
100011100
010101010
001110001
100100011
010010101
001001110
Table 2.2
Acknowledgments: The First author is grateful to UGC for providing financial assistance to carry out this work
under BSR RFSMS. Authors are thankful to the referee for improving the final version of the manuscript.
References
[1] Bose R.C. (1950): Partially balanced incomplete block designs with two associate classes involving only two replications,
Calcutta Stat. Assoc. Bull., Vol 3, pp 120-125.
[2] Mener D.M. (1967): A new family of partially balanced incomplete block designs with some Latin square designs properties,
Ann. Math. Stat., Vol38, pp 571-581.
[3] Raghava Rao D (1980): Construction and combinatorial problems in design of experiments, Dover, NewYork.
[4] Bose R.C. and Simamoto T (1952): Classification and Analysis of partially balanced incomplete block designs with two
associate classes, Jour. Amer. Stat., Vol 47, pp 151-184.](https://image.slidesharecdn.com/b04109011-160912064843/85/Construction-of-Balanced-Incomplete-Block-Designs-3-320.jpg)
Construction of Balanced Incomplete Block Designs
- 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 1 || January. 2016 || PP-09-11 www.ijmsi.org 9 | Page Construction of Balanced Incomplete Block Designs T. Shekar Goud and N.Ch. Bhatra Charyulu Department of Statistics, University College of Science, Osmania University, Hyderabad-7 ABSTRACT: Incomplete block designs were introduced to eliminate heterogeneity to a greater extent is possible with randomized block design and latin square design when the number of treatments is large. Some of the incomplete block designs are, balanced incomplete block design, partially balanced incomplete block design, youden square design, lattice design etc. In this paper an attempt is made to propose two new balanced incomplete block design construction methods using two associate class partially balanced incomplete block designs. Keywords: BIBD, Triangular type PBIBD, PBIBD. I. INTRODUCTION The arrangement of „v‟ treatments in „b‟ blocks, each of sizes k1, k2, … , kb, each of the treatment appears r1, r2, … , rv blocks such that some pairs of treatments occur in 1 blocks, some pairs of treatments occur in 2 blocks, so on and the rest of pairs of treatments occur in m blocks then the design is said to be a “General Incomplete Block Design”. The Incomplete block designs were introduced to eliminate heterogeneity to a greater extent is possible with randomized blocks and latin squares when the number of treatments is large. Some of the incomplete block designs are, balanced incomplete block design, partially balanced incomplete block design, youden square design, lattice design etc. DEFINITION 1.1: The arrangement of „v‟ treatments in „b‟ blocks each of size „k‟, each treatment appears exactly in „r‟ blocks and every pair of treatments occurs exactly „‟ times, then the design is said to „Balanced Incomplete Block Design (BIBD)‟. The parameters v, b, r, k and satisfies the parametric relations: vr = bk; (v-1) = r(k-1) and b v. DEFINITION 1.2: The arrangement of „v‟ treatments in „b‟ blocks each of size „k‟ (<v) each treatment appears in „r‟ blocks is said to be an m-associate class partially balanced incomplete block design (PBIBD) if each pair of treatments appears in 1 or 2, ... or m blocks. The parameters v, b, r, k, λ1, λ2, … λm , n1, n2, ... nm satisfies the parametric relations: vr = bk; ni = v-1; niλi = r(k-1); Pij k = Pji k ; ni pi jk = nj pj ik; Pij k = ni if ik and Pij k = ni-1 if i=k; where i, j, k = 1,2,….m. If the number of associations is two then it is called a two-associate class PBIBD. DEFINITION 1.3: The arrangement of v= ½ s(s-1) treatments in a square array of order s, such that the positions in the principal diagonal are left blank, and the ½ s(s-1) positions above the principal diagonal are filled up with the ½ s(s-1) treatment symbols and the positions below the principal diagonal are filled up by the ½ s(s-1) treatment symbols in such a manner that the resultant arrangement is symmetrical about the principal diagonal, the resulting design is said to be a triangular type PBIBD. In this design any two treatments are said to be first associates if they belong to same row or same column and are said to be second associates otherwise. The parameters of triangular type PBIBD are v = ½ s(s-1), b=s, r=2, k=s-1, 1=0, 2=1, n1=2(s-2), n2= ½ (s-2)(s-3) and the two association matrices are )4)(3(3 32 2 1 )1( sss ss Pij and )5)(4()4(2 824 2 1 )2( sss s Pij DEFINITION 1.4: The arrangement of v=s2 treatments in „s‟ rows, „s‟ columns such that each treatment occurs once in each row and in each column (Latin square of order s) and, the resulting design is a latin square type PBIBD. In this design any two treatments in the same row or column are first associates and otherwise they are said to be second associate. II. CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS In this section an attempt is made to propose two new methods for the construction of balanced incomplete block designs using partially balanced incomplete block designs. THEOREM 2.1: A BIBD with parameters v′ = n, b′= ½ n(n-1), r′= n-1, k′= n-3, λ′=1can be constructed with the existence of a triangular type PBIBD with parameters v= ½ n(n-1), b=n, r=2, k=n-1, λ0 = 0, λ1 = 1.
- 2. Construction of Balanced Incomplete… www.ijmsi.org 10 | Page Proof: Consider a two associate class triangular type partially balanced incomplete block design with parameters v = ½ n (n-1), b=n, r=2, k = n-1, λ1 = 1, λ0 = 0, whose incidence matrix is N. Construct a design with number of blocks as number of treatments (v′=b), generate the blocks corresponding to each treatment of PBIBD (b′=v) such that the block containing block numbers of each treatment as blocks with block size (k′=r) and each treatment replicated „k‟ times ( r′= k ). The resulting design is a BIBD with the parameters v′ = n , b′= ½ n(n-1), r′= n-1, k′= 2, λ′=1. The step by step procedure for the construction of BIBD is presented below. Step 1: Consider a triangular type PBIBD with parameters v = n (n-1)/2, b=n, r=2, k = n-1, λ1 = 1, λ0 = 0 whose incidence matrix is N. Step 2: Consider each block number of PBIBD as treatment, corresponding to each treatment list the block number of its occurrence. Step 3: The resulting is a BIBD is with parameters v′ = n, b′= ½ n(n-1), r′= n-1, k′= 2, λ′=1. The procedure is illustrated in the example 2.1. EXAMPLE 2.1: Consider a triangular type PBIBD with parameters v= 10, b=5, r=2, k=4, λ0=0, λ1=1 whose incidence matrix is N. The resultant BIBD with parameters v′=5, b′=10, r′=4, k′=2, λ′=1 is presented in Table 2.1. The efficiency of the BIBD is 0.625. Triangular type PBIBD Incidence Matrix of PBIBD Incidence Matrix of BIBD BIBD Blocks I 1 2 3 4 II 1 5 6 7 III 2 5 8 9 IV 3 6 8 10 V 4 7 9 10 1101001000 1010100100 0110010010 0001110001 0000001111 11000 10100 01100 10010 01010 00110 10001 01001 00101 00011 54 53 43 52 42 32 51 41 31 21 Table 2.1 THEOREM 2.2: A BIBD with parameters v′=v= s2 , b′= 2s2 , k′= ½(s2 -1), r′= s2 -1 and ′= 3 can be constructed with the existence of a two associate class partially balanced incomplete block design with parameters v=s2 , b=2s, r=2, k=s, λ0 = 0, λ1 = 1, n1=n2. Proof: Consider a partially balanced incomplete block design with parameters v=s2 , b=2s, r=2, k=s, λ0 = 0, λ1 = 1, n1 = n2 (= ½ (s2 -1) ). Corresponding to each treatment 1st associate group of treatments as one block and 2nd associate group of treatments as another block, generate 2s2 blocks with the s2 treatments such that each treatment is repeated s2 -1 times and each block contains ½ (s2 -1) treatments. The resulting design is a BIBD with parameters v′=v, b′= 2s2 , k′= ½(s2 -1), r′= s2 -1 and ′= 3. The detailed step by step procedure of construction of BIBD is presented below. Step 1: Consider a simple lattice design with parameters v=s2 , b=2s, r=2, k=s, λ0 = 0, λ1 = 1, n1=n2, whose incidence matrix is N. Step 2: Identify the treatments belong to 1st and 2nd association schemes of each treatment. Step 3: Constructing the 2s2 blocks with the s2 treatments such that corresponding to each treatment 1st association treatments as one block and 2nd associate treatments as another block. Step 4: The resulting BIBD is with parameters v′=v, b′= 2s2 , k′= ½(s2 -1), r′= s2 -1 and ′= 3. It can be noted that the design is existing when s=3 and the procedure is illustrated in the example 2.2.
- 3. Construction of Balanced Incomplete… www.ijmsi.org 11 | Page EXAMPLE 2.2: Consider a PBIBD with parameters v = 9, b= 6, k=3, r=2, λ0= 0, λ1=1. The resulting BIBD with parameters v=9, b=18, r=8, k=4, n1=4, n2=4, λ=3 is presented below. The efficiency of the BIBD is PBIBD Incidence Matrix of PBIBD BIBD Incidence Matrix of BIBD 963 852 741 987 654 321 100100100 010010010 001001001 111000000 000111000 000000111 5421 6431 6532 8721 9531 9832 8754 9764 9865 6387 5297 9841 9354 8264 7165 9621 8531 7432 000011011 000101101 000110110 011000011 100010101 110000110 011011000 101101000 110110000 011100100 101010010 110001001 100011100 010101010 001110001 100100011 010010101 001001110 Table 2.2 Acknowledgments: The First author is grateful to UGC for providing financial assistance to carry out this work under BSR RFSMS. Authors are thankful to the referee for improving the final version of the manuscript. References [1] Bose R.C. (1950): Partially balanced incomplete block designs with two associate classes involving only two replications, Calcutta Stat. Assoc. Bull., Vol 3, pp 120-125. [2] Mener D.M. (1967): A new family of partially balanced incomplete block designs with some Latin square designs properties, Ann. Math. Stat., Vol38, pp 571-581. [3] Raghava Rao D (1980): Construction and combinatorial problems in design of experiments, Dover, NewYork. [4] Bose R.C. and Simamoto T (1952): Classification and Analysis of partially balanced incomplete block designs with two associate classes, Jour. Amer. Stat., Vol 47, pp 151-184.