The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs. Developed by Don Edwards, John Grego and James Lynch Center for Reliability and Quality Sciences Department of Statistics University of South Carolina 803-777-7800. - PowerPoint PPT Presentation
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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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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.3 The Essentials of 2-Cubed DesignsPart I.3 The Essentials of 2-Cubed Designs
MethodologyMethodology– Cube PlotsCube Plots– Estimating Main EffectsEstimating Main Effects– Estimating Interactions (Interaction Tables and Graphs)Estimating Interactions (Interaction Tables and Graphs)
Statistical Significance (Effects Probability Plots)Statistical Significance (Effects Probability Plots) Example With InteractionsExample With Interactions A U-Do-It Case StudyA U-Do-It Case Study
Methodology Methodology Example 2Example 2
Looking For Patterns In The Data To Discover Looking For Patterns In The Data To Discover How The Factors Affect The Response, y.How The Factors Affect The Response, y.
Factors ResponseA B C yLo Lo Lo 66Hi Lo Lo 70Lo Hi Lo 66Hi Hi Lo 71Lo Lo Hi 66Hi Lo Hi 73Lo Hi Hi 68Hi Hi Hi 72
Factors ResponseA B C yLo Lo Lo 66Hi Lo Lo 70Lo Hi Lo 66Hi Hi Lo 71Lo Lo Hi 66Hi Lo Hi 73Lo Hi Hi 68Hi Hi Hi 72
A +_
66
66
68
66
7070
73
72
71
Methodology Methodology Example 2 - Estimating the Main Effect of AExample 2 - Estimating the Main Effect of A
Take the Difference of the Average Response for A+ and A-Take the Difference of the Average Response for A+ and A-– Eliminate Those Edges of the Cube Where Going from One Cube Eliminate Those Edges of the Cube Where Going from One Cube
Corner to Another Involves Changing A fromCorner to Another Involves Changing A from- to +- to +
– Average the Corners on the Common Faces and DifferenceAverage the Corners on the Common Faces and Difference A=[(70+71+72+73)/4]A=[(70+71+72+73)/4]
The Signs of the Main Effects Give The Recipes The Signs of the Main Effects Give The Recipes For For the 8 Runs in the Design the 8 Runs in the Design
Actual Run Corresponds to the Order of the Actual Run Corresponds to the Order of the Experimental Runs (Recipes) Experimental Runs (Recipes)
Main Effects Interaction EffectsActual
Run y A B C5 66 -1 -1 -12 70 1 -1 -11 66 -1 1 -14 71 1 1 -13 66 -1 -1 16 73 1 -1 18 68 -1 1 17 72 1 1 1
SumDivisor 8 4 4 4 4 4 4Effect
MethodologyMethodologyExample 2 - Signs TableExample 2 - Signs TableUsed to Calculate EffectsUsed to Calculate Effects
To Estimate the Main EffectsTo Estimate the Main Effects– Multiply the Response y by the Corresponding Sign ColumnMultiply the Response y by the Corresponding Sign Column– Sum the ColumnSum the Column– Divide the Sum by the Divisor to Get the Estimated Main EffectDivide the Sum by the Divisor to Get the Estimated Main Effect
U-Do-ItU-Do-It– Calculate the Main Effects Due to B and to CCalculate the Main Effects Due to B and to C
Main Effects Interaction EffectsActual
Run y A B C5 66 -1 -1 -12 70 1 -1 -11 66 -1 1 -14 71 1 1 -13 66 -1 -1 16 73 1 -1 18 68 -1 1 17 72 1 1 1
Sum 552 20Divisor 8 4 4 4 4 4 4Effect 69 5
C
B
A
+
+
+
_
_
_
66
66
68
66
7070
73
72
71
Methodology Methodology Example 2 - Estimating the Effect of A Another WayExample 2 - Estimating the Effect of A Another Way
A +_
66
66
68
66
7070
73
72
71
Methodology Methodology Example 2 - Estimating the Effect of A Example 2 - Estimating the Effect of A
Another WayAnother Way
Average the Differences of Average the Differences of A+and A- Over All the A+and A- Over All the Combinations of B and C.Combinations of B and C.– Retain Those Edges of Retain Those Edges of
the Cube Where Going the Cube Where Going from One Cube Corner from One Cube Corner to Another Involves to Another Involves Changing A from - to +Changing A from - to +
– Difference These Difference These Corners and AverageCorners and Average
MethodologyMethodologyExample 2 - Estimating the Effect of the AB Example 2 - Estimating the Effect of the AB
InteractionInteraction AverageAverage
– The Four Values in the Shaded CornersThe Four Values in the Shaded Corners– The Four Values in the Unshaded CornersThe Four Values in the Unshaded Corners
Difference the AveragesDifference the Averages [(71+72 +66+66)/4][(71+72 +66+66)/4]
MethodologyMethodologyExample 2 - Estimating the Effect of the AB Interaction Another WayExample 2 - Estimating the Effect of the AB Interaction Another Way
The Second Way Shows that the AB Interaction is Comparing the The Second Way Shows that the AB Interaction is Comparing the Differences in going from A- to A+ at B- and B+.Differences in going from A- to A+ at B- and B+.
If there is a “Significant” Difference, then A and B are said to InteractIf there is a “Significant” Difference, then A and B are said to Interact [(71-66)+(72-68)-(73-66)-(70-66)]/4[(71-66)+(72-68)-(73-66)-(70-66)]/4
= -.5 = -.5 In this example, there are no significant interactions. Thus, the In this example, there are no significant interactions. Thus, the
interpretation is straightforward.interpretation is straightforward. WARNING: When a higher order interaction is “significant,” the direct WARNING: When a higher order interaction is “significant,” the direct
interpretation of lower order interactions and main effects is interpretation of lower order interactions and main effects is misleading.misleading.
Calculate the Signs for Interactions AC, BC and ABCCalculate the Signs for Interactions AC, BC and ABC Calculate These Interaction EffectsCalculate These Interaction Effects
Main Effects Interaction EffectsActual
Run y A B C AB AC BC ABC5 66 -1 -1 -1 12 70 1 -1 -1 -11 66 -1 1 -1 -14 71 1 1 -1 13 66 -1 -1 1 16 73 1 -1 1 -18 68 -1 1 1 -17 72 1 1 1 1
Only the Main Effect A is SignificantOnly the Main Effect A is Significant Set A Hi to Maximize ySet A Hi to Maximize y Set A Lo to Minimize ySet A Lo to Minimize y
This Data is Real and Will be Considered in Later Sections
For This Data Minimizing y is the Objective
MethodologyMethodology Example 2 - Estimating the ResponseExample 2 - Estimating the Response
Since Only the Main Effect A is Significant The Since Only the Main Effect A is Significant The Estimated Mean Response (EMR) is given byEstimated Mean Response (EMR) is given by
EMR = y + (Sign of A)(Effect of A)/2 = 69 + (Sign of A)5/2 EMR = y + (Sign of A)(Effect of A)/2 = 69 + (Sign of A)5/2 For A Lo, EMR = 69 + (-1)(2.5) = 66.5For A Lo, EMR = 69 + (-1)(2.5) = 66.5 For A Hi, EMR = 69 + (+1)(2.5) =71.5For A Hi, EMR = 69 + (+1)(2.5) =71.5
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
MethodologyMethodology Example 2 - Estimating the ResponseExample 2 - Estimating the Response
Why the One-Half?Why the One-Half? The Formula Gives You Just What You Expect:The Formula Gives You Just What You Expect:
The Average Response at that Level of the AThe Average Response at that Level of the A For A Lo, EMR = 69 + (-1)(2.5) = 66.5 = (66 +66 +66 +68)/4For A Lo, EMR = 69 + (-1)(2.5) = 66.5 = (66 +66 +66 +68)/4 For A Hi, EMR = 69 + (+1)(2.5) =71.5 = (70 + 71 + 73 +72)/4For A Hi, EMR = 69 + (+1)(2.5) =71.5 = (70 + 71 + 73 +72)/4
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
Use Sign Tables to Estimate EffectsUse Sign Tables to Estimate Effects Use Probability Plots to Identify Use Probability Plots to Identify
Significant EffectsSignificant Effects Interaction Tables and Graphs are Interaction Tables and Graphs are
Used to Analyze Significant Used to Analyze Significant InteractionsInteractions(To be explained later)(To be explained later)
II. SummaryII. SummaryConcluding CommentsConcluding Comments
A Main Effect Is Easy To Interpret When There A Main Effect Is Easy To Interpret When There Are No Significant Interactions Involving ItAre No Significant Interactions Involving It
In The Presence of a Significant Higher-Order In The Presence of a Significant Higher-Order Interaction, the Lower-Order Interactions and Interaction, the Lower-Order Interactions and Corresponding Main Effects Are Hard To Corresponding Main Effects Are Hard To Interpret by Themselves. (You Still Can Interpret by Themselves. (You Still Can Figure Out What to Do, Though)Figure Out What to Do, Though)
The Size of the Effects You are Trying to The Size of the Effects You are Trying to Detect and the Noise of the Process (How Detect and the Noise of the Process (How Much Variation It Has) Will Dictate How Much Much Variation It Has) Will Dictate How Much Replication Is NeededReplication Is Needed
