Partially Balance Incomplete Block Design with Two Associate Classes Lecture - 44

 Partially Balance Incomplete Block Design 

with 

Two Associate Classes

Lecture - 44

Definition: A partially balanced incomplete block design with two associate classes is an arrangement of t treatments in b blocks, such that:

1.      Each of the t Treatments occurs r times in the arrangement, which consists of b blocks each of which contains k experimental units. No treatment appears more than once in any block.

2.      Every pair among the t treatments occurs together in either λ1 or λ2 blocks (and are said to be ith associates, if they occur together λi in blocks, i=l , 2).

3.      There exists a relationship of association between every pair of the t treatments satisfying the following conditions:

i. Any two treatments are either first or second associates.

ii. Each treatment has n1  first and n2 second associates.

iii. Given any two treatments that are ith associates, the number of treatments common to the jth associates of the first and the kth associates of the second is ρi jk  and this number is independent of the pair of treatments with which we start. Furthermore,

A partially balanced incomplete block design with two associate classes consists 8 parameters (t, b, r, k, ,λ1, λ2, n1, n2 ) of the first kind and 

are the parameters of the second kind, satisfying the following relationships.


The parameters of the second kind can be arrange as elements of two symmetric matrices.

Example: Find the parameters of the PBIBD is given below:

Block

Treatment

I

*

A

B

C

D

II

A

*

E

F

G

III

B

E

*

H

I

IV

C

F

H

*

J

V

D

G

I

J

*

Consider treatment A:

First Associate: The treatments appear in the same block with treatment A

Second Associate: The treatment not appear in the same block with treatment A.

 In block – I: B, C, D

In block – II: E, F, G

The treatments B, C, D, E, F, G are the first associates.

The other treatments (H, I, J) that not appear in the same block are the second associates.

The first and second associates of all treatments are given below:

Treatment

First Associates

Second Associates

A

B, C, D, E, F, G

H, I, J

B

A, C, D, E, H, I

F, G, J

C

A, B, D, F, H, J

E, G, I

D

A, B, C, G, I, J

E, F, H

E

A, F, G, B, H, I

C, D, J

F

A, E, G, C, H, J

B, D, I

G

A, E, F, D, I, J

B, C, H

H

B, E, I, C, F, J

A, D, G

I

B, E, H, D, G, J

A, C, F

J

C, F, H, D, G, I

A, B, E
................................
..........................................
.............................................
..............................

Treatment A

Treatment B

First associates

Second associates

First associates

3

2

Second associates

2

1



The treatment A and treatment H are the second associate of each other. 
..........................
..........................

.......................................

Treatment A

Treatment H

First associates

Second associates

First associates

4

2

Second associates

2

0

Statistical Model

Let Yij is the response of t treatments in b blocks in a PBIBD with two associates is given by:


Assumptions of PIBD model

Analysis

Consider a PBIBD under two associate schemes. The parameters of the PBIBD under two associates are



ANOVA Table


Difference between BIBD & PBIBD

* In BIBD, the pair of treatment occur λ and b >= t. While in PIBD, the pairs of treatment occur λi (i =1, 2,...) and do not hold  b >= t..

* The BIBD have five primary parameters that's t, b, r, k and λ. While PBIBD have two types of parameters based on association scheme.

* In BIBD all pairs of treatment difference estimated with the accuracy. While in PBIBD all pairs of treatment difference are not estimated with the accuracy.





 

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Partially Incomplete Block Design Lecture - 43

 

Partially Incomplete Block Design

(PBIBD)

Lecture - 43

The BIBD are connected designs and compare all pairs of treatments with the same variance, called balance variance.  The primary limitation of BIBD, if the number of treatments is increases, it required a large number of blocks or large number of experimental units per block (replications). The primary limitation of BIBD is that it is not compatible with all parameter combinations. To overcome this problem by Bose and Nair and introduced the concept of partially balance incomplete block design. In BIBD, all pairs of treatment replicated at the number of times, while in PBIBD, some pairs of treatments replicated λ1 times and some pairs of treatments λ2 times and so on.

For example, t = 8 and k = 2


The partially balanced incomplete block designs (PBIBD) compromise on this property up to some extent and help in reducing the number of replications. The partially balanced incomplete block designs remain connected like BIBD and partially balanced in the sense that some pairs of treatments have the same efficiency whereas some other pairs of treatments have the same efficiency but different from the efficiency of earlier pairs of treatments.

Association Scheme

The pair of treatment appear in the same block is called first associate and the treatment do not appear in the same block is called second associate.

Before describing the setup of PBIBD, first, we need to understand the concept of “Association Scheme”. Let we have 9 (A, B, C, D, E, F, G, H, I) treatments

A PBIBD with t = 9, b = 9, k = 3, r = 3

  

Treatment

Blocks

I

II

III

IV

V

VI

VII

VIII

IX

A

A

 

 

A

 

 

A

 

 

B

B

 

 

 

B

 

 

B

 

C

C

 

 

 

 

C

 

 

C

D

 

D

 

D

 

 

 

 

D

E

 

E

 

 

E

 

E

 

 

F

 

F

 

 

 

F

 

F

 

G

 

 

G

G

 

 

 

G

 

H

 

 

H

 

H

 

 

 

H

I

 

 

I

 

 

I

I

 

 

Or we can write as:

Blocks

I

II

III

IV

V

VI

VII

VIII

IX

A

D

G

A

B

C

A

B

C

B

E

H

D

E

F

E

F

D

C

F

I

G

H

I

I

G

H

Consider treatment A

In block – I: B and C

In block – IV: D and G

In block – VII: E and I

First Associate of treatment A: B, C, D, G

The second Associate of treatment A: F, H

In the same way consider any other treatment.

Features of Partially Balance Incomplete Block Design

In BIBD all pairs of treatments replicated at the same number of times say λ. Now consider an incomplete block design in which some pair of treatments is replicated  λ1 times and some pairs of treatments is replicated  λ2 times, and so on. Then we use partially balance incomplete block design (PBIBD).

Properties of PBIBD

i.        Every block in a PBIBD contains equal number of experimental units say k.

ii.      Every treatment appears r times in the design.

iii.    No treatment appears more than once in a block.

iv.    Every pair of treatments together in either λ1 or λ2 blocks.

v.      Any two treatments which appear to λ1 times, or λ2 times is called first associate, second associate or third associate respectively.


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