1. 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

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 Review Rope 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 Review Rope 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
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,
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)/ = ( -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