The Essentials of 2-Level Design of Experiments Part II: The

Report
The Essentials of 2-Level Design of Experiments
Part II: The Essentials of Fractional Factorial Designs
Developed by Don Edwards, John Grego and James Lynch
Center for Reliability and Quality Sciences
Department of Statistics
University of South Carolina
803-777-7800
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

A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
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:
C
-1
-1
-1
-1
1
1
1
1
D
-1
1
1
-1
1
-1
-1
1
AB AC AD BC
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
1
1
1
1
BD CD ABC
1
1
-1
-1 -1
1
1 -1
1
-1
1
-1
-1
1
1
1 -1
-1
-1 -1
-1
1
1
1
ABD
-1
-1
-1
-1
1
1
1
1
ACD
-1
-1
1
1
-1
-1
1
1
BCD ABCD
-1
1
1
1
-1
1
1
1
-1
1
1
1
-1
1
1
1
II.3 Screening Designs in Eight Runs:
Aliasing for 4 Factors in 8 Runs


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
II.3 Screening Designs in Eight Runs:
Aliasing for 4 Factors in 8 Runs
•
•
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
1 1 1
 1 1  1
1 1 1
 1 1  1
     
1
1
1
     
1
1
1
     
1 1 1
 1 1  1
II.3 Screening Designs in Eight Runs:
Aliasing for 4 Factors in 8 Runs

Rule 2: Any effect multiplied by itself is
equal to I (E.g., AxA=I)
A 
A 
I
1 1 1
 1  1 1
1 1 1
 1  1 1
     
1
1
1
     
1
1
1
     
1 1 1
 1  1 1
II.3 Screening Designs in Eight Runs:
Aliasing for 4 Factors in 8 Runs
•
•
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 find
A=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
II.3 Screening Designs in Eight Runs:
Aliasing for 4 Factors in 8 Runs

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.
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 highorder interactions are probably negligible, anyway.

In the rest of section II, we will ignore three-way and higher
interactions!
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs

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:
Stan dard
Orde r
1
2
3
4
5
6
7
8
A
B
-1
1
-1
1
-1
1
-1
1
C
-1
-1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
1
D=AB
E=AC
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
*Example based on experiment by Pankaj Arora, a student in Statistics 506
BC
1
1
-1
-1
-1
-1
1
1
ABC
-1
1
1
-1
1
-1
-1
1
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs




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).
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs
•
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
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs



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
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs

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.
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs

U-Do-It Exercise Solution.
I=ABD=ACE=BCDE
A=BD=CE=ABCDE
B=AD=ABCE=CDE
C=ABCD=AE=BDE
D=AB=ACDE=BCE
E=ABDE=AC=BCD
BC=ACD=ABE=DE
BE=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
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs

U-Do-It Exercise Solution.
I=ABD=ACE=BCDE
A=BD=CE
B=AD
C=AE
D=AB
E=AC
BC=DE
BE=CD
We often exclude higher
order terms from the alias
structure (except for the
design generator).
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs

The corrosion experiment generated the following data:
Stan dard
Orde r
1
2
3
4
5
6
7
8
C orrosion
Rate
2.71
0.93
4.80
2.53
4.89
3.35
12.29
9.92
A
B
-1
1
-1
1
-1
1
-1
1
C
-1
-1
1
1
-1
-1
1
1
D
-1
-1
-1
-1
1
1
1
1
E
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs
Computation of Factor Effects
y
2.71
0.93
4.80
2.53
4.89
3.35
12.29
9.92
41.42
8
5.178
A+BD+ B+AD C+AE D+AB E+AC BC+DE BE+CD
CE
-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
-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
-7.96
17.66
19.48
-1.32
.14
10.28
-.34
4
4
4
4
4
4
4
-1.99
4.415
4.87
-.33
.035
2.57
-.085
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs
The interaction
is probably due
to BC rather
than DE
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs


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
1
2
1
2.71
.93
1.82
4.89
3.35
4.12
2
4.80
2.53
3.67
12.29
9.92
11.11
B
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs

B and C at their
high levels
greatly increase
corrosion
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs
U-Do-It Exercise: What is the EMR if the
experimenter wishes to minimize the
corrosion rate?
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs
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
II.3 Screening Designs in Eight Runs:
Five Factors in 8 Runs



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 3factor 8-run signs table.
Stan dard
Orde r
1
2
3
4
5
6
7
8
A
B
-1
1
-1
1
-1
1
-1
1
C
-1
-1
1
1
-1
-1
1
1
AB
-1
-1
-1
-1
1
1
1
1
*Suggested by a STAT 506 project, Spring 2000
1
-1
-1
1
1
-1
-1
1
AC
1
-1
1
-1
-1
1
-1
1
D=BC
1
1
-1
-1
-1
-1
1
1
E=ABC
-1
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.
Standard
Orde r
1
2
3
4
5
6
7
8
Growth
9.7
14.7
12.3
12.7
11.2
13.1
10.1
15.0
A
B
-1
1
-1
1
-1
1
-1
1
C
-1
-1
1
1
-1
-1
1
1
D
-1
-1
-1
-1
1
1
1
1
E
1
1
-1
-1
-1
-1
1
1
-1
1
1
-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
II.3 Screening Designs in Eight Runs:
U-Do-It Solution
ALIAS STRUCTURE
 Working from the design generator, the remaining rows of
the design structure will be:
A=DE=BCE=ABCD
B=CD=ACE=ABDE
C=BD=ABE=ACDE
The first two interaction
D=BC=ABCDE=AE
terms we would normally
E=BCDE=ABC=AD
try (AB and AC) had not
AB=ACD=CE=BDE
yet appeared in the alias
AC=ABD=BE=CDE
structure, which made the
last two rows of the table
easy to obtain.
II.3 Screening Designs in Eight Runs:
U-Do-It Solution
ALIAS STRUCTURE
 Eliminating higher order interactions, the alias structure is
I=BCD=ABCE=ADE
A=DE
B=CD
Main effects are
C=BD
confounded with two way
D=BC=AE
effects, making this a
E=AD
Resolution III design.
AB=CE
AC=BE
II.3 Screening Designs in Eight Runs:
U-Do-It Solution
ANALYSIS--Computation of Factor Effects
Grams
9.7
14.7
12.3
12.7
11.2
13.1
10.1
15.0
98.8
8
12.35
A+DE
-1
1
-1
1
-1
1
-1
1
12.2
4
3.05
B+CD
-1
-1
1
1
-1
-1
1
1
1.40
4
.35
C+BD AB+CE AC+BE D+BC+ E+AD
AE
-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
-1
-1
1
-1
-1
1
-1
1
1
1
1
1
0.0
-1.60
1.40
.20
7.60
4
4
4
4
4
0.0
-.40
.35
.05
1.90
II.3 Screening Designs in Eight Runs:
U-Do-It Solution
ANALYSIS--Plot of Factor Effects
Normal Plot of the Effects
(response is Growth, Alpha = .05)
99
Effect Ty pe
Not Significant
Significant
95
A
90
Percent
80
70
60
50
40
30
20
10
5
1
-1
Lenth's PSE = 0.525
0
1
Effect
2
3
F actor
A
B
C
D
E
N ame
A
B
C
D
E
II.3 Screening Designs in Eight Runs:
U-Do-It Solution
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!

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