1. Lecture 13 Today: 4.3-4.6 Next day: Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question on D-optimality

2. Optimal Design Approach (4.4.2) Algorithm:

3. Assignment Question Suppose in the cable shrinkage example, effects A, E and AC=BE are identified as signifincat To resolve the aliasing of the interaction effects, a follow-up experiment with 4 trials is to be performed What 4 trails should be performed? Use the D-optimality criterion and report the value of D max

4. Additional Features of a Fractional Factorial Main effect or two-factor interactions (2fi) is clear if it is not aliased with other main effects or 2fi’s Main effect or 2fi is strongly clear if it is not aliased with other main effects, 2fi’s or 3fi’s

5. Blocking Fractional Factorial Designs Can perform a 2 k-p fractional factorial design in 2 q blocks That is, k factors are investigated in 2 k-p runs with 2 q blocks The design is constructed by assigning p treatment factors and q blocking factors to interactions between (k-p) of the factors

6. Example An experimenter wishes to explore the impact of 6 factors (A-F) on the response of a system There exists enough resources to run 16 experiment trials in 4 blocks A 2 6-2 fraction factorial design in 2 2 blocks is required

7. Example Design: –Fractional factorial: E=ABC; F=ABD –Blocking: b 1 =ACD; b 2 =BCD Defining Contrast sub-group:

8. Example

9. Comment Must be careful when choosing the interactions to assign the factors –Fractional factorial: E=AB; F=ABD –Blocking: b 1 =ACD; b 2 =BCD Defining Contrast sub-group:

10. Additional Features Main effect or two-factor interactions (2fi) is clear if it is not aliased with other main effects, 2fi’s or block effects Main effect or 2fi is strongly clear if it is not aliased with other main effects, 2fi’s, 3fi’s or block effects As before, block by factor interactions are negligible Analysis is same as before Appendix 4 has blocked fractional factoria designs ranked by number of clear effects

11. Fractional Factorial Split-Plot Designs It is frequently impractical to perform the fractional factorial design in a completely randomized manner Can run groups of treatments in blocks Sometimes the restrictions on randomization take place because some factors are hard to change or the process takes place in multiple stages Fractional factorial split-plot (FFSP) design may be a practical option

12. Performing FFSP Designs Randomization of FFSP designs different from fractional factorial designs Have hard to change factors (whole-plot or WP factors) and easy to change factors (sub-plot or SP factors) Experiment performed by: –selecting WP level setting, at random. –performing experimental trials by varying SP factors, while keeping the WP factors fixed.

13. Example Would like to explore the impact of 6 factors in 16 trials The experiment cannot be run in a completely random order because 3 of the factors (A,B,C) are very expensive to change Instead, several experiment trials are performed with A, B, and C fixed…varying the levels of the other factors

14. Design Matrix

15. Impact of the Randomization Restrictions Two Sources of randomization  Two sources of error –Between plot error: e w (WP error) –Within plot error: (SP error) Model: The WP and SP error terms have mutually independent normal distributions with standard deviations σ w and σ s

16. The Design Situation: –Have k factors: k 1 WP factors and k 2 SP factors –Wish to explore impact in 2 k-p trials –Have a 2 k1-p1 fractional factorial for the WP factors –Require p=p 1 +p 2 generators –Called a 2 (k 1 + p 2 )-(k 1 + p 2 ) FFSP design

17. Constructing the Design For a 2 (k 1 + p 2 )-(k 1 + p 2 ) FFSP design, have generators for WP and SP designs Rules: –WP generators (e.g., I=ABC ) contain ONLY WP factors –SP generators (e.g., I=Apqr ) must contain AT LEAST 2 SP factors Previous design: I=ABC=Apqr=BCpqr

18. Analysis of FFSP Designs Two Sources of randomization  Two sources of error –Between plot error: σ w (WP error). –Within plot error: σ s (SP error). WP Effects compared to: aσ s 2 + bσ s 2 SP effects compared to : bσ s 2 df for SP df for WP. Get more power for SP effects!!!

19. WP Effect or SP Effect? Effects aliased with WP main effects or interactions involving only WP factors tested as a WP effect. E.g., pq=ABCD tested as a WP effect. Effects aliased only with SP main effects or interactions involving at least one SP factors tested as a SP effect. E.g., pq=ABr tested as a SP effect.

20. Ranking the Designs Use minimum aberration (MA) criterion