Lecture 11

1. Lecture 11 • Today: 4.2 • Next day: 4.3-4.6

2. Analysis of Unreplicated 2k Factorial Designs • For cost reasons, 2k factorial experiments are frequently unreplicated • Can assess significance of the factorial effects using a normal or half-normal probability plot • May prefer a formal significance test procedure • Cannot use an F-test or t-test because there are no degrees of freedom for error

3. Lenth’s Method • Situation: • have performed an unreplicated 2k factorial experiment • have 2k-1 factorial effects • want to see which effects are significantly different from 0 • If none of the effects is important of the factorial effects is an independent realization of a N( , ) • Can use this fact to develop an estimator of the effect variance based on the median of the absolute effects

4. Lenth’s Method • s0= • PSE= • tPSE,i=

5. Example: Original Growth Layer Experiment • Effect Estimates and QQ-Plot:

6. Lenth’s Method • s0= • PSE= • tPSE,i=

7. Fractional Factorial Designs at 2-Levels • 2k factorial experiments can be very useful in exploring a relatively large number of factors in relatively few trials • When k is large, the number of trials is large • Suppose have enough resources to run only a fraction of the 2k unique treatments • Which sub-set of the 2k treatments should one choose?

8. Example • Suppose have 5 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 an interaction column from the first 4 factors to set the levels of the 5th factor

9. Example

10. 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

11. 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

12. 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

13. Example • The 3 relations imply other relations • Defining contrast sub-group • Word-length pattern

14. How can we compare designs? • Resolution • Minimum aberration

15. 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