Chapter 8 Two-Level Fractional Factorial Designs

Transcript Chapter 8 Two-Level Fractional Factorial Designs

Chapter 8 Two-Level Fractional Factorial Designs

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.

• 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 3

8.2 The One-half Fraction of the 2

Design

• Consider three factor and each factor has two levels. • A one-half fraction of 2 3 design is called a 2 3-1 design 4

• 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: 

 1 2 





abc

  



  1 2  







abc

  

1 2  





abc

  

• A = BC, B = AC, C = AB 5

• Aliases:

B AC

C AB

• Aliases can be found from the defining relation I

= ABC

by multiplication:

AI = A(ABC) = A 2 BC = BC BI =B(ABC) = AC CI = C(ABC) = AB

• Principal fraction: I = ABC 6

• The Alternate Fraction of the 2 3-1 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 7

• The de-aliased estimates of all effects by analyzing the eight runs as a full 2 3 design in two blocks. Hence 1   2 1 2  

A A

  '

    '

  1 2 





 

1 2 







 

• 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 2 3 ABC is a design

III

1 design with I = 8

• Resolution III Designs: –

me =

– Example: A 2 3-1 design with I = ABC • Resolution IV Designs: – 2

= 2

– Example: A 2 4-1 design with I = ABCD • Resolution V Designs: – 2

= 3

– Example: A 2 5-1 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.

• 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 2 k-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.

• 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

– 1 of the original factors 12

• Example 8.1: – Example 6.2: A, C, D, AC and AD are important. – Use 2 4-1 design with I = ABCD 13

• Using the defining relation, – A = BCD, B=ACD, C=ABD, D=ABC – AB=CD, AC=BD, BC=AD 14

• 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 2 3 design in factors, A, C and D. (see Figure 8.4 and Page 310) 15

• Example 8.2: – 5 factors – Use 2 5-1 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 2 3 design 16

• Sequences of fractional factorial: Both one-half fractions represent blocks of the complete design with the highest order interaction confounded with blocks.

• Example 8.3: – Reconsider Example 8.1

– Run the alternate fraction with I = – ABCD – Estimates of effects – Confirmation experiment 18

8.3 The One-Quarter Fraction of the 2

Design

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

• 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 2 6-2 with I = ABCE and I = BCDF. The complete defining relation is I = ABCE = BCDF = ADEF 21

• Another way to construct such design is to derive the four blocks of the 2 6 design with ABCE and BCDF confounded , and then choose the block with treatment combination that are + on ABCE and BCDF • The 2 6-2 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 22

– A single replicate of a 2 4 design in any subset of four factors that is not a word in the defining relation.

– A replicate one-half fraction of a 2 4 in any subset of four factors that is a word in the defining relation.

• In general, any 2 k-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.

• 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) 24

8.4 The General 2

k-p

Fractional Factorial Design

• A 1/ 2 p fraction of the 2 k design • Need p independent generators, and there are 2 p – p – 1 generalized interactions • Each effect has 2 p – 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.

• 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

• Example 8.5

– Estimate all main effects and get some insight regarding the two-factor interactions.

– Three-factor and higher interactions are – – negligible.

2 7  2

and 2 7  3

designs in Appendix Table XII (Page 666) 2 7  3

16-run design: main effects are aliased with three-factor interactions and two-factor interactions are aliased with two-factor – interactions 2 7  2

32-run design: all main effects and 15 of 21 two-factor interactions 27

• Analysis of 2 k-p Fractional Factorials: – For the ith effect: 

 2 (

Contrast i N

) 

Contrast i N

/ 2 ,

 2



• Projection of the 2 k-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 2 four of seven factors. Then 7 of 35 subsets are appearing in complete defining relations.

• Blocking Fractional Factorial: – Appendix Table XII 2 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 – 2 8  4

or 2 8

 3 – Four blocks – Effect estimates and sum of squares (Table 8.17) – Normal probability plot of the effect estimates (see Figure 8.19) 29

• 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 – 30

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.

III

1 • Another example is shown in Table 8.19: design

III

I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG = ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG 31

• 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.

• Assume that three-factor and higher interactions are negligible. 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) 33

• When

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 2 3 design.

• 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

• Example 8.7

– Seven factors to study eye focus time – Three large effects – Projection?

– The second fraction is run with all the signs reversed – B, D and BD are important effects 36

• 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 37

• 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

= 12, 20, 24, etc. are called

nongeometric

PB designs – Construction: • N = 12, 20, 24 and 36 (Table 8.24) • N = 28 (Table 8.23) 38

• The alias structure is

complex

in the PB designs • For example, with

= 12 and

= 11, every main effect is aliased with every 2FI not involving itself • • Every 2FI alias chain has

terms

Partial

aliasing can greatly complicate interpretation • Interactions can be particularly disruptive • Use very, very carefully (maybe never) 39

• Projection: Consider the 12-run PB design – 3 replicates of a full 2 2 design – A full 2 3 3  1 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.

• 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.

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

III

1 design (see Table 8.27) 42

• 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 43

• Resolution V designs: main effects and the two factor interactions do not alias with the other main effects and two-factor interactions. 44