Transcript Design of Engineering Experiments Part 4 – Introduction to
Design of Engineering Experiments Part 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
response curves and surfaces factors – DOX 6E Montgomery 1
Some Basic Definitions Definition of a factor effect: The change in the mean response when the factor is changed from low to high
A
y A
y A
2 2
B
y B
y B
AB
2 2 1 2 DOX 6E Montgomery 2 21 11 2
The Case of Interaction:
A
y A
y A
2 2
B
y B
y B
AB
2 2 29 2 DOX 6E Montgomery 2 1 9 3
Regression Model & The Associated Response Surface
y
0
x
1 1 2
x
2
x x
12 1 2 The least squares fit is
x
1 5.5
x
2 0.5
x x
1 2 DOX 6E Montgomery
x
1 5.5
x
2 4
The Effect of Interaction on the Response Surface
Suppose that we add an interaction term to the model:
x
1 5.5
x
2 8
x x
1 2
Interaction
is actually a form of
curvature
DOX 6E Montgomery 5
Example 5-1 The Battery Life Experiment Text 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)?
DOX 6E Montgomery 6
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:
y ijk j
( )
ij
ijk
k i j
1, 2,..., 1, 2,..., 1, 2,...,
a b n
Other models (means model, regression models) can be useful DOX 6E Montgomery 8
Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 177
i a
1
j b
1
k n
1 (
y ijk
y
...
) 2
bn i a
1 (
y i
..
y
...
) 2
an j b
1 (
y
n a b
1 1
i j
(
y ij
.
y i
..
y
y
...
) 2
y
...
) 2
i a b n
1 1 1
j k
(
y ijk
y ij
.
) 2
SS T
SS A
SS B
SS AB
SS E df
breakdown:
abn a
1
b
1 (
a
1)(
b
1) DOX 6E Montgomery 9
ANOVA Table – Fixed Effects Case Design-Expert
will perform the computations Text gives details of
manual computing
(ugh!) – see pp. 169 & 170 DOX 6E Montgomery 10
Design-Expert Output – Example 5-1
DOX 6E Montgomery 11
Residual Analysis – Example 5-1
DOX 6E Montgomery 12
Residual Analysis – Example 5-1
DOX 6E Montgomery 13
DESIGN-EXPERT Plot Life X = B: Temperature Y = A: Material A1 A1 A2 A2 A3 A3
Interaction Plot
Interaction Graph A: Material 188 146 104 62 20 15 70 B: Tem perature DOX 6E Montgomery 125 14
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 DOX 6E Montgomery 15
Quantitative and Qualitative Factors
DOX 6E Montgomery 16
Quantitative and Qualitative Factors
A
= Material type
B
= Linear effect of Temperature
B
2 = Quadratic effect of Temperature
AB
= Material type – Temp Linear
AB
2 = Material type - Temp Quad
B
3 = Cubic effect of Temperature (Aliased) Candidate model terms from Design Expert: Intercept A B B 2 AB B 3 AB 2 DOX 6E Montgomery 17
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:
SS T
SS A
SS ABC SS B
SS AB SS AB K
SS SS E AC
• Complete three-factor example in text, Example 5-5 DOX 6E Montgomery 20