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

In addition to 5 factors in 8 runs, Resolution III In addition to 5 factors in 8 runs, Resolution III designs can be used to study 6 factors in 8 runs or designs can be used to study 6 factors in 8 runs or 7 factors in 8 runs.7 factors in 8 runs.

For 6 factors in 8 runs, we could use the For 6 factors in 8 runs, we could use the assignment D=AB, E=AC, F=BC as shown below. assignment D=AB, E=AC, F=BC as shown below. The design generator would be The design generator would be

I=ABD=ACE=BCF=BCDE=ACDF=ABEF=DEFI=ABD=ACE=BCF=BCDE=ACDF=ABEF=DEF

StandardOrder

A B C D=AB E=AC F=BC ABC

1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 1

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

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

StandardOrder

A B C D=AB E=AC F=BC G=ABC

1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 1

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

A female shot-put athlete tested factors that A female shot-put athlete tested factors that affected the distance of her throws (measured in affected the distance of her throws (measured in feet). She assigned 7 factors to the columns of the feet). She assigned 7 factors to the columns of the 3-factor 8-run signs table as shown below.3-factor 8-run signs table as shown below.

StandardOrder

A B C D=AB E=AC F=BC G=ABC

1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 1

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

The alias structure for this design (eliminating all The alias structure for this design (eliminating all higher-order interactions) is:higher-order interactions) is:

I=ABD=ACE=BCF=ABCG=BCDE=ACDF=CDG=ABEF=I=ABD=ACE=BCF=ABCG=BCDE=ACDF=CDG=ABEF=

BEG=AFG=DEF=ADEG=BDFG=CEFG=ABCDEFGBEG=AFG=DEF=ADEG=BDFG=CEFG=ABCDEFG

A=BD=CE=FGA=BD=CE=FG

B=AD=CF=EGB=AD=CF=EG

C=AE=BF=DGC=AE=BF=DG

D=AB=CG=EFD=AB=CG=EF

E=AC=BG=DFE=AC=BG=DF

F=AG=BC=DEF=AG=BC=DE

G=AF=BE=CDG=AF=BE=CD

This is a Resolution III designThis is a Resolution III design

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

After randomizing runs and completing 8 throws, After randomizing runs and completing 8 throws, the following distances were obtained:the following distances were obtained:

StandardOrder

Distance A B C D E F G

1 46.9’ -1 -1 -1 1 1 1 -12 49.8’ 1 -1 -1 -1 -1 1 13 50.2’ -1 1 -1 -1 1 -1 14 49.1’ 1 1 -1 1 -1 -1 -15 49.8’ -1 -1 1 1 -1 -1 16 47.6’ 1 -1 1 -1 1 -1 -17 48.9’ -1 1 1 -1 -1 1 -18 50.3’ 1 1 1 1 1 1 1

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

Computation of Factor EffectsComputation of Factor Effects

Feet 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

46.9 -1 -1 -1 1 1 1 -149.8 1 -1 -1 -1 -1 1 150.2 -1 1 -1 -1 1 -1 149.1 1 1 -1 1 -1 -1 -149.8 -1 -1 1 1 -1 -1 147.6 1 -1 1 -1 1 -1 -148.9 -1 1 1 -1 -1 1 -150.3 1 1 1 1 1 1 1392.6 1.0 4.4 .60 -.4 -2.6 -.8 7.60

8 4 4 4 4 4 4 449.08 .25 1.1 .15 -.1 -.65 -.2 1.9

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

210

Effects

E

BG

Effects Plot for Shot Put Experiment

Recall that:Recall that:

B=EGB=EG

E=BGE=BG

G=BEG=BE

Recall that:Recall that:

B=EGB=EG

E=BGE=BG

G=BEG=BE

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

The preceding effects plot suggests 4 The preceding effects plot suggests 4 reasonable scenarios:reasonable scenarios:– Three main effects, B, E and G are presentThree main effects, B, E and G are present– Two main effects and an interaction are Two main effects and an interaction are

presentpresent B, E and BEB, E and BE B, G and BGB, G and BG E, G and EGE, G and EG

A correct interpretation cannot be made from A correct interpretation cannot be made from the data at hand, but runs can be added to the data at hand, but runs can be added to resolve this problem.resolve this problem.

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

We cannot make a correct We cannot make a correct interpretation because the current 8-interpretation because the current 8-run design is Resolution III. run design is Resolution III.

By adding additional runs, the By adding additional runs, the combined design will be Resolution IV. combined design will be Resolution IV. Main effects will no longer be Main effects will no longer be confounded with two-way effects and confounded with two-way effects and we should be able to choose from we should be able to choose from among the four scenarios listed earlier.among the four scenarios listed earlier.

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

ImplementationImplementation– Factor levels for the 8 additional Factor levels for the 8 additional

runs are found by reversing the runs are found by reversing the levels of all factors from the levels of all factors from the original 8 runs. original 8 runs. StandardOrder

A B C D E F G

1 -1 -1 -1 1 1 1 -1

StandardOrder

A B C D E F G

9 1 1 1 -1 -1 -1 1

Original Run 1Original Run 1

Followup Run 1Followup Run 1

II.3 Screening Designs in Eight Runs: II.3 Screening Designs in Eight Runs: Foldover DesignsFoldover Designs

Implementation--Follow-up RunsImplementation--Follow-up Runs

StandardOrder

A B C D E F G

1 1 1 1 -1 -1 -1 12 -1 1 1 1 1 -1 -13 1 -1 1 1 -1 1 -14 -1 -1 1 -1 1 1 15 1 1 -1 -1 1 1 -16 -1 1 -1 1 -1 1 17 1 -1 -1 1 1 -1 18 -1 -1 -1 -1 -1 -1 -1

Note that D=-AB, E=-AC,F=-BC, G=ABC

Note that D=-AB, E=-AC,F=-BC, G=ABC

Don’t forget to randomize the runs!

Don’t forget to randomize the runs!

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

ImplementationImplementation– Note that the factor levels for Note that the factor levels for allall

factors have been reversed. E.g., factors have been reversed. E.g., D is constructed by reversing the D is constructed by reversing the levels of D in the original design, levels of D in the original design, not by computing AB from not by computing AB from followup design columns A and B.followup design columns A and B.

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

After eight runs with the seven factors, we After eight runs with the seven factors, we had experienced some uncertainty in had experienced some uncertainty in interpretation. Eight runs were added to interpretation. Eight runs were added to these according to the fold-over these according to the fold-over instructions.instructions.

StandardOrder

Distance A B C D E F G

1 50.2’ 1 1 1 -1 -1 -1 12 46.8’ -1 1 1 1 1 -1 -13 46.7’ 1 -1 1 1 -1 1 -14 48.1’ -1 -1 1 -1 1 1 15 47.1’ 1 1 -1 -1 1 1 -16 50.4’ -1 1 -1 1 -1 1 17 48.5’ 1 -1 -1 1 1 -1 18 47.5’ -1 -1 -1 -1 -1 -1 -1

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

Computation of Factor Effects for Follow-up RunsComputation of Factor Effects for Follow-up Runs

Feet 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

50.2 1 1 1 -1 -1 -1 146.8 -1 1 1 1 1 -1 -146.7 1 -1 1 1 -1 1 -148.1 -1 -1 1 -1 1 1 147.1 1 1 -1 -1 1 1 -150.4 -1 1 -1 1 -1 1 148.5 1 -1 -1 1 1 -1 147.5 -1 -1 -1 -1 -1 -1 -1385.3 -.3 3.7 -1.7 -.5 -4.3 -.7 9.1

8 4 4 4 4 4 4 448.16 -.08 .93 -.43 -.13 -1.08 -.18 2.28

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

ImplementationImplementation– Conduct the second experiment Conduct the second experiment

and compute factor effects. and compute factor effects. – Enter the effects from Experiment Enter the effects from Experiment

1 (the original design) and 1 (the original design) and Experiment 2 in the first two rows Experiment 2 in the first two rows of the following table. of the following table.

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

Implementation--Effects TableImplementation--Effects Table

Average A B C D E F G

Experiment 1Effects (E1)Experiment 2Effects (E2)Main Effects(E1+ E2)/ 2Interactions(E1- E2)/ 2

The last two rows compute effects for the combined experiment

The last two rows compute effects for the combined experiment

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

Implementation--Effects Table Implementation--Effects Table InterpretationInterpretation

Average A B C D E F G

Experiment 1Effects (E1)Experiment 2Effects (E2)Main Effects(E1+ E2)/ 2

Average A B C D E F G

Interactions(E1- E2)/ 2

Block BD+CE+FG

AD+CF+EG

AE+BF+DG

AB+CG+EF

AC+BG+DF

AG+BC+DE

AF+BE+CD

Main effects are not aliased with two-way effects.

Main effects are not aliased with two-way effects.

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

Implementation--Effects Table Implementation--Effects Table InterpretationInterpretation

– The last two rows contain estimates The last two rows contain estimates of the overall average, a block effect of the overall average, a block effect and 14 effects. The 7 main effects and 14 effects. The 7 main effects are no longer aliased with two-way are no longer aliased with two-way effects.effects.

– The block effect is the difference in The block effect is the difference in the averages between the two the averages between the two experiments. If it is large, experiments. If it is large, experimental conditions may have experimental conditions may have changed between experiments.changed between experiments.

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

Computation of Factor Effects for Foldover DesignComputation of Factor Effects for Foldover Design

Average A B C D E F G

Experiment 1Effects (E1)

49.08 .25 1.1 .15 -.1 -.65 -.2 1.9

Experiment 2Effects (E2)

48.16 -.08 .93 -.43 -.13 -1.08 -.18 2.28

Main Effects(E1+ E2)/ 2

48.62 .09 1.02 -.14 -.12 -.87 -.19 2.09

Interactions(E1- E2)/ 2

.46 .17 .09 .29 .02 .22 -.01 -.19

Only main effectsare large!

Only main effectsare large!She threw better on average

in the original experiment

She threw better on averagein the original experiment

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

None of the two-way effects are large. We would None of the two-way effects are large. We would conclude that main effects B, E and G are conclude that main effects B, E and G are important. important.

We can confirm the analysis with a 15-effects We can confirm the analysis with a 15-effects normal probability plot. We can include the Block normal probability plot. We can include the Block as an effect, provided the Block effect is similar in as an effect, provided the Block effect is similar in scale to the remaining effects. Do not interpret the scale to the remaining effects. Do not interpret the Block effect as significant or insignificant based on Block effect as significant or insignificant based on its position in the probability plot--its levels were its position in the probability plot--its levels were not randomized. not randomized.

The high levels of B and G improve shot put The high levels of B and G improve shot put distance, while the low level of E improves shot put distance, while the low level of E improves shot put distance.distance.

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

210-1

Effects

E

GB

Block

Effects Plot for Foldover Shot Put Experiment

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

ImplementationImplementation– When folding over 5-factor and 6-When folding over 5-factor and 6-

factor designs, the interpretation of factor designs, the interpretation of columns with no main effects is columns with no main effects is difficult, since the effects the columns difficult, since the effects the columns estimate change in the follow-up estimate change in the follow-up design. design.

– If we had folded over a 5-factor design If we had folded over a 5-factor design with D=AB and E=AC, then we would with D=AB and E=AC, then we would refer to the 6th column as “Column 6” refer to the 6th column as “Column 6” and not as BC (and the 7th column as and not as BC (and the 7th column as “Column 7” and not as ABC).“Column 7” and not as ABC).

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

Implementation--Effects Table Implementation--Effects Table InterpretationInterpretation

– For a 5-factor design, (D=AB, For a 5-factor design, (D=AB, E=AC), the entries in the foldover E=AC), the entries in the foldover effects table would be interpreted effects table would be interpreted as shown on the following table. as shown on the following table. Entries with * denote estimates Entries with * denote estimates involving only three-way effects or involving only three-way effects or higher. Again, note that the higher. Again, note that the design is Resolution IV.design is Resolution IV.

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

Average A B C D E Col.6

Col.7

Experiment 1Effects (E1)Experiment 2Effects (E2)Main Effects(E1+ E2)/ 2

Average A B C D E * *

Interactions(E1- E2)/ 2

Block BD+CE

AD AE AB AC BC+DE

BE+CD

Average A B C D E F G

Experiment 1Effects (E1)Experiment 2Effects (E2)Main Effects(E1+ E2)/ 2

Average A B C D E F G

Interactions(E1- E2)/ 2

Block BD+CE+FG

AD+CF+EG

AE+BF+DG

AB+CG+EF

AC+BG+DF

AG+BC+DE

AF+BE+CD

5 Factors5 Factors

7 Factors7 Factors

II.3 Screening Designs in Eight Runs: II.3 Screening Designs in Eight Runs: Foldover of Resoloution III DesignsFoldover of Resoloution III Designs

Implementation--Effects Table Implementation--Effects Table InterpretationInterpretation

– The same exercise can be The same exercise can be repeated for a 6-factor design, repeated for a 6-factor design, (D=AB, E=AC, F=BC). Again, (D=AB, E=AC, F=BC). Again, entries with * denote estimates entries with * denote estimates involving only three-way effects or involving only three-way effects or higher. higher.

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

Implementation for 6-factor Implementation for 6-factor Design--Effects Table Design--Effects Table

InterpretationInterpretationAverage A B C D E F Col.

7Experiment 1Effects (E1)Experiment 2Effects (E2)Main Effects(E1+ E2)/ 2

Average A B C D E F *

Interactions(E1- E2)/ 2

Block BD+CE

AD+CF

AE+BF

AB+EF

AC+DF

BC+DE

AF+BE+CD