Chapter 4 Fractional factorial Experiments at two levels Chapter 4 Fractional factorial Experiments...

Chapter 4Chapter 4

Fractional factorial Experiments Fractional factorial Experiments at two levelsat two levels

22k-pk-p factorial designs factorial designs

Full factorial designs are rarely used in practice for Full factorial designs are rarely used in practice for large large kk ( ( kk 7) 7)

Fractional factorial designs are Fractional factorial designs are subsetssubsets of full factorial designs of full factorial designs

The main practical motivation for choosing fractional factorial designs is The main practical motivation for choosing fractional factorial designs is

run size economyrun size economy

In return a In return a priceprice is paid is paid


25 Design matrix, leaf spring experiment

Out of 31 d.f., 16 are used for Out of 31 d.f., 16 are used for estimating 3-factor or higher estimating 3-factor or higher interactioninteraction

Is it practical to commit half of the degrees of freedom (d.f.) to estimate 3-factor or higher interaction effects ?

Lower order effects are more likely Lower order effects are more likely to be important than higher order to be important than higher order effects ( 3-factor and higher intera-effects ( 3-factor and higher intera-ctions are usually not significantctions are usually not significant ) )

Effects of the same order are equally likely to be important

FACTORn y s2Runs

B C D E Q1 - + + - - 3 7.6780 0.00132 + + + + - 3 7.9580 0.02143 - - + + - 3 7.6625 0.00214 + - + - - 3 8.0700 0.0273

29 - + - + + 3 7.3254 0.003530 + + - - + 3 8.1054 0.024231 - - - - + 3 7.2125 0.001932 + - - + + 3 7.5233 0.0221

LevelFactor

- +B: high heat temp. (oF) 1840 1880C: heating time (seconds) 23 25D: transfer time (seconds) 10 12E: hold down time (seconds) 2 3Q: quench oil temp. (oF) 130-150 150-170

.


InteractionMainEffect 2-factor 3-factor 4-factor 5-factor

5 10 10 5 1

22k-pk-p factorial designfactorial design (general description) (general description)

22-p-p-th fraction (subset) of -th fraction (subset) of 22kk full factorial designfull factorial design

Consists of Consists of 22k-pk-p runs runs

The fraction is determined by The fraction is determined by pp defining words defining words

The group formed by the The group formed by the pp defining words is called defining words is called the definingthe defining

contrast subgroupcontrast subgroup, ( consists of 2, ( consists of 2pp–1 words plus the identity element –1 words plus the identity element II ) )


Generating a Generating a 22k-pk-p designdesign

Let 1,…,Let 1,…,kk--pp denote the denote the kk--pp independent columns of the +’s and –’s that independent columns of the +’s and –’s that

generate the generate the 22k-pk-p runsruns

Generate the remaining Generate the remaining pp columns columns kk--pp+1,…,+1,…,kk as interactions of the first as interactions of the first

columnscolumns

Choice of these Choice of these pp columns determines columns determines the defining contrast subgroupthe defining contrast subgroup

Example: aExample: a 226-26-2 design design dd ( A,B,C,D,E,F) with ( A,B,C,D,E,F) with E = ABE = AB & & F = ACDF = ACD has a defining has a defining

contrast subgroup contrast subgroup I = ABE= ACDF = BCDEFI = ABE= ACDF = BCDEF

A, B, C, DA, B, C, D are the independent columns and are the independent columns and E, F E, F are generated as interactionsare generated as interactions


Design matrix and free height data, leaf spring experiment

The experimental design chosen The experimental design chosen

was a half fraction of a 2was a half fraction of a 255 (2 (25-15-1) full ) full

factorial design.factorial design.

The choice was based on the

relation

E=BCD ( aliasing relation ) liasing relation )

or

I=BCDE (defining contrast subgroup )defining contrast subgroup )

FACTORRuns B C D E Q n y s2

1 - + + - - 3 7.7900 0.00032 + + + + - 3 8.0700 0.02733 - - + + - 3 7.5200 0.00124 + - + - - 3 7.6333 0.01045 - + - + - 3 7.9400 0.00366 + + - - - 3 7.9467 0.04967 - - - - - 3 7.5400 0.00848 + - - + - 3 7.6867 0.01569 - + + - + 3 7.2900 0.037310 + + + + + 3 7.2900 0.064511 - - + + + 3 7.5200 0.001212 + - + - + 3 7.6467 0.009213 - + - + + 3 7.4000 0.004814 + + - - + 3 7.6233 0.004215 - - - - + 3 7.2033 0.001616 + - - + + 3 7.6333 0.0254

LevelFactor

- +B: high heat temp. (oF) 1840 1880C: heating time (seconds) 23 25D: transfer time (seconds) 10 12E: hold down time (seconds) 2 3Q: quench oil temp. (oF) 130-150 150-170

.


B = CDE, C= BDE, D = BCE, E = BCD Q = BCDEQ,

BQ = CDEQ, CQ = BDEQ, DQ = BCEQ EQ = BCDQ,

BC = DE, BD = CE, CD = BE, BCQ = DEQ, BDQ = CEQ,BEQ = CDQ

Clear and strongly clear effects

FACTORB C D E Q- + + - -+ + + + -- - + + -+ - + - -

- + - + ++ + - - +- - - - ++ - - + +

BCD = E or I=BCDE

A A main effect or 2-factor interaction is called main effect or 2-factor interaction is called

clearclear if non of its aliases are main effects or if non of its aliases are main effects or

2-factor interactions and, 2-factor interactions and,

strongly clearstrongly clear if non of its aliases are main if non of its aliases are main

effects, 2-factor or 3-factor interactionseffects, 2-factor or 3-factor interactions

Pairs of aliased effectsPairs of aliased effects

Clear or strongly clear effectsClear or strongly clear effects


Fractional factorial effects for the leaf spring Fractional factorial effects for the leaf spring experimentexperiment

-0.3-0.25-0.2

-0.15-0.1

-0.050

0.050.1

0.150.2

0.25

B C D E Q

BQ

CQ

DQ EQ BC

BD

CD

BCQ

BDQ

BEQ


.

B = CDE, C= BDE, D = BCE, E = BCD Q = BCDEQ,

BQ = CDEQ, CQ = BDEQ, DQ = BCEQ EQ = BCDQ,

BC = DE, BD = CE, CD = BE, BCQ = DEQ, BDQ = CEQ,BEQ = CDQ

.

Design matrix and alternative choice, leaf spring experiment

FACTORB C D E Q- + + - -+ + + + -- - + + -+ - + - -

- + - + ++ + - - +- - - - ++ - - + +

FACTORB C D E Q- + + + -+ + + - -- - + - -+ - + + -

- + - + ++ + - - +- - - - ++ - - + +

BCD = E or I=BCDE Q = BCDE or Q = BCDE or I=BCDEQI=BCDEQ

(2(25-15-1)) Design Design dd11 (2(25-15-1)) Alternative design Alternative design dd22

All main effects are All main effects are strongly clear and all strongly clear and all 2-factor interaction 2-factor interaction effects are cleareffects are clear

Any 2-factor Any 2-factor interaction involving interaction involving Q is strongly clearQ is strongly clear

It is less desirable to It is less desirable to choose designs choose designs

which aliases lower which aliases lower order effectsorder effects

Preferred if 2-factor Preferred if 2-factor interactions involving interactions involving Q are more importantQ are more important

The Common choiceThe Common choice

.


Resolution IVResolution IV Resolution VResolution V

Example: TheExample: The 26-2 design design dd ( A,B,C,D,E,F) with ( A,B,C,D,E,F) with E = ABE = AB & & F = ACDF = ACD that has a that has a defining contrast subgroup defining contrast subgroup

I = ABE = ACDF = BCDEFI = ABE = ACDF = BCDEFItIt has a has a wordlength pattern wordlength pattern

WW = ( = ( 11,1,1,0),1,1,0)

and is a resolution and is a resolution IIIIII design denoted by design denoted by 262 III

Resolution ( designs)Resolution ( designs)

For a For a 22k-pk-p design, the vector design, the vector

WW = ( = ( AA33,,AA44,…,,…,AAKK))

Is called the Is called the wordlength patternwordlength pattern of the design, where of the design, where AAii denote the denote the

number of defining words of length number of defining words of length ii

The The resolutionresolution is defined to be the length of the shortest word in the is defined to be the length of the shortest word in the

defining contrast subgroup ( the smallest defining contrast subgroup ( the smallest rr such that such that AArr 1 ) 1 )

pkR2


Resolution Resolution

The The maximum resolution criterionmaximum resolution criterion chooses chooses 22k-pk-p design with the design with the maximum resolution ( Box and Hunter, 1961)maximum resolution ( Box and Hunter, 1961)

Resolution Resolution RR implies that no effect involving implies that no effect involving ii factors is aliased with factors is aliased with effects involving less than effects involving less than R-iR-i factors factors

The projection of a resolution The projection of a resolution RR design onto any design onto any R-1R-1 factors is a full factors is a full factorial design in the factorial design in the R-1R-1 factors factors

If there are at most If there are at most R-1R-1 important factors out of k factors, the fractional important factors out of k factors, the fractional design yields a full factorial design in design yields a full factorial design in R-1R-1 factorsfactors


Properties for resolution IV and VProperties for resolution IV and V

In any resolution In any resolution IVIV design, the main effects are clear design, the main effects are clear

In any resolution In any resolution VV design, the main effects are strongly clear and the design, the main effects are strongly clear and the2-factor interactions are clear2-factor interactions are clear

Among the resolution Among the resolution IVIV designs with designs with kk and and pp, those with the largest , those with the largest number of clear 2-factor interactions are the bestnumber of clear 2-factor interactions are the best


Consider the following designs ( Consider the following designs ( k k = 7, = 7, p p = 2 )= 2 )

which is an example where we need to apply the minimum aberration which is an example where we need to apply the minimum aberration

criterion to select a better designcriterion to select a better design Fries & Hunter (1980)Fries & Hunter (1980)

Minimum aberration criterionMinimum aberration criterion

Design d1 d2

Defining contrastsubgroup

II== 44556677 == 1122334466 == 1122335577 II== 11223366 == 11445577 == 223344556677

Wordlength pattern WW == ((00,,11,,22,,00,,00)) WW == ((00,,22,,00,,11,,00))

Resolution IV IV


272

Minimum aberration criterionMinimum aberration criterion

For any two For any two 22k-pk-p designs designs dd11 and and dd22, , dd11 is said to have less aberration than is said to have less aberration than

dd22 if if

AArr((dd11) < ) < AArr((dd22))

where r is the smallest integer such that where r is the smallest integer such that AArr((dd11) ) AArr((dd22))

If there is no design with less aberration than If there is no design with less aberration than dd11 , then , then dd1 1 is said to have is said to have

minimum aberrationminimum aberration

Minimum aberration automatically implies maximum resolutionMinimum aberration automatically implies maximum resolution

Sometimes it is supplemented by the criterion of maximizing the total Sometimes it is supplemented by the criterion of maximizing the total number of clear effects ( main effects and 2-factor interactions )number of clear effects ( main effects and 2-factor interactions )

Chen, Sun, and Wu (1993)Chen, Sun, and Wu (1993)


Estimation of factorial effectsEstimation of factorial effects

An effect is estimable if its alias is negligible An effect is estimable if its alias is negligible

The analysis is the same as in the full factorial experiments except that,The analysis is the same as in the full factorial experiments except that,the observed significance of an effect should be attributed to thethe observed significance of an effect should be attributed to thecombination of the effect and all its aliased effectscombination of the effect and all its aliased effects


Approaches for resolving the ambiguities in Approaches for resolving the ambiguities in aliased effectsaliased effects

Use a prior knowledge to dismiss some of the aliased effectsUse a prior knowledge to dismiss some of the aliased effects Prior knowledge may suggest that some of the aliased effects are less Prior knowledge may suggest that some of the aliased effects are less

important.important.

In some situations the factors of the whole system can be grouped by In some situations the factors of the whole system can be grouped by subsystems. In this case interactions between factors within the same subsystems. In this case interactions between factors within the same subsystem may be more important than interactions between factors subsystem may be more important than interactions between factors between subsystemsbetween subsystems

Run a follow-up experimentRun a follow-up experiment

provides additional information that can be used to de-alias the aliased provides additional information that can be used to de-alias the aliased

effectseffects


Follow up experiment techniques Follow up experiment techniques Method of adding orthogonal effectsMethod of adding orthogonal effects

Orthogonality provides maximum separation between the aliased effects Orthogonality provides maximum separation between the aliased effects

Chooses additional factor settings to make the aliased effects in the Chooses additional factor settings to make the aliased effects in the original experiment orthogonaloriginal experiment orthogonal

It takes additional work to find the orthogonality constraintsIt takes additional work to find the orthogonality constraints

the settings for the remaining effects are chosen on an the settings for the remaining effects are chosen on an ad hoc basis ad hoc basis and and requires some ingenuity on the part of the experimenterrequires some ingenuity on the part of the experimenter

useful when a small number of effects needs to be de-aliaseduseful when a small number of effects needs to be de-aliased

It is not so generally usedIt is not so generally used


The additional runs settings

B is orthogonal to BCQ & DEQ

E=DQDEQ,

Augmented design matrix and model matrixAugmented design matrix and model matrixMethod of adding orthogonal effects, Method of adding orthogonal effects, leaf spring experiment

Four additional runs are added to de-Four additional runs are added to de-alias the aliased effectalias the aliased effect BCQ=DEQBCQ=DEQ andand DQ= BCEQ DQ= BCEQ

A potential block effect should be A potential block effect should be accounted, since the additional runs accounted, since the additional runs are performed at a different timeare performed at a different time

The augmented design is no longer The augmented design is no longer orthogonalorthogonal

Regression analysis should be Regression analysis should be applied to the model matrixapplied to the model matrix

Standard factorial effect estimates Standard factorial effect estimates will not be valid and should not be will not be valid and should not be usedused


Run B C D E Q Block BCQ DEQ DQ1 - + + - - - + + -2 + + + + - - - - -3 - - + + - - - - -4 + - + - - - + + -

13 - + - + + - - - -14 + + - - + - + + -15 - - - - + - + + -16 + - - + + - - - -17 + + - - + + + + -18 - + - - - + + - +19 - + + + + + - + +20 + + + + - + - - -

More flexible and economicalMore flexible and economical

Apply an optimal design criterion like Apply an optimal design criterion like D-optimal criterionD-optimal criterion andand DDss-optimal -optimal

criterioncriterion

The working model for design optimization consists of the grand mean, a The working model for design optimization consists of the grand mean, a

block effect, and the variables identified to be significant from the original block effect, and the variables identified to be significant from the original

experimentexperiment

The approach is driven by the best model identified by the original The approach is driven by the best model identified by the original experiment as well as the optimal design criterionexperiment as well as the optimal design criterion

It works for any model and shape of experimental regionIt works for any model and shape of experimental region

It can be used to solve large design problem by using fast optimal design It can be used to solve large design problem by using fast optimal design algorithmalgorithm as OPTEX program in SAS/QC as OPTEX program in SAS/QC


Follow up experiment techniques Follow up experiment techniques Optimal design approach for follow up experimentsOptimal design approach for follow up experiments

Consider Consider NN runs in the original experiments and runs in the original experiments and nn runs are to be added. runs are to be added. Assume that this model has Assume that this model has pp columns, and the settings in the columns, and the settings in the nn runs are runs are

defined as defined as dd XXdd = [ = [ XX11, , XX22 ] is ] is ((N+nN+n) ) pp matrix , where matrix , where XX2 2 is is ((N+nN+n) ) q q submatrixsubmatrix of of XXdd

The The n n pp submatrix of submatrix of XXd d is selected so that is selected so that

thethe D-criterionD-criterion

oror

thethe DDss-criterion-criterion

where: where: represents a set of candidate represents a set of candidate n n pp submatrices submatrices


Optimal design approach for follow up Optimal design approach for follow up experimentsexperiments

,XXmax dTd

Dd

,XX)XX(XXXXmax TTTT

d21

1111222

Effective for Effective for de-aliasing de-aliasing

all the main effects or,all the main effects or,

one main effect and all the 2-factor interactions involving this main effect one main effect and all the 2-factor interactions involving this main effect

in in resolution resolution IIIIII designs for the original experiment ( designs for the original experiment ( narrow objectivesnarrow objectives).).

The augmented design is still orthogonal and the analysis follow the The augmented design is still orthogonal and the analysis follow the standard methods for fractional factorial design.standard methods for fractional factorial design.

Requires that the follow up experiment has the same run size as in the Requires that the follow up experiment has the same run size as in the original experiment ( original experiment ( not usefulnot useful))

Less flexible than the optimal design approach.Less flexible than the optimal design approach.


Follow up experiment techniques Follow up experiment techniques Fold-over technique for follow up experimentsFold-over technique for follow up experiments

..


Augmented design matrix using Fold-over Augmented design matrix using Fold-over techniquestechniques

d1 isRun 1 2 3 4=12 5=13 6=23 7=123 81 - - - + + + - +2 - - + + - - + +3 - + - - + - + +4 - + + - - + - +5 + - - - - + + +6 + - + - + - - +7 + + - + - - - +8 + + + + + + + +

d2

Run -1 -2 -3 -4 -5 -6 -7 -89 + + + - - - + -10 + + - - + + - -11 + - + + - + - -12 + - - + + - + -13 - + + + + - - -14 - + - + - + + -15 - - + - + + + -16 - - - - - - - -

d: I=1237=1256=1346=1457=2345=1467=3567

472 III

)( IV372

Blocking in full factorial designsBlocking in full factorial designs

We need We need qq independent variables independent variables BB11, , BB22, … , , … , BBqq for defining 2 for defining 2qq blocks. blocks.

Select the factorial effects Select the factorial effects vv11, , vv22, … , , … , vvqq that shall be confounded with that shall be confounded with BB11, ,

BB22, … , , … , BBqq . .

Define the remaining block effects by multiplying the Define the remaining block effects by multiplying the BBii’s’s

The 2The 2qq-1 products of the -1 products of the BBii’s and the column ’s and the column II form the so-called form the so-called block-block-

defining contrast subgroupdefining contrast subgroup


Blocking in fractional factorial designsBlocking in fractional factorial designs

Blocking is more complicated due to the presence of two defining Blocking is more complicated due to the presence of two defining contrast subgroupscontrast subgroups

block defining contrast subgroupblock defining contrast subgroup ( defines the blocking scheme ) ( defines the blocking scheme )

treatment defining contrast subgrouptreatment defining contrast subgroup ( defines the fraction of the design ) ( defines the fraction of the design )

In the context of blockingIn the context of blocking

A main effect or 2-factor interaction is said to be,A main effect or 2-factor interaction is said to be,

clearclear if non of its aliases are main effects or 2-factor interactions as well as if non of its aliases are main effects or 2-factor interactions as well as

not confounded with any block effects and, not confounded with any block effects and,

stronglystrongly clearclear if in addition non of itsif in addition non of its aliases are 3-factor interactionsaliases are 3-factor interactions


Blocking in fractional factorial designsBlocking in fractional factorial designs

There is no natural extension of the minimum aberration criterion for There is no natural extension of the minimum aberration criterion for

blocked blocked 22k-pk-p designs designs

Total number of clear effectsTotal number of clear effects is used to compare and rank order different is used to compare and rank order different

blocked blocked 22k-pk-p designs ( should not be the sole criterion for selection ) designs ( should not be the sole criterion for selection )

The choice can depend on what set of main effects and 2-factor interactions The choice can depend on what set of main effects and 2-factor interactions

is believed to be more importantis believed to be more important


God JulGod Jul

Next timeNext time

Chapter 6Chapter 6

Other design and analysis techniques for Other design and analysis techniques for experiments at more than two levelsexperiments at more than two levels

Erik löfvingErik löfvingThursday, January 15Thursday, January 15thth 2004 2004

13.15 - 15.0013.15 - 15.00

Chapter 4 Fractional factorial Experiments at two levels Chapter 4 Fractional factorial Experiments...

Documents

Transcript of Chapter 4 Fractional factorial Experiments at two levels Chapter 4 Fractional factorial Experiments...