Factorial Experiments
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•
•
•
•
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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
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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
B  yB  yB 

 11
2
2
52  20 30  40
AB 

 1
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
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Regression Model & The Associated Response Surface
y  0  1 x1   2 x2  12 x1 x2  
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  10.5x1  5.5x2  8x1 x2
Interaction is actually a form of curvature
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Example 5.1 The Battery Life Experiment
Text reference pg. 167
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     i   j  ( )ij   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. 168
a
b
n
a
b
i 1
j 1
2
2
2
(
y

y
)

bn
(
y

y
)

an
(
y

y
)
 ijk ...
 i.. ...
 . j. ...
i 1 j 1 k 1
a
b
a
b
n
 n ( yij .  yi..  y. j .  y... )  ( yijk  yij . ) 2
2
i 1 j 1
i 1 j 1 k 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
Design-Expert will perform the computations
Text gives details of manual computing (ugh!) –
see pp. 171
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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
Interaction Graph
A: Material
188
X = B: Temperature
Y = A: Material
A1 A1
A2 A2
A3 A3
Life
146
104
2
2
62
2
20
15
70
125
B: Tem perature
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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
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Quantitative and Qualitative Factors
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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:
SST  SS A  SS B 
 SS ABC 
 SS AB  SS AC 
 SS AB
K
 SS E
• Complete three-factor example in text, Example 5.5
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