The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs The...
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The Essentials of 2-Level Design of The Essentials of 2-Level Design of Experiments Experiments Part I: The Essentials of Full Factorial Part I: The Essentials of Full Factorial Designs Designs Developed by Don Edwards, John Grego and Developed by Don Edwards, John Grego and James Lynch James Lynch Center for Reliability and Quality Sciences Center for Reliability and Quality Sciences Department of Statistics Department of Statistics University of South Carolina University of South Carolina 803-777-7800 803-777-7800
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Transcript of The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs The...
The Essentials of 2-Level Design of The Essentials of 2-Level Design of ExperimentsExperiments
Part I: The Essentials of Full Factorial DesignsPart I: The Essentials of Full Factorial Designs
Developed by Don Edwards, John Grego and James Developed by Don Edwards, John Grego and James LynchLynch
Center for Reliability and Quality SciencesCenter for Reliability and Quality SciencesDepartment of StatisticsDepartment of Statistics
University of South CarolinaUniversity of South Carolina803-777-7800803-777-7800
Part I. Full Factorial DesignsPart I. Full Factorial Designs
2244 Designs Designs– IntroductionIntroduction– Analysis ToolsAnalysis Tools– ExampleExample– Violin ExerciseViolin Exercise
22kk Designs Designs
2244 Designs Designs IntroductionIntroduction
Suppose the effects of four factors, each having Suppose the effects of four factors, each having two levels, are to be investigated.two levels, are to be investigated.
How many combinations of factor levels are How many combinations of factor levels are there?there?– With 16 runs, one per each treatment With 16 runs, one per each treatment
combination, we can estimate:combination, we can estimate: four main effects - (A,B,C,D)four main effects - (A,B,C,D) six two-way interactions - six two-way interactions -
(AB,AC,AD,BC,BD,CD)(AB,AC,AD,BC,BD,CD) four three-way interactions - four three-way interactions -
(ABC,ABD,ACD,BCD)(ABC,ABD,ACD,BCD) one four-way interaction (ABCD).one four-way interaction (ABCD).
2244 Designs Designs
Analysis ToolsAnalysis Tools - - Design MatrixDesign Matrix
StandardOrder A B C D
1 1 1 1 1 2 2 1 1 1 3 1 2 1 1 4 2 2 1 1 5 1 1 2 1 6 2 1 2 1 7 1 2 2 1 8 2 2 2 1 9 1 1 1 210 2 1 1 211 1 2 1 212 2 2 1 213 1 1 2 214 2 1 2 215 1 2 2 216 2 2 2 2
2244 Designs Designs
Analysis ToolsAnalysis Tools - - Design Matrix Signs TableDesign Matrix Signs Table
StandardOrder A B C D
1 -1 -1 -1 -1 2 1 -1 -1 -1 3 -1 1 -1 -1 4 1 1 -1 -1 5 -1 -1 1 -1 6 1 -1 1 -1 7 -1 1 1 -1 8 1 1 1 -1 9 -1 -1 -1 110 1 -1 -1 111 -1 1 -1 112 1 1 -1 113 -1 -1 1 114 1 -1 1 115 -1 1 1 116 1 1 1 1
2244 Designs Designs
Analysis ToolsAnalysis Tools - Signs Table- Signs Table
Main Effects Interaction EffectsActualOrder y A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
-1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1-1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1
1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1-1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1-1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1
1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1-1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1
1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1-1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1
1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1-1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1
1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1-1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Sum
Divisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect
2244 Designs Designs
Analysis ToolsAnalysis Tools - Fifteen Effects Paper- Fifteen Effects Paper
2244 Designs Designs Example 4Example 4**
Response: Computer throughput Response: Computer throughput (kbytes/sec), (large y’s are desirable)(kbytes/sec), (large y’s are desirable)
Factors: A, B, C and D were various Factors: A, B, C and D were various performance tuning parameters such as performance tuning parameters such as – number of buffersnumber of buffers– size of unix inode tables for file handlingsize of unix inode tables for file handling
**Data courtesy of Greg DobbinsData courtesy of Greg Dobbins
2244 Designs Designs Example 4 - Experimental Report FormExample 4 - Experimental Report Form
RunOrder
StandardOrder
A B C D y
1 14 1 -1 1 1 752 8 1 1 1 -1 743 3 -1 1 -1 -1 644 16 1 1 1 1 725 6 1 -1 1 -1 736 11 -1 1 -1 1 667 13 -1 -1 1 1 668 2 1 -1 -1 -1 709 1 -1 -1 -1 -1 66
10 7 -1 1 1 -1 6811 15 -1 1 1 1 6712 10 1 -1 -1 1 7013 12 1 1 -1 1 7014 4 1 1 -1 -1 7115 9 -1 -1 -1 1 6716 5 -1 -1 1 -1 67
2244 Designs Designs Example 4 - Signs TableExample 4 - Signs Table
U-Do-ItU-Do-It Fill Out the Signs Table to Fill Out the Signs Table to
Estimate the Factor EffectsEstimate the Factor Effects
Fill Out the Signs Table to Fill Out the Signs Table to Estimate the Factor EffectsEstimate the Factor Effects
Main Effects Interaction EffectsRun
Order y A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
9 66 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 18 70 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -13 64 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -114 71 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 116 67 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -15 73 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 110 68 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 12 74 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -115 67 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -112 70 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 16 66 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 113 70 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -17 66 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 11 75 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -111 67 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -14 72 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
SumDivisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect
2244 Designs Designs Example 4 - Completed Signs TableExample 4 - Completed Signs Table
Main Effects Interaction EffectsRun
Order y A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
9 66 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 18 70 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -13 64 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -114 71 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 116 67 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -15 73 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 110 68 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 12 74 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -115 67 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -112 70 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 16 66 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 113 70 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -17 66 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 11 75 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -111 67 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -14 72 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Sum 1106 44 -2 18 0 0 8 -2 2 -4 -4 -1 -6 6 -4 -2Divisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 69.125 5.5 -.25 2.25 0.00 0.00 1.00 -.25 0.25 -.50 -.50 -1 -.75 0.75 -0.50 -0.25
2244 Designs Designs Example 4 - Effects Normal Probability PlotExample 4 - Effects Normal Probability Plot
Factors A and C Stand OutFactors A and C Stand Out Choose Hi Settings of Both A and C Choose Hi Settings of Both A and C
since the response is throughputsince the response is throughput
Factors A and C Stand OutFactors A and C Stand Out Choose Hi Settings of Both A and C Choose Hi Settings of Both A and C
since the response is throughputsince the response is throughput
2244 Designs Designs Example 4 - EMR at A=Hi, C=HiExample 4 - EMR at A=Hi, C=Hi
EMR = 69.125 +(5.5 + EMR = 69.125 +(5.5 + 2.25)/22.25)/2
= 73 = 73
EMR = 69.125 +(5.5 + EMR = 69.125 +(5.5 + 2.25)/22.25)/2
= 73 = 73
y A B C D66 -1 -1 -1 -170 1 -1 -1 -164 -1 1 -1 -171 1 1 -1 -167 -1 -1 1 -173 1 -1 1 -168 -1 1 1 -174 1 1 1 -167 -1 -1 -1 170 1 -1 -1 166 -1 1 -1 170 1 1 -1 166 -1 -1 1 175 1 -1 1 167 -1 1 1 172 1 1 1 1
1106 44 -2 18 016 8 8 8 8
69.125 5.5 -.25 2.25 0.00
2244 Designs Designs Examples 2 and 4 DiscussionExamples 2 and 4 Discussion
Examples 2 (from Lecture 6.2) and 4 are Examples 2 (from Lecture 6.2) and 4 are RelatedRelated– Original Data Was In TenthsOriginal Data Was In Tenths– The Numbers were Rounded Off for Ease of The Numbers were Rounded Off for Ease of
CalculationCalculation Example 2Example 2
– Half Fraction (2Half Fraction (24-14-1, 8 Runs) of the Data in , 8 Runs) of the Data in Example 4. Example 4.
– The Runs in Example 4 when ABCD=1 were The Runs in Example 4 when ABCD=1 were the runs used in Example 2.the runs used in Example 2.
2244 Designs Designs Examples 2 and 4 DiscussionExamples 2 and 4 Discussion
Main Effects Interaction EffectsActual
Run y A B C AB AC BC ABC5 66 -1 -1 -1 1 1 1 -12 70 1 -1 -1 -1 -1 1 11 66 -1 1 -1 -1 1 -1 14 71 1 1 -1 1 -1 -1 -13 66 -1 -1 1 1 -1 -1 16 73 1 -1 1 -1 1 -1 -18 68 -1 1 1 -1 -1 1 -17 72 1 1 1 1 1 1 1
Sum 552 20 2 6 -2 2 0 -4Divisor 8 4 4 4 4 4 4 4Effect 69 5 .5 1.5 -.5 .5 0 -1
Main Effects Interaction EffectsActual
Run y A B C AB AC BC ABC5 66 -1 -1 -1 1 1 1 -12 70 1 -1 -1 -1 -1 1 11 66 -1 1 -1 -1 1 -1 14 71 1 1 -1 1 -1 -1 -13 66 -1 -1 1 1 -1 -1 16 73 1 -1 1 -1 1 -1 -18 68 -1 1 1 -1 -1 1 -17 72 1 1 1 1 1 1 1
Sum 552 20 2 6 -2 2 0 -4Divisor 8 4 4 4 4 4 4 4Effect 69 5 .5 1.5 -.5 .5 0 -1
Example 4
y A B C D ABCD
66 -1 -1 -1 -1 170 1 -1 -1 -1 -164 -1 1 -1 -1 -171 1 1 -1 -1 167 -1 -1 1 -1 -173 1 -1 1 -1 168 -1 1 1 -1 174 1 1 1 -1 -167 -1 -1 -1 1 -170 1 -1 -1 1 166 -1 1 -1 1 170 1 1 -1 1 -166 -1 -1 1 1 175 1 -1 1 1 -167 -1 1 1 1 -172 1 1 1 1 1
2244 Designs Designs Example 2 and 4 - Effects Normal Probability Example 2 and 4 - Effects Normal Probability
PlotsPlots Factor A Still Stands OutFactor A Still Stands Out The (Hidden) Replication in the Additional Runs Teased Out A The (Hidden) Replication in the Additional Runs Teased Out A
Significant Effect Due to Factor C.Significant Effect Due to Factor C.
543210-1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Effects
AC
Example 4 - Full 16 Run Design
543210-1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Effects
AC
Example 4 - Full 16 Run Design
543210-1
1.0
0.5
0.0
-0.5
-1.0
-1.5
Effects
A
Example 2 - Half Fraction
543210-1
1.0
0.5
0.0
-0.5
-1.0
-1.5
Effects
A
Example 2 - Half Fraction
