This document describes the construction of three-associate class partially balanced incomplete block designs (PBIBDs) in two replicates. It defines triangular, superimposed triangular, and tetrahedral association schemes for allocating treatments to blocks in PBIBDs. Examples are provided to illustrate the construction of three-associate class PBIBDs using a triangular association scheme with 12 treatments and 6 blocks, and a superimposed triangular association scheme with 24 treatments and 4 blocks. The document provides information on the parameters and associations between treatments for the different design constructions.

1. Indian Agricultural Research Institute
New Delhi
THREE-ASSOCIATE CLASS PARTIALLY BALANCED
INCOMPLETE BLOCK DESIGNS IN TWO REPLICATES
SUMEET SAURAV
Roll No. : 20389
M.Sc.(Agricultural Statistics)

2. OVERVIEW
 Introduction
 Definition
 Association scheme
&
 Construction of design
 Comparison
 Summary and Conclusions
 References
Three Associate Triangular
PBIB Designs
Three Associate Tetrahedral
PBIB Designs
Three Associate Circular
Lattice PBIB Designs
24 November 2014 2

3. Introduction
 When a large number of treatments are to be tested in an
experiment, incomplete block designs with smaller block size can be
adopted to maintain the homogeneity within blocks.
 In the class of incomplete block designs the balanced incomplete
block (BIB) design, is the simplest one.
 These designs estimate all possible treatment paired comparisons
with same variance and hence are variance balanced.
 But balanced incomplete block designs are not available for every
parametric combination. Also, even if a BIB design exists for a
given number of treatments (v) and block size (k), it may require too
many replications.
24 November 2014 3

4. Introduction…
 To overcome this problem another class of binary, equi-replicate and
proper designs, called partially balanced incomplete block (PBIB)
designs were introduced by Bose and Nair (1939).
 In these designs, the variance of every estimated elementary contrast
among treatment effects is not the same.
 If the experimenter is constrained of resources, PBIB designs with
three-associate classes are an alternative to BIB designs or PBIB
designs with two-associate classes.
24 November 2014 4

5. Definition
Following Bose et al. (1954), an incomplete block design for v treatments is
said to be partially balanced with 3-associate classes, if the experimental
material can be divided into b blocks each of size k (<v) such that
(i) each of the treatments occurs in r blocks,
(ii) there exists an abstract relation between treatments satisfying the
following:
• two treatments are either 1st, 2nd or 3rd associates, the relation of
association being symmetrical.
• each treatment has exactly ni ith associates, and
• given any two treatments that are mutually ith associates, the number
of treatments common to the jth associates of the first and kth
associates of the second is Pi
jk (i,j,k = 1,2,3).
(iii) two treatments that are mutually ith associates occur together in exactly
i blocks.

6. Three Associate Triangular Designs
Kipkemoi et al.(2013) defined
a three-class triangular
association scheme for v=n(n-
2)/2 treatments.
Consider a square array of n
rows and n columns (n is even
and > 4) with both diagonal
entries nij (i = j and i + j = n +
1) in array having no
treatments allocated.
* n12 n 13 n14 n15 *
n21 * n23 n24 * n26
n31 n32 * * n35 n36
n41 n42 * * n45 n46
n51 * n53 n54 * n56
* n62 n63 n64 n65 *
24 November 2014 6

7. Triangular…
The treatment entries are allocated to these positions by following steps:
(i) The initial set of n(n-2)/2 positions are first filled by v treatments on
the upper side of the principle diagonal in a natural order starting
from right to left from the top row.
(ii) The second set of n(n-2)/2 positions are then filled by v treatment
entries from left to right starting from the bottom row.
Thus, the final arrangement has every treatment appears twice in
the array.
24 November 2014 7

8. Association Scheme
Two treatments are said to be
i. First associates, if they both occur in the same row and same column.
ii. Second associates, if they either occur in the same row or the same
column but not both.
iii. Third associates, if they neither occur in the same row nor in the same
column.
The parameters are n 1 =1, n 2 = 2(n-4),
n(n  6) 
12
2
n3










0 0 0
 



0 2(n 4) 0
n(n  10) 
24
2
0 0
1 P









0 1 0
  
1 n 6 n 4



n(n  8) 
20
0 n 
4
2
2 P

0 0 1


 
0 0 2(n 4)

n(n 6) 12
24 November 2014 8








 

2
1 2(n 4)
3 P

9. Example
Treatment 1st
Associates
2nd
Associates
3rd
Associates
1 4 2,3,9,11 5,6,7,8,10,12
4 1 2 , 3, 9, 11 5,6,7,8,10,12
2 3 1, 4 ,6, 7 5,8,9,10,11,12
5 12 6,7,8,10 1,2,3,4,9,11
9
Let n = 6  v = 12




24 November 2014
* 4 3 2 1 *
12 * 7 6 * 5
10 11 * * 9 8
8 9 * * 11 10
5 * 6 7 * 12
* 1 2 3 4 *
 
 
 
 
 
 
 
 
 







0 0 0
0 4 0
0 0 0
1 P











0 1 0
1 0 2
0 2 4
2 P











0 0 1
0 0 4
1 4 6
3 P

10. Taking each row and column to constitute a block, n distinct blocks with
parameters v=n(n-2)/2, b = n, k = n-2, r = 2, λ1 = 2, λ2 = 1, λ3 = 0 is
obtained.
10
Construction of Designs …
24 November 2014
Blocks
I 1,2,3,4
II 5,6,7,12
III 8,9,11,10
IV 5,8,10,12
V 1,9,11,4
VI 2,6,7,3
Example
v = 12, b = 6, k = 4, r = 2, λ1= 2, λ2 = 1, λ3 = 0
* 4 3 2 1 *
12 * 7 6 * 5
10 11 * * 9 8
8 9 * * 11 10
5 * 6 7 * 12
* 1 2 3 4 *
 
 
 
 
 
 
 
 
 

11. Triangular (Superimposed) Association
Scheme
 For n  8, even positive integer, triangular (superimposed) association
scheme is obtained by transposing and then superimposing the array of
triangular association scheme on the original array.
 Parameter are: n1=3, n2= n(n-4),
n(n  10) 
24
2
n3










2 0 0
 



0 4(n 4) 0
n(n  10) 
24
2
0 0
1 P









0 3 0
  



3 2(n 4) 2(n 6)
n(n  14) 
48
0 2(n 
6)
2
2 P



0 0 3
 

0 16 4(n 8)
n(n 18) 80
24 November 2014 11








 

2
3 4(n 8)
3 P

12. Example
* 6 5 4 3 2 1 *
24 * 11 10 9 8 * 7
22 23 * 15 14 * 13 12
19 20 21 * * 18 17 16
16 17 18 * * 21 20 19
12 13 * 14 15 * 23 22
7 * 8 9 10 11 * 24
* 1 2 3 4 6 6 *
 
 
For n=8, v=24
 
 
 
 
 
 
 
 
 
 
By transposing and then
superimposing the array of
triangular association scheme


























5
* (6,24) (5,22) (4,19) (3,16) (2,12) (1,7) *
(24,6) * (11,23) (10,20) (9,17) (8,13) * (7,1)
(22,5) (23,11) * (15,21) (14,18) * (13,8) (12,2)
(19,4) (20,10) (21,15) * * (18,14) (17,9) (16,3)
(16,3) (17,9) (18,14) * * (21,15) (20,10) (19,4)
(12,2) (13,8) * (14,18) (15,21) * (23,11) (22,5)
(7,1) * (8,13) (9,17) (10,20) (11,23) * (24,6)
* (1,7) (2,12) (3,16) (4,19) (5,22) (6,24) *

13. Various associates of treatments
Treatment 1st associates 2nd associates 3rd associates
1 6, 7, 24
2,3,4,5,8,9,10,11,12,13,
16,17,19,20,22,23
14,15,18,21
7 1, 6, 24
2,3,4,5,8,9,10,11,12,13,
16,17,19,20,22,23
14,15,18,21
2 5, 12, 22
1,3,4,6,7,8,11,13,14,15,
16,18,19,21,23,24
9,10,17,20
14 15,18,21
2,3,4,5,8,9,10,11,12,13,
16,17,19,20,22,23
1,6,7,24











2 0 0
0 16 0
0 0 4
1 P











0 3 0
3 8 4
0 4 0
2 P











0 0 3
0 16 0
3 0 0
3 P
24 November 2014 13

14. Construction of Designs
n
b 
, , k=2(n-2), r =2, λ1= 2, λ2= 1, λ3= 0.
Blocks
Parameters of this series of designs are:
I 1,2,3,4,5,6,7,12,16,19,22,24
II 1,6,7,8,9,10,11,13,17,20,23,24
III 2,5,8,11,12,13,14,15,18,21,22,23
IV 3,4,9,10,14,15,16,17,18,19,20,21
n(n 
2)
2
v

2
Example
v = 24, b = 4, k
= 12, r = 2, λ1=
2, λ2 = 1, λ3 = 0
24 November 2014 14

15. Three Associate Tetrahedral PBIB Designs
 Sharma et al. (2009) defined tetrahedral association scheme for
number of treatments be v = 6n (n ≥ 2).
 A tetrahedron has four triangular faces and six edges, arrange these
treatments on the edges of a tetrahedron such that each edge contains
exactly n distinct treatments.
Association Scheme
Treatment  is the,
• first associate of , if  lies on the same edge of ;
• the second associate, if  lies on any of the edges that pass through the
two vertices located on the edge of ; and
• third associate, otherwise
24 November 2014 15

16. Parameters
The parameters of the association scheme are:
v = 6n, n1 = n-1, n2 = 4n, n3 = n,





 





n 2 0 0
0 4n 0
0 0 n
1 P





0 n 1 0








n 1 2n n
0 n 0
2 P
0 0 n 1










0 4n 0
n 
1 0 0


3 P
24 November 2014 16

17. Example
Let v = 24 (= 6×4). An
arrangement of these
treatments on the six
edges of a tetrahedron
such that each edge
contains 4 distinct
treatments
24 November 2014 17

18. Example…
Treatment 1st Associates 2nd Associates 3rd Associates
1 2, 3, 4 5,6,7,8,9,10,11,12,13,14,
15,16,17,18,19,20
21,22,23,24
2 1, 3, 4 5,6,7,8,9,10,11,12,13,14,
15,16,17,18,19,20
21,22,23,24
5 6, 7, 8 1,2,3,4,9,10,11,12,13,14,
15,16,21,22,23,24
17,18,19,20
21 22,23,24 5,6,7,8,9,10,11,12,13,14,
15,16,17,18,19,20
1, 2, 3, 4











2 0 0
0 16 0
0 0 4
P1











0 3 0
3 8 4
0 4 0
2 P











0 0 3
0 16 0
3 0 0
3 P
,
,
24 November 2014 18

19. Construction of Designs
Four blocks of the designs are obtained, each one corresponding to a
triangular face, by taking together the treatments that lie on the three
edges of the face as the block contents.
The parameters of the design are: v = 6n, b = 4, r = 2, k = 3n, 1 = 2, 2
= 1 and 3 = 0.
Blocks
I ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
II (1, 2, 3, 4, 13, 14, 15, 16, 17, 18, 19, 20)
III (5, 6, 7, 8, 13, 14, 15, 16, 21, 22, 23, 24)
IV (9, 10, 11, 12, 17, 18, 19, 20, 21, 22, 23, 24)
Example
v = 24 (= 6×4),
b = 4, r = 2,
k = 12, 1 = 2,
2 = 1, 3 = 0
24 November 2014 19

20. Particular Case
For n = 1, this scheme reduces to a two class Group Divisible (GD)
association scheme with parameter v = 6, b = 4, r = 2, k = 3, 1 = 1, 2 = 0,
n1 = 4, n2 = 1 which is not reported in Clatworthy (1973).
 Association schemes for treatments 1, 2 and 5 are:
Treatment 1st associates 2nd associates
1 2,6,3,4 5
2 1,5,3,4 6
5 2,6,3,4 1
 Blocks of the design obtained are:
Blocks
I 1, 2, 3
II 1, 4, 6
III 2, 4, 5
IV 3, 5, 6
24 November 2014 20

21. Circular Lattice PBIB(3) Designs
 These designs were introduced by Rao (1956).
 Consider n concentric circles and n diameters, giving rise to 2n2
lattice points on the circles.
Association Scheme
Corresponding to any treatment, the first associate is that treatment
which is on the same circle and same diameter, second associates are
those which are either on the same circle or on the same diameter, and
the rest are third associates.
Parameters are n1=1, n2=4(n-1), n3=2(n-1)2,
 0 0 0   0 1 0   0 0 1

     
     
           
= 0 4(n-1) 0 , = 1 2(n-2) 2(n-1) , = 0 4 4(n-2) .
P P P
1 2 3
2 2
   
0 0 2(n-1) 0 2(n-1) 2(n-1) n-2 1 4(n-2) 2 n-2
24 November 2014 21

22. Example
For n=3, v=18; we have 3
concentric circles and 3
diameters such that each
point contains one treatment
The associates of treatment 1 are:
1st Associate 2nd Associates 3rd Associates
4 2, 3, 5, 6, 7, 10, 13, 16 8, 9, 11, 12, 14, 15, 17, 18
24 November 2014 22

23. Construction of Designs
Identifying the points as treatments lies on the circles and diameters
as blocks, one gets a series of PBIB(3) with parameters v=2n2, b=2n,
r=2, k=2n 1 = 2, 2 = 1 and 3 = 0.
Replication Blocks Treatments
I
1 (1, 2, 3, 4, 5, 6)
2 (7, 8, 9, 10, 11, 12)
3 (13, 14, 15, 16, 17, 18)
II
4 (1, 4, 7, 10, 13, 16)
5 (2, 5, 8, 11, 14, 17)
6 (3, 6, 9, 12, 15, 18)
Example
v=18, b=6, r=2 , k=6
1 = 2, 2 = 1, 3 = 0.
24 November 2014 23

24. Generalized Circular Lattice Designs
 Generalized circular lattice designs were introduced by (Varghese and
Sharma, 2004) which covers more number of treatments.
 Let the number of treatments be v = 2sn2, n ≥2.
 Draw n concentric circles and n diameters.
Association Scheme
 The parameters of the association scheme are:
v=2sn2, n1=2s-1, n2=4s(n-1), n3=2s(n-1)2, n≥2.
2s-1 0 0 0 2s-1 0 0 0 2s-1
     
     
     
     
= 0 4s(n-1) 0 , = 2s-1 2s(n-2) 2s(n-1) and = 0 4s 4s(n-2) .
P P P
1 2 3
2 2
0 0 2s(n-1) 0 2s(n-1) 2s(n-1)(n-2) 2s-1 4s(n-2) 2s(n-2)
24 November 2014 24

25. Example
Let v = 36 (=2×2×32).
Arrange 36 treatments on
the 18 intersecting points
of 3 concentric circles
and 3 diameters such that
each point contains two
treatments.
24 November 2014 25

26. Example…
Treatments 1stAssociates 2nd Associates 3rd Associates
1 2, 7, 8
3,4,5,6,9,10,11,12,13,1
4,19,20,25,26,31,32
15,16,17,18,21,22,23,24,2
7,28,29,30,33,34,35,36
2 1, 7, 8
3,4,5,6,9,10,11,12,13,1
4,19,20,25,26,31,32
15,16,17,18,21,22,23,24,2
7,28,29,30,33,34,35,36
3 4, 9, 10
1,2,5,6,7,8,11,12,15,16
,21,22,27,28,33,34
13,14,17,18,19,20,23,24,2
6,29,30,31,32,35,,36
15 16, 21, 22
3,4,9,10,13,14,17,18,1
9,20,23,24,27,28,33,34
1,2,5,6,7,8,11,12,25,26,29
,30,31,32,35,36,




3 0 0
0 16 0
0 0 16







1 P











0 3 0
3 4 8
0 8 8
2 P











0 0 3
0 8 8
3 8 4
3 P
24 November 2014 26

27. Construction of Designs
The design with v = 36, b = 6, r = 2, k = 12, 1 = 2, 2 = 1, 3 = 0 is:
Replications Blocks Treatments
I
1 (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
2 (13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24)
3 (25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36)
II
4 (1, 2, 7, 8, 13, 14, 19, 20, 25, 26, 31, 32)
5 (3, 4, 9, 10, 15, 16, 21, 22, 27, 28, 33, 34)
6 (5, 6, 11, 12, 17, 18, 23, 24, 29, 30, 35, 36)
24 November 2014 27

28. Comparison of Designs
All the four classes of designs exist for v = 24 and r = 2.
Type of PBIB(3)
Design
v b r k nnnVVVE
1 2 3 1 2 3 1 2 3 V
Triangular 24 8 2 6 2 1 0 1 8 14 1 1.1875 1.5000 1.2119 0.8251
Triangular
(Superimposed)
24 4 2 12 2 1 0 3 16 4 1 1.0625 1.1250 1.0652 0.9387
Tetrahedral 24 4 2 12 2 1 0 3 16 4 1 1.0625 1.1250 1.0652 0.9387
Circular Lattice 24 4 2 12 2 1 0 5 12 6 1 1.0833 1.1666 1.0869 0.9200
24 November 2014 28

29. Summary and Conclusions
 PBIB designs with three-associate classes in less number of
replications can be used advantageously when there is a constraint
of resources.
 Four classes of three-associate class association schemes viz., two
classes of triangular designs, tetrahedral and circular lattice and
general methods of construction of PBIB(3) designs based on
these association schemes were discussed here.
 The first two series of designs are for the same treatment structure
v = n(n-2)/2, but the number of blocks and block size varies as per
the association scheme whereas last two series are for v = 6n and v
= 2sn2.
24 November 2014 29

30. Summary and Conclusions
 A comparison among these designs for same number of treatments (v
= 24) showed that PBIB(3) designs based on tetrahedral association
scheme and triangular (superimposed) association scheme have the
maximum efficiency among these four classes of designs
 Designs based on circular lattice association scheme are resolvable
and hence its replications can be used over space or time.
24 November 2014 30

31. References
Bose, R.C, and Nair, K.R. (1939). Partially balanced incomplete block
designs, Sankhya, 4, 337-372.
Bose, R.C, Clatworthy, W.H. and Shrikhande, S.S.(1954). Tables of
partially balanced designs with two associate classes. North Carolina
Agricultural Experiment Station Technical Bulletin No. 107. Raleigh.
N.C.
Clatworthy, W.H. (1973). Tables of two-associate partially balanced
designs. National Bureau of Standards, Applied Maths. Series No.63,
Washington D.C.
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33. THANK YOU
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