DESAIN EKSPERIMEN BLOCKING FACTORS - …debrina.lecture.ub.ac.id/files/2018/02/5-Blocking-Factors.pdf · The Latin Square Design • The Latin square design systematically allows

DESAINEKSPERIMENBLOCKINGFACTORS

SemesterGenap2017/2018JurusanTeknikIndustriUniversitasBrawijaya

Outline

•  TheRandomizedCompleteBlockDesign•  TheLatinSquareDesign•  TheGraeco-LatinSquareDesign•  BalancedIncompleteBlockDesign

2/33 28/02/18 Desain Eksperimen - Semester Genap 2017/2018

•  In some experiment, the variabilitymay arise fromfactorsthatwearenotinterestedin.

•  Anuisancefactorisafactorthatprobablyhassomeeffectontheresponse,butit’sofnointeresttotheexperimenter…however,thevariabilityittransmitstotheresponseneedstobeminimized

•  These nuisance factor could be unknown anduncontrolledàuserandomization

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TheRandomizedCompleteBlockDesign

28/02/18 Desain Eksperimen - Semester Genap 2017/2018

•  Ifthenuisancefactorareknownbutuncontrollableàusetheanalysisofcovariance.

•  Ifthenuisancefactorareknownbutcontrollableàusetheblockingtechnique

•  Typical nuisance factors include batches of rawmaterial, operators,piecesof testequipment, time(shifts,days,etc.),differentexperimentalunits

4



•  Many industrial experiments involveblocking(orshould)

•  Failure to block is a common flaw indesigninganexperiment(consequences?)

5



•  Wewishdeterminewhetherornotfourdifferenttips produce different readings on a hardnesstestingmachine.

•  Onefactortobeconsideràtiptype•  Completely Randomized Design could be usedwith one potential problemà the testing blockcouldbedifferent

•  The experiment error could include both therandomandcouponerrors.

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TheRandomizedCompleteBlockDesign-example


•  Toreducetheerrorfromtestingcoupon,randomizecompleteblockdesign(RCBD)isused

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•  Eachcouponiscalleda“block”;that is, it’samorehomogenousexperimentalunitonwhichtotestthetips

•  “complete” indicates each testing coupon (BLOCK)containsalltreatments

•  Variabilitybetween blocks can be large, variabilitywithinablockshouldberelativelysmall

•  Ingeneral,ablockisaspecificlevelofthenuisancefactor

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•  A complete replicate of the basic experiment isconductedineachblock

•  Ablockrepresentsarestrictiononrandomization•  Allrunswithinablockarerandomized•  Onceagain,weareinterestedintestingtheequalityof treatment means, but now we have to removethe variability associated with the nuisance factor(theblocks)

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•  Supposethatthereareatreatments(factorlevels)andbblocks

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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA


•  Suppose that there are a treatments (factorlevels)andbblocks

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1,2,...,1, 2,...,ij i j ij

i ay

j bµ τ β ε

=⎧= + + + ⎨

=⎩

0 1 2 1: where (1/ ) ( )b

a i i j ijH bµ µ µ µ µ τ β µ τ

== = = = + + = +∑L


•  Astatisticalmodel(effectsmodel)fortheRCBDis

•  Therelevant(fixedeffects)hypothesesare

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1,2,...,1,2,...,ij i j ij

i ay

j bµ τ β ε

=⎧= + + + ⎨

=⎩

0 1 2 1: where (1/ ) ( )b

a i i j ijH bµ µ µ µ µ τ β µ τ

== = = = + + = +∑L


jiH µµ ≠ oneleast at :1


•  Or

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0 oneleast at :0:

1

210

≠

====

i

a

HH

τ

τττ !


•  Partitioningthetotalvariability

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2.. . .. . ..

1 1 1 1

2. . ..

2 2. .. . ..

1 1

2. . ..

1 1

( ) [( ) ( )

( )]

( ) ( )

( )

a b a b

ij i ji j i j

ij i j

a b

i ji j

a b

ij i ji j

T Treatments Blocks E

y y y y y y

y y y y

b y y a y y

y y y y

SS SS SS SS

= = = =

= =

= =

− = − + −

+ − − +

= − + −

+ − − +

= + +

∑∑ ∑∑

∑ ∑

∑∑


Thedegreesoffreedomforthesumsofsquaresin

areasfollows:Therefore,ratiosofsumsofsquarestotheirdegreesof

freedomresultinmeansquaresandtheratioofthemeansquarefortreatmentstotheerrormeansquareisanFstatisticthatcanbeusedtotestthehypothesisofequaltreatmentmeans

15

T Treatments Blocks ESS SS SS SS= + +

1 1 1 ( 1)( 1)ab a b a b− = − + − + − −



•  Meansquares

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( )

( )

( ) 2

1

2

2

1

2

2

1

1

σ

βσ

τσ

=−

+=

−+=

∑

∑

=

=

E

a

ij

Block

a

ii

treatment

MSEb

aMSE

a

bMSE


•  F-testwith(a-1),(a-1)(b-1)degreeoffreedom

•  RejectthenullhypothesisifF0>Fα,a-1,(a-1)(b-1)

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E

Treatments

MSMSF =0


•  ANOVATable

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19

Manual computing:


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•  MeaningofF0=MSBlocks/MSE?– TherandomizationinRBCDisappliedonlytotreatmentwithinblocks

– TheBlockrepresentsarestrictiononrandomization

– Twokindsofcontroversialtheories

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•  MeaningofF0=MSBlocks/MSE?– Generalpractice,theblockfactorhasalargeeffectandthenoisereductionobtainedbyblockingwasprobablyhelpfulinimprovingtheprecisionofthecomparisonoftreatmentmeansiftherationislarge

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TheRandomizedCompleteBlockDesign–Example

•  ToconductthisexperimentasaRCBD,assignall4pressurestoeachofthe6batchesofresin

•  Each batch of resin is called a “block”; that is, it’s a morehomogenous experimental unit on which to test the extrusionpressures

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TheRandomizedCompleteBlockDesign–Example


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TheRandomizedCompleteBlockDesign–Example—Minitab

•  StatàANOVAàTwo-way

Vascular-Graft.MTW

Two-way ANOVA: Yield versus Pressure, Batch Source DF SS MS F P Pressure 3 178.171 59.3904 8.11 0.002 Batch 5 192.252 38.4504 5.25 0.006 Error 15 109.886 7.3258 Total 23 480.310 S = 2.707 R-Sq = 77.12% R-Sq(adj) = 64.92%

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Vascular-Graft.MTW

•  Basicresidualplotsindicatethatnormality,constantvarianceassumptionsaresatisfied

•  Noobviousproblemswithrandomization•  Nopatternsintheresidualsvs.block•  Canalsoplotresidualsversusthepressure(residualsbyfactor)

•  Theseplotsprovidemoreinformationabouttheconstantvarianceassumption,possibleoutliers

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TheRandomizedCompleteBlockDesign–Example—ResidualAnalysis


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Vascular-Graft.MTW

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TheRandomizedCompleteBlockDesign–Example—NoBlocking

•  StatàANOVAàOne-way

Vascular-Graft.MTW

One-way ANOVA: Yield versus Pressure Source DF SS MS F P Pressure 3 178.2 59.4 3.93 0.023 Error 20 302.1 15.1 Total 23 480.3 S = 3.887 R-Sq = 37.10% R-Sq(adj) = 27.66%

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TheRandomizedCompleteBlockDesign–Example—NoBlocking-Residual

Vascular-Graft.MTW

•  TheRCBDutilizesanadditivemodel–nointeractionbetweentreatmentsandblocks

•  Treatmentsand/orblocksasrandomeffects•  Missingvalues•  Whataretheconsequencesofnotblockingifweshouldhave?

TheRandomizedCompleteBlockDesign–Other Aspects

1,2,...,1,2,...,ij i j ij

i ay

j bµ τ β ε

=⎧= + + + ⎨

=⎩

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•  SamplesizingintheRCBD?TheOCcurveapproachcanbeusedtodeterminethenumberofblockstorun..seepage133

TheRandomizedCompleteBlockDesign–Other Aspects

1,2,...,1,2,...,ij i j ij

i ay

j bµ τ β ε

=⎧= + + + ⎨

=⎩

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TheLatinSquareDesign

•  Thesedesignsareusedtosimultaneouslycontrol(oreliminate)twosourcesofnuisancevariability

•  Those two sources of nuisance factors haveexactlysamelevelsoffactortobeconsidered

•  Asignificantassumption is that thethree factors(treatments,nuisancefactors)donotinteract

•  If this assumption is violated, the Latin squaredesignwillnotproducevalidresults

•  Latinsquaresarenotusedasmuchas theRCBDinindustrialexperimentation

32 28/02/18 Desain Eksperimen - Semester Genap 2017/2018

TheLatinSquareDesign

•  TheLatinsquaredesignsystematicallyallowsblockingintwodirections

•  Ingeneral,aLatinsquareforpfactorsisasquarecontainingprowsandpcolumns.

•  Eachcellcontainoneandonlyoneofplettersthatrepresentthetreatments.


ALatinSquareDesign–TheRocketPropellant

•  Thisisa

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5 5 Latin square design×

StatisticalAnalysisoftheLatinSquareDesign

•  Thestatistical(effects)modelis

•  Thestatisticalanalysis(ANOVA)ismuchliketheanalysisfortheRCBD.

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1,2,...,1,2,...,1,2,...,

ijk i j k ijk

i py j p

k pµ α τ β ε

=⎧⎪

= + + + + =⎨⎪ =⎩


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TheStandardLatinSquareDesign•  Asquarewithfirstrowandcolumninalphabeticalorder.


OtherTopics

•  Missingvaluesinblockeddesigns– RCBD– Latinsquare– Estimatedby

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)1)(1(2)( '

...'..

'..

'..

−−

−++=

ppyyyyp

y kjiijk


OtherTopics

•  ReplicationofLatinSquares– Toincreasetheerrordegreesoffreedom– Threemethods

1.Usethesamebatchesandoperatorsineachreplicate2.Usethesamebatchesbutdifferentoperatorsineachreplicate

3.Usedifferentbatchesanddifferentoperator


OtherTopics

•  ReplicationofLatinSquares– ANOVAinCase1


OtherTopics

•  ReplicationofLatinSquares– ANOVAnCase2


OtherTopics•  ReplicationofLatinSquares

– ANOVAnCase3


OtherTopics•  Crossoverdesign

–  ptreatmentstobetestedinptimeperiodsusingnpexperimentunits.

–  Ex:20subjectstobeassignedtotwoperiods

–  Firsthalfofthesubjectsareassignedtoperiod1(inrandom)andtheotherhalfareassignedtoperiod2.

–  Taketurnafterexperimentsaredone.


OtherTopics•  Crossoverdesign

– ANOVA


Graeco-LatinSquare

•  For a pxp Latin square, one can superimpose asecond pxp Latin square that treatments aredenotedbyGreekletters.


Graeco-LatinSquare

•  IfthetwosquareshavethepropertythateachGreekletterappearsonceandonlyoncewitheachLatinletter,thetwoLatinsquaresaretobeorthogonalandthisdesignisnamedasGraeco-LatinSquare.

•  Itcancontrolthreesourcesofextraneousvariability.


Graeco-LatinSquare•  ANOVA


Graeco-LatinSquare--Example

•  Intherocketpropellantproblem,batchofmaterial,operators,andtestassembliesareimportant.

•  If5ofthemareconsidered,aGraeco-Latnsquarecanbeused.






Documents

DESAIN EKSPERIMEN BLOCKING FACTORS - …debrina.lecture.ub.ac.id/files/2018/02/5-Blocking-Factors.pdf · The Latin Square Design • The Latin square design systematically allows