Introduction to Design of Experiments

University of California at BerkeleyMechanical Engineering DepartmentSummer, 2001

by Michael Montero

Part 1Full Factorial Design and Analysis (2 levels)

Part 2Fractional Factorial Design and Analysis (2 levels)

Part 3Software Introduction and 3-Level or Higher Designs

Introduction to DOE - Part 3

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Available DOE Software

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Example: Minitab v11.21

• 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...

SAS JMP MixsoftS-Plus Nutek Qualitek-4Genstat StatSoftMinitab Adept Scientific DOE_PC IVState-Ease, Design-Expert Process Builder STRATEGYEchip S-Matrix CARDStatgraphics Qualitron Systems DoESSystat RSD Associates MatrexUmetrics MODDE 6

Commercial Software for Experimental Design

DOE Specific

General Statistical Package

Minitab Example: Injection Molding Experiment

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Injection Molding Experiment(Box, G. E. P., Hunter, W. G., and Hunter J.S., “Statistics for Experimenters: AnIntroduction to Design, Data Analysis, and Model Building”, Wiley Interscience, p. 413,1978.)

Problem: Identify important factors effecting part shrinkage.Less shrinkage is better.

DOE: 48IV2 −

Design Generators

Minitab: Create Factorial Design Step by Step

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1

Minitab: Factorial Designs Dialog Box

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Predefined Designs

Custom Designs

Screening Design

Select # of Factors

Summary of Possible 2-Level Designs

Design Selection

2

534

Minitab: Summary of 2-Level Designs

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Minitab: Design Selection

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Minitab: Factorial Designs Dialog Box Cont’d

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6

7 8

Define Factors

Output Selection

Additional Options

Minitab: Define Factors and Actual Level Values

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Minitab: Additional Design Options

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Choose if you wantto fold on certainfactors

Randomize orderof tests

Store design incurrent worksheet

Choose whichfraction to use

AB

C

AB

C

I = + ABC I = - ABC

Minitab: Output Selection

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Display confounding patternup to selected order

Generators, defining relation,and design matrix displayed

Minitab: Command Session Window

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Minitab: Worksheet or Data Window

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Design Matrix

Type in or paste in response values

Minitab: Analyze Factorial Design

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Minitab: Select Response

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Select terms inmodel by order

10

1112

Response, effectsand residual plots

Store effects, residuals,etc. in worksheet

Minitab: Select Terms for Effects Calculation and StoreResults in Worksheet

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11

Store effects inworksheet

Store residualsand fits inworksheet

Minitab: Graphical Options

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Normal ProbabilityPlot of Effects Pareto Chart Select Confidence

Residual plots

Minitab: Effect Calculations

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Minitab: Plots

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Normal Plot of Effects

Pareto Chart of Effects

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

Minitab: C

alculator

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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

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Laser-assisted Composite Mfg. Experiment(Mazumdar and Hoa, 1995.)

Problem: Verify if laser power truly effects compositestrength (measured by short-beam-shear test)

DOE: 31

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 islinear or quadratic over the three levels.

Replicates will allow forestimate of error

αααα = 0.10

Linear and Quadratic Contrasts

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linear contrast = y3 - y1 = -1y1 + 0y2 + 1y3

quadratic contrast = (y3 - y2) - (y2 - y1) = 1y1 - 2y2 + 1y3 =

To define the quadratic contrast, one can use the following argument. If relationshipis linear, then (y3 - y2) and (y2 - y1) should approximately be the same:

= 1y1 - 2y2 + 1y3≈≈≈≈ 0 if relationship is linear

So, in vector form:

linear contrast = (-1 0 1)(y1 y2 y3)T

quad. contrast = (1 -2 1)(y1 y2 y3)T

Linear Contrast Vector (u)

Quad. Contrast Vector (v)

Where dot product of contrast vectors equals 0. Contrast vectors are orthogonal to oneanother ensuring that the contrasts are independent of one another:

u • v = (-1 0 1) • (1 -2 1) = (-1)(1) + (0)(-2) + (1)(1) = 0

Response Vector

0.0034

Linear and Quadratic Effects

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Length (u) = [(-1)2 + (0)2 + (1)2]1/2=

Scale vectors so that they both have unit lengths. Hence, divide contrast vectorby length of vector:

2

Length (v) = [(1)2 + (-2)2 + (1)2]1/2= 6

Linear and Quadratic Effect Estimates:

( ) ( ) 636.8=•−= T321l yyy101

21A

( ) ( ) 109.0−=•−= T321q yyy121

61A

αααα = 0.10ANOVA:

Linear term is significant (Fcalc>Fcrit) whilequadratic term is not (Fcalc<Fcrit)

AVE = 30.921

21.2490.003

Predictive Model and Orthogonal Polynomials

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To be able to predict composite strength through the range of the design space (40 to 60Watts), we must extend the notion of orthogonal contrasts to orthogonal polynomials:

ε+++=6

)(2

)( 21 xPAxPAAVEy ql

Where:

( )10110

501 −=−=−= x

∆mx(x)P

( )12132

10503

323

22

2 −=��

�

�

��

�

�−�

�

�

� −=��

�

�

��

�

�−�

�

�

� −= x∆

mx(x)P

, when x ={40,50,60} respectively

, when x ={40,50,60} respectively

x ≡ laser power∆ ≡ distance between consecutive levelsm ≡ middle level

Example: What is composite strength when laser is powered at 55 Watts?

( )[ ]6

321050)(553

0.1092

1050)(558.63630.921y2 −−

−−+=

34.03.05563.05330.921y =++=

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)�Al x Bl�Al x Bq�Aq x Bl�Aq x Bq

• Also in mixed-level designs (For example 2131)

�A x Bl�A x Bq

• Polynomials with fourth and higher degrees, however, should be avoidedunless response’s behavior can be justified by a physical model

�Data can be well fitted by using higher-degree polynomial model butthe resulting fitted model will lack predictive power

� In regression analysis, this is referred to as overfitting�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

Further Topics: 3-Level Fractional Factorial Designs

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3k-p designs rely on generators and defining relation not based onmultiplicative column but modulus calculus:

Example: 34-1

(Generator: D = ABC) where xD = xA + xB + xC (mod 3)

So:3/3 = 1 remainder 01/3 = 0.3 remainder 12/3 = 0.6 remainder 2

Where: x = coded value (0, 1, or 2)

Column D’s coded pattern is generated by the xA + xB + xC (mod 3) relation

Further Topics: 2m4n Mixed Designs

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2m4n designs can be generated from fractional factorial 2k-p designs bymethod of column replacement:

Example: 27-4 (Generators Not Shown)

2441

Statistical Literature

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Box, G. E. P., Hunter, W. G. and Hunter, J.S., Statistics for Experimenters: An Introduction toDesign, 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.

Wu, C. F. J. and Hamada, M., Experiments: Planning, Analysis, and Parameter DesignOptimization, Wiley Series in Probability and Statistics, 2000.

Myers, R. H. and Montgomery, D. C., Response Surface Methodology: Process and ProductOptimization Using Designed Experiments, Wiley Series in Probability and Statistics, 1995

Experimental Design and Optimization

Statistics and Multiple Linear RegressionWalpole, R. E., Myers and R. H., Myers, S. L., Probability and Statistics for Engineers andScientists, Prentice Hall, 6th edition, 1998.

Sen, A. and Srivastava, M., Regression Analysis: Theory, Methods, and Applications, Springer-Verlag, 1990.