Balance Incomplete Block Design (BIBD) Lecture - 40

Balance Incomplete Block Design 

(BIBD) 

Lecture - 40 

An incomplete block design is said to be incomplete block design (BIBD), if 

i.          Each treatment is replicated at the same number of times in the (entire) design, (say rij = r ).

rt = bk = n 

r = bk / t = n/t 

ii.         Each pair of treatments must appear together in the number of blocks (say λij = λ).

Thus in BIBD, we have t treatments distributed over b blocks, each of size k (k < t) such that each treatment occurs in r block not more than once in a block and each pair of treatments occurs together in  blocks.

Thus the symbols t, b, r, k, and  are the parameters of the balance incomplete block design and can be expressed as:

BIBD (t, b, r , k, λ)

If t = b the design is said to be symmetric design.

 

Properties of BIBD

i.          The total number of observation = rt = bk = n

ii.         r( k -1 ) = λ ( t - 1 )

iii.  b => t

A BIBD exist: A necessary but not sufficient condition is

     

Example: Whether it is BIBD (4, 3, 2, 2, 1)

      Solution: Here t = 4, b = 3, r = 2, k = 2, and

Check condition – 1:

r×t = b×k

2×4 ≠ 3×2

Condition – 1 is not satisfied. It is not a BIBD.

Example: BIBD (4, 4, 3, 3, 2)

Blocks	Treatments
	T1	T2	T3	T4
I	*	*	*
II	*	*		*
III	*		*	*
IV		*	*	*

alternatively

Blocks	Treatments
I	A	B	C
II	A	B		D
III	A		C	D
IV		B	C	D

Example: BIBD (3, 3, 2, 2, 1)

Blocks	Treatments
I	A	B
II	B	C
III	A	C

Example: BIBD (4, 6, 3, 2, 1)

Blocks	A	B	C	D
I	*	*
II	*		*
III		*	*
IV		*		*
V			*	*
VI	*			*

Alternatively

Blocks	Treatments
I	A	B
II	A	C
III	B	C
IV	B	D
V	C	D
VI	A	D

Types of BIBD

Symmetrical BIBD

A BIBD is symmetrical if the number of blocks is equal to the number of treatments that’s t = b.

We know that

r x t = b x k

as b = t

r x t = t x k

r = k

Properties

i.                    Each block contains r = k treatments.

ii.                  Each treatment occurs in r = k blocks.

iii.        Each pair of treatment occurs in  λ blocks.

iv.        Each pair of blocks intersects on  λ treatments.

 Resolvable Design

A block design of b blocks in which each of t treatments is a replicated r time. OR

A BIBD is said to be resolvable design, if b / r is an integer.

Properties

A BIBD (t, b, r, k, λ) will be resolvable BIBD

i.  b ≥ t + r -1

ii.  t = Rank (N Nt ) = Rank (N) ≤ b - (r-1)

or 

Rank (N) ≤ b

Example: A BIBD (t = 4, b = 9, r = 3, k = 2,  λ= 1 ) is resolvable design?

Solution: A BIBD will be resolvable if

 b ≥ t + r -1

 b ≥ 4 + 3 -1

 b ≥ 6

The given BIBD is a resolvable design.

Affine Resolvable Design

A resolvable design is said to affine resolvable design if b = t + r -1 and any two blocks form different set have k^2 / t  treatments common. 

where:

Properties:

i. b/ r is an integer.

ii.  b = t + r -1

iii. k^2 / t is an integer.

Example: A BIBD (t = 4, b = 6, r = 3, k = 2, λ = 1) is resolvable design?

Solution:

Condition – 1: k^2 / t = 4/4 = 1 is an integer

Condition – 2: b = t + r - 1

b = 4 + 3 - 1

b = 6

The given BIBD is affine resolvable design.

• Read More: Statistical & Analysis of BIBD