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The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs

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.

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The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs

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  1. The Essentials of 2-Level Design of ExperimentsPart I: The Essentials of Full Factorial Designs Developed by Don Edwards, John Grego and James LynchCenter for Reliability and Quality SciencesDepartment of StatisticsUniversity of South Carolina803-777-7800

  2. Part I.3 The Essentials of 2-Cubed Designs • Methodology • Cube Plots • Estimating Main Effects • Estimating Interactions (Interaction Tables and Graphs)

  3. Part I.3 The Essentials of 2-Cubed Designs • Statistical Significance:When is an Effect “Real”? • An Example With Interactions • A U-Do-It Case Study • Replication • Rope Pull Exercise

  4. Statistical SignificanceWhen is an Effect “Real”?(As Opposed to Being “Due to Error”)Introduction • The Effects (Main and Interactions) We Compute are Really Estimates of the “True Effects” (Remember MAE). • All the True Effects are Probably Nonzero, but Some are Very Small - It is More Correct to Ask If an Effect is “Distinguishable from Error” or “Indistinguishable from Error”.

  5. Statistical SignificanceWhen is an Effect “Real”?(As Opposed to Being “Due to Error”)Introduction • We will Discuss Tools to Help in This Decision • Normal Probability Plots of Estimated Effects • Replication • ANOVA

  6. Statistical SignificanceWhen is an Effect “Real”?Normal Probability Plots of Estimated Effects • What if all the true effects were zero, so that estimated effects represented only random error?

  7. Statistical SignificanceWhen is an Effect “Real”?Normal Probability Plots - Background • In 1959, Cuthbert Daniel Found a Way to Plot the Estimated Effects so that Effects Due to Random Error Fall (Roughly) on a Straight Line in the Plot • To Construct a Normal Probability Plot of the Effects • 1. Order the Estimated Effects from Smallest to Largest (Minus Signs Count: -1 is Less Than 2, For Example). • 2. Plot the Points (Ei,Pi), i = 1,..,m on Normal Probability Paper, Where m = Number of Effects, Ei is the ith Smallest Effect (Put the E’s on the Horizontal Axis), and Pi = 100(i-0.5)/m. • 3. Normal Probability Paper is on the next Slide for m = 7. • To Use This Paper • Scale the x-axis (Horizontal Axis) to Cover the Range of the Effects • Plot the Smallest Value on Line 1, the Next Smallest on Line 2, etc.

  8. Statistical SignificanceWhen is an Effect “Real”?Normal Probability Plots - Seven Effects Paper

  9. Statistical SignificanceWhen is an Effect “Real”?Normal Probability Plots - Interpretation • If There are Enough Effects Plotted, and Some are Due to Random Error, These Will Lie Approximately on a Straight Line Centered at 0. Sketch in the Line. • Identify Any Effects That Fall off the Line to the Upper Right and Lower Left. These Effects are Probably Not Due to Noise; They are “Distinguishable from Error”.

  10. Statistical SignificanceWhen is an Effect “Real”?Normal Probability Plots - Example 2

  11. Methodology Example 3 - PC Response Time • Objective Reduce Company’s PC Response Time • Factors • A: Cache (Two Levels Lo, Hi) • B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM • C: Line (Lo - 56K modem, Hi - LAN)

  12. Methodology Example 3 - PC Response Time • Response: PC Response Time • Factors • A: Cache (Two Levels Lo,Hi) • B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) • C: Line (Lo - 56K modem, Hi - LAN)

  13. Methodology Example 3 - PC Response TimeCube Plot • Response: PC Response Time • Factors • A: Cache (Two Levels Lo,Hi) • B: Machine (Lo - 200MHz, 64 MB • RAM, Hi - 400MHz, 1GB RAM) • C: Line (Lo - 56K modem, Hi - LAN)

  14. MethodologyExample 3 - PC Response TimeEstimating the Effects - Signs Tables

  15. Methodology Example 3 - PC Response TimeEffects Normal Probability Plot

  16. MethodologyInteraction Tables and Graphs • Tools for Aiding Interpretation of SIGNIFICANT Two-Way Interactions • At the Right is a Blank AB Interaction Table • In the Table, 1 Corresponds to the Lo Level and 2 to the Hi Level

  17. Methodology Example 3 - PC Response TimeAC Interaction Table

  18. Methodology Interaction Tables and GraphsInteraction Plots - Construction • 1. For a Given Pair of Factors (Say A and B) Find the Average Response at Each of Their Four Level Combinations. • 2. Plot These with Response on the Vertical Axis, Using One of the Factor’s Levels (Say B) on the Horizontal Axis. Connect and Label the Averages with the Same Level of the Other Factor (A).

  19. MethodologyInteraction Tables and GraphsInteraction Plots - Interpretation • 1. If the Lines are Roughly Parallel, There is No Strong Interaction. • 2. If There is Interaction, the Plot Shows Clearly the Effect of a Factor at Each of the Levels of the Other Factor. • 3. Maximizing and Minimizing Combinations of the Factors are Easily Identified on the Plot and in the Table.

  20. MethodologyExample 3AC Interaction Table and Graph • Response: PC Response Time • Factors • A: Cache (Two Levels Lo,Hi) • B: Machine (Lo - 200MHz, 64 MB • RAM, Hi - 400MHz, 1GB RAM) • C: Line (Lo - 56K modem, Hi - LAN)

  21. MethodologyExample 3 - AC Interaction Graph • Response: PC Response Time • Factors • A: Cache (Two Levels Lo,Hi) • B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) • C: Line (Lo - 56K modem, Hi - LAN)

  22. MethodologyExample 3 - AC Interaction InterpretationNoise Factors versus Control Factors • Response: PC Response Time • Factors • A: Cache (Two Levels Lo,Hi) • B: Machine (Lo - 200MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM) • C: Line (Lo - 56K modem, Hi - LAN) • To minimize the response, choose B Hi and C Hi. When C is Hi, the effect of A is negligible. • Refer back to the cube plot—the (B Hi, C Hi) edge had the two lowest readings. Our analysis shows that this was not due to a BC interaction, but to a significant B main effect and the particular form of the significant AC interaction.

  23. MethodologyExample 3 - Estimating the Mean Response: A = +1, B = -1, C = +1 • Estimated Mean Response (EMR) = y + [(Sign of A)(Effect of A)+(Sign of B)(Effect of B) +(Sign of C)(Effect of C)+(Sign of AC)(Effect of AC)]/2 • For A = +1, B = -1, C = +1, EMR = 24.9 + [(+1)(-11.9)+(-1)(-16.2)+(+1)(-17.2)+(1)(12)]/2 = 24.4 • Notice that for A = +1 and C = +1, [(Sign of A)(Effect of A)+(Sign of C)(Effect of C)+(Sign of AC)(Effect of AC)]/2 = [(+1)(-11.9)+(+1)(-17.2)+(1)(12)]/2 = -17.1/2 = -8.55 = A2C2 – y; so EMR=24.9- 8.55=16.35 • For Calculating EMR Include: • Significant Main Effects • Significant Interactions, and All Their Lower Order Interactions and Main Effects

  24. MethodologyU-Do-It: Example 3 - Estimate the Response A = +1, B = +1, C = +1 and A = +1, B = +1, C = -1

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