Design of Experiments Review
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![Page 1: Design of Experiments Review](https://reader037.fdocuments.in/reader037/viewer/2022102710/54befc124a795941758b45c5/html5/thumbnails/1.jpg)
Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
IXCUSTOMER & COMPETITIVE INTELLIGENCE
FOR SYSTEMS INNOVATION & DESIGN
S IGMAS DEPARTMENT OF
STATISTICSDR. RICK EDGEMAN, PROFESSOR & CHAIR – SIX SIGMA BLACK BELT
[email protected] OFFICE: +1-208-885-4410
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
IXS IGMAS DEPARTMENT OF
STATISTICS
Design of Experimentsand
2k Factorial Designs
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
IXS IGMAS
DEPARTMENT OF
STATISTICS
a highly structured strategy for acquiring, assessing, and applying customer, competitor, and enterprise intelligence for the purposes of product, system or enterprise innovation and
design.
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
The Scientific Context of Quality
Improvement
The Scientific Method & Informed Observation;Data Driven Decision Making;
Directed Experimentation;
2k Factorial Experiments, Interaction and Scree Plots;
2k-p Fractional Factorial Designs;Central Composite Designs & Response Surface Methods;
Process Optimization - Selecting Your Settings.
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
The Scientific Method of Informed Observation
Design ofExperiments
Sin
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Soliciting, Hearing & Acting Upon theVoice of the Process
“In God we trust. ... all others must bring data.”
Thank you Dr. Freud.It is key to effective
DECISION-MAKING!
DATA!You are DRIVEN!
Data-Driven Decision Making
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six SigmaWhere and What Do We Measure
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Directed Experimentation
Regression data is commonly observational in nature, having arisen simply from observing the response variable, Y, and noting the values of the driver variables which led to the response.
In contrast, data in an experimental design setting usually arise from planning or setting the values of interest of the driver variables and then observing the response variable.
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Experimental DesignsThe scheme used to determine the settings of the
drivervariables and of collecting the data is referred to as anexperimental design. Three very useful classes ofexperimental designs are:
2k factorial designs,
2k-p fractional factorial designs, and
Central Composite Designs (CCD)
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
2k Factorial Designs2k factorial designs are experimental designs for which there are k factors (or driver variables) and each of these factors will be investigated at 2 levels,“high” and “low” or, symbolically, “+” and “-”.
All possible combinations of factor levels are used inthe investigation. That is, if there are k= three driver variables, then the data that would be collected would be represented as follows:
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
23 Factorial Design Data
X1 X2 X3 Y
- - - Y1
- - + Y2
- + - Y3
- + + Y4
+ - - Y5
+ - + Y6
+ + - Y7
+ + + Y8
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
The 23 Full Factorial Model
The full factorial model when k = 3 is given by:
Y = 0 + 1X1 + 2X2 + 3X3 + 12X1X2 + 13X1X3 + 23X2X3 + 123X1X2X3 +
It is rare to investigate the “three-factor interaction” term. A replicated 2k design (r2k)would gather r observations under each of the2k (factorial) combinations previously listed.
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Modified 23 Full Factorial Data Set
X1 X2 X3 X1X2 X1X3 X2X3 X1X2X3 Y
- - - + + + - Y1
- - + + - - + Y2
- + - - + - + Y3
- + + - - + - Y4
+ - - - + + + Y5
+ - + - + - - Y6
+ + - + - - - Y7
+ + + + + + + Y8
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
IXS IGMAS DEPARTMENT OF
STATISTICS
Example:Optimization of a
Flexible Packaging Material Process
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
A 24 Factorial Design Example:Seal Strength of Flexible Packaging Material
A key characteristic of a flexible packaging material is its seal strength, measured in grams / square inch.
This is the force required to separate the seal once it has been made.
A flexible packaging material manufacturer has identified four variables which are believed to influence the seal strength (Y) of a particular material and has specified operating ranges for these variables which, it is thought, are broad enough to identify the impact of the variable if, in fact, there is an impact. These variables follow.
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six SigmaFlexible Packaging Material
VariablesResponse Variable: Y = Seal Strength (gm/si)Driver Variables: High = +1 Low = -1
Temperature in Degrees 300 250Pressure psi 100 80Material Thickness (inch) .03 .02Dwell in Seconds .20 .10
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Flexible Packaging Material Data
Degrees Pressure Gage Dwell Strength
-1 -1 -1 -1 150 -1 -1 -1 1 158 -1 -1 1 -1 141 -1 -1 1 1 163 -1 1 -1 -1 160 -1 1 -1 1 164 -1 1 1 -1 147 -1 1 1 1 168 1 -1 -1 -1 153 1 -1 -1 1 159 1 -1 1 -1 149 1 -1 1 1 160 1 1 -1 -1 170 1 1 -1 1 163 1 1 1 -1 171 1 1 1 1 178
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
Factor Main Effects for Seal Strength of aFlexible Packaging Material
DwellGagePressureDegrees
165.0
162.5
160.0
157.5
155.0
Str
engt
hMain Effects for Flexible Packaging Material
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
DegreesLow = -1 High = 1
High = 1 Gage Low = -1
PressureLow = -1 High = 1
High = 1
Dwell
Low = -1
163 168 160 178
158 164 159 163
141 147 149 171
150 160 153 170
Graphic Representation:Flexible Packaging Material Example
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma DegreesLow = -1 High = 1
High = 1 Gage Low = -1
PressureLow = -1 High = 1
High = 1
Dwell
Low = -1
163 168 160 178
158 164 159 163
141 147 149 171
150 160 153 170
Main Effect for Degrees
Right Cube vs. Left Cube:(153 + 149 + … + 178)/8 - (150 + 141 + … + 168)/8
= 1303/8 - 1251/8 = 162.875 - 156.375 = 6.50
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma DegreesLow = -1 High = 1
High = 1 Gage Low = -1
PressureLow = -1 High = 1
High = 1
Dwell
Low = -1
163 168 160 178
158 164 159 163
141 147 149 171
150 160 153 170
Right Faces vs. Left Faces [(160 + … + 168 + 170 + … + 178)/8] - [(150 + … + 163 + 153 + … + 160)/8]
= 165.125 - 154.125 = 11.0
Main Effect for Pressure
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma DegreesLow = -1 High = 1
High = 1 Gage Low = -1
PressureLow = -1 High = 1
High = 1
Dwell
Low = -1
163 168 160 178
158 164 159 163
141 147 149 171
150 160 153 170
Main Effect for Gage
Back Faces vs. Front Faces (141 + 163 + ….. + 178)/8 - (150 + 158 + ….. + 163)/8 =
159.625 - 159.625 = 0.0
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma DegreesLow = -1 High = 1
High = 1 Gage Low = -1
PressureLow = -1 High = 1
High = 1
Dwell
Low = -1
163 168 160 178
158 164 159 163
141 147 149 171
150 160 153 170
Main Effect for Dwell
Top Faces vs. Bottom Faces
(158 + 163 + …. + 163 + 178)/8 - (150 + 141 + … + 170 + 171)/8 =
164.125 - 155.125 = 9.0
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
1
1
-1-1
1
1-1
-1
1 1
-1-1
1
1-1
-1
1
1
-1-1
1
1
-1
-1
Degrees
Pressure
Gage
Dwell
Two Factor Interactions for Strength Data
Interaction Effects for Seal Strength of a
Flexible Packaging Material
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma DegreesLow = -1 High = 1
High = 1 Gage Low = -1
PressureLow = -1 High = 1
High = 1
Dwell
Low = -1
163 168 160 178
158 164 159 163
141 147 149 171
150 160 153 170
Interaction Effect for Dwell with Degrees
(150 + 141 + 160 + 147 + 159 + 160 + 163 + 178)/8 -(158 + 163 + 164 + 168 + 153 + 149 + 170 + 171)/8 =
157.25 - 162.00 = -4.75
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma Analysis of Variance for Seal Strength
Source DF SS MS Fcalc Pvalue
Degrees 1 169.00 169.00 12.95 0.016Pressure 1 484.00 484.00 37.09 0.002Gage 1 0.00 0.00 0.00 1.000Dwell 1 324.00 324.00 24.83 0.004
Degrees*Pressure 1 72.25 72.25 5.54 0.065Degrees*Gage 1 42.25 42.25 3.24 0.132Degrees*Dwell 1 90.25 90.25 6.92 0.047Pressure*Gage 1 12.25 12.25 0.94 0.377Pressure*Dwell 1 30.25 30.25 2.32 0.188Gage*Dwell 1 156.25 156.25 11.97 0.018
Error 5 65.25 13.05
Total 15 1445.75
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six SigmaWhich Terms are Important?An Application of Scree Plots
10987654321
500
400
300
200
100
0
Index
Sum
_Sqr
s
Sum of Squares Scree Plot for Flexible Packaging Material
Sums of Squares Above the LineMay be Associated with ActiveEffects and Interactions.
SS Below the LineMay be Associated with Inactive Effects & Interactions.
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma Regression AnalysisStrength = 160 + 3.25 Degrees + 5.50 Pressure + 0.000 Gage + 4.50 Dwell + 2.12 temp*pr + 1.62 temp*gag - 2.38 temp*dw + 0.875 pres*gag - 1.37 pres*dw + 3.13 gage*dw
Predictor Coefficient Effect Std.Dev. Tcalc
Pvalue
Constant 159.625 ----- 0.9031 176.75 0.000
Degrees 3.250 6.500 0.9031 3.60 0.016 Pressure 5.500 11.000 0.9031
6.09 0.002 Gage 0.000 0.000 0.9031 0.00 1.000 Dwell 4.500 9.000 0.9031 4.98 0.004 temp*pr 2.125 4.250 0.9031
2.35 0.065 temp*gag 1.625 3.250 0.9031
1.80 0.132 temp*dw -2.375 -4.750 0.9031 -2.63 0.047 pres*gag 0.875 1.750 0.9031
0.97 0.377 pres*dw -1.375 -2.750 0.9031
-1.52 0.188 gage*dw 3.125 6.250 0.9031
3.46 0.018
S = 3.612 R2 = 95.5% R2adj =
86.5%
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six SigmaAnalysis of Variance:Overall Model
Source DF SS MS Fcalc Pvalue
Regression 10 1380.50 138.05 10.58 0.009Error 5 65.25 13.05
Total 15 1445.75
Unusual ObservationsObs Degrees Strength Fit StDev Fit Residual St Resid 10 1.00 159.000 154.875 2.995 4.125 2.04 R
R denotes an observation with a large standardized residual
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Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation
Dr. Rick L. Edgeman, University of Idaho
Six Sigma
IXS IGMAS DEPARTMENT OF
STATISTICS
End of Session