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DOX 6E Montgomery 1
Design of Engineering ExperimentsPart 4 – 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
DOX 6E Montgomery 2
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 3021
2 230 52 20 40
112 2
52 20 30 401
2 2
A A
B B
A y y
B y y
AB
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The Case of Interaction:
50 12 20 401
2 240 12 20 50
92 2
12 20 40 5029
2 2
A A
B B
A y y
B y y
AB
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Regression Model & The Associated Response Surface
0 1 1 2 2 12 1 2
1 2 1 2 1 2
The least squares fit is
ˆ 35.5 10.5 5.5 0.5 35.5 10.5 5.5
y x x x x
y x x x x x x
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The Effect of Interaction on the Response Surface
Suppose that we add an interaction term to the model:
1 2 1 2ˆ 35.5 10.5 5.5 8y x x x x
Interaction is actually a form of curvature
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Example 5-1 The Battery Life ExperimentText reference pg. 165
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:
1,2,...,
( ) 1,2,...,
1, 2,...,ijk i j ij ijk
i a
y j b
k n
Other models (means model, regression models) can be useful
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Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 177
2 2 2... .. ... . . ...
1 1 1 1 1
2 2. .. . . ... .
1 1 1 1 1
( ) ( ) ( )
( ) ( )
a b n a b
ijk i ji j k i j
a b a b n
ij i j ijk iji j i j k
y y bn y y an y y
n y y y y y y
breakdown:
1 1 1 ( 1)( 1) ( 1)
T A B AB ESS SS SS SS SS
df
abn a b a b ab n
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ANOVA Table – Fixed Effects Case
Design-Expert will perform the computations
Text gives details of manual computing (ugh!) – see pp. 169 & 170
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Design-Expert 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
X = B: TemperatureY = A: Material
A1 A1A2 A2A3 A3
A: MaterialInteraction Graph
Life
B: Temperature
15 70 125
20
62
104
146
188
2
2
22
2
2
DOX 6E Montgomery 15
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
DOX 6E Montgomery 17
Quantitative and Qualitative Factors
Candidate model terms from Design- Expert: Intercept
A B B2
AB B3
AB2
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)
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Regression Model Summary of Results
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Regression Model Summary of Results
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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:
• Complete three-factor example in text, Example 5-5
T A B AB AC
ABC AB K E
SS SS SS SS SS
SS SS SS
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