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Design and Analysis of Experiments 8E 2012 Montgomery
1Chapter 8
Design of Engineering Experiments The 2k-p Fractional Factorial Design
• Text reference, Chapter 8• Motivation for fractional factorials is obvious; as the
number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly
• Emphasis is on factor screening; efficiently identify the factors with large effects
• There may be many variables (often because we don’t know much about the system)
• Almost always run as unreplicated factorials, but often with center points
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Design and Analysis of Experiments 8E 2012 Montgomery
2Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
3Chapter 8
Why do Fractional Factorial Designs Work?
• The sparsity of effects principle– There may be lots of factors, but few are important– System is dominated by main effects, low-order
interactions• The projection property
– Every fractional factorial contains full factorials in fewer factors
• Sequential experimentation– Can add runs to a fractional factorial to resolve
difficulties (or ambiguities) in interpretation
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Design and Analysis of Experiments 8E 2012 Montgomery
4Chapter 8
The One-Half Fraction of the 2k
• Section 8.2, page 321• Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1
• Consider a really simple case, the 23-1 • Note that I =ABC
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Design and Analysis of Experiments 8E 2012 Montgomery
5Chapter 8
The One-Half Fraction of the 23
For the principal fraction, notice that the contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction.
This phenomena is called aliasing and it occurs in all fractional designs
Aliases can be found directly from the columns in the table of + and - signs
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Design and Analysis of Experiments 8E 2012 Montgomery
6Chapter 8
Aliasing in the One-Half Fraction of the 23
A = BC, B = AC, C = AB (or me = 2fi)
Aliases can be found from the defining relation I = ABC by multiplication:
AI = A(ABC) = A2BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
Textbook notation for aliased effects:
[ ] , [ ] , [ ]A A BC B B AC C C AB
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Design and Analysis of Experiments 8E 2012 Montgomery
7Chapter 8
The Alternate Fraction of the 23-1
• I = -ABC is the defining relation• Implies slightly different aliases: A = -BC,
B= -AC, and C = -AB• Both designs belong to the same family, defined
by
• Suppose that after running the principal fraction, the alternate fraction was also run
• The two groups of runs can be combined to form a full factorial – an example of sequential experimentation
I ABC
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Design and Analysis of Experiments 8E 2012 Montgomery
8Chapter 8
Design Resolution
• Resolution III Designs:– me = 2fi – example
• Resolution IV Designs:– 2fi = 2fi– example
• Resolution V Designs:– 2fi = 3fi– example
3 12III
4 12IV
5 12V
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Design and Analysis of Experiments 8E 2012 Montgomery
9Chapter 8
Construction of a One-half Fraction
The basic design; the design generator
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Design and Analysis of Experiments 8E 2012 Montgomery
10Chapter 8
Projection of Fractional Factorials
Every fractional factorial contains full factorials in fewer factors
The “flashlight” analogy
A one-half fraction will project into a full factorial in any k – 1 of the original factors
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Design and Analysis of Experiments 8E 2012 Montgomery
11Chapter 8
Example 8.1
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Design and Analysis of Experiments 8E 2012 Montgomery
12Chapter 8
Example 8.1Interpretation of results often relies on making some assumptions
Ockham’s razor
Confirmation experiments can be important
Adding the alternate fraction – see page 322
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Design and Analysis of Experiments 8E 2012 Montgomery
13Chapter 8
The AC and AD interactions can be verified by inspection of the cube plot
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Design and Analysis of Experiments 8E 2012 Montgomery
14Chapter 8
Confirmation experiment for this example: see page 332
Use the model to predict the response at a test combination of interest in the design space – not one of the points in the current design.
Run this test combination – then compare predicted and observed.
For Example 8.1, consider the point +, +, -, +. The predicted response is
Actual response is 104.
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Design and Analysis of Experiments 8E 2012 Montgomery
15Chapter 8
Possible Strategies for
Follow-Up Experimentation
Following a Fractional
Factorial Design
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Design and Analysis of Experiments 8E 2012 Montgomery
16Chapter 8
The One-Quarter Fraction of the 2k
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Design and Analysis of Experiments 8E 2012 Montgomery
17Chapter 8
The One-Quarter Fraction of the 26-2
Complete defining relation: I = ABCE = BCDF = ADEF
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Design and Analysis of Experiments 8E 2012 Montgomery
18Chapter 8
The One-Quarter Fraction of the 26-2
• Uses of the alternate fractions
• Projection of the design into subsets of the original six variables
• Any subset of the original six variables that is not a word in the complete defining relation will result in a full factorial design– Consider ABCD (full factorial)– Consider ABCE (replicated half fraction)– Consider ABCF (full factorial)
, E ABC F BCD
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Design and Analysis of Experiments 8E 2012 Montgomery
19Chapter 8
A One-Quarter Fraction of the 26-2:Example 8.4, Page 336
• Injection molding process with six factors• Design matrix, page 338• Calculation of effects, normal probability
plot of effects• Two factors (A, B) and the AB interaction
are important• Residual analysis indicates there are some
dispersion effects (see page 307)
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Design and Analysis of Experiments 8E 2012 Montgomery
20Chapter 8
The General 2k-p Fractional Factorial Design
• Section 8.4, page 340• 2k-1 = one-half fraction, 2k-2 = one-quarter fraction,
2k-3 = one-eighth fraction, …, 2k-p = 1/ 2p fraction• Add p columns to the basic design; select p
independent generators• Important to select generators so as to maximize
resolution, see Table 8.14• Projection – a design of resolution R contains full
factorials in any R – 1 of the factors• Blocking
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Design and Analysis of Experiments 8E 2012 Montgomery
21Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
22Chapter 8
The General 2k-p Design: Resolution may not be Sufficient
• Minimum abberation designs
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Design and Analysis of Experiments 8E 2012 Montgomery
23Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
24Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
25Chapter 8
Main effects aliased with the 2fis
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Design and Analysis of Experiments 8E 2012 Montgomery
26Chapter 8
Resolution III Designs: Section 8.5, page 351
• Designs with main effects aliased with two-factor interactions
• Used for screening (5 – 7 variables in 8 runs, 9 - 15 variables in 16 runs, for example)
• A saturated design has k = N – 1 variables• See Table 8.19, page 351 for a 7 42III
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Design and Analysis of Experiments 8E 2012 Montgomery
27Chapter 8
Resolution III Designs
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Design and Analysis of Experiments 8E 2012 Montgomery
28Chapter 8
Resolution III Designs• Sequential assembly of fractions to separate aliased effects
(page 354)• Switching the signs in one column provides estimates of
that factor and all of its two-factor interactions• Switching the signs in all columns dealiases all main
effects from their two-factor interaction alias chains – called a full fold-over
• Defining relation for a fold-over (page 356)• Be careful – these rules only work for Resolution III
designs• There are other rules for Resolution IV designs, and other
methods for adding runs to fractions to dealias effects of interest
• Example 8.7, eye focus time, page 354
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Design and Analysis of Experiments 8E 2012 Montgomery
29Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
30Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
31Chapter 8
Remember that the full fold-over technique illustrated in this example (running a “mirror image” design with all signs reversed) only works in a Resolution III design.
Defining relation for a fold-over design – see page 356.
Blocking can be an important consideration in a fold-over design – see page 356.
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Design and Analysis of Experiments 8E 2012 Montgomery
32Chapter 8
Plackett-Burman Designs
• These are a different class of resolution III design• The number of runs, N, need only be a multiple of
four• N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …• The designs where N = 12, 20, 24, etc. are called
nongeometric PB designs• See text, page 357-358 for comments on
construction of Plackett-Burman designs
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Design and Analysis of Experiments 8E 2012 Montgomery
33Chapter 8
Plackett-Burman Designs
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Design and Analysis of Experiments 8E 2012 Montgomery
34Chapter 8
This is a nonregular design
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Design and Analysis of Experiments 8E 2012 Montgomery
35Chapter 8
Projection of the 12-run design into 3 and 4 factors
All PB designs have projectivity 3 (contrast with other resolution III fractions)
Plackett-Burman Designs
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Design and Analysis of Experiments 8E 2012 Montgomery
36Chapter 8
Plackett-Burman Designs
• The alias structure is complex in the PB designs• For example, with N = 12 and k = 11, every main
effect is aliased with every 2FI not involving itself• Every 2FI alias chain has 45 terms• Partial aliasing can potentially greatly complicate
interpretation if there are several large interactions• Use carefully – but there are some excellent
opportunities
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Design and Analysis of Experiments 8E 2012 Montgomery
37Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
38Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
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Design and Analysis of Experiments 8E 2012 Montgomery
40Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
41Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
42Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
43Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
44Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
45Chapter 8
Resolution IV and V Designs (Page 366)
A resolution IV design must have at least 2k runs.
“optimal” designs may often prove useful.
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Design and Analysis of Experiments 8E 2012 Montgomery
46Chapter 8
Sequential Experimentation with Resolution IV Designs – Page 367
We can’t use the full fold-over procedure given previously for Resolution III designs – it will result in replicating the runs in the original design.
Switching the signs in a single column allows all of the two-factor interactions involving that column to be separated.
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Design and Analysis of Experiments 8E 2012 Montgomery
47Chapter 8
The spin coater experiment – page 368
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Design and Analysis of Experiments 8E 2012 Montgomery
48Chapter 8
[AB] = AB + CE
We need to dealias these interactions
The fold-over design switches the signs in column A
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Design and Analysis of Experiments 8E 2012 Montgomery
49Chapter 8
The aliases from the complete design following the fold-over (32 runs) are as follows:
Finding the aliases involves using the alias matrix. Aliases can also be found from computer software.
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Design and Analysis of Experiments 8E 2012 Montgomery
50Chapter 8
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Design and Analysis of Experiments 8E 2012 Montgomery
51Chapter 8
A full fold-over of a Resolution IV design is usually not necessary, and it’s potentially very inefficient.
In the spin coater example, there were seven degrees of freedom available to estimate two-factor interaction alias chains.
After adding the fold-over (16 more runs), there are only 12 degrees of freedom available for estimating two-factor interactions (16 new runs yields only five more degrees of freedom).
A partial fold-over (semifold) may be a better choice of follow-up design. To construct a partial fold-over:
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Design and Analysis of Experiments 8E 2012 Montgomery
52Chapter 8
Not an orthogonal design – but that’s not such a big deal
Correlated parameter estimates
Larger standard errors of regression model coefficients or effects
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Design and Analysis of Experiments 8E 2012 Montgomery
53Chapter 8
There are still 12 degrees of freedom available to estimate
two-factor interactions
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Design and Analysis of Experiments 8E 2012 Montgomery
54Chapter 8
Resolution V Designs – Page 373
We used a Resolution V design (a 25-2) in Example 8.2
Generally, these are large designs (at least 32 runs) for six or more factors
Non-regular designs can be found using optimal design construction methods
JMP has excellent capability
Examples for k = 6 and 8 factors are illustrated in the book
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Design and Analysis of Experiments 8E 2012 Montgomery
55Chapter 8
Supersaturated Designs
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Design and Analysis of Experiments 8E 2012 Montgomery
56Chapter 8