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

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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.

• $Read\; More: <a\; href="https://collagestatistics.blogspot.com/2024/09/partially-balance-incomplete-block.html"\; target="\_blank">PBIBD\; with\; Two\; Associates</a>$

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