1. 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

2. Part I.3 The Essentials of 2-Cubed Designs Methodology Methodology –Cube Plots – Estimating Main Effects – Estimating Interactions (Interaction Tables and Graphs) Statistical Significance (Effects Probability Plots) Statistical Significance (Effects Probability Plots) Example With Interactions Example With Interactions A U-Do-It Case Study A U-Do-It Case Study

3. Methodology Example 2 Looking For Patterns In The Data To Discover How The Factors Affect The Response, y. Looking For Patterns In The Data To Discover How The Factors Affect The Response, y.

4. Methodology Example 2 - Cube Plot

5. Methodology Example 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 Corner to Another Involves Changing A from - to + –Average the Corners on the Common Faces and Difference A=[(70+71+72+73)/4] -[(66+66+68+66)/4] = 71.5-66.5 = 5 A=[(70+71+72+73)/4] -[(66+66+68+66)/4] = 71.5-66.5 = 5

6. Methodology Example 2 - Estimating the Main Effect of B

7. Methodology Example 2 - Estimating the Main Effect of C

8. Methodology Example 2 - Signs Table The Signs of the Main Effects Give The Recipes For the 8 Runs in the Design The Signs of the Main Effects Give The Recipes For the 8 Runs in the Design Actual Run Corresponds to the Order of the Experimental Runs (Recipes) Actual Run Corresponds to the Order of the Experimental Runs (Recipes)

9. Methodology Example 2 - Signs Table Used to Calculate Effects To Estimate the Main Effects To Estimate the Main Effects –Multiply the Response y by the Corresponding Sign Column –Sum the Column –Divide the Sum by the Divisor to Get the Estimated Main Effect U-Do-It U-Do-It –Calculate the Main Effects Due to B and to C

10. Methodology Example 2 - Estimating the Effect of A Another Way

11. Average the Differences of A+and A- Over All the Combinations of B and C. Average the Differences of A+and A- Over All the Combinations of B and C. –Retain Those Edges of the Cube Where Going from One Cube Corner to Another Involves Changing A from - to + –Difference These Corners and Average [(72-68)+(71-66)+(73-66) +(70-66)]/4 = 20/4 = 5 [(72-68)+(71-66)+(73-66) +(70-66)]/4 = 20/4 = 5

12. Methodology Example 2 - Estimating the Effect of the AB Interaction Average Average –The Four Values in the Shaded Corners –The Four Values in the Unshaded Corners Difference the Averages Difference the Averages [(71+72 +66+66)/4] -[(70+73+66+68)/4] =68.75-69.25=-.5 [(71+72 +66+66)/4] -[(70+73+66+68)/4] =68.75-69.25=-.5

13. Methodology Example 2 - Estimating the Effect of the AB Interaction Another Way The Second Way Shows that the AB Interaction is Comparing the Differences in going from A- to A+ at B- and B+. The Second Way Shows that the AB Interaction is Comparing the Differences in going from A- to A+ at B- and B+. If there is a “Significant” Difference, then A and B are said to Interact If there is a “Significant” Difference, then A and B are said to Interact [(71-66)+(72-68)-(73-66)-(70- 66)]/4 = -.5 [(71-66)+(72-68)-(73-66)-(70- 66)]/4 = -.5 In this example, there are no significant interactions. Thus, the interpretation is straightforward. In this example, there are no significant interactions. Thus, the interpretation is straightforward. WARNING: When a higher order interaction is “significant,” the direct interpretation of lower order interactions and main effects is misleading. WARNING: When a higher order interaction is “significant,” the direct interpretation of lower order interactions and main effects is misleading.

14. Methodology Example 2 - Signs Table Calculating the Signs and the Effect of Interaction AB

15. Methodology Example 2 - Signs Table U-Do-It Calculate the Signs for Interactions AC, BC and ABC Calculate the Signs for Interactions AC, BC and ABC Calculate These Interaction Effects Calculate These Interaction Effects

16. Methodology Example 2 - ANOVA Table

17. Methodology Example 2 - Effects Normal Probability Plot

18. Methodology Example 2 - Discussion Only the Main Effect A is Significant Only the Main Effect A is Significant Set A Hi to Maximize y Set A Hi to Maximize y Set A Lo to Minimize y Set A Lo to Minimize y Only the Main Effect A is Significant Only the Main Effect A is Significant Set A Hi to Maximize y Set A Hi to Maximize y Set A Lo to Minimize y Set A Lo to Minimize y o This Data is Real and Will be Considered in Later Sections o For This Data Minimizing y is the Objective o This Data is Real and Will be Considered in Later Sections o For This Data Minimizing y is the Objective

19. Methodology Example 2 - Estimating the Response Since Only the Main Effect A is Significant The Estimated Mean Response (EMR) is given by EMR = y + (Sign of A)(Effect of A)/2 = 69 + (Sign of A)5/2 Since Only the Main Effect A is Significant The Estimated Mean Response (EMR) is given by EMR = y + (Sign of A)(Effect of A)/2 = 69 + (Sign of A)5/2 For A Lo, EMR = 69 + (-1)(2.5) = 66.5 For A Lo, EMR = 69 + (-1)(2.5) = 66.5 For A Hi, EMR = 69 + (+1)(2.5) =71.5 For A Hi, EMR = 69 + (+1)(2.5) =71.5 Since Only the Main Effect A is Significant The Estimated Mean Response (EMR) is given by EMR = y + (Sign of A)(Effect of A)/2 = 69 + (Sign of A)5/2 Since Only the Main Effect A is Significant The Estimated Mean Response (EMR) is given by EMR = y + (Sign of A)(Effect of A)/2 = 69 + (Sign of A)5/2 For A Lo, EMR = 69 + (-1)(2.5) = 66.5 For A Lo, EMR = 69 + (-1)(2.5) = 66.5 For A Hi, EMR = 69 + (+1)(2.5) =71.5 For A Hi, EMR = 69 + (+1)(2.5) =71.5

20. Methodology Example 2 - Estimating the Response Why the One-Half? The Formula Gives You Just What You Expect: The Average Response at that Level of the A The Formula Gives You Just What You Expect: The Average Response at that Level of the A For A Lo, EMR = 69 + (-1)(2.5) = 66.5 = (66 +66 +66 +68)/4 For 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)/4 For A Hi, EMR = 69 + (+1)(2.5) =71.5 = (70 + 71 + 73 +72)/4 The Formula Gives You Just What You Expect: The Average Response at that Level of the A The Formula Gives You Just What You Expect: The Average Response at that Level of the A For A Lo, EMR = 69 + (-1)(2.5) = 66.5 = (66 +66 +66 +68)/4 For 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)/4 For A Hi, EMR = 69 + (+1)(2.5) =71.5 = (70 + 71 + 73 +72)/4

21. II. Summary Key Ideas Use Sign Tables to Estimate Effects Use Sign Tables to Estimate Effects Use Probability Plots to Identify Significant Effects Use Probability Plots to Identify Significant Effects Interaction Tables and Graphs are Used to Analyze Significant Interactions (To be explained later) Interaction Tables and Graphs are Used to Analyze Significant Interactions (To be explained later)

22. II. Summary Concluding Comments A Main Effect Is Easy To Interpret When There Are No Significant Interactions Involving It A Main Effect Is Easy To Interpret When There Are No Significant Interactions Involving It In The Presence of a Significant Higher-Order Interaction, the Lower-Order Interactions and Corresponding Main Effects Are Hard To Interpret by Themselves. (You Still Can Figure Out What to Do, Though) In The Presence of a Significant Higher-Order Interaction, the Lower-Order Interactions and Corresponding Main Effects Are Hard To Interpret by Themselves. (You Still Can Figure Out What to Do, Though) The Size of the Effects You are Trying to Detect and the Noise of the Process (How Much Variation It Has) Will Dictate How Much Replication Is Needed The Size of the Effects You are Trying to Detect and the Noise of the Process (How Much Variation It Has) Will Dictate How Much Replication Is Needed