The successful use of fractional factorial designs is based on three key ideas: 1)The sparsity of...

$: The successful use of fractional factorial designs is based on three key ideas: 1)The sparsity of effects principle. When there are several variables,$
The 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:


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 FactorialHence, our design with ABCD completely confounded is as follows:

a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1

ABCD0 Y00000 Y00011 Y00101 Y01001

ABCD1 Y00001 Y00010 Y00100 Y01000


ABCD0 Y00110 Y01010 Y01100 Y01111

ABCD1 Y00111 Y01011 Y01101 Y01110

The fractional factorial design


ABCD1 Y00001 Y00010 Y00100 Y01000


ABCD1 Y00111 Y01011 Y01101 Y01110


Each calculated sum of squares will be associated with two sources of variation.

Source

Prin. Frac.

Alias Source Prin. Frac.

Alias

A ABCD A2BCD BCD BC ABCD AB2C2D AD

B 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


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


Lets reorganize:

Source Alias

A BCD

B ACD

C ABD

AB CD

AC BD

BC AD

ABC D

Complete 23 Design


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.


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 2III

3-1 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, 2IV

4-1 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 two-factor interaction, but two-factor interactions are aliased with three factor interactions. Example, 2V

5-1 with ABCDE as the principle fraction.


Example:


ABCD0 3 4 7 2

ABCD0 6 3 6 2


ABCD0 7 2 5 9

ABCD0 8 3 6 5

Assuming all factors are fixed, the linear model is as follows:


1,..., ; 1,..., ; 1,..., ; 1,..., ; 1,...,

ijklm i j k l m ijklij kl ik jl jk ilY

i a j b k c l d m n

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.


Partial confounding:

Group-Interaction Confounded designs

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

a0b0c0 a0b0c1 a0b1c0 a1b0c0 a0b1c1 a1b0c1 a1b1c0 a1b1c1

Block0 Y0000 Y0011 Y0101 Y0110

Block1 Y0001 Y0010 Y0100 Y0111

Partial confounding:

Confounding ABC replicate 1:

Group-Interaction Confounded designs


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


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


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


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

1 1 0 0 0 1 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



ABCD0(00) Y00000 Y01001

ABCD1(11) Y10001 Y11000

ABCD2(01) Y20100 Y20011

ABCD3(10) Y30010 Y30101


ABCD0(00) Y00111 Y01110

ABCD1(11) Y10110 Y21111

ABCD2(01) Y21010 Y21101

ABCD3(10) Y31100 Y31011


One of the possible one-quarter designs is:


ABCD2(01) Y20100 Y20011


ABCD2(01) Y21010 Y21101


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




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


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!!!


What happens after analyzing the data:

Can do a confirmatory experiment, complete the block!!


L1 x1 x2 0 x4 mod 2

L2 x1 0 x3 x4 mod 2

0000 0000

0001 0001

0010 0010

0100 0100

1000 1000

0011 0011

0101 0101

1001 1001


L1 x1 x2 0 x4 mod 2

L2 x1 0 x3 x4 mod 2

0110 0110

1010 1010

1100 1100

0111 0111

1011 1011

1101 1101

1110 1110

1111 1111


L1 L2 a b c d L1 L2 a b c d

0 0 1 0

1 1 0 1



ABCD0(00)

ABCD1(11)

ABCD2(01)

ABCD3(10)


ABCD0(00)

ABCD1(11)

ABCD2(01)

ABCD3(10)




A ABD,ACD,BC

B ABD,ACD,BC

C ABD,ACD,BC

D ABD,ACD,BC

AB ABD,ACD,BC

AC ABD,ACD,BC




AD ABD,ACD,BC

BD ABD,ACD,BC

CD ABD,ACD,BC

ABC ABD,ACD,BC

BCD ABD,ACD,BC

ABCD ABD,ACD,BC

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