3. Fractional Factorial Designs

• Two-level fractional factorial designs

• Confounding

• Blocking

E. Barrios Design and Analysis of Engineering Experiments 3–1E. Barrios Design and Analysis of Engineering Experiments 3–1

Session 3 Fractional Factorial Designs 2Session 3 Fractional Factorial Designs 2

Two-Level Fractional Factorial Designs

24 Full Factorial Design

I 1 2 3 4 12 13 14 23 24 34 123 124 134 234 1234 conversion+ − − − − + + + + + + − − − − + 70

+ + − − − − − − + + + + + + − − 60

+ − + − − − + + − − + + + − + − 89

+ + + − − + − − − − + − − + + + 81

+ − − + − + − + − + − + − + + − 69

+ + − + − − + − − + − − + − + + 62

+ − + + − − − + + − − − + + − + 88

+ + + + − + + − + − − + − − − − 81

+ − − − + + + − + − − − + + + − 60

+ + − − + − − + + − − + − − + + 49

+ − + − + − + − − + − + − + − + 88

+ + + − + + − + − + − − + − − − 82

+ − − + + + − − − − + + + − − + 60

+ + − + + − + + − − + − − + − − 52

+ − + + + − − − + + + − − − + − 86

+ + + + + + + + + + + + + + + + 79

We can accommodate 16 estimates: 1 mean; 4 main effects; 6 two-factors interactioneffects; 4 three-factors interaction effects; 1 four-factors interaction effect

E. Barrios Design and Analysis of Engineering Experiments 3–2E. Barrios Design and Analysis of Engineering Experiments 3–2

Session 3 Fractional Factorial Designs 3Session 3 Fractional Factorial Designs 3

Two-Level Fractional Factorial Designsa

Modification of a Bearing Example

Two-Level Eight Run Orthogonal Array

a b c ab ac bc abc Failure Raterun A B C D y1 − − − + + + − 162 + − − − − + + 73 − + − − + − + 144 + + − + − − − 55 − − + + − − + 116 + − + − − + − 77 − + + − − + − 138 + + + + + + + 4

Last column,abc estimates two effects:D + ABC.FactorsD andABC areconfounded: lD → D + ABC, andD andABC arealiases.

aBHH2e Chapter 6.

E. Barrios Design and Analysis of Engineering Experiments 3–3E. Barrios Design and Analysis of Engineering Experiments 3–3

Session 3 Fractional Factorial Designs 4Session 3 Fractional Factorial Designs 4

Two-Level Fractional Factorial Designs

Modification of a Bearing Example

E. Barrios Design and Analysis of Engineering Experiments 3–4E. Barrios Design and Analysis of Engineering Experiments 3–4

Session 3 Fractional Factorial Designs 5Session 3 Fractional Factorial Designs 5

Two-Level Fractional Factorial Designs

Modification of a Bearing Example

Two-Level Eight Run Orthogonal Array

a b c ab ac bc abc

run A B C D

1 − − − + + + −

2 + − − − − + +

3 − + − − + − +

4 + + − + − − −

5 − − + + − − +

6 + − + − − + −

7 − + + − − + −

8 + + + + + + +

Confounding pattern:

lA → A + (BCD) lAB → AB + CDlB → B + (ACD) lAC → AC + BDlC → C + (ABD) lBC → BC + ADlD → D + (ABC)

Sometimes 3rd and higher order interactions are small enough to be ignored.

E. Barrios Design and Analysis of Engineering Experiments 3–5E. Barrios Design and Analysis of Engineering Experiments 3–5

Session 3 Fractional Factorial Designs 6Session 3 Fractional Factorial Designs 6

Two-Level Fractional Factorial Designs

The Anatomy of the Half Replicate

Generation:A = a; B = b; C = c andD = abc. Thus

D = ABC

is thegenerating relationof the design. Note that

I = D × D = D2 = ABC · D

Thus, for this design:

I = ABCD, B = ACD, C = ABD, D = ABCAB = CD, AC = BD, AD = BC

This design isresolution 4, since the length of the defining relation has four letters(factors)I = ABCD. It is denoted as

24−1

IV

where, the 2 means that factors of the design have 2 levels each; 4-1 because there are4 factors and we are running only one have of the full factorial: 8 = 24−1 = 16/2; andIV because the design is resolution 4.E. Barrios Design and Analysis of Engineering Experiments 3–6E. Barrios Design and Analysis of Engineering Experiments 3–6

Session 3 Fractional Factorial Designs 7Session 3 Fractional Factorial Designs 7

Two-Level Fractional Factorial Designs

The Anatomy of the Half Replicate

If instead, we use the columnab to accommodate factorD, thenAB = D and there-fore I = ABD. Then, for this design,

A = BD, B = AD, C = ABCD, D = ABAC = BCD, BC = ACD, CD = ABC

Note:lA → A + BD; lB → B + AD; lD → D + AB

The defining relation contains three lettersI = ABD, thus the design is of resolutionIII, 24−1

III .

E. Barrios Design and Analysis of Engineering Experiments 3–7E. Barrios Design and Analysis of Engineering Experiments 3–7

Session 3 Fractional Factorial Designs 8Session 3 Fractional Factorial Designs 8

Two-Level Fractional Factorial Designs

Justification for the use of fractional factorials:

• Redundancy:When high order interactions are considered negligible lower ordereffects are arranged to be confounded with them and thus are estimable.

• Parsimony:Effect sparsity;vital few, trivial many; Pareto effect.

• Projectivity: A 3D design project onto a22 factorial design inall 3 subspacesof dimension 2. Then 3D designs are ofprojectivity 2. Similarly, 24−1 designsare ofprojectivity 3 since after dropping any factor a full23 design is left for theremaining three factors.

In general, for fractional factorials designs of resolution R, the projectivityP =R − 1. Every subset ofP = R − 1 factors is a complete factorial (possiblyreplicated) inP factors.

E. Barrios Design and Analysis of Engineering Experiments 3–8E. Barrios Design and Analysis of Engineering Experiments 3–8

Session 3 Fractional Factorial Designs 9Session 3 Fractional Factorial Designs 9

Two-Level Fractional Factorial Designs

3D Projectivity:

A 23−1

III design showing projections into three22 factorials.

E. Barrios Design and Analysis of Engineering Experiments 3–9E. Barrios Design and Analysis of Engineering Experiments 3–9

Session 3 Fractional Factorial Designs 10Session 3 Fractional Factorial Designs 10

Two-Level Fractional Factorial Designs

Note

It is recommended to dedicate just a modest amount of the budget to the first stages ofthe experimentation.

• Find or determine which factors to consider and appropriateresponses

• Determine proper experimental region and factor ranges.

Then you can dedicate to study deeper your experiment

• Estimate better factor effects

• Confirmatory experimentation

• Optimize product or process.

E. Barrios Design and Analysis of Engineering Experiments 3–10E. Barrios Design and Analysis of Engineering Experiments 3–10

Session 3 Fractional Factorial Designs 11Session 3 Fractional Factorial Designs 11

Two-Level Fractional Factorial Designs

Sequential Experimentation

In sequential experimentation, unless the total number of runs is necessary to achievea desired level of precision, it is usually best to start witha fractional factorial. Thedesign could be later augmented if necessary

• To cover “more interesting” regions.

• To resolve ambiguities.

E. Barrios Design and Analysis of Engineering Experiments 3–11E. Barrios Design and Analysis of Engineering Experiments 3–11

Session 3 Fractional Factorial Designs 12Session 3 Fractional Factorial Designs 12

Two-Level Fractional Factorial DesignsEight-run Designsa

aBHH2e Chapter 6,BHH Chapter 10.

E. Barrios Design and Analysis of Engineering Experiments 3–12E. Barrios Design and Analysis of Engineering Experiments 3–12

Session 3 Fractional Factorial Designs 13Session 3 Fractional Factorial Designs 13

Two-Level Fractional Factorial DesignsEight-run nodal designs

E. Barrios Design and Analysis of Engineering Experiments 3–13E. Barrios Design and Analysis of Engineering Experiments 3–13

Session 3 Fractional Factorial Designs 14Session 3 Fractional Factorial Designs 14

Two-Level Fractional Factorial DesignsA Bicycle Example

E. Barrios Design and Analysis of Engineering Experiments 3–14E. Barrios Design and Analysis of Engineering Experiments 3–14

Session 3 Fractional Factorial Designs 15Session 3 Fractional Factorial Designs 15

Two-Level Fractional Factorial DesignsSign switching, Foldover and Sequential Assembly

After running a fractional factorial further runs may be necessary to resolve ambigui-ties.

Folding over (changing signs) one column (main effect, sayD) provide unaliased es-timates of the main effect and all two-factor interactions involving factorD.

E. Barrios Design and Analysis of Engineering Experiments 3–15E. Barrios Design and Analysis of Engineering Experiments 3–15

Session 3 Fractional Factorial Designs 16Session 3 Fractional Factorial Designs 16

Two-Level Fractional Factorial DesignsA Bicycle Example. Second fraction

E. Barrios Design and Analysis of Engineering Experiments 3–16E. Barrios Design and Analysis of Engineering Experiments 3–16

Session 3 Fractional Factorial Designs 17Session 3 Fractional Factorial Designs 17

Two-Level Fractional Factorial DesignsA Bicycle Example. Resulting 16-run design

Main effect D and two-factor interactions involvingD are free of aliasing.

For any given fraction one-column foldover will “dealias” aparticular main effect andall its interactionsE. Barrios Design and Analysis of Engineering Experiments 3–17E. Barrios Design and Analysis of Engineering Experiments 3–17

Session 3 Fractional Factorial Designs 18Session 3 Fractional Factorial Designs 18

Two-Level Fractional Factorial DesignsAn Investigation Using Multiple-Column Foldover

Filtration Example

E. Barrios Design and Analysis of Engineering Experiments 3–18E. Barrios Design and Analysis of Engineering Experiments 3–18

Session 3 Fractional Factorial Designs 19Session 3 Fractional Factorial Designs 19

Two-Level Fractional Factorial DesignsFiltration Example

E. Barrios Design and Analysis of Engineering Experiments 3–19E. Barrios Design and Analysis of Engineering Experiments 3–19

Session 3 Fractional Factorial Designs 20Session 3 Fractional Factorial Designs 20

Two-Level Fractional Factorial DesignsFiltration Example

E. Barrios Design and Analysis of Engineering Experiments 3–20E. Barrios Design and Analysis of Engineering Experiments 3–20

Session 3 Fractional Factorial Designs 21Session 3 Fractional Factorial Designs 21

Two-Level Fractional Factorial DesignsSecond Fraction: a27−3

III . (foldover all columns [mirror image])

E. Barrios Design and Analysis of Engineering Experiments 3–21E. Barrios Design and Analysis of Engineering Experiments 3–21

Session 3 Fractional Factorial Designs 22Session 3 Fractional Factorial Designs 22

Two-Level Fractional Factorial DesignsAnalysis of the resulting sixteen-run design: a27−3

IV fractional factorial

E. Barrios Design and Analysis of Engineering Experiments 3–22E. Barrios Design and Analysis of Engineering Experiments 3–22

Session 3 Fractional Factorial Designs 23Session 3 Fractional Factorial Designs 23

Two-Level Fractional Factorial Designs2D Projection over the [AE] subspace. A22 design replicated 4 times

E. Barrios Design and Analysis of Engineering Experiments 3–23E. Barrios Design and Analysis of Engineering Experiments 3–23

Session 3 Fractional Factorial Designs 24Session 3 Fractional Factorial Designs 24

Two-Level Fractional Factorial DesignsIncreasing Design Resolution from III to IV by Foldover

In general, any design of resolution III plus its mirror image becomes a design ofresolution IV.

Consider for example, the28−5

III (= 18 : 27−4

III ) design and its mirror image. Their gener-ation relations are respectively:

I8 = 8 = 124 = 135 = 236 = 1237 (1)

andI8 = −8 = −124 = −135 = −236 = 1237 (2)

then, combining (1) and (2)I16 = 1237

Also, from (1),I8 = (8)(124) = 1248, and from (2),I8 = (−8)(−124) = 1248. Thus,I16 = 1248. The four generators for this28−4

III design are:

I16 = 1237 = 1248 = 1358 = 2368

E. Barrios Design and Analysis of Engineering Experiments 3–24E. Barrios Design and Analysis of Engineering Experiments 3–24

Session 3 Fractional Factorial Designs 25Session 3 Fractional Factorial Designs 25

Two-Level Fractional Factorial DesignsSixteen-Run DesignsNodal Designs:

E. Barrios Design and Analysis of Engineering Experiments 3–25E. Barrios Design and Analysis of Engineering Experiments 3–25

Session 3 Fractional Factorial Designs 26Session 3 Fractional Factorial Designs 26

Two-Level Fractional Factorial DesignsDesign Matrix and Alias Patterns

E. Barrios Design and Analysis of Engineering Experiments 3–26E. Barrios Design and Analysis of Engineering Experiments 3–26

Session 3 Fractional Factorial Designs 27Session 3 Fractional Factorial Designs 27

Two-Level Fractional Factorial DesignsThe 25−1

V Nodal Design. Reactor Example

E. Barrios Design and Analysis of Engineering Experiments 3–27E. Barrios Design and Analysis of Engineering Experiments 3–27

Session 3 Fractional Factorial Designs 28Session 3 Fractional Factorial Designs 28

Two-Level Fractional Factorial DesignsAlias Pattern

E. Barrios Design and Analysis of Engineering Experiments 3–28E. Barrios Design and Analysis of Engineering Experiments 3–28

Session 3 Fractional Factorial Designs 29Session 3 Fractional Factorial Designs 29

Two-Level Fractional Factorial DesignsNormal plots for full and half fraction factorial designs.

−10 0 10 20

−2

−1

01

2

Full Factorial

effects

norm

al s

core

A

B

C

D

E

A:B

A:C B:C

A:D

B:D

C:D

A:E

B:E

C:E

D:E

A:B:C A:B:D

A:C:D

B:C:D

A:B:E A:C:E

B:C:E A:D:E

B:D:E

C:D:E A:B:C:D

A:B:C:E

A:B:D:E

A:C:D:E

B:C:D:E A:B:C:D:E

−10 −5 0 5 10 15 20 25

−1

01

Fractional Factorial

effects

norm

al s

core

A

B

C

D

E

A:B

A:C

B:C

A:D

B:D

C:D

A:E B:E

C:E

D:E

E. Barrios Design and Analysis of Engineering Experiments 3–29E. Barrios Design and Analysis of Engineering Experiments 3–29

Session 3 Fractional Factorial Designs 30Session 3 Fractional Factorial Designs 30

Two-Level Fractional Factorial Designs

3D Projectivity of 25−1

V design.

E. Barrios Design and Analysis of Engineering Experiments 3–30E. Barrios Design and Analysis of Engineering Experiments 3–30

Session 3 Fractional Factorial Designs 31Session 3 Fractional Factorial Designs 31

Two-Level Fractional Factorial DesignsThe 28−4

IV Nodal Design. Paint Trial Example

E. Barrios Design and Analysis of Engineering Experiments 3–31E. Barrios Design and Analysis of Engineering Experiments 3–31

Session 3 Fractional Factorial Designs 32Session 3 Fractional Factorial Designs 32

Two-Level Fractional Factorial DesignsNormal plot of effects for glossiness and abrasion.

E. Barrios Design and Analysis of Engineering Experiments 3–32E. Barrios Design and Analysis of Engineering Experiments 3–32

Session 3 Fractional Factorial Designs 33Session 3 Fractional Factorial Designs 33

Two-Level Fractional Factorial DesignsContour plots for glossiness and abrasion.

E. Barrios Design and Analysis of Engineering Experiments 3–33E. Barrios Design and Analysis of Engineering Experiments 3–33

Session 3 Fractional Factorial Designs 34Session 3 Fractional Factorial Designs 34

Two-Level Fractional Factorial DesignsThe 215−11

III Nodal Design. Shrinkage of Speedometer Examplea.

aQuinlan (1985)

E. Barrios Design and Analysis of Engineering Experiments 3–34E. Barrios Design and Analysis of Engineering Experiments 3–34

Session 3 Fractional Factorial Designs 35Session 3 Fractional Factorial Designs 35

Two-Level Fractional Factorial DesignsNormal plot of effects for location and dispersion responses.

−10 0 10 20 30

−1

0

1

Location Effects

effects

norm

al s

core A

B

C

D

E

F

G

H

J

K

L

M

N

O

P

−1.0 −0.5 0.0 0.5 1.0 1.5

−1

0

1

Dispersion Effects

effects

norm

al s

core

A

B

C D

E

F

G

H

J

K

L

M

N

O

P

Location Effects

factors

effe

cts

A C E G J L N P

−20

−10

0

10

20

ME

ME

Dispersion Effects

factors

effe

cts

A C E G J L N P

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5ME

ME

E. Barrios Design and Analysis of Engineering Experiments 3–35E. Barrios Design and Analysis of Engineering Experiments 3–35

Session 3 Fractional Factorial Designs 36Session 3 Fractional Factorial Designs 36

Elimination of Block EffectsBoys Shoes Example

2 4 6 8 10

108

110

112

114

boys

wea

r

material Amaterial B

Two−sample Comparison

2 4 6 8 10

−1.0

−0.5

0.0

0.5

1.0

boys

wea

r di

ffere

nce

Paired Comparison

10 + 10 observations2 sample means

18 degrees of freedom

10 differences1 sample mean

9 degrees of freedom

E. Barrios Design and Analysis of Engineering Experiments 3–36E. Barrios Design and Analysis of Engineering Experiments 3–36

Session 3 Fractional Factorial Designs 37Session 3 Fractional Factorial Designs 37

Elimination of Block Effects

“Block what you can, randomize what you cannot”

• Identify important extraneous factorswithin blocks and eliminate them.

• Representative variationbetween blocks should be encourage

E. Barrios Design and Analysis of Engineering Experiments 3–37E. Barrios Design and Analysis of Engineering Experiments 3–37

Session 3 Fractional Factorial Designs 38Session 3 Fractional Factorial Designs 38

Two-Level Factorial DesignsBlocking Arrangements for 2k Factorial Designsa

Number of Number of BlockVariables Runs Size Block Interactions Confounded with Blocks

3 8 4 B1 = 123 123

2 B1 = 12,B2 = 13 12, 13, 23

4 16 8 B1 = 1234 1234

4 B1 = 124,B2 = 134 124, 134, 23

2 B1 = 12,B2 = 23, 12, 23, 34, 13, 1234, 24, 14B3 = 34

5 32 16 B1 = 12345 12345

8 B,= 123,B2 = 345 123, 345, 1245

4 B1 = 125,B2 = 235, 125, 235, 345, 13, 1234, 24, 145

B3 = 345

2 B1 = 12,B2 = 13, 12, 13, 34, 45, 23, 1234, 1245, 14,B3 = 34,B4 = 45 1345, 35, 24, 2345, 1235, 15, 25,

i.e., all 2fi and 4fi

6 64 32 B1 = 123456 123456

16 B1 = 1236,B2 = 3456 1236, 3456, 1245

8 B1 = 135,B2 = 1256, 135, 1256, 1234, 236, 245, 3456, 146B3 = 1234

4 B1 = 126,B2 = 136, 126, 136, 346, 456, 23, 1234, 1245,

B3 = 346,B4 = 456 14, 1345, 35, 246, 23456, 12356, 156, 252 B1 = 12,B2 = 23, All 2fi, 4fi, and 6fi

B3 = 34,B4 = 45,B5 = 56

aBHH2e Table 5A.1

E. Barrios Design and Analysis of Engineering Experiments 3–38E. Barrios Design and Analysis of Engineering Experiments 3–38

Session 3 Fractional Factorial Designs 39Session 3 Fractional Factorial Designs 39

Elimination of Block Effects25−1

V Design Matrix

E. Barrios Design and Analysis of Engineering Experiments 3–39E. Barrios Design and Analysis of Engineering Experiments 3–39

Session 3 Fractional Factorial Designs 40Session 3 Fractional Factorial Designs 40

Two-Level Fractional Factorial Designs

Minimal Aberration Two-Level Fractional Factorial Design for k Variables andN Runsa

(Number in Parentheses Represent Replication).

aBHH2e Table 6.22

E. Barrios Design and Analysis of Engineering Experiments 3–40E. Barrios Design and Analysis of Engineering Experiments 3–40