Chapter 5
Incomplete Block Designs
If the number of treatments to be compared is large, then we need a large number of blocks to
accommodate all the treatments. This requires more experimental material and so the cost of
experimentation becomes high which may be in terms of money, labour, time etc. The completely
randomized design and randomized block design may not be suitable in such situations because they will
require a large number of experimental units to accommodate all the treatments. In such cases, when the
sufficient number of homogeneous experimental units are not available to accommodate all the treatments
in a block, then incomplete block designs can be used. In incomplete block designs, each block receives
only some of the selected treatments and not all the treatments. Sometimes it is possible that the available
blocks can accommodate only a limited number of treatments due to several reasons. For example, the
goodness of a car is judged by different features like fuel efficiency, engine performance, body structure
etc. Each of this factor depends on many other factors, e.g., the engine consists of many parts and the
performance of every part combined together will result in the final performance of the engine. These
factors can be treated as treatment effects. If all these factors are to be compared, then we need a large
number of cars to design a complete experiment. This may be an expensive affair. The incomplete block
designs overcome such problems. It is possible to use much less number of cars with the set up of
incomplete block design and all the treatments need not be assigned to all the cars. Rather some
treatments will be implemented in some cars and remaining treatments in other cars. The efficiency of
such designs is, in general, not less than the efficiency of a complete block design. In another example,
consider a situation of destructive experiments, e.g., testing the life of television sets, LCD panels, etc. If
there are a large number of treatments to be compared, then we need a large number of television sets or
LCD panels. The incomplete block designs can use a lesser number of television sets or LCD panels to
conduct the test of the significance of treatment effects without losing, in general, the efficiency of the
design of the experiment. This also results in the reduction of experimental cost. Similarly, in any
experiment involving animals like as biological experiments, one would always like to sacrifice fewer
animals. Moreover, the government guidelines also restrict the experimenter to use a smaller number of
animals. In such cases, either the number of treatments to be compared can be reduced depending upon
the number of animals in each block or to reduce the block size. In such cases when the number of
treatments to be compared is larger than the number of animals in each block, then the block size is
reduced and the setup of incomplete block designs can be used. This will result in a lower cost of
experimentation. The incomplete block designs need less number of observations in a block than the
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
1
,observations in a complete block design to conduct the test of hypothesis without losing the efficiency of
the design of experiment, in general.
Complete and incomplete block designs:
The designs in which every block receives all the treatments are called the complete block designs.
The designs in which every block does not receive all the treatments but only some of the treatments are
called incomplete block design.
The block size is smaller than the total number of treatments to be compared in the incomplete block
designs.
There are three types of analysis in the incomplete block designs
intrablock analysis,
interblock analysis and
recovery of interblock information.
Intrablock analysis:
In intrablock analysis, the treatment effects are estimated after eliminating the block effects and then the
analysis and the test of significance of treatment effects are conducted further. If the blocking factor is not
marked, then the intrablock analysis is sufficient enough to provide reliable, correct and valid statistical
inferences.
Interblock analysis:
There is a possibility that the blocking factor is important and the block totals may carry some important
information about the treatment effects. In such situations, one would like to utilize the information on
block effects (instead of removing it as in the intrablock analysis) in estimating the treatment effects to
conduct the analysis of design. This is achieved through the interblock analysis of an incomplete block
design by considering the block effects to be random.
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
2
,Recovery of interblock information:
When both the intrablock and the interblock analysis have been conducted, then two estimates of
treatment effects are available from each of the analysis. A natural question then arises -- Is it possible to
pool these two estimates together and obtain an improved estimator of the treatment effects to use it for
the construction of test statistic for testing of hypothesis? Since such an estimator comprises of more
information to estimate the treatment effects, so this is naturally expected to provide better statistical
inferences. This is achieved by combining the intrablock and interblock analysis together through the
recovery of interblock information.
Intrablock analysis of incomplete block design:
We start here with the usual approach involving the summations over different subscripts of y’s. Then
gradually, we will switch to a matrix-based approach so that the reader can compare both the approaches.
They can also learn the one-to-one relationships between the two approaches for better understanding.
Notations and normal equations:
Let
- v treatments have to be compared.
- b blocks are available.
- ki : Number of plots in ith block (i = 1,2,…,b).
- rj : Number of plots receiving jth treatment ( j = 1,2,…,v).
- n: Total number of plots.
n r1 r2 ... rv k1 k2 ... kb .
- Each treatment may occur more than once in each block
or
may not occur at all.
- nij denotes the number of times the jth treatment occurs in ith block
For example, nij 1 or 0 for all i , j means that no treatment occurs more than once in a block and
treatment may not occur in some blocks at all. Similarly, nij 1 means that jth treatment occurs in ith
block and nij 0 means that jth treatment does not occurs in ith block.
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
3
, It may be noticed that
v
n
j 1
ij ki i 1,..., b
ni
ij rj j 1,..., v
n nij
i j
Model:
Let yijm denotes the response (yield) from the mth replicate of jth treatment in ith block and
yijm i j ijm i 1, 2,..., b, j 1, 2,.., v , m 1, 2,..., nij
[Note: We are not considering here the general mean effect in this model for better understanding of the
issues in the estimation of parameters. Later, we will consider it in the analysis.]
Following notations are used in further description.
Block totals : B1 , B2 ,..., Bb where Bi yijm .
j m
Treatment totals: V1 ,V2 ,...,Vv where V j yijm
i m
Grand total : Y yijm
i j m
Generally, a design is denoted by D(v, b, r , k , n) where v, b, r, k and n are the parameters of the design.
Example:
Let us consider an example to understand the meaning of these notations. Suppose there are 3 blocks
(Block 1, Block 2 and Block 3) and 5 treatments ( 1 , 2 , 3 , 4 , 5 ). So b = 3 and v = 5. These
treatments are arranged in different plots in blocks as follows:
Block 1: 5 plots Plot 1 1 Plot 2 1 Plot 3 2 Plot 4 2 Plot 5 3
Block 2: 4 plots Plot 1 2 Plot 2 4 Plot 3 5 Plot 4 5
Block 3: 2 plots Plot 1 2 Plot 2 2
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
4
Incomplete Block Designs
If the number of treatments to be compared is large, then we need a large number of blocks to
accommodate all the treatments. This requires more experimental material and so the cost of
experimentation becomes high which may be in terms of money, labour, time etc. The completely
randomized design and randomized block design may not be suitable in such situations because they will
require a large number of experimental units to accommodate all the treatments. In such cases, when the
sufficient number of homogeneous experimental units are not available to accommodate all the treatments
in a block, then incomplete block designs can be used. In incomplete block designs, each block receives
only some of the selected treatments and not all the treatments. Sometimes it is possible that the available
blocks can accommodate only a limited number of treatments due to several reasons. For example, the
goodness of a car is judged by different features like fuel efficiency, engine performance, body structure
etc. Each of this factor depends on many other factors, e.g., the engine consists of many parts and the
performance of every part combined together will result in the final performance of the engine. These
factors can be treated as treatment effects. If all these factors are to be compared, then we need a large
number of cars to design a complete experiment. This may be an expensive affair. The incomplete block
designs overcome such problems. It is possible to use much less number of cars with the set up of
incomplete block design and all the treatments need not be assigned to all the cars. Rather some
treatments will be implemented in some cars and remaining treatments in other cars. The efficiency of
such designs is, in general, not less than the efficiency of a complete block design. In another example,
consider a situation of destructive experiments, e.g., testing the life of television sets, LCD panels, etc. If
there are a large number of treatments to be compared, then we need a large number of television sets or
LCD panels. The incomplete block designs can use a lesser number of television sets or LCD panels to
conduct the test of the significance of treatment effects without losing, in general, the efficiency of the
design of the experiment. This also results in the reduction of experimental cost. Similarly, in any
experiment involving animals like as biological experiments, one would always like to sacrifice fewer
animals. Moreover, the government guidelines also restrict the experimenter to use a smaller number of
animals. In such cases, either the number of treatments to be compared can be reduced depending upon
the number of animals in each block or to reduce the block size. In such cases when the number of
treatments to be compared is larger than the number of animals in each block, then the block size is
reduced and the setup of incomplete block designs can be used. This will result in a lower cost of
experimentation. The incomplete block designs need less number of observations in a block than the
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
1
,observations in a complete block design to conduct the test of hypothesis without losing the efficiency of
the design of experiment, in general.
Complete and incomplete block designs:
The designs in which every block receives all the treatments are called the complete block designs.
The designs in which every block does not receive all the treatments but only some of the treatments are
called incomplete block design.
The block size is smaller than the total number of treatments to be compared in the incomplete block
designs.
There are three types of analysis in the incomplete block designs
intrablock analysis,
interblock analysis and
recovery of interblock information.
Intrablock analysis:
In intrablock analysis, the treatment effects are estimated after eliminating the block effects and then the
analysis and the test of significance of treatment effects are conducted further. If the blocking factor is not
marked, then the intrablock analysis is sufficient enough to provide reliable, correct and valid statistical
inferences.
Interblock analysis:
There is a possibility that the blocking factor is important and the block totals may carry some important
information about the treatment effects. In such situations, one would like to utilize the information on
block effects (instead of removing it as in the intrablock analysis) in estimating the treatment effects to
conduct the analysis of design. This is achieved through the interblock analysis of an incomplete block
design by considering the block effects to be random.
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
2
,Recovery of interblock information:
When both the intrablock and the interblock analysis have been conducted, then two estimates of
treatment effects are available from each of the analysis. A natural question then arises -- Is it possible to
pool these two estimates together and obtain an improved estimator of the treatment effects to use it for
the construction of test statistic for testing of hypothesis? Since such an estimator comprises of more
information to estimate the treatment effects, so this is naturally expected to provide better statistical
inferences. This is achieved by combining the intrablock and interblock analysis together through the
recovery of interblock information.
Intrablock analysis of incomplete block design:
We start here with the usual approach involving the summations over different subscripts of y’s. Then
gradually, we will switch to a matrix-based approach so that the reader can compare both the approaches.
They can also learn the one-to-one relationships between the two approaches for better understanding.
Notations and normal equations:
Let
- v treatments have to be compared.
- b blocks are available.
- ki : Number of plots in ith block (i = 1,2,…,b).
- rj : Number of plots receiving jth treatment ( j = 1,2,…,v).
- n: Total number of plots.
n r1 r2 ... rv k1 k2 ... kb .
- Each treatment may occur more than once in each block
or
may not occur at all.
- nij denotes the number of times the jth treatment occurs in ith block
For example, nij 1 or 0 for all i , j means that no treatment occurs more than once in a block and
treatment may not occur in some blocks at all. Similarly, nij 1 means that jth treatment occurs in ith
block and nij 0 means that jth treatment does not occurs in ith block.
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
3
, It may be noticed that
v
n
j 1
ij ki i 1,..., b
ni
ij rj j 1,..., v
n nij
i j
Model:
Let yijm denotes the response (yield) from the mth replicate of jth treatment in ith block and
yijm i j ijm i 1, 2,..., b, j 1, 2,.., v , m 1, 2,..., nij
[Note: We are not considering here the general mean effect in this model for better understanding of the
issues in the estimation of parameters. Later, we will consider it in the analysis.]
Following notations are used in further description.
Block totals : B1 , B2 ,..., Bb where Bi yijm .
j m
Treatment totals: V1 ,V2 ,...,Vv where V j yijm
i m
Grand total : Y yijm
i j m
Generally, a design is denoted by D(v, b, r , k , n) where v, b, r, k and n are the parameters of the design.
Example:
Let us consider an example to understand the meaning of these notations. Suppose there are 3 blocks
(Block 1, Block 2 and Block 3) and 5 treatments ( 1 , 2 , 3 , 4 , 5 ). So b = 3 and v = 5. These
treatments are arranged in different plots in blocks as follows:
Block 1: 5 plots Plot 1 1 Plot 2 1 Plot 3 2 Plot 4 2 Plot 5 3
Block 2: 4 plots Plot 1 2 Plot 2 4 Plot 3 5 Plot 4 5
Block 3: 2 plots Plot 1 2 Plot 2 2
Analysis of Variance | Chapter 5 | Incomplete Block Designs | Shalabh, IIT Kanpur
4
