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 Designs24 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.Factors D and ABC are confounded: lD → D + ABC, and D and ABC are aliases.

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 DesignsModification 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 DesignsModification of a Bearing Example

Two-Level Eight Run Orthogonal Arraya 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 DesignsThe Anatomy of the Half Replicate

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

D = ABC

is the generating relation of the design. Note that

I = D ×D = D2 = ABC ·DThus, for this design:

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

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

24−1IV

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 DesignsThe Anatomy of the Half Replicate

If instead, we use the column ab to accommodate factor D, then AB = 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 letters I = 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 DesignsJustification 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 a 22 factorial design in all 3 subspacesof dimension 2. Then 3D designs are of projectivity 2. Similarly, 24−1 designsare of projectivity 3 since after dropping any factor a full 23 design is left for theremaining three factors.

In general, for fractional factorials designs of resolution R, the projectivity P =R − 1. Every subset of P = R − 1 factors is a complete factorial (possiblyreplicated) in P 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 Designs3D Projectivity:

A 23−1III design showing projections into three 22 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 DesignsNote

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 appropriate responses

• 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 DesignsSequential Experimentation

In sequential experimentation, unless the total number of runs is necessary to achievea desired level of precision, it is usually best to start with a 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, say D) provide unaliased es-timates of the main effect and all two-factor interactions involving factor D.

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 involving D are free of aliasing.

For any given fraction one-column foldover will “dealias” a particular 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 FoldoverFiltration 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: a 27−3

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: a 27−3

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. A 22 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, the 28−5III (= 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 this 28−4

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 Designs3D 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 factors within blocks and eliminate them.

• Representative variation between 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 1232 B1 = 12,B2 = 13 12, 13, 23

4 16 8 B1 = 1234 12344 B1 = 124,B2 = 134 124, 134, 232 B1 = 12,B2 = 23, 12, 23, 34, 13, 1234, 24, 14

B3 = 34

5 32 16 B1 = 12345 123458 B, = 123,B2 = 345 123, 345, 12454 B1 = 125,B2 = 235, 125, 235, 345, 13, 1234, 24, 145

B3 = 3452 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 12345616 B1 = 1236,B2 = 3456 1236, 3456, 12458 B1 = 135,B2 = 1256, 135, 1256, 1234, 236, 245, 3456, 146B3 = 12344 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 Effects25−1V 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 DesignsMinimal 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