Fractional Factorial
The successful use of fractional factorial designs is based on three
key ideas:
1) The sparsity of effects principle. When there are several
variables, the system or process is likely to be driven primarily
by some of the main effects an low order interactions.
2) The projection property. Fractional factorial designs can be
projected into stronger designs in the subset of significant
factors.
3) Sequential experimentation.
Fractional Factorial
For a 24 design (factors A, B, C and D) a one-half fraction, 24-1,
can be constructed as follows:
Choose an interaction term to completely confound, say ABCD.
Using the defining contrast L = x1 + x2 + x3 + x4 like we did before
we get:
Fractional Factorial
L
x1
x2
x3
x4
mod
2
L
x1
x2
x3
x4
mod
2
0000
0
0
0
0
0
0110
0
1
1
0
0
0001
0
0
0
1
1
1010
0
1
0
1
0
0010
0
0
1
0
1
1100
1
1
0
0
0
0100
0
1
0
0
1
0111
0
1
1
1
1
1000
1
0
0
0
1
1011
1
0
1
1
1
0011
0
0
1
1
0
1101
1
1
0
1
1
0101
0
1
0
1
0
1110
1
1
1
0
1
1001
1
0
0
1
0
1111
1
1
1
1
0
Fractional Factorial
Hence, our design with ABCD completely confounded is as follows:
a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1
ABCD0
Y00000
ABCD1
Y00011
Y00001
Y00010
Y00100
Y00101
Y01001
Y01000
a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1
ABCD0
Y00110
Y01010
Y01100
ABCD1
Y01111
Y00111
Y01011
Y01101
Y01110
The fractional factorial design
a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1
ABCD1
Y00001
Y00010
Y00100
Y01000
a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1
ABCD1
Y00111
Y01011
Y01101
Y01110
Fractional Factorial
Each calculated sum of squares will be associated with two
sources of variation.
Sourc
e
Prin.
Frac.
Alias
Source
Prin.
Frac.
A
ABCD
B
Alias
A2BCD
BCD
BC
ABCD
AB2C2D
AD
ABCD
AB2CD
ACD
BD
ABCD
AB2CD2
AC
C
ABCD
ABC2D
ABD
CD
ABCD
ABC2D2
AB
D
ABCD
ABCD2
ABC
ABC
ABCD
A2B2C2D
D
AB
ABCD A2B2CD
CD
ABD
ABCD
A2B2CD2
C
AC
ABCD A2BC2D
BD
ACD
ABCD
A2BC2D2
B
AD
ABCD A2BCD2
BC
BCD
ABCD
AB2C2D2
A
Fractional Factorial
Lets clean a bit:
Source
Alias
Source
Alias
A
BCD
BC
AD
B
ACD
BD
AC
C
ABD
CD
AB
D
ABC
ABC
D
AB
CD
ABD
C
AC
BD
ACD
B
AD
BC
BCD
A
Fractional Factorial
Lets reorganize:
Complete 23 Design
Source
Alias
A
BCD
B
ACD
C
ABD
AB
CD
AC
BD
BC
AD
ABC
D
Fractional Factorial
So to analyze a 24-1 fractional factorial design we need to run a complete
23 factorial design (ignoring one of the factors) and analyze the data
based on that design and re-interpret it in terms of the 24-1 design.
Fractional Factorial
Resolution:
Many resolutions the three listed in the book are:
1. Resolution III designs: No main effect is aliased with any
other main effect, they are aliased with two factor
interactions and two factor interactions are aliased with each
other. Example 2III3-1 with ABC as the principle fractions.
2. Resolution IV designs: No main effect is aliased with any
other main effect or any two factor interaction, but two factor
interactions are aliased with each other. Example, 2IV4-1 with
ABCD as the principle fraction.
3. Resolution V designs. No main effect or two-factor
interactions is aliased with any other main effect or twofactor interaction, but two-factor interactions are aliased with
three factor interactions. Example, 2V5-1 with ABCDE as the
principle fraction.
Fractional Factorial
Example:
a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1
ABCD0
3
4
7
2
ABCD0
6
3
6
2
a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1
ABCD0
7
2
5
9
ABCD0
8
3
6
5
Fractional Factorial
Assuming all factors are fixed, the linear model is as follows:
Yijklm     i   j   k   l   ij   kl    ik    jl      jk  il    m ijkl 

 
 

i  1,..., a; j  1,..., b; k  1,..., c; l  1,..., d ; m  1,..., n
Fractional Factorial
If we still cant run this design all at once, we can block; that is we can
implement a group-interaction confounding step. We can confound
the highest level interaction of the 23 design, as we did before.
Group-Interaction Confounded designs
Partial confounding:
Confounding ABC replicate 1:
L
x1
x2
x3
mod 2
000
0
0
0
0
001
0
0
1
1
010
0
1
0
1
100
1
0
0
1
011
0
1
1
0
101
1
0
1
0
110
1
1
0
0
111
1
1
1
1
Group-Interaction Confounded designs
Partial confounding:
Confounding ABC replicate 1:
a0b0c0
Block0
Block1
a0b0c1
a0b1c0
a1b0c0
Y0000
Y0001
Y0010
Y0100
a0b1c1
a1b0c1
a1b1c0
Y0011
Y0101
Y0110
a1b1c1
Y0111
Fractional Factorial
For a 24 design (factors A, B, C and D) a one-quarter fraction,
24-2, can be constructed as follows:
Choose two interaction terms to confound, say ABD and ACD,
these will serve as our principle fractions. The third interaction,
called the generalized interaction, that we confounded in the
way is: A2BCD2 = BC.
Need two defining contrasts
L1 = x1 + x2 + 0 + x4
and
L2 = x1 + 0 + x3 + x4
Fractional Factorial
L1
x1
x2
0
x4
mod
2
L2
x1
0
x3
x4
mod
2
0000
0
0
0
0
0
0000
0
0
0
0
0
0001
0
0
0
1
1
0001
0
0
0
1
1
0010
0
0
0
0
0
0010
0
0
1
0
1
0100
0
1
0
0
1
0100
0
0
0
0
0
1000
1
0
0
0
1
1000
1
0
0
0
1
0011
0
0
0
1
1
0011
0
0
1
1
0
0101
0
1
0
1
0
0101
0
0
0
1
1
1001
1
0
0
1
0
1001
1
0
0
1
0
Fractional Factorial
L1
x1
x2
0
x4
mod
2
L2
x1
0
x3
x4
mod
2
0110
0
1
0
0
1
0110
0
0
1
0
1
1010
1
0
0
0
1
1010
1
0
1
0
0
1100
1
1
0
0
0
1100
1
0
0
0
1
0111
0
1
0
1
0
0111
0
0
1
1
0
1011
1
0
0
1
0
1011
1
0
1
1
1
1101
1
1
0
1
1
1101
1
0
0
1
0
1110
1
1
0
0
0
1110
1
0
1
0
0
1111
1
1
0
1
1
1111
1
0
1
1
1
Fractional Factorial
L1
L2
a
b
c
d
L1
L2
a
b
c
d
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
1
0
1
1
1
1
0
1
0
1
1
1
0
1
1
0
1
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
1
0
1
1
0
1
1
0
0
1
1
1
1
1
0
1
1
1
1
0
1
Fractional Factorial
Hence, our design with ABCD completely confounded is as follows:
a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1
ABCD0(00)
Y00000
ABCD1(11)
Y01001
Y10001
Y11000
ABCD2(01)
Y20100
ABCD3(10)
Y20011
Y30010
Y30101
a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1
ABCD0(00)
ABCD1(11)
ABCD2(01)
ABCD3(10)
Y00111
Y01110
Y10110
Y21111
Y21010
Y21101
Y31100
Y31011
Fractional Factorial
One of the possible one-quarter designs is:
a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1
ABCD2(01)
Y20100
Y20011
a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1
ABCD2(01)
Y21010
Y21101
Fractional Factorial
Each calculated sum of squares will be associated with four sources of
variation.
Source
Prin. Frac.
Alias
A
ABD,ACD,BC
A2BD, A2CD, ABC
BD,CD,ABC
B
ABD,ACD,BC
AB2D, ABCD, B2C
AD,ABCD,C
C
ABD,ACD,BC
ABCD, AC2D, BC2
ABCD,AD,B
D
ABD,ACD,BC
ABD2, ACD2, BCD
AB,AC,BCD
AB
ABD,ACD,BC
A2B2D, A2BCD, AB2C
D,BCD,AC
AC
ABD,ACD,BC
A2BCD, A2C2D, ABC2
BCD,D,AB
Fractional Factorial
Each calculated sum of squares will be associated with four sources of
variation.
Source
Prin. Frac.
Alias
AD
ABD,ACD,BC
A2BD2, A2CD2, ABC
B,C,ABC
BD
ABD,ACD,BC
AB2D2, ACD2, B2CD
A,AC,CD
CD
ABD,ACD,BC
ABCD2, AC2D2, BC2D
ABC,A,BD
ABC
ABD,ACD,BC
A2B2CD, A2BC2D, AB2C2
CD,BD,A
BCD
ABD,ACD,BC
AB2CD2, ABC2D2, B2C2D
AC,B,D
ABCD
ABD,ACD,BC
A2B2CD2, A2BC2D2, AB2C2D
C,B,AD
Fractional Factorial
The above is not quite satisfactory because we are aliasing some of the
main effects with other main effects;
i.e. the resolution is not good enough!!!
Fractional Factorial
What happens after analyzing the data:
Can do a confirmatory experiment, complete the block!!
Fractional Factorial
L1
x1
x2
0
x4
mod
2
L2
0000
0000
0001
0001
0010
0010
0100
0100
1000
1000
0011
0011
0101
0101
1001
1001
x1
0
x3
x4
mod
2
Fractional Factorial
L1
x1
x2
0
x4
mod
2
L2
0110
0110
1010
1010
1100
1100
0111
0111
1011
1011
1101
1101
1110
1110
1111
1111
x1
0
x3
x4
mod
2
Fractional Factorial
L1
L2
0
1
a
b
c
d
L1
L2
0
1
0
1
0
1
a
b
c
d
Fractional Factorial
Hence, our design with ABCD completely confounded is as follows:
a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1
ABCD0(00)
ABCD1(11)
ABCD2(01)
ABCD3(10)
a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1
ABCD0(00)
ABCD1(11)
ABCD2(01)
ABCD3(10)
Fractional Factorial
Each calculated sum of squares will be associated with four sources of
variation.
Source
Prin. Frac.
A
ABD,ACD,BC
B
ABD,ACD,BC
C
ABD,ACD,BC
D
ABD,ACD,BC
AB
ABD,ACD,BC
AC
ABD,ACD,BC
Alias
Fractional Factorial
Each calculated sum of squares will be associated with four sources of
variation.
Source
Prin. Frac.
AD
ABD,ACD,BC
BD
ABD,ACD,BC
CD
ABD,ACD,BC
ABC
ABD,ACD,BC
BCD
ABD,ACD,BC
ABCD
ABD,ACD,BC
Alias