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: When is an Effect “Real”? Statistical Significance: When is an Effect “Real”? An Example With Interactions An Example With Interactions A U-Do-It Case Study A U-Do-It Case Study Replication Replication Rope Pull Exercise Rope Pull Exercise

3. U-Do-It Case Study Ball Bearing Example* Purpose of the Design Purpose of the Design –Test (Under Accelerated Conditions) New Bearing Prototypes for Use in a Specific Application for Which the Current Design’s Performance Was “Unsatisfactory”. –Response of Interest: y - Bearing Life (h). Design Factors: Design Factors: –FactorLevels (Lo,Hi) A: Cage DesignCurrent, New B: Outer Ring OsculationCurrent, New C: Inner Ring Heat Treatment.Current, New The 8 Standard Runs of the 2 3 Design Were Randomly Ordered, and Each Prototype Bearing Tested. The 8 Standard Runs of the 2 3 Design Were Randomly Ordered, and Each Prototype Bearing Tested. * Empirical Basis for this data was motivated by C. Hellstrand’s article “The necessity of modern quality improvement and some experiences with implications in the manufacture of ball bearings (1989, Philos. Trans. Royal Society London, A 327, 529-537)

4. U-Do-It Case Study Ball Bearing Example - A Typical Ball Bearing

5. U-Do-It Case Study Ball Bearing Example - Operator Report Form

6. U-Do-It Case Study Ball Bearing Example - Exercise Instructions In Class In Class –Put the results of the experiment in standard order and enter the data into a cube plot (in Minitab—see handout) –Estimate the factor effects (in Minitab) –Construct and interpret a normal probability plot of the factor effects (in Minitab)

7. U-Do-It Case Study Ball Bearing Example - Exercise Instructions In Class In Class –Construct BC interaction graph in Minitab; use table and graph to interpret BC interaction –Determine the factor settings that maximize bearing life and estimate the Mean Response (EMR) at these settings. How close is your answer to the observed mean response at your optimal settings? –If you would like to do hand calculations, blank signs tables, cube plots, etc. are provided over the next several slides

8. U-Do-It Case Study Ball Bearing Example - Signs Table

9. U-Do-It Case Study Ball Bearing Example - Cube Plot

10. U-Do-It Case Study Ball Bearing Example - Seven Effects Paper

11. U-Do-It Case Study Ball Bearing Example - Interaction Table

12. U-Do-It Case Study Solution Ball Bearing Example - Cube Plot Bearing Lifetimes (h) Shown Factor A: Cage Design B: Outer Ring Osculation C: Inner Ring Heat Treatment Factor A: Cage Design B: Outer Ring Osculation C: Inner Ring Heat Treatment Levels: Lo = Current, Hi = New Levels: Lo = Current, Hi = New Factor A: Cage Design B: Outer Ring Osculation C: Inner Ring Heat Treatment Factor A: Cage Design B: Outer Ring Osculation C: Inner Ring Heat Treatment Levels: Lo = Current, Hi = New Levels: Lo = Current, Hi = New

13. U-Do-It Case Study Solution Ball Bearing Example - Signs Table

14. U-Do-It Case Study Solution Ball Bearing Example - Probability Plot

15. U-Do-It Case Study Solution Ball Bearing Example - Completed BC Interaction Table

16. U-Do-It Case Study Solution Ball Bearing Example - BC Interaction Plot Interpretation Interpretation –Choose the Hi Level for both B and C to Maximize the Bearing Life Interpretation Interpretation –Choose the Hi Level for both B and C to Maximize the Bearing Life o Factors B: Outer Ring Osculation C: Inner Ring Heat Treatment o Levels: Lo = Current, Hi = New o Factors B: Outer Ring Osculation C: Inner Ring Heat Treatment o Levels: Lo = Current, Hi = New

17. U-Do-It Case Study Solution Ball Bearing Example - Expected Mean Response Since the BC Interaction is Significant, the Main Effects B and C and the BC Interaction are Included Since the BC Interaction is Significant, the Main Effects B and C and the BC Interaction are Included Factor A is NOT Included Since it was Not Significant Factor A is NOT Included Since it was Not Significant Since the BC Interaction is Significant, the Main Effects B and C and the BC Interaction are Included Since the BC Interaction is Significant, the Main Effects B and C and the BC Interaction are Included Factor A is NOT Included Since it was Not Significant Factor A is NOT Included Since it was Not Significant o For B = +1, C = +1, EMR = 41.5 + [(+1)(45.5)+(+1)(43)+(+1)(39.5)]/2 = 105.5 (vs (99+112)/2 = 105.5 Observed MR)

18. U-Do-It Case Study Solution Ball Bearing Example - Interpretation of the Experiment Unexpected Interaction Discovered (Would Not Have Been Discovered Using “One-at-a-Time” Experimentation). Results May Carry Over to Other Bearing Designs. Unexpected Interaction Discovered (Would Not Have Been Discovered Using “One-at-a-Time” Experimentation). Results May Carry Over to Other Bearing Designs. Contrary to Existing Beliefs, the Two Cage Designs had Very Similar Lifetimes. This was Very Important Since Bearings Were Much Cheaper to Produce Under One of the Two Cage Designs. Contrary to Existing Beliefs, the Two Cage Designs had Very Similar Lifetimes. This was Very Important Since Bearings Were Much Cheaper to Produce Under One of the Two Cage Designs. New Design’s Performance (In the Specific Application Under Investigation) Far Superior to That of the Current Bearing Being Used. New Design’s Performance (In the Specific Application Under Investigation) Far Superior to That of the Current Bearing Being Used.