Balanced Incomplete Block Design (BIBD), Analysis of Variance and Design of Experiments-II - Computer Science Engineering (CSE) PDF Download

 Page 1


MODULE - II
LECTURE - 9
BALANCED INCOMPLETE BLOCK 
DESIGN (BIBD)
Dr. Shalabh
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur
Analysis of Variance and 
Design of Experiments-II
Page 2


MODULE - II
LECTURE - 9
BALANCED INCOMPLETE BLOCK 
DESIGN (BIBD)
Dr. Shalabh
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur
Analysis of Variance and 
Design of Experiments-II
The designs like CRD and RBD are the complete block designs. We now discuss the balanced incomplete block design
(BIBD) and the partially balanced incomplete block design (PBIBD) which are the incomplete block designs.
2
A balanced incomplete block design (BIBD) is an incomplete block design in which
? b blocks have the same number k of plots each and
? every treatment is replicated r times in the design.
? Each treatment occurs at most once in a block, i.e., or 1  where is the number of times the j
th
treatment 
occurs in i
th
block,  i = 1, 2,…, b;  j = 1, 2,…, v.
? Every pair of treatments occurs together is of the b blocks . 
0
ij
n =
ij
n
?
Such design is denoted by 5 parameters                      .
The parameters                         are not chosen arbitrarily.
They satisfy the following relations:
(, , , ; ) Db k v r ?
, , , and bk v r ?
()
( ) ( 1) ( 1)
( ) ( ).
   
  
 andhence
I bk vr
II v r k
III b v r k
?
=
-= -
= >
,
,
ij
j
ij
j
nk i
nr j
=
=
?
?
Hence for all
for all
     
    
and   for all   Obviously         cannot be a constant for all j.  So the design is not 
orthogonal.
'
1' 2 '
...
jj
j ij j ij b b
nn n n n n ? + ++ =
' 1,2,..., . jj v ?=
ij
n
r
Page 3


MODULE - II
LECTURE - 9
BALANCED INCOMPLETE BLOCK 
DESIGN (BIBD)
Dr. Shalabh
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur
Analysis of Variance and 
Design of Experiments-II
The designs like CRD and RBD are the complete block designs. We now discuss the balanced incomplete block design
(BIBD) and the partially balanced incomplete block design (PBIBD) which are the incomplete block designs.
2
A balanced incomplete block design (BIBD) is an incomplete block design in which
? b blocks have the same number k of plots each and
? every treatment is replicated r times in the design.
? Each treatment occurs at most once in a block, i.e., or 1  where is the number of times the j
th
treatment 
occurs in i
th
block,  i = 1, 2,…, b;  j = 1, 2,…, v.
? Every pair of treatments occurs together is of the b blocks . 
0
ij
n =
ij
n
?
Such design is denoted by 5 parameters                      .
The parameters                         are not chosen arbitrarily.
They satisfy the following relations:
(, , , ; ) Db k v r ?
, , , and bk v r ?
()
( ) ( 1) ( 1)
( ) ( ).
   
  
 andhence
I bk vr
II v r k
III b v r k
?
=
-= -
= >
,
,
ij
j
ij
j
nk i
nr j
=
=
?
?
Hence for all
for all
     
    
and   for all   Obviously         cannot be a constant for all j.  So the design is not 
orthogonal.
'
1' 2 '
...
jj
j ij j ij b b
nn n n n n ? + ++ =
' 1,2,..., . jj v ?=
ij
n
r
Example of BIBD
3
1 10 1 6
( , ; , ; ) : 10 ( , ,..., ), 6 ( , ,..., ), 3, 5, 2 Db k v r b B B v T T k r ? ? = = = = = In the design consider say say       
1 12 3
2 12 4
3 13 5
4 14 6
5 1 56
6 2 36
7 2 45
8 256
9 345
10 3 4 6
Blocks  Treatments
B
B
B
B
B
B
B
B
B
B
Now we see how the conditions of BIBD are satisfied.
( ) 10 3 30 6 5 30
( ) ( 1) 2 5 10 ( 1) 5 2 10
( 1) ( 1)
( ) 10 6
i bk vr
bk vr
ii v r k
v rk
iii b
?
?
= ×= = × =
? =
-= × = -= × =
? -= -
= =
and
and
    
  
   
  
 
Page 4


MODULE - II
LECTURE - 9
BALANCED INCOMPLETE BLOCK 
DESIGN (BIBD)
Dr. Shalabh
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur
Analysis of Variance and 
Design of Experiments-II
The designs like CRD and RBD are the complete block designs. We now discuss the balanced incomplete block design
(BIBD) and the partially balanced incomplete block design (PBIBD) which are the incomplete block designs.
2
A balanced incomplete block design (BIBD) is an incomplete block design in which
? b blocks have the same number k of plots each and
? every treatment is replicated r times in the design.
? Each treatment occurs at most once in a block, i.e., or 1  where is the number of times the j
th
treatment 
occurs in i
th
block,  i = 1, 2,…, b;  j = 1, 2,…, v.
? Every pair of treatments occurs together is of the b blocks . 
0
ij
n =
ij
n
?
Such design is denoted by 5 parameters                      .
The parameters                         are not chosen arbitrarily.
They satisfy the following relations:
(, , , ; ) Db k v r ?
, , , and bk v r ?
()
( ) ( 1) ( 1)
( ) ( ).
   
  
 andhence
I bk vr
II v r k
III b v r k
?
=
-= -
= >
,
,
ij
j
ij
j
nk i
nr j
=
=
?
?
Hence for all
for all
     
    
and   for all   Obviously         cannot be a constant for all j.  So the design is not 
orthogonal.
'
1' 2 '
...
jj
j ij j ij b b
nn n n n n ? + ++ =
' 1,2,..., . jj v ?=
ij
n
r
Example of BIBD
3
1 10 1 6
( , ; , ; ) : 10 ( , ,..., ), 6 ( , ,..., ), 3, 5, 2 Db k v r b B B v T T k r ? ? = = = = = In the design consider say say       
1 12 3
2 12 4
3 13 5
4 14 6
5 1 56
6 2 36
7 2 45
8 256
9 345
10 3 4 6
Blocks  Treatments
B
B
B
B
B
B
B
B
B
B
Now we see how the conditions of BIBD are satisfied.
( ) 10 3 30 6 5 30
( ) ( 1) 2 5 10 ( 1) 5 2 10
( 1) ( 1)
( ) 10 6
i bk vr
bk vr
ii v r k
v rk
iii b
?
?
= ×= = × =
? =
-= × = -= × =
? -= -
= =
and
and
    
  
   
  
 
Even if the parameters satisfy the relations, it is not always possible to arrange the treatments in blocks to get the
corresponding design.
The necessary and sufficient conditions to be satisfied by the parameters for the existence of a BIBD are not known.
The conditions (I)-(III) are some necessary condition only. The construction of such design depends on the actual
arrangement of the treatments into blocks and this problem is addressed in combinatorial mathematics. Tables are available
giving all the designs involving at most 20 replications and their methods of construction.
4
Page 5


MODULE - II
LECTURE - 9
BALANCED INCOMPLETE BLOCK 
DESIGN (BIBD)
Dr. Shalabh
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur
Analysis of Variance and 
Design of Experiments-II
The designs like CRD and RBD are the complete block designs. We now discuss the balanced incomplete block design
(BIBD) and the partially balanced incomplete block design (PBIBD) which are the incomplete block designs.
2
A balanced incomplete block design (BIBD) is an incomplete block design in which
? b blocks have the same number k of plots each and
? every treatment is replicated r times in the design.
? Each treatment occurs at most once in a block, i.e., or 1  where is the number of times the j
th
treatment 
occurs in i
th
block,  i = 1, 2,…, b;  j = 1, 2,…, v.
? Every pair of treatments occurs together is of the b blocks . 
0
ij
n =
ij
n
?
Such design is denoted by 5 parameters                      .
The parameters                         are not chosen arbitrarily.
They satisfy the following relations:
(, , , ; ) Db k v r ?
, , , and bk v r ?
()
( ) ( 1) ( 1)
( ) ( ).
   
  
 andhence
I bk vr
II v r k
III b v r k
?
=
-= -
= >
,
,
ij
j
ij
j
nk i
nr j
=
=
?
?
Hence for all
for all
     
    
and   for all   Obviously         cannot be a constant for all j.  So the design is not 
orthogonal.
'
1' 2 '
...
jj
j ij j ij b b
nn n n n n ? + ++ =
' 1,2,..., . jj v ?=
ij
n
r
Example of BIBD
3
1 10 1 6
( , ; , ; ) : 10 ( , ,..., ), 6 ( , ,..., ), 3, 5, 2 Db k v r b B B v T T k r ? ? = = = = = In the design consider say say       
1 12 3
2 12 4
3 13 5
4 14 6
5 1 56
6 2 36
7 2 45
8 256
9 345
10 3 4 6
Blocks  Treatments
B
B
B
B
B
B
B
B
B
B
Now we see how the conditions of BIBD are satisfied.
( ) 10 3 30 6 5 30
( ) ( 1) 2 5 10 ( 1) 5 2 10
( 1) ( 1)
( ) 10 6
i bk vr
bk vr
ii v r k
v rk
iii b
?
?
= ×= = × =
? =
-= × = -= × =
? -= -
= =
and
and
    
  
   
  
 
Even if the parameters satisfy the relations, it is not always possible to arrange the treatments in blocks to get the
corresponding design.
The necessary and sufficient conditions to be satisfied by the parameters for the existence of a BIBD are not known.
The conditions (I)-(III) are some necessary condition only. The construction of such design depends on the actual
arrangement of the treatments into blocks and this problem is addressed in combinatorial mathematics. Tables are available
giving all the designs involving at most 20 replications and their methods of construction.
4 5
()
( ) ( 1) ( 1)
() .
I bk vr
II v r k
III b v
?
=
-= -
=
Proof: (I) 
Let incidence matrix.
Observing that the quantities  are the scalars and the transpose of each other, we find their values. 
Consider
( ):
ij
N n bv = ×
Theorem
11 1 1
'
bv v b
E NE E N E and
11 21 1
12 22 2
11
12
1
2
1
1
1
(1,1,...,1)
  
1
(1,1,...,1)
(1,1,...,1)
 .
b
b
bv
v v bv
j
j
j
j
bj
j
b
nn n
nn n
E NE
nn n
n
n
n
k
k
k
bk
×
? ?? ?
? ?? ?
? ?? ?
=
? ?? ?
? ?? ?
? ? ? ?
? ?
? ?
? ?
? ?
=
? ?
? ?
? ?
? ?
? ?
??
??
??
=
??
??
??
=
?
?
?
?
?
? ? ? ??
?
?
?