II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs

1. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs • In addition to 5 factors in 8 runs, Resolution III designs can be used to study 6 factors in 8 runs or 7 factors in 8 runs. • For 6 factors in 8 runs, we could use the assignment D=AB, E=AC, F=BC as shown below. The design generator would be I=ABD=ACE=BCF=BCDE=ACDF=ABEF=DEF

2. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs • For 7 factors in 8 runs, we could use the assignment D=AB, E=AC, F=BC, G=ABC. The full design generator has 15 terms so we will not record it. Each main effect is confounded with 3 2-way effects.

3. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs • A female shot-put athlete tested factors that affected the distance of her throws (measured in feet). She assigned 7 factors to the columns of the 3-factor 8-run signs table as shown below.

4. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs • The alias structure for this design (eliminating all higher-order interactions) is: I=ABD=ACE=BCF=ABCG=BCDE=ACDF=CDG=ABEF= BEG=AFG=DEF=ADEG=BDFG=CEFG=ABCDEFG A=BD=CE=FG B=AD=CF=EG C=AE=BF=DG D=AB=CG=EF E=AC=BG=DF F=AG=BC=DE G=AF=BE=CD This is a Resolution III design

5. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs • After randomizing runs and completing 8 throws, the following distances were obtained:

6. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs Computation of Factor Effects

7. II.3 Screening Designs in Eight Runs: Other Screening Designs in 8 runs • Recall that: • B=EG • E=BG • G=BE

8. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs • The preceding effects plot suggests 4 reasonable scenarios: • Three main effects, B, E and G are present • Two main effects and an interaction are present • B, E and BE • B, G and BG • E, G and EG • A correct interpretation cannot be made from the data at hand, but runs can be added to resolve this problem.

9. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs • We cannot make a correct interpretation because the current 8-run design is Resolution III. • By adding additional runs, the combined design will be Resolution IV. Main effects will no longer be confounded with two-way effects and we should be able to choose from among the four scenarios listed earlier.

10. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation • Factor levels for the 8 additional runs are found by reversing the levels of all factors from the original 8 runs. OriginalRun 1 Followup Run 1

11. II.3 Screening Designs in Eight Runs: Foldover Designs Implementation--Follow-up Runs Don’t forget to randomize the runs! Note that D=-AB, E=-AC, F=-BC, G=ABC

12. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation • Note that the factor levels for all factors have been reversed. E.g., D is constructed by reversing the levels of D in the original design, not by computing AB from followup design columns A and B.

13. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs • After eight runs with the seven factors, we had experienced some uncertainty in interpretation. Eight runs were added to these according to the fold-over instructions.

14. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Computation of Factor Effects for Follow-up Runs

15. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation • Conduct the second experiment and compute factor effects. • Enter the effects from Experiment 1 (the original design) and Experiment 2 in the first two rows of the following table.

16. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation--Effects Table The last two rows compute effects for the combined experiment

17. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation--Effects Table Interpretation Main effects are not aliased with two-way effects.

18. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation--Effects Table Interpretation • The last two rows contain estimates of the overall average, a block effect and 14 effects. The 7 main effects are no longer aliased with two-way effects. • The block effect is the difference in the averages between the two experiments. If it is large, experimental conditions may have changed between experiments.

19. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Computation of Factor Effects for Foldover Design Only main effects are large! She threw better on average in the original experiment

20. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs • None of the two-way effects are large. We would conclude that main effects B, E and G are important. • We can confirm the analysis with a 15-effects normal probability plot. We can include the Block as an effect, provided the Block effect is similar in scale to the remaining effects. Do not interpret the Block effect as significant or insignificant based on its position in the probability plot--its levels were not randomized. • The high levels of B and G improve shot put distance, while the low level of E improves shot put distance.

21. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs

22. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation • When folding over 5-factor and 6-factor designs, the interpretation of columns with no main effects is difficult, since the effects the columns estimate change in the follow-up design. • If we had folded over a 5-factor design with D=AB and E=AC, then we would refer to the 6th column as “Column 6” and not as BC (and the 7th column as “Column 7” and not as ABC).

23. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation--Effects Table Interpretation • For a 5-factor design, (D=AB, E=AC), the entries in the foldover effects table would be interpreted as shown on the following table. Entries with * denote estimates involving only three-way effects or higher. Again, note that the design is Resolution IV.

24. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs 5 Factors 7 Factors

25. II.3 Screening Designs in Eight Runs: Foldover of Resoloution III Designs Implementation--Effects Table Interpretation • The same exercise can be repeated for a 6-factor design, (D=AB, E=AC, F=BC). Again, entries with * denote estimates involving only three-way effects or higher.

26. II.3 Screening Designs in Eight Runs: Foldover of Resolution III Designs Implementation for 6-factor Design--Effects Table Interpretation