The Essentials of 2-Level Design of Experiments Part II: The Essentials of Fractional Factorial Designs

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

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

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

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

6. II.1 Introduction: A Quick ReviewRope Pull Study -Completed Seven Effects Normal Plot • 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 • FactorsA: 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, • Two Factors Þ 4 runs • Three Factors Þ 8 runs • Four Factors Þ 16 runs • What if we have seven factors? • 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.

9. II.1 Introduction: How Many Runs? • 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 26 in These Factors, Each at Two Levels, Would Require 64 Runs

10. II.1 Introduction: The Fractional Factorial Idea • In the 23 design, look at the computation of C using the y's in standard order C = ( -y1 -y2 -y3 -y4 +y5+y6 +y7 +y8)/4

11. II.1 Introduction: The Fractional Factorial Idea • Now, look at the same thing for AB: AB = ( +y1 -y2 -y3 +y4 +y5-y6 -y7 +y8)/4

12. II.1 Introduction: The Fractional Factorial Idea • So, C = ( -y1 -y2 -y3 -y4 +y5+y6 +y7 +y8)/4 AB = ( +y1 -y2 -y3 +y4 +y5-y6 -y7 +y8)/4 • Add These Together to get C+AB: C+AB = ( -2y2 -2y3+2y5+2y8)/4 = ( -y2 -y3+y5+y8)/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).

13. II.1 Introduction: The Fractional Factorial Idea Figure 1 - 23 Design Signs Table • C+AB = ( -y2 -y3+y5+y8)/2 • Use Runs 2, 3, 5, and 8 (i.e., When ABC = I)

14. II.1 Introduction: The Fractional Factorial Idea • 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 • We Will See How This Is Done In This Module