Some Basic Definitions
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Chapter 5
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Design & Analysis of Experiments
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B yB yB 2
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Definition of a factor effect: The change in the mean response
when the factor is changed from low to high
Chapter 5
Chapter 5
Design & Analysis of Experiments
8E 2012 Montgomery
y A y A
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The Case of Interaction:
Design & Analysis of Experiments
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Text reference, Chapter 5
General principles of factorial experiments
The two-factor factorial with fixed effects
The ANOVA for factorials
Extensions to more than two factors
Quantitative and qualitative factors –
response curves and surfaces
Chapter 5
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•
•
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Design of Engineering Experiments
– Introduction to Factorials
E 0 E1 x1 E 2 x2 E12 x1 x2 H
Design & Analysis of Experiments
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What effects do material type & temperature have on life?
Chapter 5
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2. Is there a choice of material that would give long life regardless of
temperature (a robust product)?
1.
A = Material type; B = Temperature (A quantitative variable)
Example 5.1 The Battery Life Experiment
Text reference pg. 187
Chapter 5
The least squares fit is
yˆ 35.5 10.5 x1 5.5 x2 0.5 x1 x2 # 35.5 10.5 x1 5.5 x2
y
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Regression Model & The Associated Response Surface
35.5 10.5 x1 5.5 x2 8 x1 x2
The General Two-Factor
Factorial Experiment
Design & Analysis of Experiments
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Chapter 5
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This is a completely randomized design
a levels of factor A; b levels of factor B; n replicates
Chapter 5
Interaction is actually a form of curvature
yˆ
Suppose that we add an interaction term to the model:
8
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The Effect of Interaction on the Response Surface
Design & Analysis of Experiments
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Chapter 5
Design & Analysis of Experiments
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Text gives details of manual computing (ugh!) – see pp.
192
JMP and Design-Expert will perform the computations
ANOVA Table – Fixed Effects Case
Chapter 5
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Other models (means model, regression models) can be useful
yijk
­ i 1, 2,..., a
°
P W i E j (WE )ij H ijk ® j 1, 2,..., b
°k 1, 2,..., n
¯
Statistical (effects) model:
b
n
b
j 1
b
a
b
n
SS A SS B SS AB SS E
i 1 j 1
i 1 j 1 k 1
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Design & Analysis of Experiments
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n¦¦ ( yij . yi.. y. j . y... ) 2 ¦¦¦ ( yijk yij . ) 2
a
i 1
a
bn¦ ( yi.. y... ) 2 an¦ ( y. j . y... ) 2
df breakdown:
abn 1 a 1 b 1 (a 1)(b 1) ab(n 1)
SST
i 1 j 1 k 1
¦¦¦ ( yijk y... )2
a
Extension of the ANOVA to Factorials
(Fixed Effects Case) – pg. 189
Chapter 5
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Residual Analysis – Example 5.1
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Design-Expert Output – Example 5.1
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Residual Analysis – Example 5.1
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JMP output – Example 5.1
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B : T e m p e ra tu re
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A : M a te ria l
Inte ra c tio n G ra p h
Design & Analysis of Experiments
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104
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Candidate model
terms from DesignExpert:
Intercept
A
B
B2
AB
B3
AB2
Design & Analysis of Experiments
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B3 = Cubic effect of
Temperature (Aliased)
AB2 = Material type - TempQuad
AB = Material type – TempLinear
B2 = Quadratic effect of
Temperature
B = Linear effect of Temperature
A = Material type
Quantitative and Qualitative Factors
Chapter 5
A1 A1
A2 A2
A3 A3
X = B: Temperature
Y = A: Material
Life
DESIGN-EXPERT Plot
Interaction Plot
L i fe
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Design & Analysis of Experiments
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Chapter 5
Design & Analysis of Experiments
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Quantitative and Qualitative Factors
Chapter 5
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• The basic ANOVA procedure treats every factor as if it
were qualitative
• Sometimes an experiment will involve both quantitative
and qualitative factors, such as in Example 5.1
• This can be accounted for in the analysis to produce
regression models for the quantitative factors at each level
(or combination of levels) of the qualitative factors
• These response curves and/or response surfaces are often
a considerable aid in practical interpretation of the results
Quantitative and Qualitative Factors
Design & Analysis of Experiments
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Design & Analysis of Experiments
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Chapter 5
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Regression Model Summary of Results
Chapter 5
Chapter 5
Design & Analysis of Experiments
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Design & Analysis of Experiments
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Regression Model Summary of Results
Design & Analysis of Experiments
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Design & Analysis of Experiments
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Chapter 5
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SS ABC SS ABK SS E
SS A SS B SS AB SS AC Chapter 5
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• Complete three-factor example in text, Example 5.5
SST
• Basic procedure is similar to the two-factor case; all
abc…kn treatment combinations are run in random
order
• ANOVA identity is also similar:
Factorials with More Than
Two Factors