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Fractional Factorial

- The successful use of fractional factorial

designs is based on three key ideas - 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. - The projection property. Fractional factorial

designs can be projected into stronger designs in

the subset of significant factors. - 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 Y00011 Y00101 Y01001

ABCD1 Y00001 Y00010 Y00100 Y01000

a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1

ABCD0 Y00110 Y01010 Y01100 Y01111

ABCD1 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.

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

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 - 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. - 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. - 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, 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

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 a0b0c1 a0b1c0 a1b0c0 a0b1c1 a1b0c1 a1b1c0 a1b1c1

Block0 Y0000 Y0011 Y0101 Y0110

Block1 Y0001 Y0010 Y0100 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

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

Fractional Factorial

Hence, our design with ABCD completely confounded

is as follows

a0b0c0d0 a0b0c0d1 a0b0c1d0 a0b1c0d0 a1b0c0d0 a0b0c1d1 a0b1c0d1 a1b0c0d1

ABCD0(00) Y00000 Y01001

ABCD1(11) Y10001 Y11000

ABCD2(01) Y20100 Y20011

ABCD3(10) Y30010 Y30101

a0b1c1d0 a1b0c1d0 a1b1c0d0 a0b1c1d1 a1b0c1d1 a1b1c0d1 a1b1c1d0 a1b1c1d1

ABCD0(00) Y00111 Y01110

ABCD1(11) Y10110 Y21111

ABCD2(01) Y21010 Y21101

ABCD3(10) 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 x1 0 x3 x4 mod 2

0000 0000

0001 0001

0010 0010

0100 0100

1000 1000

0011 0011

0101 0101

1001 1001

Fractional Factorial

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

Fractional Factorial

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

0 0 1 0

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)

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. Alias

A ABD,ACD,BC

B ABD,ACD,BC

C ABD,ACD,BC

D ABD,ACD,BC

AB ABD,ACD,BC

AC ABD,ACD,BC

Fractional Factorial

Each calculated sum of squares will be associated

with four sources of variation.

Source Prin. Frac. Alias

AD ABD,ACD,BC

BD ABD,ACD,BC

CD ABD,ACD,BC

ABC ABD,ACD,BC

BCD ABD,ACD,BC

ABCD ABD,ACD,BC