THREE-ASSOCIATE CLASS PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS IN TWO REPLICATES BY SUMEET...
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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.
24 November 2014 28
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.
24 November 2014 32
24 November 2014 33
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