Transcript Design and Analysis of Multi

Design and Analysis of
Multi-Factored Experiments
Fractional Factorials Not Based on
the Powers of 2 – Irregular Designs
L. M. Lye
DOE Course
1
Plackett-Burman Designs
• The standard two-level designs provide the choice
of 4, 8, 16, 32, or more runs, but only to the power
of 2.
• In 1946, Plackett and Burman invented alternative
2 level designs that are multiples of 4.
• The 12-, 20-, 24-, and 28-run PB designs are
particular interest because they fill gaps in the
standard designs.
• Unfortunately, these designs have very messy alias
structures.
L. M. Lye
DOE Course
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PB Designs (continued)
• For example, the 11 factor in the 12-run choice,
which is very popular, causes the main effect to be
aliased with 45 two-factor interactions.
• In theory, if you are willing to accept that
interactions are zero, you may get away with it.
BUT, this is a very dangerous assumption.
• Best to stay away from PB designs – better to use
standard FFDs or those recently developed
Minimum run Resolution V designs.
• PB designs are available in Design-Expert but
avoid it!.
L. M. Lye
DOE Course
3
More Irregular Fraction Designs
• It is possible to do other “irregular” fractions and
still maintain a relatively high resolution.
However, these designs are not orthogonal.
• An example of this design is the ¾ replication for
4 factors. It can be created by identifying the
standard quarter-fraction, and then selecting two
more quarter-fractions. i.e. 4 + 4 + 4 = 12 runs.
• This is a 12-run resolution V design. See next few
pages on the design and alias structure.
• These designs were developed by Peter John
(1961, 1962, 1971).
L. M. Lye
DOE Course
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John’s ¾ Four Factor Screening Design
Std
A
B
C
D
1
-1
-1
-1
-1
2
1
1
-1
-1
3
-1
-1
1
-1
4
1
-1
1
-1
5
-1
1
1
-1
6
1
1
1
-1
7
-1
-1
-1
1
8
1
-1
-1
1
9
-1
1
-1
1
10
1
1
-1
1
11
1
-1
1
1
12
-1
1
1
1
L. M. Lye
DOE Course
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Alias Structure for Factorial Model
Intercept = intercept – ABD
[A] = A – ACD
[B] = B – BCD
[C] = C – ABCD
[D] = D – ABCD
[AB] = AB – ABCD
[AC] = AC –BCD
[AD] = AD –BCD
[BC] = BC – ACD
[BD] = BD – ACD
[CD] = CD – ABD
[ABC] = ABC -ABD
L. M. Lye
DOE Course
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Alias structure for factorial maineffect model
[Intercept] = intercept – 0.333 CD – 0.333 ABC + 0.333 ABD
[A] = A – 0.333 BC – 0.333 BD – 0.333 ACD
[B] = B – 0.333 AC – 0.333 AD - 0.333 ACD
[C] = C – 0.5 AB
[D] = D – 0.5 AB
L. M. Lye
DOE Course
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Warning: Irregular fractions may produce
irregularities in effect estimates
• Irregular fractions have somewhat peculiar alias structures.
E.g. when evaluated for fitting a two-factor interaction
model, they exhibit good properties: main effect aliased
with three-factor interaction, etc.
• But, if you fit only the main effects, they become partially
aliased with one or more two-factor interactions. Main
effects can get inflated by any large 2 factor interactions.
Insignificant main effects may be selected as a result.
• Check the p-values in ANOVA for the selected model
terms. If there are no interactions, or they are relatively
small, then no anomaly.
• Normally not a problem because you would never restrict
yourself to main effects only.
L. M. Lye
DOE Course
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Example: Best set up for using RGB projectors
•
•
•
•
•
•
Factors:
Low
A: Font size
10 pt
B: Font Style
Arial
C: Background
Black
D: Lighting
Off
Response: Readability (seconds)
High
18 pt
Times
White
On
• Readability – time to transcribe a series of random
numbers displayed on the screen by a group of students.
• We will use a irregular fraction design with 12 runs.
L. M. Lye
DOE Course
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Effects Plot
Half Normal plot
DESIGN-EXPERT Plot
Readability
A:
B:
C:
D:
Font Size
Font Sty le
Background
Lighting
99
97
A
Half Normal %probability
95
90
C
85
80
AD
70
60
D
40
20
0
0.00
4.71
9.42
14.13
18.83
|Effect|
L. M. Lye
DOE Course
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ANOVA
Analysis of variance table [Partial sum of squares]
Sum of
Mean F
Source Squares DF
Square Value Prob > F
Model 1501.58 4
375.40 60.64 < 0.0001
A
1064.08 1
1064.08 171.89 < 0.0001
C
266.67 1
266.67 43.08 0.0003
D
16.67
1
16.67 2.69
0.1448
AD
168.75 1
168.75 27.26 0.0012
Residual
43.33
7
6.19
Cor Total 1544.92 11
L. M. Lye
DOE Course
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Results
Interaction Graph
DESIGN-EXPERT Plot
Readability
D: Li ghti ng
58.0187
X = A: Font Size
Y = D: Lighting
Design Points
48.264
Readability
D1 Of f
D2 On
Actual Factors
B: Font Sty le = Arial 38.5093
C: Background = Black
28.7547
19
10.00
12.00
14.00
16.00
18.00
A: Font Si ze
L. M. Lye
DOE Course
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Conclusion
• Bigger font – better readability in general
• Lights on is better with 18 pt but lights off is
better if Font is size 10.
• Saved 4 runs by using irregular fraction design.
• Design-Expert can construct ¾ fraction when the
number of factors is 4, 5, or 6. For 7 factors the
fraction is 3/8; for 8 factors the fraction is 3/16;
and for 9, 10, and 11 factors the fraction is 1/8,
1/16, and 3/64, respectively.
L. M. Lye
DOE Course
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Newer Irregular designs
• There are also newer minimum run Resolution IV
and V designs available in Design-Expert 7. E.g. 6
Factors in 22 runs, 10 factors in 56 runs, etc. These
are generated by computer. Alias structure is
complicated and the designs are slightly nonorthogonal.
• Another approach to obtain irregular fractions is by
use of a semi-foldover where only half the number
of runs are necessary compared to a full foldover.
• E.g. 24-1 = 8 runs + semi-foldover = 12 runs.
• See case study of Hawkins and Lye (2006)
• Semi-foldovers can be done using DX-7.
L. M. Lye
DOE Course
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Recommendations
• Avoid the use of low resolution (Res III) minimum
run designs such as Plackett-Burman designs.
Unless you can assume all interactions are zero
and that time and budget is really tight.
• Irregular fraction design can be used with some
caution. This is usually not too serious a problem.
But check alias structure.
• New min run Res V designs can be used to save
on runs without compromising too much.
L. M. Lye
DOE Course
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