Stat 470-14

1. Stat 470-14 • Today: Finish Chapter 3; 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 • Important Sections in Chapter 3: • 3.1-3.7 (Read these) • 3.8 (don’t bother reading), • 3.11-3.13 (Read these)

2. Blocking in 2k Experiments • The factorial experiment is an example of a completely randomized design • Often wish to block such experiments • As an example, you may wish to use the same paper helicopter for more than one trial • But which treatments should appear together in a block?

3. Blocking in 2k Experiments • Consider a 23 factorial experiment in 2 blocks

4. Blocking in 2k Experiments • Presumably, blocks are important. • Effect hierarchy suggest we sacrifice higher order interactions • Which is better: • b=ABC • b=AB • Can write as:

5. Blocking in 2k Experiments • Suppose wish to run the experiment in 4 blocks • b1=AB and b2=BC • These imply a third relation • Group is called the defining contrast sub-group

6. Blocking in 2k Experiments • Identifies which effects are confounded with blocks • Cannot tell difference between these effects and the blocking effects

7. Suppose wish to run the 23 experiment in 4 blocks b1=AB and b2=BC I=ABb1=BCb2=ACb1b2 Suppose wish to run the 23 experiment in 4 blocks b1=ABC and b2=BC I=ABCb1=BCb2=Ab1b2 Which design is better?

8. Ranking the Designs • Let D denote a blocking design • gi(D) is the number of i-factor interactions confounded in blocks (i=1,2,…k) • For any 2 blocking schemes (D1 and D2) , let r be the smallest i such that

9. Ranking the Designs • Effect hierarchy suggests that the design that confounds the fewest lower order terms is best • So, if then D1 has less aberration • A design has minimum aberration (MA) if no design has less aberration

10. 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?

11. 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

12. Example

13. 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

14. 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

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

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

17. How can we compare designs? • Resolution

18. Minimum aberration:

19. 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