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 i^th 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

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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. 

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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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