Stat 470-15

1. Stat 470-15 • Today: Start Chapter 4 • Assignment 3: 3.14 a, b (do normal qq-plots only),c, 3.16, 3.17 • Additional questions: 3.14 b (also use the IER version of Lenth’s method and compare to the qq-plot conclusions), 3.19

2. Fractional Factorial Designs at 2-Levels • Use a 2k-p fractional factorial design to explore k factors in 2k-p trials • In general, can construct a 2k-p fractional factorial design from the full factorial design with 2k-p trials • Set the levels of the first (k-p) factors similar to the full factorial design with 2k-p trials • Next, use the interaction columns between the first (k-p) factors to set levels of the remaining factors

3. Example • Suppose have 7 factors, each at 2-levels, but only enough resources to run 16 trials • Can use a 16-run full factorial to design the experiment • Use the 16 unique treatments for 4 factors to set the levels of the first 4 factors (A-D) • Use interaction columns from the first 4 factors to set the levels of the remaining 3 factors

4. Example

5. Example • Would like to have as few short words as possible • Why?

6. How can we compare designs? • Resolution

7. Minimum aberration (MA):

8. Example • Suppose have 7 factors, each at 2-levels, but only enough resources to run 32 trials • Can use a 27-2 fractional factorial design • Which one is better? • D1: I=ABCDF=ABCEG=DEFG • D2: I=ABCF=ADEG=BCDEG

9. Example • Suppose have 8 factors (A-H), each at 2-levels, but only enough resources to run 32 trials • Can use a 28-3 fractional factorial design • Table 4A gives the minimum aberration (MA) designs for 8, 16, 32 and 64 runs • From Table 4A.3, MA design gives: • 6=123 • 7=124 • 8=1345

10. Example • Table 4A.3, MA design is: • 6=123 • 7=124 • 8=1345 • Design for our factors: • Word length pattern:

11. Example • Speedometer cables can be noisy because of shrinkage in the plastic casing material • An experiment was conducted to find out what caused shrinkage • Engineers started with 6 different factors: • A braiding tension • B wire diameter • C liner tension • D liner temperature • E coating material • F melt temperature

12. Example • Response is percentage shrinkage per specimen • There were two levels of each factor • A 26-2 fractional factorial • The purpose of such an experiment is to determine which factors impact the response

13. Example • Constructing the design • Write down the 16 run full factorial • Use interaction columns to set levels of the other 2 factors • Which interaction columns do we use? • Table 4A.2 gives 16 run MA designs • E=ABC; F=ABD

14. Example

15. Example • Results

16. Example • Which effects can we estimate? • Defining Contrast Sub-Group: I=ABCE=ABDF=CDEF • Word-Length Pattern: • Resolution:

17. Example • Effect Estimates and QQ-Plot:

18. Comments • Use defining contrast subgroup to determine which effects to estimate • Can use qq-plot or Lenth’s method to evaluate the significance of the effects • Fractional factorial designs allow you to explore many factors in relatively few trials • Trade-off run-size for information about interactions