Introduction to Design of Experiments
by Michael Montero
University of California at Berkeley
Mechanical Engineering Department
Summer, 2001
Introduction to DOE - Part 3
Part 1
Full Factorial Design and Analysis (2 levels)
Part 2
Fractional Factorial Design and Analysis (2 levels)
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Part 3
Software Introduction and 3-Level or Higher Designs
M. G. Montero
Available DOE Software
Commercial Software for Experimental Design
SAS JMP
S-Plus
Genstat
Minitab
State-Ease, Design-Expert
Echip
Statgraphics
Systat
Umetrics MODDE 6
Example: Minitab v11.21
DOE Specific
Windows Based (Windows 9x, NT, and 2000)
Spreadsheet-like interface and command line interface
User-friendly menus
2k full and fractional factorial designs (regular and non-regular)
Response surface building
Analysis of Variance (ANOVA)
Multiple linear regression
Statistical Process Control (SPC), time-series analysis (autoregression)
Reproducibility and Repeatability (R&R)
And more...
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•
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•
•
•
•
•
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Mixsoft
Nutek Qualitek-4
StatSoft
General Statistical Package
Adept Scientific DOE_PC IV
Process Builder STRATEGY
S-Matrix CARD
Qualitron Systems DoES
RSD Associates Matrex
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Minitab Example: Injection Molding Experiment
Injection Molding Experiment
(Box, G. E. P., Hunter, W. G., and Hunter J.S., “Statistics for Experimenters: An
Introduction to Design, Data Analysis, and Model Building”, Wiley Interscience, p. 413,
1978.)
Problem: Identify important factors effecting part shrinkage.
Less shrinkage is better.
8 −4
2IV
Design Generators
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DOE:
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Minitab: Create Factorial Design Step by Step
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1
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Minitab: Factorial Designs Dialog Box
Summary of Possible 2-Level Designs
Predefined Designs
2
Custom Designs
Screening Design
4
5
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3
Select # of Factors
Design Selection
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Minitab: Summary of 2-Level Designs
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4
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Minitab: Design Selection
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5
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Minitab: Factorial Designs Dialog Box Cont’d
Define Factors
7
8
Output Selection
Additional Options
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6
Minitab: Define Factors and Actual Level Values
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6
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Minitab: Additional Design Options
I = - ABC
I = + ABC
C
C
Choose which
fraction to use
A
B
A
B
7
Choose if you want
to fold on certain
factors
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Randomize order
of tests
Store design in
current worksheet
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Minitab: Output Selection
Generators, defining relation,
and design matrix displayed
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8
Display confounding pattern
up to selected order
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Minitab: Command Session Window
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Minitab: Worksheet or Data Window
Design Matrix
Type in or paste in response values
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Minitab: Analyze Factorial Design
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9
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Minitab: Select Response
9
Select terms in
model by order
12
Response, effects
and residual plots
11
Store effects, residuals,
etc. in worksheet
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10
Minitab: Select Terms for Effects Calculation and Store
Results in Worksheet
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10
Store effects in
worksheet
11
Store residuals
and fits in
worksheet
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Minitab: Graphical Options
Normal Probability
Pareto Chart
Plot of Effects
Select Confidence
12
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Residual plots
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Minitab: Effect Calculations
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Minitab: Plots
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Normal Plot of Effects
Pareto Chart of Effects
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Minitab: Factorial Plots
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Minitab: Setup of Factorial Plots
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Minitab: Main and Interaction Plots
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All Main Effects Plot
AE Interaction Plot
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Minitab: Calculator
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Minitab: Calculator Used to Construct AE Column
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Minitab: Multiple Regression
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Minitab: Linear Fit
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Three level or Higher Factor Levels
Laser-assisted Composite Mfg. Experiment
(Mazumdar and Hoa, 1995.)
DOE: 31
Replicates will allow for
estimate of error
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Problem: Verify if laser power truly effects composite
strength (measured by short-beam-shear test)
ANOVA: Analysis of variance indicates that laser power does significantly effect (Fcalc > Fcrit)
composite strength. Next, look into whether the relationship between the factor and response is
linear or quadratic over the three levels.
α = 0.10
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Linear and Quadratic Contrasts
linear contrast = y3 - y1 = -1y1 + 0y2 + 1y3
To define the quadratic contrast, one can use the following argument. If relationship
is linear, then (y3 - y2) and (y2 - y1) should approximately be the same:
quadratic contrast = (y3 - y2) - (y2 - y1) = 1y1 - 2y2 + 1y3 =
= 1y1 - 2y2 + 1y3≈ 0 if relationship is linear
linear contrast = (-1 0 1)(y1 y2 y3)T
Linear Contrast Vector (u)
Response Vector
quad. contrast = (1 -2 1)(y1 y2 y3)T
Quad. Contrast Vector (v)
Where dot product of contrast vectors equals 0. Contrast vectors are orthogonal to one
another ensuring that the contrasts are independent of one another:
u • v = (-1 0 1) • (1 -2 1) = (-1)(1) + (0)(-2) + (1)(1) = 0
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So, in vector form:
Linear and Quadratic Effects
Scale vectors so that they both have unit lengths. Hence, divide contrast vector
by length of vector:
Length (u) = [(-1)2 + (0)2 + (1)2]1/2= 2
Length (v) = [(1)2 + (-2)2 + (1)2]1/2= 6
Al =
Aq =
1
2
1
6
(− 1
(1
0 1) • ( y1
y2
y3 )T = 8.636
− 2 1) • ( y1
y2
y3 )T = −0.109
AVE = 30.921
ANOVA:
α = 0.10
21.249
0.0034
0.003
Linear term is significant (Fcalc>Fcrit) while
quadratic term is not (Fcalc<Fcrit)
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Linear and Quadratic Effect Estimates:
Predictive Model and Orthogonal Polynomials
To be able to predict composite strength through the range of the design space (40 to 60
Watts), we must extend the notion of orthogonal contrasts to orthogonal polynomials:
y = AVE + Al
P1 ( x)
2
+ Aq
P2 ( x)
6
+ε
Where:
x − m x − 50
=
= (− 1 0 1) , when x ={40,50,60} respectively
∆
10
éæ x − 50 ö 2 2 ù
éæ x − m ö 2 2 ù
P2(x) = 3êç
÷ − ú = (1 − 2 1) , when x ={40,50,60} respectively
÷ − ú = 3êç
3 úû
3 úû
êëè 10 ø
êëè ∆ ø
P1(x) =
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x ≡ laser power
∆ ≡ distance between consecutive levels
m ≡ middle level
Example: What is composite strength when laser is powered at 55 Watts?
y = 30.921 + 8.636
(55 − 50) 10
− 0.109
2
y = 30.921 + 3.053 + .0556 = 34.03
[
3 ((55 − 50) 10 )2 − 2
3
]
6
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Extending Orthogonal Polynomials
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• Model can be extended for any level equally spaced (4, 5, 6, etc.)
• Analysis is the same using factorial plots, normal plot of effects, and
confidence intervals or ANOVA for statistical testing
• Analysis of equal level DOEs with more than one factor is the same but we
must also consider interaction estimates (For example 32)
Þ Al x Bl
Þ Al x Bq
Þ Aq x Bl
Þ Aq x Bq
• Also in mixed-level designs (For example 2131)
Þ A x Bl
Þ A x Bq
• Polynomials with fourth and higher degrees, however, should be avoided
unless response’s behavior can be justified by a physical model
Þ Data can be well fitted by using higher-degree polynomial model but
the resulting fitted model will lack predictive power
Þ In regression analysis, this is referred to as overfitting
Þ Average variance of the regression parameter estimates is proportional
to the number of regression parameters in the model.
→ Overfitting inflates variance and lowers accuracy of predictive
model (Draper and Smith, 1998)
• Higher degree polynomials become harder to interpret
M. G. Montero
Further Topics: 3-Level Fractional Factorial Designs
3k-p designs rely on generators and defining relation not based on
multiplicative column but modulus calculus:
Example: 34-1
(Generator: D = ABC) where xD = xA + xB + xC (mod 3)
Where:
x = coded value (0, 1, or 2)
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So:
3/3 = 1 remainder 0
1/3 = 0.3 remainder 1
2/3 = 0.6 remainder 2
Column D’s coded pattern is generated by the
xA + xB + xC (mod 3) relation
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Further Topics: 2m4n Mixed Designs
2m4n designs can be generated from fractional factorial 2k-p designs by
method of column replacement:
Example: 27-4 (Generators Not Shown)
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2441
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Statistical Literature
Experimental Design and Optimization
Box, G. E. P., Hunter, W. G. and Hunter, J.S., Statistics for Experimenters: An Introduction to
Design, Data Analysis, and Model Building, Wiley Series in Probability and Statistics, 1978.
Devor, R. E., Chang, T. and Sutherland, J. W., Statistical Quality Design and Control:
Contemporary Concepts and Methods, Macmillan, 1992.
Ross, P. J., Taguchi Techniques for Quality Engineering, McGraw Hill, 2nd Edition, 1996.
Myers, R. H. and Montgomery, D. C., Response Surface Methodology: Process and Product
Optimization Using Designed Experiments, Wiley Series in Probability and Statistics, 1995
Statistics and Multiple Linear Regression
Walpole, R. E., Myers and R. H., Myers, S. L., Probability and Statistics for Engineers and
Scientists, Prentice Hall, 6th edition, 1998.
Sen, A. and Srivastava, M., Regression Analysis: Theory, Methods, and Applications, SpringerVerlag, 1990.
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Wu, C. F. J. and Hamada, M., Experiments: Planning, Analysis, and Parameter Design
Optimization, Wiley Series in Probability and Statistics, 2000.