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)

OVERVIEW

Introduction

Definition

Association scheme

&

Construction of design

Comparison

Summary and Conclusions

References

224 November 2014

Three Associate Triangular

PBIB Designs

Three Associate Tetrahedral

PBIB Designs

Three Associate Circular

Lattice PBIB Designs

Introduction

324 November 2014

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.

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

DefinitionFollowing 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 Pijk (i,j,k = 1,2,3).

(iii) two treatments that are mutually ith associates occur together in exactly

i blocks.

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.

624 November 2014

* 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 *

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.

724 November 2014

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),

824 November 2014

2

126)n(nn3

2

2410)n(n00

04)2(n0

000

1P

2

208)n(n4n0

4n6n1

010

2P

2

126)n(n4)2(n1

4)2(n00

100

3P

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 *

000

040

000

1P

420

201

010

2P

641

400

100

3P

10

Construction of Designs …

24 November 2014

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.

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 *

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),

24 November 2014 11

2

2410)n(nn3

2

2410)n(n00

04)4(n0

002

1P

2

4814)n(n6)2(n0

6)2(n4)2(n3

030

2P

2

8018)n(n8)4(n3

8)4(n160

300

3P

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

*)24,6()22,5()19,4()16,3()12,2()7,1(*

)6,24(*)23,11()20,10()17,9()13,8(*)1,7(

)5,22()11,23(*)21,15()18,14(*)8,13()2,12(

)4,19()10,20()15,21(**)14,18()9,17()3,16(

)3,16()9,17()14,18(**)15,21()10,20()4,19(

)2,12()8,13(*)18,14()21,15(*)11,23()5,22(

)1,7(*)13,8()17,9()20,10()23,11(*)6,24(

*)7,1()12,2()16,3()19,4()22,5()24,6(*

By transposing and thensuperimposing the array oftriangular association scheme

5

Various associates of treatments

24 November 2014 13

Treatment 1st associates 2nd associates 3rd associates

1 6, 7, 242,3,4,5,8,9,10,11,12,13,

16,17,19,20,22,2314,15,18,21

7 1, 6, 242,3,4,5,8,9,10,11,12,13,

16,17,19,20,22,2314,15,18,21

2 5, 12, 221,3,4,6,7,8,11,13,14,15,

16,18,19,21,23,249,10,17,20

14 15,18,212,3,4,5,8,9,10,11,12,13,

16,17,19,20,22,231,6,7,24

400

0160

002

1P

040

483

030

2P

003

0160

300

3P

Construction of Designs

Blocks

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

24 November 2014 14

Parameters of this series of designs are:

, , k=2(n-2), r =2, λ1= 2, λ2= 1, λ3= 0. 2

2)n(nv

2

nb

Example

v = 24, b = 4, k

= 12, r = 2, λ1=

2, λ2 = 1, λ3 = 0

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.

24 November 2014 15

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

Association Scheme

Parameters

The parameters of the association scheme are:

v = 6n, n1 = n-1, n2 = 4n, n3 = n,

24 November 2014 16

n00

04n0

002n

1P

0n0

n2n1n

01n0

2P

001n

04n0

1n00

3P

Example

24 November 2014 17

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

Example…

24 November 2014 18

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

400

0160

002

1P

040

483

030

2P

003

0160

300

3P

,

,

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.

24 November 2014 19

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

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:

Blocks of the design obtained are:

24 November 2014 20

Treatment 1st associates 2nd associates

1 2,6,3,4 5

2 1,5,3,4 6

5 2,6,3,4 1

Blocks

I 1, 2, 3

II 1, 4, 6

III 2, 4, 5

IV 3, 5, 6

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.

24 November 2014 21

1 2 3

2 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) .

0 0 2(n-1) 0 2(n-1) 2(n-1) n-2 1 4(n-2) 2 n-2

P P P

Parameters are n1=1, n2=4(n-1), n3=2(n-1)2,

Example

For n=3, v=18; we have 3

concentric circles and 3

diameters such that each

point contains one treatment

24 November 2014 22

1st Associate 2nd Associates 3rd Associates

4 2, 3, 5, 6, 7, 10, 13, 16 8, 9, 11, 12, 14, 15, 17, 18

The associates of treatment 1 are:

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.

24 November 2014 23

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.

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.

24 November 2014 24

1 2 3

2 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) .

0 0 2s(n-1) 0 2s(n-1) 2s(n-1)(n-2) 2s-1 4s(n-2) 2s(n-2)

P P P

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

Example…

Treatments 1stAssociates 2nd Associates 3rd Associates

1 2, 7, 83,4,5,6,9,10,11,12,13,1

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, 83,4,5,6,9,10,11,12,13,1

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, 101,2,5,6,7,8,11,12,15,16

,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, 223,4,9,10,13,14,17,18,1

9,20,23,24,27,28,33,34

1,2,5,6,7,8,11,12,25,26,29

,30,31,32,35,36,

24 November 2014 26

1600

0160

003

1P

880

843

030

2P

483

880

300

3P

Construction of Designs

The design with v = 36, b = 6, r = 2, k = 12, 1 = 2, 2 = 1, 3 = 0 is:

24 November 2014 27

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)

Comparison of Designs

All the four classes of designs exist for v = 24 and r = 2.

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Type of PBIB(3)

Designv b r k 1 2 3 n1 n2 n3 V1 V2 V3 E

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

V

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

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

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.

Das, M.N. (1960). Circular designs, Journal of Indian Society

Agricultural Statistics, 12, 45-56.

Dey, A. (1986). Theory of Block Designs, Wile Eastern Limited, New

Delhi, 41-53.

24 November 2014 31

References

Kipkemoi, E.C., Koske, J.K. and Mutiso, J.M. (2013). Construction of

three-associate class partially balanced incomplete block designs in

two replicates, American Journal of Mathematical Science and

Applications, 1(1), 61-65.

Rao, C.R. (1956). A general class of quasifactorial and related designs,

Sankhya 17, 165-174.

Saha, G.M., Kulshrestha, A.C. and Dey, A. (1973). On a new type of m-

class cyclic association Scheme and designs based on the scheme,

Annals of Statistics, 1, 985-990.

Sharma, V.K., Varghese, C. and Jaggi, S. (2010). Tetrahedral and cubical

association schemes with related PBIB(3) designs, Model Assisted

Statistics and Applications, 5(2), 93-99.

Varghese, C. and Sharma, V.K. (2004). A series of resolvable PBIB(3)

designs with two replicates, Metrika, 60, 251-254.

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THANK YOU