Blocking can be used in the 2 designs to deal with cases when replicates
cannot be run under identical conditions. Randomized complete block designs
that were discussed in Chapter 6
for factorial experiments are also applicable here. At times, even with
just two levels per factor, it is not possible to run all treatment combinations
for one replicate of the experiment under homogeneous conditions. For
example, each replicate of the 2 design requires four runs. If each
run requires two hours and testing facilities are available for only four
hours per day, two days of testing would be required to run one complete
replicate. Blocking can be used to separate the treatment runs on the
two different days. Blocks that do not contain all treatments of a replicate
are called incomplete blocks. In incomplete block designs, the
block effect is confounded with certain effect(s) under investigation.
[Note]
For the 2 design assume that treatments and were run on the first day and treatments
and were run on the second day. Then,
the incomplete block design for this experiment is:
For this design the block effect may be calculated as:(9)
The interaction effect is:(10)
Eqns. (9) and (10) show
that, in this design, the interaction effect cannot be distinguished
from the block effect because the formulas to calculate these effects
are the same. In other words, the interaction is said to be confounded
with the block effect and it is not possible to say if the effect calculated
based on these equations is due to the interaction effect, the block effect
or both. In incomplete block designs some effects are always confounded
with the blocks. Therefore, it is important to design these experiments
in such a way that the important effects are not confounded with the blocks.
In most cases, the experimenter can assume that higher order interactions
are unimportant. In this case, it would better to use incomplete block
designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts
as shown next:(11)
where the s are the exponents for the factors
in the effect that is to be confounded with the block effect and the s are values based on the level of
the th factor (in a treatment that is to
be allocated to a block). For 2 designs the s are either 0 or 1 and the s have a value of 0 for the low level
of the th factor and a value of 1 for the high
level of the factor in the treatment under consideration. As an example,
consider the 2 design where the interaction effect
is confounded with the block. Since
there are two factors, , with representing factor and representing factor . Therefore:
The value of is one because the exponent of factor
in the confounded interaction is one. Similarly, the value of is one because the exponent of factor
in the confounded interaction is also one. Therefore, the defining
contrast for this design can be written as:
Once the defining contrast is known, it can be used to allocate treatments
to the blocks.For the 2 design, there are four treatments
, , and . Assume that represents block 2 and represents block 1. In order to decide
which block the treatment belongs to, the levels of factors
and for this run are used. Since factor
is at the low level in this treatment,
. Similarly, since factor is also at the low level in this treatment,
. Therefore:
Note that the value of used to decide the block allocation
is "mod 2" of the original value. [Note]
This value is obtained by taking the value of 1 for odd numbers and 0
otherwise. Based on the value of , treatment is assigned to block 1. Other treatments
can be assigned using the following calculations:
Therefore, to confound the interaction with the block effect in the 2 incomplete block design, treatments
and (with ) should be assigned to block 2 and
treatment combinations and (with ) should be assigned to block 1.
Example 7.5
This example illustrates how treatments can be allocated to two blocks
for an unreplicated 2 design. Consider the unreplicated 2 design to investigate the four factors
affecting the defects in automobile vinyl panels discussed in Chapter
7, Normal
Probability Plot of Effects. Assume that the 16 treatments required
for this experiment were run by two different operators with each operator
conducting 8 runs. This experiment is an example of an incomplete block
design. The analyst in charge of this experiment assumed that the interaction
was not significant and decided to
allocate treatments to the two operators so that the interaction was confounded with the
block effect (the two operators are the blocks). The allocation scheme
to assign treatments to the two operators can be obtained as follows.
The defining contrast for the 2 design where the interaction is confounded with the
blocks is:
The treatments can be allocated to the two operators using the values
of the defining contrast. Assume that represents block 2 and represents block 1. Then the value
of the defining contrast for treatment is:
Therefore, treatment should be assigned to Block 1 or the
first operator. Similarly, for treatment we have:
Therefore, should be assigned to Block 2 or the
second operator. Other treatments can be allocated to the two operators
in a similar manner to arrive at the allocation scheme shown in Figure
7.22.
Figure 7.22: Allocation of treatments to two blocks for
the 2 design in Example 7.5 by confounding interaction
with the blocks.
In DOE++, to confound the interaction for the 2 design into two blocks, the number
of blocks are specified as shown in Figure 7.23.
Then the interaction is entered in the Block Generator
window (Figure 7.24) which is available using the Block Generator button
in Figure 7.23. The design generated by DOE++ is
shown in Figure 7.25. This design matches the allocation
scheme of Figure 7.22.
Figure 7.23:
Adding block properties for the experiment in Example 7.5.
Figure: 7.24: Specifying the interaction as the interaction to be confounded with
the blocks for Example 7.5.
Figure 7.25:
Two block design for the experiment in Example 7.5.
For the analysis of this design, the sum of squares for all effects
are calculated assuming no blocking. Then, to account for blocking, the
sum of squares corresponding to the interaction is considered as the sum
of squares due to blocks and . In DOE++ this is done by displaying
this sum of squares as the sum of squares due to the blocks. This is shown
in Figure 7.26 where the sum of squares in question
is obtained as 72.25 and is displayed against Block. The interaction ABCD,
which is confounded with the blocks, is not displayed. Since the design
is unreplicated, any of the methods to analyze unreplicated designs mentioned
in Chapter 7, Unreplicated
2k
Designs, have to be used to identify significant effects.
Figure 7.26:
ANOVA table for the experiment of Example 7.5.
A single replicate of the 2 design can be run in up to 2 blocks where . The number of effects confounded
with the blocks equals the degrees of freedom associated with the block
effect. If two blocks are used (the block effect has two levels), then
one ( effect is confounded with the blocks.
If four blocks are used, then three () effects are confounded with the blocks
and so on. [Note]
For example an unreplicated 2 design may be confounded in 2 (four) blocks using two contrasts,
and . Let and be the effects to be confounded with
the blocks. Corresponding to these two effects, the contrasts are respectively:Based on the values of and the treatments can be assigned to
the four blocks as follows:
Since the block effect has three degrees of freedom, three effects are
confounded with the block effect. In addition to and , the third effect confounded with
the block effect is their generalized interaction,.
In general, when an unreplicated 2 design is confounded in 2 blocks, contrasts are needed (). effects are selected to define these
contrasts such that none of these effects are the generalized interaction
of the others. The 2 blocks can then be assigned the treatments
using the contrasts. effects, that are also confounded
with the blocks, are then obtained as the generalized interaction of the
effects. In the statistical analysis
of these designs, the sum of squares are computed as if no blocking were
used. Then the block sum of squares is obtained by adding the sum of squares
for all the effects confounded with the blocks.
Example 7.6
This example illustrates how DOE++ obtains the sum of squares when treatments
for an unreplicated 2 design are allocated among four blocks.
Consider again the unreplicated 2 design used to investigate the defects
in automobile vinyl panels presented in Chapter 7, Normal
Probability Plot of Effects. Assume that the 16 treatments needed
to complete the experiment were run by four operators. Therefore, there
are four blocks. Assume that the treatments were allocated to the blocks
using the generators mentioned in the previous section, i.e.
treatments were allocated among the four operators by confounding the
effects, and with the blocks. These effects can
be specified as Block Generators as shown in Figure 7.27.
(The generalized interaction of these two effects, interaction , will also get confounded with the
blocks.) The resulting design is shown in Figure 7.28
and matches the allocation scheme obtained in the previous section.
Figure 7.27: Specifying the interactions and as block generators for Example 7.6.
The sum of squares in this case can be obtained by calculating the sum
of squares for each of the effects assuming there is no blocking. Once
the individual sum of squares have been obtained, the block sum of squares
can be calculated. The block sum of squares is the sum of the sum of squares
of effects, , and , since these effects are confounded
with the block effect. As shown in Figure 7.29,
this sum of squares is 92.25 and is displayed against Block. [Note
]
The interactions , and , which are confounded with the blocks,
are not displayed. [Note
]
Since the present design is unreplicated any of the methods to analyze
unreplicated designs mentioned in Chapter 7, Unreplicated
2k
Designs, have to be used to identify significant effects.
Figure 7.28:
Design for the experiment in Example 7.6.
Figure 7.29:
ANOVA table for the experiment in Example 7.6.
designs to deal with cases when replicates
cannot be run under identical conditions. Randomized complete block designs
that were discussed in Chapter 6
for factorial experiments are also applicable here. At times, even with
just two levels per factor, it is not possible to run all treatment combinations
for one replicate of the experiment under homogeneous conditions. For
example, each replicate of the 2
design requires four runs. If each
run requires two hours and testing facilities are available for only four
hours per day, two days of testing would be required to run one complete
replicate. Blocking can be used to separate the treatment runs on the
two different days. Blocks that do not contain all treatments of a replicate
are called incomplete blocks. In incomplete block designs, the
block effect is confounded with certain effect(s) under investigation.
[Note]
For the 2
design assume that treatments 
and 
were run on the first day and treatments
and 
were run on the second day. Then,
the incomplete block design for this experiment is:
(9)
(10)
(11)
design in Example 7.5 by confounding interaction
with the blocks.
as the interaction to be confounded with
the blocks for Example 7.5.
and 
as block generators for Example 7.6.
