 ```Design and Analysis of
Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
1/33
Blocking and
Confounding in TwoLevel Factorial Designs
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
2/33
Outline
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Introduction
Blocking Replicated 2k factorial Design
Confounding in 2k factorial Design
Blocks
Why Blocking is Important
Blocks
Partial Confounding
Introduction
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Sometimes it is impossible to perform all of
runs in one batch of material
Or to ensure the robustness, one might
deliberately vary the experimental conditions
to ensure the treatment are equally effective.
Blocking is a technique for dealing with
controllable nuisance variables
Introduction
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Two cases are considered
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Replicated designs
Unreplicated designs
Blocking a Replicated 2k
Factorial Design
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A 2k design has been replicated n times.
Each set of nonhomogeneous conditions
defines a block
Each replicate is run in one of the block
The runs in each block would be made in
random order.
Blocking a Replicated 2k
Factorial Design -- example
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Only four experiment trials can be made from
a single batch. Three batch of raw material are
required.
Blocking a Replicated 2k
Factorial Design -- example
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Sum of Squares in Block
Bi2 y...2


12
i 1 4
 6.50
3
SS Blocks
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ANOVA
Confounding in The 2k
Factorial Design
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Problem: Impossible to perform a complete
Confounding is a design technique for
blocks, where block size is smaller than the
number of treatment combinations in one
replicate.
9
Confounding in The 2k
Factorial Design
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certain treatment effects (usually high order
interactions ) to be indistinguishable from, or
confounded with, blocks.
If the case is to analyze a 2k factorial design
in 2p incomplete blocks, where p&lt;k, one can
use runs in two blocks (p=1), four blocks
(p=2), and so on.
10
Confounding the 2k Factorial
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
Suppose we want to run a single replicate of
the 22 design. Each of the 22=4 treatment
combinations requires a quantity of raw
material, for example, and each batch of raw
material is only large enough for two
treatment combinations to be tested.
Two batches are required.
11
Confounding the 2k Factorial
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One can treat batches as blocks
One needs assign two of the four treatment
combinations to each blocks
12
Confounding the 2k Factorial
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The order of the treatment combinations are
run within one block is randomly selected.
For the effects, A and B:
A=1/2[ab+a-b-(1)]
B=1/2[ab-a+b-(1)]
Are unaffected
13
Confounding the 2k Factorial
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For the effects, AB:
AB=1/2[ab-a-b+(1)]
is identical to block effect
 AB is confounded with blocks
14
Confounding the 2k Factorial
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We could assign the block effects to
confounded with A or B.
However we usually want to confound with
higher order interaction effects.
15
Confounding the 2k Factorial
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Three factors example
16
Confounding the 2k Factorial
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ABC is confounded with blocks
It is a random order within one block.
17
Confounding the 2k Factorial
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Multiple replicates are required to obtain the
estimate error when k is small.
For example, 23 design with four replicate in two
blocks
18
Confounding the 2k Factorial
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ANOVA
32 observations
19
Confounding the 2k Factorial
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Same as example 6.2
Four factors: Temperature, pressure,
concentration, and stirring rate.
Response variable: filtration rate.
Each batch of material is nough for 8 treatment
combinations only.
This is a 24 design n two blocks.
20
Confounding the 2k Factorial
21
Confounding the 2k Factorial
Factorial Fit: Filtration versus Block, Temperature, Pressure, ...
Estimated Effects and Coefficients for Filtration (coded units)
Term
Effect
Coef
Constant
60.063
Block
-9.313
Temperature
21.625
10.812
Pressure
3.125
1.563
Conc.
9.875
4.938
Stir rate
14.625
7.313
Temperature*Pressure
0.125
0.063
Temperature*Conc.
-18.125
-9.063
Temperature*Stir rate
16.625
8.313
Pressure*Conc.
2.375
1.188
Pressure*Stir rate
-0.375
-0.188
Conc.*Stir rate
-1.125
-0.562
Temperature*Pressure*Conc.
1.875
0.938
Temperature*Pressure*Stir rate
4.125
2.063
Temperature*Conc.*Stir rate
-1.625
-0.812
Pressure*Conc.*Stir rate
-2.625
-1.312
S = * PRESS = *
22
Confounding the 2k Factorial
Factorial Fit: Filtration versus Block, Temperature, Pressure, ...
Analysis of Variance for Filtration (coded units)
Source
Blocks
Main Effects
2-Way Interactions
3-Way Interactions
Residual Error
Total
DF
1
4
6
4
0
15
Seq SS
1387.6
3155.2
2447.9
120.2
*
7110.9
1387.6
3155.2
2447.9
120.2
*
1387.56
788.81
407.98
30.06
*
F
*
*
*
*
P
*
*
*
*
23
Confounding the 2k Factorial
24
Confounding the 2k Factorial
25
Confounding the 2k Factorial
ABCD
Factorial Fit: Filtration versus Block, Temperature, Conc., Stir rate
Estimated Effects and Coefficients for Filtration (coded units)
Term
Effect
Coef
SE Coef
T
Constant
60.063
1.141
52.63
Block
-9.313
1.141
-8.16
Temperature
21.625
10.812
1.141
9.47
Conc.
9.875
4.938
1.141
4.33
Stir rate
14.625
7.313
1.141
6.41
Temperature*Conc.
-18.125
-9.062
1.141
-7.94
Temperature*Stir rate
16.625
8.312
1.141
7.28
P
0.000
0.000
0.000
0.002
0.000
0.000
0.000
S = 4.56512 PRESS = 592.790
R-Sq = 97.36% R-Sq(pred) = 91.66% R-Sq(adj) = 95.60%
Analysis of Variance for Filtration (coded units)
Source
DF
Seq SS
Blocks
1
1387.6
Main Effects
3
3116.2
2-Way Interactions
2
2419.6
Residual Error
9
187.6
Total
15
7110.9
1387.6
3116.2
2419.6
187.6
1387.56
1038.73
1209.81
20.84
F
66.58
49.84
58.05
P
0.000
0.000
0.000
26
Another Illustration

Assuming we don’t have blocking in previous
example, we will not be able to notice the effect
Now the
first eight
runs (in run
order) have
filtration
rate reduced
by 20 units
27
Another Illustration
28
four blocks
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2k factorial design confounded in four blocks of
2k-2 observations.
Useful if k ≧ 4 and block sizes are relatively
small.
with eight runs.
Select two factors to be confound with, say ADE
and BCE.
29
four blocks
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L1=x1+x4+x5
L2=x2+x3+x5
Pairs of L1 and L2 group into four blocks
30
four blocks
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Example: L1=1, L2=1  block 4
abcde: L1=x1+x4+x5=1+1+1=3(mod 2)=1
L2=x2+x3+x5=1+1+1=3(mod 2)=1
31
blocks

2k factorial design confounded in 2p blocks
of 2k-p observations.
32
blocks
33
Partial Confounding
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In Figure 7.3, it is a completely confounded
case
ABC s confounded with blocks in each
replicate.
34
Partial Confounding
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Consider the case below, it is partial
confounding.
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ABC is confounded in replicate I and so on.
35
Partial Confounding
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As a result, information on ABC can be
obtained from data in replicate II, II, IV,
and so on.
We say &frac34; of information can be obtained on
the interactions because they are
unconfounded in only three replicates.
&frac34; is the relative information for the
confounded effects
36
Partial Confounding
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ANOVA
37
Partial Confounding-- example
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From Example 6.1
Response variable: etch rate
Factors: A=gap, B=gas flow, C=RF power.
Only four treatment combinations can be
tested during a shift.
There is shift-to-shift difference in etch
performance. The experimenter decide to
use shift as a blocking factor.
38
Partial Confounding-- example
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Each replicate of the 23 design must be run
in two blocks. Two replicates are run.
ABC is confounded in replicate I and AB is
confounded in replicate II.
39
Partial Confounding-- example
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How to create partial confounding in
Minitab?
40
Partial Confounding-- example
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Replicate I is confounded with ABC
STAT&gt;DOE&gt;Factorial &gt;Create Factorial
Design
41
Partial Confounding-- example
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Design &gt;Full Factorial
Number of blocks  2  OK
42
Partial Confounding-- example
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Factors &gt; Fill in appropriate information
 OK  OK
43
Partial Confounding-- example
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Result of Replicate I (default is to confound
with ABC)
44
Partial Confounding-- example
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Replicate II is confounded with AB
STAT&gt;DOE&gt;Factorial &gt;Create Factorial
Design
2 level factorial (specify generators)
45
Partial Confounding-- example
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Design &gt;Full Factorial
46
Partial Confounding-- example
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Generators …&gt; Define blocks by listing …
 AB
OK
47
Partial Confounding-- example
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Result of Replicate II
48
Partial Confounding-- example
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Change block number 3 -&gt; 1, 2 -&gt; 4 in Replicate II
Copy columns of CenterPt, Gap, …RF Power from
Replicate II to below the corresponding columns in
Replicate I.
49
Partial Confounding-- example
50
Partial Confounding-- example
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STAT&gt; DOE&gt; Factorial&gt; Define Custom Factorial
Design
Factors  Gap, Gas Flow, RF Power
51
Partial Confounding-- example
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Low/High &gt; OK
Designs &gt;Blocks&gt;Specify by column  Blocks
OK
52
Partial Confounding-- example
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Now you can fill in collected data.
53
Partial Confounding-- example

ANOVA
Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF
Estimated Effects and Coefficients for Etch Rate (coded units)
Term
Effect
Coef
SE Coef
Constant
776.06
12.63
Block 1
-22.94
28.23
Block 2
-8.19
28.23
Block 3
32.69
28.23
Gap
-101.62
-50.81
12.63
Gas Flow
7.38
3.69
12.63
RF
306.13
153.06
12.63
Gap*Gas Flow
-42.00
-21.00
17.86
Gap*RF
-153.63
-76.81
12.63
Gas Flow*RF
-2.13
-1.06
12.63
Gap*Gas Flow*RF
-1.75
-0.87
17.86
T
61.46
-0.81
-0.29
1.16
-4.02
0.29
12.12
-1.18
-6.08
-0.08
-0.05
P
0.000
0.453
0.783
0.299
0.010
0.782
0.000
0.293
0.002
0.936
0.963
S = 50.5071 PRESS = 130609
R-Sq = 97.60% R-Sq(pred) = 75.42% R-Sq(adj) = 92.80%
54
Partial Confounding-- example

ANOVA
Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF
Analysis of Variance for Etch Rate (coded units)
Source
Blocks
Main Effects
2-Way Interactions
3-Way Interactions
Residual Error
Total
DF
3
3
3
1
5
15
Seq SS
4333
416378
97949
6
12755
531421
5266
416378
97949
6
12755
1755
138793
32650
6 0.00
2551
F
0.69
54.41
12.80
0.963
P
0.597
0.000
0.009
* NOTE * There is partial confounding, no alias table was printed.
55
Partial Confounding-- example

ANOVA
56
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