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Chapter 8 Two-Level Fractional

Factorial Designs

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8.1 Introduction

• The number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly

• After assuming some high-order interactions are negligible, we only need to run a fraction of the complete factorial design to obtain the information for the main effects and low-order interactions

• Fractional factorial designs

• Screening experiments: many factors are considered and the objective is to identify those factors that have large effects.

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• Three key ideas:

1. The sparsity of effects principle

– There may be lots of factors, but few are important

– System is dominated by main effects, low-order interactions

2. The projection property

– Every fractional factorial contains full factorials in fewer factors

3. Sequential experimentation

– Can add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation

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8.2 The One-half Fraction of the 2k

Design

• Consider three factor and each factor has two

levels.

• A one-half fraction of 23 design is called a 23-1

design

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• In this example, ABC is called the generator of

this fraction (only + in ABC column). Sometimes

we refer a generator (e.g. ABC) as a word.

• The defining relation:

I = ABC

• Estimate the effects:

• A = BC, B = AC, C = AB

( )

( )

( )ABC

ACB

BCA

abccba

abccba

abccba

ll

ll

ll

=++−−=

=+−+−=

=+−−=

2

1

2

1

2

1

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• Aliases:

• Aliases can be found from the defining relation I

= ABC by multiplication:

AI = A(ABC) = A2BC = BC

BI =B(ABC) = AC

CI = C(ABC) = AB

• Principal fraction: I = ABC

, , A B CA BC B AC C AB→ + → + → +l l l

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• The Alternate Fraction of the 23-1 design:

I = - ABC

• When we estimate A, B and C using this design,

we are really estimating A – BC, B – AC, and C –

AB, i.e.

• Both designs belong to the same family, defined

by

I = � ABC

• Suppose that after running the principal fraction,

the alternate fraction was also run

• The two groups of runs can be combined to form a

full factorial – an example of sequential

experimentation

ABCACBBCACBA

−→−→−→'''

,, lll

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• The de-aliased estimates of all effects by

analyzing the eight runs as a full 23 design in two

blocks. Hence

• Design resolution: A design is of resolution R if

no p-factor effect is aliased with another effect

containing less than R – p factors.

• The one-half fraction of the 23 design with I =

ABC is a design

( ) ( )

( ) ( ) BCBCABCA

ABCABCA

AA

AA

→+−+=−

→−++=+

2

1

2

1

2

1

2

1

'

'

ll

ll

132 −

III

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• Resolution III Designs:

– me = 2fi

– Example: A 23-1 design with I = ABC

• Resolution IV Designs:

– 2fi = 2fi

– Example: A 24-1 design with I = ABCD

• Resolution V Designs:

– 2fi = 3fi

– Example: A 25-1 design with I = ABCDE

• In general, the resolution of a two-level fractional factorial design is the smallest number of letters in any word in the defining relation.

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• The higher the resolution, the less restrictive the

assumptions that are required regarding which

interactions are negligible to obtain a unique

interpretation of the data.

• Constructing one-half fraction:

– Write down a full 2k-1 factorial design

– Add the kth factor by identifying its plus and

minus levels with the signs of ABC…(K – 1)

– K = ABC…(K – 1) => I = ABC…K

– Another way is to partition the runs into two

blocks with the highest-order interaction

ABC…K confounded.

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• Any fractional factorial

design of resolution R

contains complete factorial

designs in any subset of R – 1

factors.

• A one-half fraction will

project into a full factorial in

any k – 1 of the original

factors

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• Example 8.1:

– Example 6.2: A, C, D, AC and AD are

important.

– Use 24-1 design with I = ABCD

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• This design is the principal fraction, I = ABCD

• Using the defining relation,

– A = BCD, B=ACD, C=ABD, D=ABC

– AB=CD, AC=BD, BC=AD

4 12IV

−

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• A, C and D are large.

• Since A, C and D are

important factors, the

significant interactions

are most likely AC and

AD.

• Project this one-half

design into a single

replicate of the 23 design

in factors, A, C and D.

(see Figure 8.4 and Page

310)

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• Example 8.2:

– 5 factors

– Use 25-1 design with I = ABCDE (Table 8.5)

– Every main effect is aliased with four-factor

interaction, and two-factor interaction is aliased

with three-factor interaction.

– Table 8.6 (Page 312)

– Figure 8.6: the normal probability plot of the

effect estimates

– A, B, C and AB are important

– Table 8.7: ANOVA table

– Residual Analysis

– Collapse into two replicates of a 23 design

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• Sequences of

fractional factorial:

Both one-half

fractions represent

blocks of the

complete design

with the highest-

order interaction

confounded with

blocks.

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• Example 8.3:

– Reconsider Example 8.1

– Run the alternate fraction with I = – ABCD

– Estimates of effects

– Confirmation experiment

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8.3 The One-Quarter Fraction of the

2k Design

• A one-quarter fraction of the 2k design is called a

2k-2 fractional factorial design

• Construction:

– Write down a full factorial in k – 2 factors

– Add two columns with appropriately chosen

interactions involving the first k – 2 factors

– Two generators, P and Q

– I = P and I = Q are called the generating

relations for the design

– All four fractions are the family.

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• The complete defining relation: I = P = Q = PQ

• P, Q and PQ are called words.

• Each effect has three aliases

• A one-quarter fraction of the 26-2 with I = ABCE

and I = BCDF. The complete defining relation is

I = ABCE = BCDF = ADEF

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• Another way to construct such design is to derive

the four blocks of the 26 design with ABCE and

BCDF confounded , and then choose the block

with treatment combination that are + on ABCE

and BCDF

• The 26-2 design with I = ABCE and I = BCDF is

the principal fraction.

• Three alternate fractions:

– I = ABCE and I = - BCDF

– I = -ABCE and I = BCDF

– I = - ABCE and I = -BCDF

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• This fractional factorial will project into

– A single replicate of a 24 design in any subset of

four factors that is not a word in the defining

relation.

– A replicate one-half fraction of a 24 in any

subset of four factors that is a word in the

defining relation.

• In general, any 2k-2 fractional factorial design can

be collapsed into either a full factorial or a

fractional factorial in some subset of r � k –2 of

the original factors.

262 −

IV

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• Example 8.4:

– Injection molding process with six factors

– Design table (see Table 8.10)

– The effect estimates, sum of squares, and

regression coefficients are in Table 8.11

– Normal probability plot of the effects

– A, B, and AB are important effects.

– Residual Analysis (Page 322 – 325)

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8.4 The General 2k-p Fractional

Factorial Design

• A 1/ 2p fraction of the 2k design

• Need p independent generators, and there are 2p –

p – 1 generalized interactions

• Each effect has 2p – 1 aliases.

• A reasonable criterion: the highest possible

resolution, and less aliasing

• Minimum aberration design: minimize the number

of words in the defining relation that are of

minimum length.

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• Minimizing aberration of resolution R ensures that

a design has the minimum # of main effects

aliased with interactions of order R – 1, the

minimum # of two-factor interactions aliased with

interactions of order R – 2, ….

• Table 8.14

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• Example 8.5

– Estimate all main effects and get some insight

regarding the two-factor interactions.

– Three-factor and higher interactions are

negligible.

– designs in Appendix Table XII

(Page 666)

– 16-run design: main effects are aliased with

three-factor interactions and two-factor

interactions are aliased with two-factor

interactions

– 32-run design: all main effects and 15 of 21

two-factor interactions

3727 2 and 2 −−

IVIV

272 −

IV

372 −

IV

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• Analysis of 2k-p Fractional Factorials:

– For the ith effect:

• Projection of the 2k-p Fractional Factorials

– Project into any subset of r � k – p of the

original factors: a full factorial or a fractional

factorial (if the subsets of factors are appearing

as words in the complete defining relation.)

– Very useful in screening experiments

– For example 16-run design: Choose any

four of seven factors. Then 7 of 35 subsets are

appearing in complete defining relations.

pNiii N

N

Contrast

N

Contrast −=== 2,2/

)(2l

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IV

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• Blocking Fractional Factorial:

– Appendix Table XII

– Consider the fractional factorial design with

I = ABCE = BCDF = ADEF. Select ABD (and

its aliases) to be confounded with blocks. (see

Figure 8.18)

• Example 8.6

– There are 8 factors

–

– Four blocks

– Effect estimates and sum of squares (Table 8.17)

– Normal probability plot of the effect estimates

(see Figure 8.19)

262 −

IV

3848 2or 2 −−

IVIV

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• A, B and AD + BG are important effects

• ANOVA table for the model with A, B, D and AD

(see Table 8.18)

• Residual Analysis (Figure 8.20)

• The best combination of operating conditions: A –

, B + and D –

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8.5 Resolution III Designs

• Designs with main effects aliased with two-factor

interactions

• A saturated design has k = N – 1 factors, where N is

the number of runs.

• For example: 4 runs for up to 3 factors, 8 runs for up

to 7 factors, 16 runs for up to 15 factors

• In Section 8.2, there is an example, design.

• Another example is shown in Table 8.19: designI = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG

= AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG

132 −

III

472 −

III

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• This design is a one-sixteenth fraction, and a

principal fraction.

I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG =

ABEF = BEG= AFG = DEF = ADEG = CEFG = BDFG =

ABCDEFG

• Each effect has 15 aliases.

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• Assume that three-factor and higher interactions

are negligible.

• The saturated design in Table 8.19 can be used

to obtain resolution III designs for studying fewer

than 7 factors in 8 runs. For example, for 6 factors

in 8 runs, drop any one column in Table 8.19 (see

Table 8.20)

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III

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• When d factors are dropped , the new defining

relation is obtained as those words in the original

defining relation that do not contain any dropped

letters.

• If we drop B, D, F and G, then the treatment

combinations of columns A, C, and E correspond

to two replicates of a 23 design.

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• Sequential assembly of fractions to separate

aliased effects:

– Fold over of the original design

– Switching the signs in one column provides

estimates of that factor and all of its two-factor

interactions

– Switching the signs in all columns dealiases all

main effects from their two-factor interaction

alias chains – called a full fold-over

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• Example 8.7

– Seven factors to study eye focus time

– Run design (see Table 8.21)

– Three large effects

– Projection?

– The second fraction is run with all the signs

reversed

– B, D and BD are important effects

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III

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• The defining relation for a fold-over design

– Each separate fraction has L + U words used as

generators.

– L: like sign

– U: unlike sign

– The defining relation of the combining designs

is the L words of like sign and the U – 1 words

consisting of independent even products of the

words of unlike sign.

– Be careful – these rules only work for

Resolution III designs

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• Plackett-Burman Designs

– These are a different class of resolution III

design

– Two-level fractional factorial designs for

studying k = N – 1 factors in N runs, where N

= 4 n

– N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

– The designs where N = 12, 20, 24, etc. are

called nongeometric PB designs

– Construction:

• N = 12, 20, 24 and 36 (Table 8.24)

• N = 28 (Table 8.23)

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• The alias structure is complex in the PB designs

• For example, with N = 12 and k = 11, every main

effect is aliased with every 2FI not involving itself

• Every 2FI alias chain has 45 terms

• Partial aliasing can greatly complicate

interpretation

• Interactions can be particularly disruptive

• Use very, very carefully (maybe never)

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• Projection: Consider the

12-run PB design

– 3 replicates of a full 22

design

– A full 23 design + a

design

– Projection into 4 factors is

not a balanced design

– Projectivity 3: collapse

into a full fractional in any

subset of three factors.

132 −

III

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• Example 8.8:

– Use a set of simulated data and the 11 factors, 12-

run design

– Assume A, B, D, AB, and AD are important

factors

– Table 8.25 is a 12-run PB design

– Effect estimates are shown in Table 8.26

– From this table, A, B, C, D, E, J, and K are

important factors.

– Interaction? (due to the complex alias structure)

– Folding over the design

– Resolve main effects but still leave the uncertain

about interaction effects.

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8.6 Resolution IV and V Designs

• Resolution IV: if three-factor and higher

interactions are negligible, the main effects may

be estimated directly

• Minimal design: Resolution IV design with 2k

runs

• Construction: The process of fold over a

design (see Table 8.27)

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III

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• Fold over resolution IV designs: (Montgomery

and Runger, 1996)

– Break as many two-factor interactions alias

chains as possible

– Break the two-factor interactions on a specific

alias chain

– Break the two-factor interactions involving a

specific factor

– For the second fraction, the sign is reversed on

every design generators that has an even

number of letters

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• Resolution V designs: main effects and the two-

factor interactions do not alias with the other main

effects and two-factor interactions.