1. The Essentials of 2-Level Design of Experiments Part II: The Essentials of Fractional Factorial Designs The Essentials of 2-Level Design of Experiments Part II: The Essentials of Fractional 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 II: The Essentials of Fractional Factorial Designs 1. Introduction to Fractional Factorials 1. Introduction to Fractional Factorials 2. Four Factors in Eight Runs 2. Four Factors in Eight Runs 3. Screening Designs in Eight Runs 3. Screening Designs in Eight Runs 4. K Factors in Sixteen Runs 4. K Factors in Sixteen Runs

3. II.1 Introduction to Fractional Factorials A Quick Review of Full Factorials A Quick Review of Full Factorials How Many Runs? How Many Runs? The Fractional Factorial Idea The Fractional Factorial Idea

4. II.1 Introduction: A Quick Review of Full Factorials Use Cube Plots to Understand Factor Effects Use Cube Plots to Understand Factor Effects 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 Interaction Tables and Graphs are Used to Analyze Significant Interactions

5. II.1 Introduction: A Quick Review Rope Pull Study - Completed Cube Plot and Signs Table Factors: Factors: –A: Vacuum Level (Lo, Hi) –B: Needle Type (EX, GB) –C: Upper Boot Speed (1000,1200) Response: Response: –Rope Pull (in inches) Factors: Factors: –A: Vacuum Level (Lo, Hi) –B: Needle Type (EX, GB) –C: Upper Boot Speed (1000,1200) Response: Response: –Rope Pull (in inches)

6. II.1 Introduction: A Quick Review Rope Pull Study -Completed Seven Effects Normal Plot Factors: Factors: –A: Vacuum Level (Lo, Hi) –B: Needle Type (EX, GB) –C: Upper Boot Speed (1000,1200) Factors: Factors: –A: Vacuum Level (Lo, Hi) –B: Needle Type (EX, GB) –C: Upper Boot Speed (1000,1200)

7. II.1 Introduction: A Quick Review Rope Pull Study - Completed AC Interaction Table and Plot o Factors A: Vacuum Level (Lo, Hi) C: Upper Boot Speed (1000,1200)

8. II.1 Introduction: How Many Runs? We have seen, for factors at two levels, We have seen, for factors at two levels, –Two Factors  4 runs –Three Factors  8 runs –Four Factors  16 runs What if we have seven factors? What if we have seven factors? What if we have fifteen? What if we have fifteen? There are ways to investigate up to seven factors using only 8 runs, or up to 15 factors using 16 runs, if it is safe to assume that high-order interactions are negligible. There are ways to investigate up to seven factors using only 8 runs, or up to 15 factors using 16 runs, if it is safe to assume that high-order interactions are negligible.

9. II.1 Introduction: How Many Runs? For Example, For Example, –We May Be Interested in Determining the Effects on Quality Characteristics of Hosiery  A: Band Speed  B: Panty Speed  C: Upper Boot Speed  D: Lower Boot Speed  E: Needle Type  F: Vacuum Level –A Full 2 6 in These Factors, Each at Two Levels, Would Require 64 Runs

10. II.1 Introduction: The Fractional Factorial Idea In the 2 3 design, look at the computation of C using the y's in standard order C = ( -y 1 -y 2 -y 3 -y 4 +y 5 +y 6 +y 7 +y 8 )/4 In the 2 3 design, look at the computation of C using the y's in standard order C = ( -y 1 -y 2 -y 3 -y 4 +y 5 +y 6 +y 7 +y 8 )/4

11. II.1 Introduction: The Fractional Factorial Idea Now, look at the same thing for AB: AB = ( +y 1 -y 2 -y 3 +y 4 +y 5 -y 6 -y 7 +y 8 )/4 Now, look at the same thing for AB: AB = ( +y 1 -y 2 -y 3 +y 4 +y 5 -y 6 -y 7 +y 8 )/4

12. II.1 Introduction: The Fractional Factorial Idea So, C = ( -y 1 -y 2 -y 3 -y 4 +y 5 +y 6 +y 7 +y 8 )/4 AB = ( +y 1 -y 2 -y 3 +y 4 +y 5 -y 6 -y 7 +y 8 )/4 So, C = ( -y 1 -y 2 -y 3 -y 4 +y 5 +y 6 +y 7 +y 8 )/4 AB = ( +y 1 -y 2 -y 3 +y 4 +y 5 -y 6 -y 7 +y 8 )/4 Add These Together to get C+AB: C+AB = ( -2y 2 -2y 3 +2y 5 +2y 8 )/4 = ( -y 2 -y 3 +y 5 +y 8 )/2 Add These Together to get C+AB: C+AB = ( -2y 2 -2y 3 +2y 5 +2y 8 )/4 = ( -y 2 -y 3 +y 5 +y 8 )/2 So, if we want to estimate C+AB, we only need 4 runs to do it! Or, if we are fairly sure that AB is negligible, we only need 4 runs to estimate C (and the same 4 runs can estimate A and B if BC and AC are negligible). So, if we want to estimate C+AB, we only need 4 runs to do it! Or, if we are fairly sure that AB is negligible, we only need 4 runs to estimate C (and the same 4 runs can estimate A and B if BC and AC are negligible).

13. II.1 Introduction: The Fractional Factorial Idea Figure 1 - 2 3 Design Signs Table C+AB = ( -y 2 -y 3 +y 5 +y 8 )/2 C+AB = ( -y 2 -y 3 +y 5 +y 8 )/2 Use Runs 2, 3, 5, and 8 (i.e., When ABC = I) Use Runs 2, 3, 5, and 8 (i.e., When ABC = I)

14. II.1 Introduction: The Fractional Factorial Idea Objectives Of Fractional Factorials Objectives Of Fractional Factorials –To Reduce the Number of Required Runs –To Screen Out Insignificant Factors In The Initial Stages of Experimentation  A Screening Design This Can Be Done Without Substantial Loss In Information If Higher-Order Interactions Can Be Assumed To Be Negligible This Can Be Done Without Substantial Loss In Information If Higher-Order Interactions Can Be Assumed To Be Negligible We Will See How This Is Done In This Module We Will See How This Is Done In This Module