Introduction to Blocking and Confounding in 2k Designs
Read Sections 7.1 – 7.4 in your text
Note: These notes have been modified from lecture notes created by Tisha Hooks and Christopher Malone.
In an _____________ world, it is ______ to replicate the ______________ experiment within each block.
Example 1:
Suppose we have a 22 factorial experiment. This experiment has 4 design points (i.e. (1), a, b, ab). Running each
design point in 3 blocks yields the following.
The outcomes observed from this experiment are given below.
Let’s set-up this design in Minitab.
Specify it’s a 22 Factorial Design
Specify 3 replicates in 3 blocks
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We will get the following output in Minitab regarding the design set-up.
Question:
1. Do you recall how the randomization was carried out?
Here is the output with the response values included.
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Analysis in Minitab
The analysis would proceed as usual…
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


Proceed with examining the estimated model
Check the residual plots thoroughly
Create any necessary model graphs (interaction plot, main effect plot, etc.)
Find optimal settings for a desired response
The ANOVA table from the Minitab output.
Questions:
2. Which effects/factors are important?
3. Why aren’t we interested in testing the blocking variable?
The other ANOVA table outputted in Minitab.
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The normal probability plot of the standardized effects is given below.
The least square means are given below.
The residual plots.
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More plots: Interaction, Main Effects, Surface, and Contour
Often, 2k experiments are carried out with a minimal number of runs to minimize cost and time to complete the
experiment. Many times, investigators cannot replicate over several blocks. For other experiments, it may be
physically impossible to avoid a blocking variable. Consider the following example.
Example 2:
Suppose we have another 22 experiment, but this time only ______ runs can be completed per block. Thus, ______
complete run of the experiment will require two blocks. How do we decide which factors to put in each block?
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Let’s take a closer look at what is gained/lost for each possible scenario.
The usual 22 design
No Loss of Information:
Possible 22 designs in two blocks
Losing Information about Effect A
The previous example (Example 1):
Note: Experiment is repeated in 3 blocks
Losing Information about Effect B
Losing Information about Effect AB
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Let’s take a close look at the possible scenarios when using two blocks.
Losing
Information
about Factor A
Losing
Information
about Factor B
Losing
Information
about Factor AB
Confounding
This occurs when treatment effects become indistinguishable from the effects of blocking.

Confounding occurs in experiments when not all experimental settings can be run in each block.
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Higher order interactions are usually the effects chosen to be confounded with blocks.
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If two treatment effects are indistinguishable (see Chapter 8), this is called _________________ and is
different from confounding.
Thus, in the previous example, it would be best to use the following configuration to run the experiment.
Note that within each block treatment combinations would be run in a random order, in addition to the random
ordering of blocks.
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Example 3:
Next, let’s consider a 23 factorial experiment in two blocks. We will want to confound the ABC interaction effect with
blocks this time. The easiest way to do this is to write out the factor combinations with the +1’s and -1’s, as shown
below.
Next, sort the ABC column by the +1’s and -1’s. Then, all the -1’s will be in Block 1 and all the +1’s will be in Block 2.
Let’s take a look at confounding in general (page 307) in your text. Confounding is based off a defining contrast.
A defining contrast is a __________________ contrast used to determine the confounding structure.
The contrast is written as follows:
where
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
αi = 0 or 1 is the exponent on the ith factor to be confounded, and
xi = 0 (low level) or 1 (high level) is the level for each factor.
The treatment combinations that produce the same value of L1 (mod 2) will be placed in the same block. This
defining contrast is used for a ________ in _____ blocks.
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Applying this method to our previous example:
The following defining contrast is used for a 23 experiment which is to be run in 2 blocks. The effect to be
confounded is the ABC interaction effect, so α1 = α2 = α3 = 1. Therefore, the defining contrast is given by
L1 = (1)x1 + (1)x2 + (1)x3
We can easily do these computations in Excel.
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Example 4:
Consider the example on page 313 in your text. A 25 factorial experiment (32 observations) with at most 8 settings
available per block is under consideration. Therefore, we need to run an experiment with 4 blocks and 8
observations per block.
In general, for a 2k factorial arranged in 2p blocks, ____ effects must be chosen to be confounded. For this example,
k = ______
p = _____.
Note that each block will contain 2k – p runs. So, we must select ______ effects to be confounded with blocks. Let
ADE and BCE be used for the defining relationship for this example.
For ADE:
L1 = α1x1 + α2x2 + α3x3 + α4x4 + α5x5
= x1 + x4 + x5
For BCE:
L2 = α1x1 + α2x2 + α3x3 + α4x4 + α5x5
= x2 + x3 + x5
These defining contrasts are again calculated mod 2 and form 4 possible combinations which are used to specify
which effects are placed in each block.
L1
0
1
0
1
L2
0
0
1
1
Pair
(0,0)
(1,0)
(0,1)
(1,1)
Block
1
2
3
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Open the file ConfoundingWithBlocking.xls from the course website. Using this file, verify the blocking scheme
below.
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You can use filtering to help to easily identify blocks.
The resulting blocking structure for this design is given below.
Generalized Interactions
It is important to note that in addition to the _____ independent effects that are chosen to be confounded, exactly
_______________ other effects will also be confounded with blocks. These are called generalized interactions of the
p independent effects initially chosen. A generalized interaction is defined as the product of two effects mod 2. For
example, we have (ADE)(BCE) = ABCDE2 = ABCD (mod 2) also confounded with blocks. To see this, consider this
partial table of the +1/-1 signs for Blocks 2 and 4 of the above design.
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Creating this Design in Minitab
We could create this design using the default generators in Minitab.
We get the following information in the session window regarding this design.
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We could create this design using our own specified generators in Minitab.
We then get the following information in the session window regarding this design.
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We can verify that this design matches what we came up with earlier.
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Your book also gives you suggested blocking arrangements.
Question:
4. Suppose we have an experiment with factors A, B, C, and D and we want to run an experiment with four
blocks each with four runs.
a. How many effects must be confounded with blocks?
b. Generate the design using Minitab.
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