INCOMPLETE BLOCK DESIGN (IBD) Lecture - 39

Introduction 

INCOMPLETE BLOCK DESIGN

(IBD)

Lecture - 39

Introduction

The classical designs are flexible designs like RCBD is not adequate a large number of treatments or treatments combinations. As the number of treatments rises, so does the number of replicates per block. The homogeneity is not preserved as the block size is increase. In addition to maintaining homogeneity, increasing the block size also increases expenses of material, time, and labour, etc. The incomplete block design can be used to get around the situation when there not sufficient homogeneous experimental units to accommodate all treatments in a block. In complete block design, each block only receives some of the treatments and not all of the treatments. Sometimes, the available blocks may not be able to accommodate all treatment for a variety of reasons.

Let's study the lifespan of ten different kinds of energy-efficient light bulbs, for instance. The cost of the experiment will increase if each brand of energy-saving bulb is replicated six times, as this will require 60 energy-saving bulbs in addition to 60 plugs. The idea of incomplete design is more effective in these circumstances than traditional designs.

There are three types of analysis in the incomplete block designs

·  Intra block analysis,

·  Inter block analysis and

· Recovery of inter block information. 

Incomplete Block Design Notations

let we have t number of treatments distributed in b number of blocks. There k number of replicates in each block and ri is the number of replicate for treatment i, in the entire design. 

that's

t: Number of treatments.

b: Number of blocks.

k: size of block (No. of experimental units in a block)

ri:  Number of replicate for treatment , in the entire design.

bk: Total number of experimental units in the entire design.

n = b x k

if all treatment is replicated at the same number of times say r, then ri = r

Then the total number of observations in the entire design is r x t = n

Thus, 

rt = bk = n

Incidence Matrix

Let we have t treatments and b blocks and nij be the number of observations for j^th treatment in i^th block.

For a complete block design:

or we can write as:

For incomplete block design:

Incidence Matrix

A matrix obtain from  is called incidence matrix.

Where:

As r appear t times 

We know that rt = bk

Incomplete Block Design Definition

An experiment in which the number of experimental units in a block is less than the number of treatments or treatments combinations is called incomplete block and a design constitute for such a block is called incomplete block design.

k < t

Where: 

k is the number of experimental units in a block and t is the number of treatment.

Let we have t treatments and each treatment is replicated r times, b is the number of blocks and k is the size (number of experimental units in a block) of each block.

 

In incomplete block design the number of experimental units in a block is less than the number of treatments and each block does not receive all the treatments but only some of the treatments. nij is the number of observation i^th block received j^th  treatment. 

The total number of observations is

n = r x t = b x k     

Where: k < t

It may be noticed that

Connected Design

An IBD in which the treatment contrasts are estimable is called connected design. Let τj and τj/ occur in the same block, the contrast τj - τj/ is estimable.

Example: whether the following design is connected

Block I = {1, 2, 3} and Block 2 = {1, 4, 5}

Solution:

Block I	Block II
1
2	1
3	4
	5

 The design is connected.

Statistical Model of IBD

Let Yijm be the yield of m^th replicate of j^th treatment in i^th block, and then it can be represented by the linear model.

Yijm = μ + βi + τj + ϵijm   

i = 1, 2, ..., b      j = 1, 2, ..., t       m =1, 2, ..., nij 

Estimation of Parameters

We know that

• Read More: Balance Incomplete Block Design