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

2. Analysis of Unreplicated 2 k Factorial Designs For cost reasons, 2 k 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 2 k factorial experiment –have 2 k -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 s 0 = PSE= t PSE,i =

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

6. Lenth’s Method s 0 = PSE= t PSE,i =

7. Fractional Factorial Designs at 2-Levels 2 k 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 2 k unique treatments Which sub-set of the 2 k 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 5 th factor

9. Example

10. Fractional Factorial Designs at 2-Levels Use a 2 k-p fractional factorial design to explore k factors in 2 k-p trials In general, can construct a 2 k-p fractional factorial design from the full factorial design with 2 k-p trials Set the levels of the first (k-p) factors similar to the full factorial design with 2 k-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 2 k-p fractional factorial design to explore k factors in 2 k-p trials In general, can construct a 2 k-p fractional factorial design from the full factorial design with 2 k-p trials Set the levels of the first (k-p) factors similar to the full factorial design with 2 k-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 2 7-2 fractional factorial design Which one is better? –D 1 : I=ABCDF=ABCEG=DEFG –D 2 : I=ABCF=ADEG=BCDEG