Chapter 5
Design & Analysis of Experiments
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Design of Engineering Experiments
– Introduction to Factorials
•
•
•
•
•
•
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
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Some Basic Definitions
Definition of a factor effect: The change in the mean response
when the factor is changed from low to high
40 + 52 20 + 30
−
= 21
2
2
30 + 52 20 + 40
−
= 11
B = yB + − yB − =
2
2
52 + 20 30 + 40
=
−1
AB = −
2
2
A = y A+ − y A − =
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The Case of Interaction:
50 + 12 20 + 40
A = y A+ − y A − =
−
=1
2
2
40 + 12 20 + 50
B=
yB + − yB − =
−
=
−9
2
2
12 + 20 40 + 50
AB = −
=
−29
2
2
Chapter 5
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Regression Model & The Associated Response Surface
y =β 0 + β1 x1 + β 2 x2 + β12 x1 x2 + ε
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
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The Effect of Interaction on the Response Surface
Suppose that we add an interaction term to the model:
yˆ =35.5 + 10.5 x1 + 5.5 x2 + 8 x1 x2
Interaction is actually a form of curvature
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Example 5.1 The Battery Life Experiment
Text reference pg. 187
A = Material type; B = Temperature (A quantitative variable)
1.
What effects do material type & temperature have on life?
2. Is there a choice of material that would give long life regardless of
temperature (a robust product)?
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The General Two-Factor
Factorial Experiment
a levels of factor A; b levels of factor B; n replicates
This is a completely randomized design
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Statistical (effects) model:
 i = 1, 2,..., a

yijk = µ + τ i + β j + (τβ )ij + ε ijk  j = 1, 2,..., b
k = 1, 2,..., n

Other models (means model, regression models) can be useful
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Extension of the ANOVA to Factorials
(Fixed Effects Case) – pg. 189
a
b
n
a
b
=i 1
=j 1
2
2
2
(
)
(
)
(
)
y
y
bn
y
y
an
y
y
−
=
−
+
−
∑∑∑ ijk ...
∑ i.. ...
∑ . j. ...
k 1
=i 1 =j 1 =
a
b
a
b
n
+ n∑∑ ( yij . − yi.. − y. j . + y... ) + ∑∑∑ ( yijk − yij . ) 2
2
=i 1 =j 1
k 1
=i 1 =j 1=
SST = SS A + SS B + SS AB + SS E
df breakdown:
abn − 1 = a − 1 + b − 1 + (a − 1)(b − 1) + ab(n − 1)
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ANOVA Table – Fixed Effects Case
JMP and Design-Expert will perform the computations
Text gives details of manual computing (ugh!) – see pp.
192
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Chapter 5
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Design-Expert Output – Example 5.1
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JMP output – Example 5.1
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Residual Analysis – Example 5.1
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Residual Analysis – Example 5.1
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Interaction Plot
DESIGN-EXPERT Plot
Life
Inte ra c tio n G ra p h
A : M a te ria l
188
X = B: Temperature
Y = A: Material
A1 A1
A2 A2
A3 A3
L i fe
146
104
2
2
62
2
20
15
70
125
B : T e m p e ra tu re
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Quantitative and Qualitative Factors
• 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
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Quantitative and Qualitative Factors
A = Material type
B = Linear effect of Temperature
B2 = Quadratic effect of
Temperature
AB = Material type – TempLinear
AB2 = Material type - TempQuad
B3 = Cubic effect of
Temperature (Aliased)
Chapter 5
Candidate model
terms from DesignExpert:
Intercept
A
B
B2
AB
B3
AB2
Design & Analysis of Experiments
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Quantitative and Qualitative Factors
Chapter 5
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Regression Model Summary of Results
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Design & Analysis of Experiments
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Regression Model Summary of Results
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Design & Analysis of Experiments
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Design & Analysis of Experiments
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Design & Analysis of Experiments
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Design & Analysis of Experiments
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Factorials with More Than
Two Factors
• Basic procedure is similar to the two-factor case; all
abc…kn treatment combinations are run in random
order
• ANOVA identity is also similar:
SST = SS A + SS B +  + SS AB + SS AC + 
+ SS ABC +  + SS ABK + SS E
• Complete three-factor example in text, Example 5.5
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