Constructions Methods of BIBD Lecture - 42

 

Constructions Methods 

of

 BIBD

 Lecture - 42

A balance incomplete block design is used when there is a lack of resources or the experimental material is not available in abundance and expressed as:

BIBD (t, b, r , k, λ) 

A BIBD consist on five parameters that’s number of treatment (t), number of blocks (b), number of replications ( r ), number of treatments in the design (k), and number of times each pair of treatments in the same block (λ).

1.      A BIBD is always exist, when k = t – 1

Let we have four treatments (A, B, C, D) and four blocks.

That’s t = 4, b = 4

we know that 

k = t - 1

k = 4 - 1

k = 3

rt = bk

4r = 4 x 3

4r =12

r = 3

	A	B	C	D
Block 1	Yes	Yes	No	Yes
Block 2	No	Yes	Yes	Yes
Block 3	Yes	No	Yes	Yes
Block 4	Yes	Yes	Yes	No

Treatment combination in each block:

Block 1	Block 2	Block 3	Block 4
AB	BC	AC	AB
AD	BD	AD	AC
BD	CD	CD	BC

2.      Un Reduced BIBD

A BIBD can be also made up by using all possible pairs of treatments that’s tC2 and it requires a large number of blocks that’s 

In this case total number of observation is

Example: Let we have five treatments (A, B, C, D, E), K = 2 and  λ = 1.

	Blocks
	1	2	3	4	5	6	7	8	9	10
A	*	*	*	*
B	*				*	*	*
C		*			*			*	*
D			*			*		*		*
E				*			*		*	*

Or it can be expressed as:

	Blocks
	1	2	3	4	5	6	7	8	9	10
A	A	A	A	A
B	B				B	B	B
C		C			C			C	C
D			D			D		D		D
E				E			E		E	E

Or it can be expressed as:

Blocks
1	2	3	4	5	6	7	8	9	10
A	A	A	A	B	B	B	C	C	D
B	C	D	E	C	D	E	D	E	E

 

3.      Complementary of BIBD

A BIBD can be obtained by the complementary of original BIBD. A commentary BIBD contains all those blocks and treatments that are not available in original BIBD.

If t, b, r, k and  λ are the parameters of a BIBD, then t*, b*, r*, k* and λ* are the parameters of the commentary BIBD.

Where:

t* = t  and  b* = b

r* = b - r

 k* = t - k

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

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

Or it can be expressed as:

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

Complementary of the BIBD (4, 6, 3, 2, 1)  (  for complementary treatments)

Or it can be expressed as:

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

Example: A BIBD (5, 10, 4, 2, 1) is given below:

Blocks
1	2	3	4	5	6	7	8	9	10
A	A	A	A	B	B	B	C	C	D
B	C	D	E	C	D	E	D	E	E

Find complementary BIBD.

Solution:

 t* = t = 5,  b* = b = 10

 r* = b* -  r

r* = 10 - 4

r* = 6

, k* = t - k 

 k* = 5 - 2

 k* = 3

 

BIBD
1	2	3	4	5	6	7	8	9	10
A	A	A	A	B	B	B	C	C	D
B	C	D	E	C	D	E	D	E	E
Complementary BIBD
C	B	B	B	A	A	A	A	A	A
D	D	C	C	D	C	C	B	B	B
E	E	E	D	E	E	D	E	D	C

4.      Resolvable Design

An incomplete-block design is resolved if the blocks are grouped into larger blocks and the large blocks form a complete-block design.

For resolvable design b ≥ t + r - 1 and k^2 / t is an integer. 

A lattice design is a special sort of resolved incomplete-block design in which  k^2 = t. The construction method is as follows.

a.       Write the treatments in a k ×k square array.

b.      For the first large block, the rows of the square are the blocks.

c.       For the second large block, the columns of the square are the blocks.

d.      For the third large block (if any), write down a k ×k Latin square and use its letters as the blocks.

e.       For the fourth large block (if any), write down a k×k Latin square orthogonal to the first one and use its letters as the blocks.

f.        And so on, using r −2 mutually orthogonal Latin squares.

Example: A BIBD (9, 12, 3, 3, 1)

t = 9, b = 12, r = 3, k = 3 and λ = 1

further 

k^2 = t

Let the treatments are 1, 2, 3, …, 9.

Using Graeco Latin square and assign treatments as:

The complementary of the above shall be also a BIBD.

Experiment

A company needs to compare the comfort score of 9 pillows. They select 12 customers to evaluate the pillows. Customers only have patience to test at most 3 pillows. The data generated by the company is given blow:

Customers	Pillows
	A	B	C	D	E	F	G	H	I
1	59	26	38
2				85	92	69
3							74	52	27
4	63			70			68
5		26			98			59
6			31			60			35
7	62				85				30
8		23				73	75
9			49	74				51
10	52					76		43
11		18		79					41
12			42		84		81

i.        What is t, b, r and k

ii.      Is it a BIBD?

    iii.   Test the hypothesis about treatments effects at 5 %.

iv.     Develop a complementary design of the above.

v..                Test the hypothesis based on complementary design at 5 %.

• Read More; PBIBD