screening design in 8 runs

The Essentials of 2-Level Design of ExperimentsPart II: The Essentials of Fractional Factorial Designs

II.3 Screening Designs in 8 runs

Aliasing for 4 Factors in 8 Runs 5 Factors in 8 runs A U-Do-It Case Study Foldover of Resolution III Designs

II.3 Screening Designs in Eight Runs: Aliasing for 4 Factors in 8 Runs

In an earlier exercise from II.2, four factors were studied in 8 runs by using only those runs from a 24 design for which ABCD was positive:

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD-1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 11 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1

-1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 11 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1

-1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 11 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1

-1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1


We use “I” to denote a column of ones and note that I=ABCD for this particular design

DEFINITION: The set of effects whose levels are constant (either 1 or -1) in a design are design generators.

E.g, the design generator for the example in II.2 with 4 factors in 8 runs is I=ABCD

The alias structure for all effects can be constructed from the design generator


To construct the confounding structure, we need two simple rules:

Rule 1: Any effect column multiplied by I is unchanged (E.g., AxI=A)

A I A

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_ Rule 2: Any effect multiplied by itself is equal to I (E.g., AxA=I)A A I

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We can now construct an alias table by multiplying both sides of the design generator by any effect.

E.g., for effect A, we have the steps: AxI=AxABCD A=IxBCD (Applying Rule 1 to the left and Rule 2 to

the right) A=BCD (Applying Rule 1 to the right)

If we do this for each effect, we findA=BCD AB=CD BD=AC ACD=B

B=ACD AC=BD CD=AB BCD=A

C=ABD AD=BC ABC=D ABCD=I

D=ABC BC=AD ABD=C


Several of these statements are redundant. When we remove the redundant statements, we obtain the alias structure (which usually starts with the design generator):

I=ABCD D=ABC

A=BCD AB=CD

B=ACD AC=BD

C=ABD AD=BC

The alias structure will be complicated for more parsimonious designs; we will add a few more guidelines for constructing alias tables later on.

The alias structure will be complicated for more parsimonious designs; we will add a few more guidelines for constructing alias tables later on.

II.3 Screening Designs in Eight Runs: Five Factors in 8 Runs

Suppose five two-level factors A, B, C, D, E are to be examined. If using a full factorial design, there would be 25=32 runs, and 31 effects estimated– 5 main effects– 10 two-way interactions– 10 three-way interactions– 5 four-way interactions– 1 five-way interaction

In many cases so much experimentation is impractical, and high-order interactions are probably negligible, anyway.

In the rest of section II, we will ignore three-way and higher interactions!


An experimenter wanted to study the effect of 5 factors on corrosion rate of iron rebar* in only 8 runs by assigning D to column AB and E to column AC in the 3-factor 8-run signs table:

StandardOrder

A B C D=AB E=AC 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


For this particular design, the experimenter used only 8 runs (1/4 fraction) of a 32 run (or 25) design (I.e., a 25-2 design).

For each of these 8 runs, D=AB and E=AC. If we multiply both sides of the first equation by D, we obtain DxD=ABxD, or I=ABD.

Likewise, if we multiply both sides of E=AC by E, we obtain ExE=ACxE, or I=ACE.

We can say the design is comprised of the 8 runs for which both ABD and ACE are equal to one (I=ABD=ACE).


There are 3 other equivalent 1/4 fractions the experimenter could have used:

ABD = 1, ACE = -1 (I = ABD = -ACE) ABD = -1, ACE = 1 (I = -ABD = ACE) ABD = -1, ACE = -1 (I = -ABD = -ACE)

The fraction the experimenter chose is called the principal fraction


I=ABD=ACE is the design generator If ABD and ACE are constant, then their

interaction must be constant, too. Using Rule 2, their interaction is ABD x ACE = BCDE

The first two rows of the confounding structure are provided below.

Line 1: I = ABD = ACE = BCDE Line 2:

AxI=AxABD=AxACE=AxBCDE A=BD=CE=ABCDE

The shortest word in the design generator has three letters, so we call this a Resolution III design

The shortest word in the design generator has three letters, so we call this a Resolution III design


U-Do-It Exercise. Complete the remaining 6 non-redundant rows of the confounding structure for the corrosion experiment. Start with the main effects and then try any two-way effects that have not yet appeared in the alias structure.


U-Do-It Exercise Solution.

I=ABD=ACE=BCDEA=BD=CE=ABCDEB=AD=ABCE=CDEC=ABCD=AE=BDED=AB=ACDE=BCEE=ABDE=AC=BCDBC=ACD=ABE=DEBE=ADE=ABC=CD

After computing the alias structure for main effects, it may require trial and error to find the remaining rows of the alias structure

After computing the alias structure for main effects, it may require trial and error to find the remaining rows of the alias structure


U-Do-It Exercise Solution.

I=ABD=ACE=BCDEA=BD=CEB=ADC=AED=ABE=ACBC=DEBE=CD

We often exclude higher order terms from the alias structure (except for the design generator).

We often exclude higher order terms from the alias structure (except for the design generator).


The corrosion experiment generated the following data:

StandardOrder

CorrosionRate

A B C D E

1 2.71 -1 -1 -1 1 12 0.93 1 -1 -1 -1 -13 4.80 -1 1 -1 -1 14 2.53 1 1 -1 1 -15 4.89 -1 -1 1 1 -16 3.35 1 -1 1 -1 17 12.29 -1 1 1 -1 -18 9.92 1 1 1 1 1


Computation of Factor Effects

y A+BD+CE

B+AD C+AE D+AB E+AC BC+DE BE+CD

2.71 -1 -1 -1 1 1 1 -10.93 1 -1 -1 -1 -1 1 14.80 -1 1 -1 -1 1 -1 12.53 1 1 -1 1 -1 -1 -14.89 -1 -1 1 1 -1 -1 13.35 1 -1 1 -1 1 -1 -112.29 -1 1 1 -1 -1 1 -19.92 1 1 1 1 1 1 141.42 -7.96 17.66 19.48 -1.32 .14 10.28 -.34

8 4 4 4 4 4 4 45.178 -1.99 4.415 4.87 -.33 .035 2.57 -.085


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Effects Plot for Corrosion Experiment

The interaction is probably due to BC rather than DE

The interaction is probably due to BC rather than DE


Factor A at its high level reduced the corrosion rate by 1.99 units

Factor B and C main effects cannot be interpreted in the presence of a significant BC interaction. C

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2.71 4.891 .93 3.35

1.82 4.12

4.80 12.292 2.53 9.92

3.67 11.11

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BC Interaction Plot for Corrosion Experiment

_ B and C at their high levels greatly increase corrosion

_ B and C at their high levels greatly increase corrosion


U-Do-It Exercise: What is the EMR if the experimenter wishes to minimize the corrosion rate?


U-Do-It Exercise Solution

_ A should be set high, B and C should be low and BC should be high, so our solution is:

EMR=5.178+(-1.99/2)-(4.415/2)-(4.87/2)+(2.57/2)

EMR=.8255


D and E could have been assigned to any of the last 4 columns (AB, AC, BC or ABC) in the 3-factor 8-run signs table.

All of the resulting designs would be Resolution III, which means that at least one main effect would be aliased with at least one two-way effect.

For a Resolution IV design (e.g., 4 factors in 8 runs)

The shortest word in the design generator has 4 letters (e.g., I=ABCD for 4 factors in 8 runs)

No main effects are aliased with two-way effects, but at least one two-way effect is aliased with another two-way effect

What qualities would a Resolution V design have?

II.3 Screening Designs in Eight Runs: U-Do-It Case Study

A statistically-minded vegetarian* studied 5 factors that would affect the growth of alfalfa sprouts. Factors included measures such as presoak time and watering regimen. The response was biomass measured in grams after 48 hours. Factor D was assigned to the BC column and factor E was assigned to the ABC column in the 3-factor 8-run signs table.

StandardOrder

A B C AB AC D=BC E=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: U-Do-It Case Study

The runs table appears below. Find the alias structure for this data and analyze the data.

StandardOrder

Growth A B C D E

1 9.7 -1 -1 -1 1 -12 14.7 1 -1 -1 1 13 12.3 -1 1 -1 -1 14 12.7 1 1 -1 -1 -15 11.2 -1 -1 1 -1 16 13.1 1 -1 1 -1 -17 10.1 -1 1 1 1 -18 15.0 1 1 1 1 1

II.3 Screening Designs in Eight Runs: U-Do-It Solution

ALIAS STRUCTURE The design generator was computed as

follows. Since D=BC, when we multiply each side of the equation by D, we obtain DxD=BCD or I=BCD. Also, since E=ABC, when we mulitply each side of this equation by E, we obtain I=ABCE. The interaction of BCD and ABCE will also be constant (and positive in this case), so we have I=BCDxABCE=AxBxBxCxCxDxE=ADE

The design generator is I=BCD=ABCE=ADE


ALIAS STRUCTURE Working from the design generator, the

remaining rows of the design structure will be:

A=DE=BCE=ABCDB=CD=ACE=ABDEC=BD=ABE=ACDED=BC=ABCDE=AEE=BCDE=ABC=ADAB=ACD=CE=BDEAC=ABD=BE=CDE

The first two interaction terms we would normally try (AB and AC) had not yet appeared in the alias structure, which made the last two rows of the table easy to obtain.

The first two interaction terms we would normally try (AB and AC) had not yet appeared in the alias structure, which made the last two rows of the table easy to obtain.


ALIAS STRUCTURE Eliminating higher order interactions, the

alias structure isI=BCD=ABCE=ADEA=DEB=CDC=BDD=BC=AEE=ADAB=CEAC=BE

Main effects are confounded with two way effects, making this a Resolution III design.

Main effects are confounded with two way effects, making this a Resolution III design.


ANALYSIS--Computation of Factor Effects

Grams A+DE B+CD C+BD AB+CE AC+BE D+BC+AE

E+AD

9.7 -1 -1 -1 1 1 1 -114.7 1 -1 -1 -1 -1 1 112.3 -1 1 -1 -1 1 -1 112.7 1 1 -1 1 -1 -1 -111.2 -1 -1 1 1 -1 -1 113.1 1 -1 1 -1 1 -1 -110.1 -1 1 1 -1 -1 1 -115.0 1 1 1 1 1 1 198.8 12.2 1.40 0.0 -1.60 1.40 .20 7.60

8 4 4 4 4 4 4 412.35 3.05 .35 0.0 -.40 .35 .05 1.90


ANALYSIS--Plot of Factor Effects

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A AB BC CD DE E

Factor Name

Not SignificantSignificant

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A

Normal Plot of the Effects(response is Growth, Alpha = .05)

Lenth's PSE = 0.525


ANALYSIS--Interpretation

Factor A at its high level increases the yield by 3.05 grams

Factor E at its high level increases the yield by 1.90 grams

Both of these effects are confounded with two way interactions, but we have used the simplest possible explanation for the significant effects we observed

Note: the most important result in the actual experiment was an insignificant main effect. The experimenter found that the recommended presoak time for the alfalfa seeds could be lowered from 16 hours to 4 hours with no deleterious effect on the yield--a significant time savings!

screening design in 8 runs

Technology

abcd

acbd

adbc

acd

run signs

fractional

high level

resolution