5
Two-Level Fractional Factorial Designs
• Because the number of runs in a 2k factorial design increases rapidly as the number of factors
increases, it is often impossible to run the full factorial design given available resources.
• If the experimenter can reasonably assume that certain high-order interactions (often 3-way
and higher order) are negligible, then it is often possible to estimate and test for the significance
of main effects and low-order interactions from only a fraction of the full factorial design.
• A design which contains a subset of factor level combinations from a full factorial design is
called a fractional factorial design.
• A fractional factorial design is often used as a screening experiment involving many factors
with the goal of identifying only those factors having large effects. Once specific factors are
identified as important, they are investigated in greater detail in subsequent experiments.
• The successful use of two-level fractional factorial designs is based on three ideas:
1. The sparsity of effects principle: When there are many variables under consideration,
it is typical for the system or process to be dominated by main effects and low-order
interactions.
2. The projective property: A fractional factorial design can be projected into stronger
designs in a subset of the significant factors.
3. Sequential experimentation: If is possible to combine the runs from two or more
fractional factorial designs to sequentially form a larger design that allows estimation of
all interaction effects of interest.
5.1
2k−1 Fractional Factorial Designs
• Situation: There are k factors of interest each having 2 levels, but there are only enough
resources to run 1/2 of the full factorial 2k design. Thus, we say we want to run a 1/2
fraction of a 2k design. This design is called a 2k−1 fractional factorial design.
– Suppose there are 5 factors of interest (A, B, C, D, and E), and there are only enough
resources for 16 experimental runs. Thus, we want to run a 1/2 fraction of a 25 design.
This design is called a 25−1 fractional factorial design.
• A 1/2 fraction can be generated from any interaction, but using the highest-order interaction
is the standard. The interaction used to generate the 1/2 fraction is called the generator of
the fractional factorial design.
– When there are 3 factors, use ABC as the generator of the 23−1 design, when there are
4 factors, use ABCD as the generator of the 24−1 design, . . ., when there are k factors,
use ABC . . . K as the generator of the 2k−1 design.
• To generate a 2k−1 fractional factorial design from the highest-order interaction:
(i) Select only those treatment combinations that have a plus (+) sign in the ABC · · · K
column of the 2k design, OR,
(ii) Select only those treatment combinations that have a minus (−) sign in the ABC · · · K
column of the 2k design.
Because the I column also contains all plus signs, we refer to I = ABC · · · K or
I = −ABC · · · K
as a defining relation for the 2k−1 design.
71
• Example When there are 3 factors, we have the following table of pluses and minuses.
Treatment
total
a
b
c
abc
ab
ac
bc
(1)
I
+
+
+
+
+
+
+
+
A
+
−
−
+
+
+
−
−
Factorial Effect
C AB AC
−
−
−
−
−
+
+
+
−
+
+
+
−
+
−
+
−
+
+
−
−
−
+
+
B
−
+
−
+
+
−
+
−
BC
+
−
−
+
−
−
+
+
ABC
+
+
+
+
−
−
−
−
Aliased
effects
I=
A=
B=
C=
I=
A=
B=
C=
Thus, for defining relation I = ABC , the treatment combinations with ABC = +1 form
the 23−1 design, or, the 1/2 fraction of the 23 design. The remaining 4 treatment combinations
with ABC = −1 are discarded (i.e, discard the ab, ac, bc, and (1) rows).
• Because only 1/2 of the full factorial design is run, each of the 2k effects (including the
intercept) is aliased with one other effect. That is, estimation of aliased effects are calculated
identically and, therefore, cannot be separated from each other.
• To find the estimate of any model effect, use the difference in means between the + and −
rows as we did with the 2k design.
• Example In the 23−1 design with defining relation I = ABC, let lA , lB , and lC represent
estimates of the main effects. Then
lA =
1
[(a + abc) − (b + c)]
2
1
[(b + abc) − (a + c)]
2
1
=
[(c + abc) − (a + b)]
2
lB =
lC
Now, let lAB , lAC , and lBC represent estimates of the two-factor interaction effects. Then
lBC =
1
[
2
]
1
[
2
1
=
[
2
lAC =
]
lAB
]
• Note that lA = lBC , lB = lAC , and lC = lAB . Hence, it is impossible to separate the effect
of A from BC, the effect of B from AC, and the effect of C from AB. Thus, the A and BC
effects are aliased, the B and AC effects are aliased, and the C and AB effects are aliased.
• In addition, note that the intercept I is aliased with ABC.
• If two effects E1 and E2 are aliased, then we cannot estimate each separately. That is, each
of the two columns produces an estimate of the sum of the two effects. For effect i, this is
denoted li −→ E1 + E2 for i = 1, 2.
72
Example: In the 23−1 design, we have
lA −→ A + BC
lBC −→ A + BC
lB −→ B + AC
and
lAC −→ B + AC
lC −→ C + AB
lAB −→ C + AB
• The alias structure for any 2k−1 design can be determined by taking the defining relation
I = ABC · · · K and multiplying it by any effect. The resulting effect is the aliased effect.
– In the 23−1 design, to find the aliases of the main effects A, B, and C, we multiply
I = ABC by A, B, and C, and reduce:
A = A · I = A · ABC = A2 BC = BC
=⇒ A = BC
B = B·I =
=
=
=⇒ B = AC
C = C ·I =
=
=
=⇒ C = AB
• Now consider the other 1/2 fraction having defining relation I = −ABC. This design contains
the treatment combinations that have minus signs for ABC · · · K. This is known as the
alternate or complementary 1/2 fraction.
– In the 23−1 design, the alternate 1/2 fraction are the 4 treatment combinations corresponding to rows (1), ab, ac, and bc.
Treatment
total
ab
ac
bc
(1)
I
+
+
+
+
A
+
+
−
−
B
+
−
+
−
Factorial Effect
C AB AC
−
+
−
+
−
+
+
−
−
−
+
+
BC
−
−
+
+
ABC
−
−
−
−
Aliased
effects
I = −ABC
A = −BC
B = −AC
C = −AB
– The defining relation is I = −ABC and the alias structure using the multiplication rule
is
A = A · I = A · (−ABC) = −A2 BC = −BC =⇒ A = −BC
B = B·I =
=
=
=⇒ B = −AC
C = C ·I =
=
=
=⇒ C = −BC
Thus, the A and −BC effects are aliased, the B and −AC effects are aliased, and the C
0
0
and −AB effects are aliased. The alternate fraction yields estimates lA
, lB
and lC0 where
0
lA
−→
0
lB
−→
lC0 −→
• Creating a 2k−1 fractional-factorial design is equivalent to forming 2 blocks from a 2k design
and then selecting one block to run. In practice, it does not matter which fraction we use
(although it is common to use the (+, +) case to avoid using the negative of an estimate.
• Because of aliased pairs of effects, only one effect from each pair can be included in the model.
73
Example: A 24−1 design (a 1/2 fraction of a 24 design)
• Using generator I = ABCD, the following table contains the 8 combinations ABCD = +1.
This is one possible 24−1 design.
I
+
+
+
+
+
+
+
+
A
−
+
+
−
+
−
−
+
B
−
+
−
+
−
+
−
+
C D
− −
− −
+ −
+ −
− +
− +
+ +
+ +
ABCD
+
+
+
+
+
+
+
+
Notation
• Using generator I = −ABCD, the following table contains the 8 combinations ABCD = −1.
This is a second possible 24−1 design.
I
+
+
+
+
+
+
+
+
A
+
−
−
+
−
+
+
−
B
−
+
−
+
−
+
−
+
C D
− −
− −
+ −
+ −
− +
− +
+ +
+ +
ABCD
−
−
−
−
−
−
−
−
Notation
• In the 24−1 design, to find the aliases of the A, B, C, D, AB, AC, . . . we multiply I = ABCD
by A, B, C, D, AB, AC, . . . and then reduce:
A = A · I = A · ABCD = A2 BCD = BCD
lA = lBCD =⇒ A + BCD
B = B·I =
=
=
lB = lACD =⇒ B + ACD
C = C ·I =
=
=
lC = lABD =⇒ C + ABD
D = D·I =
=
=
lD = lABC =⇒ D + ABC
AB = AB · I = AB · ABCD = A2 B 2 CD = CD
lAB = lCD =⇒ AB + CD
AC = AC · I = AC · ABCD = A2 BC 2 D = BD
lAC = lBD =⇒ AC + BD
AD = AD · I =
lAD = lBC =⇒
74
5.2
Design Resolution
A design is resolution R if no p-factor effect is aliased with another effect containing less than
R − p factors.
• Resolution III Designs: No main effect is aliased with any other main effect, but at least one
main effect is aliased with a two-factor interaction.
• Resolution IV Designs: No main effect is aliased with any other main effect or two-factor interaction, but at least one two-factor interaction is aliased with another two-factor interaction.
• Resolution V Designs: No main effect or two-factor interaction is aliased with any other main
effect or two-factor interaction, but at least one two-factor interaction is aliased with a threefactor interaction. For example:
– The 23−1 design with defining relation I = ABC is resolution III and is denoted 23−1
III .
– The 24−1 design with defining relation I = ABCD is resolution IV and is denoted 24−1
IV .
– The 25−1 design with defining relation I = ABCDE is resolution V and is denoted 25−1
V .
• When choosing a design, we want the highest resolution possible to minimize the number of
interactions that must be considered negligible in order to obtain a unique interpretation of
the data.
5.3
2k−2 Fractional Factorial Designs
• There are k factors of interest each having 2 levels. There are only enough resources to run
1/4 of the full factorial 2k design. Thus, we say we want to run a 1/4 fraction of a 2k
design. This design is called a 2k−2 fractional factorial design.
– Example: There are 6 factors of interest (A, B, C, D, E, F ). There are only enough
resources for 16 experimental runs which is 1/4 of a 26 design. Thus, we want to run a
1/4 fraction of a 26 design. This design is called a 26−2 fractional factorial design.
• When selecting a 1/4 fraction, we want to be sure that we select design points that will enable
us to estimate effects of interest.
• Generation of such a design (if it exists) requires carefully choosing two interactions to generate
the design of maximum resolution and then decide on the sign of each generator.
• These two signed interactions are called the generators of the 2k−2 fractional factorial design,
and with their generalized interactions form the complete defining relation for the design.
– Suppose there are 6 factors and we choose ABCE and BCDF to be the generators of
the 26−2 design. Then there are 4 ways to assign signs to these:
(+, +) I = ABCE and I = BCDF
(−, +) I = −ABCE and I = BCDF
(+, −) I = ABCE and I = −BCDF
(−, −) I = −ABCE and I = −BCDF
Each of these four assignments will generate a unique 1/4 fraction of a 26 design.
– Suppose we choose the (+,+) case: I = ABCE and I = BCDF . The generalized
interaction ABCE · BCDF =
. Thus, the complete defining
relation is given by I = ABCE = BCDF =
.
75
• The resolution of any design equals the length of the shortest ‘word’ (excluding I) in the
complete defining relation.
– In the 26−2 design with complete defining relation I = ABCE = BCDF = ADEF , the
length of the shortest ‘word’ is four. Therefore, this is a 26−2
IV fractional factorial design.
• A 2k−2 fractional factorial design is formed by selecting one of the four sign pairs ( (+, +),
(+, −), (−, +) or (−, −)) and the selecting only those treatment combinations that match
the pair of signs in the columns corresponding to the two generating effects. This can be
accomplished in two ways:
(i) List all 2k combinations and selecting the rows with plus signs in the two columns corresponding to the two generators.
(ii) Or, more simply, list all 2k−2 combinations for a factorial design having k − 2 factors.
Then create columns for the remaining two factors based on the effects in the complete
defining relation.
• Example: Generate the 26−2 design with defining relation I = ABCE = BCDF = ADEF
using method (ii).
– Generate a 24 design in the first four factors (A, B, C, and D). Then, choose aliases for
E and F that can be expressed only in terms of A, B, C, and D. Use these to determine
the columns for E and F .
E = ABC = BCDEF = ADF
F =
=
=
– Now, multiply the appropriate columns to form columns for E and F .
A
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
24 Design
B C
− −
− −
+ −
+ −
− +
− +
+ +
+ +
− −
− −
+ −
+ −
− +
− +
+ +
+ +
D
−
−
−
−
−
−
−
−
+
+
+
+
+
+
+
+
E=
−
+
+
−
+
−
−
+
−
+
+
−
+
−
−
+
F =
−
−
+
+
+
+
−
−
+
+
−
−
−
−
+
+
Treatment
Combination
(1)
ae
bef
abf
cef
acf
bc
abce
df
adef
bde
abd
cde
acd
bcd
abcdef
Sample
Data
6
10
32
60
4
15
26
60
8
12
34
60
16
5
37
52
• Because only 1/4 of the full factorial design is run, each of the 2k effects (including the intercept) is aliased with three other effects. That is, estimation of aliased effects are calculated
identically and, therefore, cannot be separated from each other.
• The alias structure for any 2k−2 design can be determined by taking the defining relation and
multiplying it by any effect. The resulting four effects are all aliased.
76
• Example revisited: In the 26−2 design with defining relation I = ABCE = BCDF = ADEF ,
the effects are aliased as follows:
A = BCE = ABCDF = DEF
B = ACE = CDF = ABDEF
C = ABE = BDF = ACDEF
D=
=
=
E=
=
=
F = ABCEF = BCD = ADE
AB = CE = ACDF = BDEF
AC = BE = ABDF = CDEF
AD = BCDE = ABCF = EF
AE =
=
=
AF =
=
=
BD = ACDE = CF = ABEF
BF = ACEF = CD = ABDE
ABD =
=
=
ACD = BDE = ABF = CEF
I = ABCE = BCDF = ADEF
• Thus, when determining a model for analysis, only one effect in each alias class is allowed
to be in the model.
• The complete estimation structure is given below with the main effects and two-factor interactions highlighted in bold-face.
l1
l2
l3
l4
l5
l6
l7
l8
= A + ABCE + ABCDF + DEF
= B + ACE + CDF + ABDEF
= C + ABE + BDF + ACDEF
= D + ABCDE + BCF + AEF
= E + ABC + BCDEF + ADF
= F + ABCEF + BCD + ADE
= AB + CE + ACDF + BDEF
= AC + BE + ABDF + CDEF
l9 = AD + BCDE + ABCF + EF
l10 = AE + BC + ABCDEF + DF
l11 = AF + BCEF + ABCD + DE
l12 = BD + ACDE + CF + ABEF
l13 = BF + ACEF + CD + ABDE
l14 = ABD + CDE + ACF + BEF
l15 = ACD + BDE + ABF + CEF
l16 = I + ABCE + BCDF + ADEF
• If we restrict consideration to only main effects and two-factor interactions (i.e., assume that
all 3, 4, 5, and 6 factor interactions are negligible) then when estimating effects, we get the
simplified estimation structure:
l1
l2
l3
l4
l5
=A
=B
=C
=D
=E
l6 = F
l7 = AB + CE
l8 = AC + BE
l9 = AD + EF
l10 = AE + BC + DF
l11
l12
l13
l16
= AF + DE
= BD + CF
= BF + CD
=I
• Thus, in l7 to l13 , we can only include one of the aliased effects in the model.
Example of a Data Analysis
• We will now analyze the data from this 26−2 design.
• The model contains only main effects and a subset of the two-factor interactions. This leaves
only 2 df for the M SE .
• You can only put one effect from each alias class in the model. For example,
– AB and CE are aliased −→ we can put AB or CE in the model (but not both).
– AE, DF , and BC are aliased −→ we can put only one of these 3 effects in the model.
• Pool together the aliased two factor interactions that have very large p-values of .8729, .9744,
and .9744 for F A + DE, BD + CF , and F B + CD, respectively.
• Reanalyze the data. You now have 5 df for the new M SE .
77
ANOVA FOR THE 2**(6-2) DESIGN
General Linear Models Procedure
Dependent Variable: Y
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
13
2
15
6564.3125
95.1250
6659.4375
504.9471
47.5625
R-Square
0.985716
C.V.
25.25055
Root MSE
6.8966
F Value
Pr > F
10.62
0.0893
<-- Only 2 df for Error
Y Mean
27.313
ANOVA FOR THE 2**(6-2) DESIGN
Source
A
B
C
D
E
F
A*B
A*C
A*D
A*E
F*A
B*D
F*B
DF
(+CE)
(+BE)
(+BF)
(+DF+BC)
(+DE)
(+CF)
(+CD)
Type III SS
1
1
1
1
1
1
1
1
1
1
1
1
1
Mean Square
F Value
Pr > F
770.0625
770.0625
5076.5625
5076.5625
3.0625
3.0625
7.5625
7.5625
0.5625
0.5625
0.5625
0.5625
564.0625
564.0625
10.5625
10.5625
115.5625
115.5625
14.0625
14.0625
1.5625 <-1.5625
0.0625 <-0.0625
0.0625 <-0.0625
16.19
106.73
0.06
0.16
0.01
0.01
11.86
0.22
2.43
0.30
0.03
0.00
0.00
0.0566
0.0092
0.8234
0.7286
0.9233
0.9233
0.0750
0.6839
0.2594
0.6411
0.8729
0.9744
0.9744
<-- A
<-- B
<-- AB+CE
+- Pool
| these
+- first
• After pooling the three terms into the M SE , we get the following ANOVA.
• The A and B main effects are highly significant.
• The other significant effect is the AB + CE effect. Because A and B are significant, we
conclude that the AB effect is the dominant effect in AB + CE.
ANOVA FOR THE 2**(6-2) DESIGN WITH POOLED TERMS
The GLM Procedure
Dependent Variable: Y
Source
DF
Sum of
Squares
Model
Error
Corrected Total
10
5
15
6562.625000
96.812500
6659.437500
R-Square
0.985462
Coeff Var
16.11088
Mean Square
F Value
Pr > F
656.262500
19.362500
33.89
0.0006
Root MSE
4.400284
78
Y Mean
27.31250
Source
A
B
C
D
E
F
A*B
A*C
A*D
A*E
(+CE)
(+BE)
(+BF)
(+DF+BC)
DF
Type III SS
Mean Square
F Value
Pr > F
1
1
1
1
1
1
1
1
1
1
770.062500
5076.562500
3.062500
7.562500
0.562500
0.562500
564.062500
10.562500
115.562500
14.062500
770.062500
5076.562500
3.062500
7.562500
0.562500
0.562500
564.062500
10.562500
115.562500
14.062500
39.77
262.19
0.16
0.39
0.03
0.03
29.13
0.55
5.97
0.73
0.0015
<.0001
0.7073
0.5594
0.8713
0.8713
0.0029
0.4933
0.0584
0.4330
<-- A
<-- B
<-- AB
<-- AD or BF
===========================================================================
DM ’LOG;CLEAR;OUT;CLEAR;’;
OPTIONS NODATE NONUMBER PS=60 LS=78;
**************************************************;
*** 2**(6-2) DESIGN WITH I= ABCE = BCDF = ADEF ***;
**************************************************;
DATA
DO
DO
DO
DO
IN;
D = -1 TO 1
C = -1 TO 1
B = -1 TO 1
A = -1 TO 1
E = A*B*C;
F = B*C*D;
INPUT Y @@;
END; END; END;
CARDS;
6 10 32 60 4 15
;
BY
BY
BY
BY
2;
2;
2;
2;
OUTPUT;
END;
26 60 8 12 34 60 16 5 37 52
PROC GLM DATA=IN;
CLASS A B C D E;
MODEL Y = A B C D E F
A*B A*C A*D A*E A*F B*D B*F / SS3;
TITLE ’ANOVA FOR THE 2**(6-2) DESIGN’;
PROC GLM DATA=IN;
CLASS A B C D E;
MODEL Y = A B C D E F A*B A*C A*D A*E / SS3;
TITLE ’ANOVA FOR THE 2**(6-2) DESIGN WITH POOLED TERMS’;
RUN;
79
80
5.4
Using SAS to Generate a Fractional Factorial Design
• Use SAS to generate a 1/4 fraction of the 26 design (a 26−2 design).
• SAS used defining relation I = BCDE = ACDF = ABEF .
********************************;
*** GENERATE A 2**6-2 DESIGN ***;
********************************;
PROC FACTEX;
FACTORS X1 X2 X3 X4 X5 X6 / NLEV=2;
SIZE FRACTION=4;
EXAMINE ALIASING(3) CONFOUNDING DESIGN;
MODEL RESOLUTION=4;
OUTPUT OUT = FF62
X1 NVALS=(100 200)
X2 NVALS=(5 10)
X3 CVALS=(’ON’ ’OFF’)
X4 CVALS=(’DAY’ ’NIGHT’)
X5 NVALS=(0 1)
X6 NVALS=(50 55);
TITLE ’GENERATION OF A 2**(6-2) RESOLUTION IV DESIGN’;
PROC PRINT DATA=FF62;
RUN;
GENERATION OF A 2**(6-2) RESOLUTION IV DESIGN
Design Points
Experiment
Number
X1
X2
X3
X4
X5
X6
---------------------------------------------------------------1
-1
-1
-1
-1
-1
-1
2
-1
-1
-1
1
1
1
3
-1
-1
1
-1
1
1
4
-1
-1
1
1
-1
-1
5
-1
1
-1
-1
1
-1
6
-1
1
-1
1
-1
1
7
-1
1
1
-1
-1
1
8
-1
1
1
1
1
-1
9
1
-1
-1
-1
-1
1
10
1
-1
-1
1
1
-1
11
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
81
Factor Confounding Rules
X5 = X2*X3*X4
X6 = X1*X3*X4
So
<-<--
I = BCDE = ACDF = ABEF
GENERATION OF A 2**(6-2) RESOLUTION IV DESIGN
Aliasing Structure
X1 = X2*X5*X6 = X3*X4*X6
X2 = X1*X5*X6 = X3*X4*X5
X3 = X1*X4*X6 = X2*X4*X5
X4 = X1*X3*X6 = X2*X3*X5
X5 = X1*X2*X6 = X2*X3*X4
X6 = X1*X2*X5 = X1*X3*X4
X1*X2 = X5*X6
X1*X3 = X4*X6
X1*X4 = X3*X6
X1*X5 = X2*X6
X1*X6 = X2*X5 = X3*X4
X2*X3 = X4*X5
X2*X4 = X3*X5
X1*X2*X3 = X1*X4*X5 = X2*X4*X6 = X3*X5*X6
X1*X2*X4 = X1*X3*X5 = X2*X3*X6 = X4*X5*X6
OBS
X1
X2
X3
X4
X5
X6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
100
100
100
100
100
100
100
100
200
200
200
200
200
200
200
200
5
5
5
5
10
10
10
10
5
5
5
5
10
10
10
10
ON
ON
OFF
OFF
ON
ON
OFF
OFF
ON
ON
OFF
OFF
ON
ON
OFF
OFF
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
50
55
55
50
50
55
55
50
55
50
50
55
55
50
50
55
82
E = BCD
F = ACD
5.5
Using SAS to Generate a Fractional Factorial Design with Blocks
*********************************************************;
*** GENERATE A 2**6-1 DESIGN WITH 2 BLOCKS OF SIZE 16 ***;
*********************************************************;
PROC FACTEX;
FACTORS X1 X2 X3 X4 X5 X6 / NLEV=2;
BLOCKS NBLOCKS=2;
SIZE FRACTION=2;
EXAMINE ALIASING(6) CONFOUNDING DESIGN;
MODEL RESOLUTION=5;
OUTPUT OUT = FF61B2
X1 NVALS=(100 200)
X2 NVALS=(5 10)
X3 CVALS=(’ON’ ’OFF’)
X4 CVALS=(’DAY’ ’NIGHT’)
X5 NVALS=(0 1)
X6 NVALS=(50 55)
BLOCKNAME=SHIFT CVALS=(’A.M.’ ’P.M.’);
TITLE ’GENERATION OF A 2**(6-1) DESIGN IN 2 BLOCKS’;
PROC PRINT DATA=FF61B2; RUN;
===========================================================
GENERATION OF A 2**(6-1) DESIGN IN 2 BLOCKS
Design Points
Experiment
Number
X1
X2
X3
X4
X5
X6
Block
------------------------------------------------------------1
-1
-1
-1
-1
-1
-1
1
2
-1
-1
-1
-1
1
1
2
3
-1
-1
-1
1
-1
1
2
4
-1
-1
-1
1
1
-1
1
5
-1
-1
1
-1
-1
1
2
6
-1
-1
1
-1
1
-1
1
7
-1
-1
1
1
-1
-1
1
8
-1
-1
1
1
1
1
2
9
-1
1
-1
-1
-1
1
1
10
-1
1
-1
-1
1
-1
2
11
-1
1
-1
1
-1
-1
2
12
-1
1
-1
1
1
1
1
13
-1
1
1
-1
-1
-1
2
14
-1
1
1
-1
1
1
1
15
-1
1
1
1
-1
1
1
16
-1
1
1
1
1
-1
2
Experiment
Number
X1
X2
X3
X4
X5
X6
Block
------------------------------------------------------------17
1
-1
-1
-1
-1
1
1
18
1
-1
-1
-1
1
-1
2
19
1
-1
-1
1
-1
-1
2
20
1
-1
-1
1
1
1
1
21
1
-1
1
-1
-1
-1
2
22
1
-1
1
-1
1
1
1
23
1
-1
1
1
-1
1
1
24
1
-1
1
1
1
-1
2
25
1
1
-1
-1
-1
-1
1
26
1
1
-1
-1
1
1
2
27
1
1
-1
1
-1
1
2
28
1
1
-1
1
1
-1
1
29
1
1
1
-1
-1
1
2
30
1
1
1
-1
1
-1
1
31
1
1
1
1
-1
-1
1
32
1
1
1
1
1
1
2
83
Factor Confounding Rules
X6 = X1*X2*X3*X4*X5
Block Pseudo-factor Confounding Rules
[B1] = X3*X4*X5
Aliasing Structure
0 = X1*X2*X3*X4*X5*X6
X1 = X2*X3*X4*X5*X6
X2 = X1*X3*X4*X5*X6
X3 = X1*X2*X4*X5*X6
X4 = X1*X2*X3*X5*X6
X5 = X1*X2*X3*X4*X6
X6 = X1*X2*X3*X4*X5
X1*X2 = X3*X4*X5*X6
X1*X3 = X2*X4*X5*X6
X1*X4 = X2*X3*X5*X6
X1*X5 = X2*X3*X4*X6
X1*X6 = X2*X3*X4*X5
X2*X3 = X1*X4*X5*X6
X2*X4 = X1*X3*X5*X6
X2*X5 = X1*X3*X4*X6
X2*X6 = X1*X3*X4*X5
X3*X4 = X1*X2*X5*X6
X3*X5 = X1*X2*X4*X6
X3*X6 = X1*X2*X4*X5
X4*X5 = X1*X2*X3*X6
X4*X6 = X1*X2*X3*X5
X5*X6 = X1*X2*X3*X4
X1*X2*X3 = X4*X5*X6
X1*X2*X4 = X3*X5*X6
X1*X2*X5 = X3*X4*X6
[B] = X1*X2*X6 = X3*X4*X5
X1*X3*X4 = X2*X5*X6
X1*X3*X5 = X2*X4*X6
X1*X3*X6 = X2*X4*X5
X1*X4*X5 = X2*X3*X6
X1*X4*X6 = X2*X3*X5
X1*X5*X6 = X2*X3*X4
GENERATION OF A 2**(6-1) DESIGN IN 2 BLOCKS
OBS
SHIFT
X1
X2
X3
X4
X5
X6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
A.M.
100
100
100
100
100
100
100
100
200
200
200
200
200
200
200
200
5
5
5
5
10
10
10
10
5
5
5
5
10
10
10
10
ON
ON
OFF
OFF
ON
ON
OFF
OFF
ON
ON
OFF
OFF
ON
ON
OFF
OFF
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
50
50
50
50
55
55
55
55
55
55
55
55
50
50
50
50
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
P.M.
100
100
100
100
100
100
100
100
200
200
200
200
200
200
200
200
5
5
5
5
10
10
10
10
5
5
5
5
10
10
10
10
ON
ON
OFF
OFF
ON
ON
OFF
OFF
ON
ON
OFF
OFF
ON
ON
OFF
OFF
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
DAY
NIGHT
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
55
55
55
55
50
50
50
50
50
50
50
50
55
55
55
55
84