DESAINEKSPERIMENBLOCKINGFACTORS
SemesterGenap2017/2018JurusanTeknikIndustriUniversitasBrawijaya
Outline
• TheRandomizedCompleteBlockDesign• TheLatinSquareDesign• TheGraeco-LatinSquareDesign• BalancedIncompleteBlockDesign
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• In some experiment, the variabilitymay arise fromfactorsthatwearenotinterestedin.
• Anuisancefactorisafactorthatprobablyhassomeeffectontheresponse,butit’sofnointeresttotheexperimenter…however,thevariabilityittransmitstotheresponseneedstobeminimized
• These nuisance factor could be unknown anduncontrolledàuserandomization
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TheRandomizedCompleteBlockDesign
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• Ifthenuisancefactorareknownbutuncontrollableàusetheanalysisofcovariance.
• Ifthenuisancefactorareknownbutcontrollableàusetheblockingtechnique
• Typical nuisance factors include batches of rawmaterial, operators,piecesof testequipment, time(shifts,days,etc.),differentexperimentalunits
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TheRandomizedCompleteBlockDesign
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• Many industrial experiments involveblocking(orshould)
• Failure to block is a common flaw indesigninganexperiment(consequences?)
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TheRandomizedCompleteBlockDesign
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• 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
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• Toreducetheerrorfromtestingcoupon,randomizecompleteblockdesign(RCBD)isused
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TheRandomizedCompleteBlockDesign-example
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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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TheRandomizedCompleteBlockDesign-example
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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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TheRandomizedCompleteBlockDesign-example
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• Supposethatthereareatreatments(factorlevels)andbblocks
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
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• Suppose that there are a treatments (factorlevels)andbblocks
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
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
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• 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
TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
jiH µµ ≠ oneleast at :1
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• Or
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
0 oneleast at :0:
1
210
≠
====
i
a
HH
τ
τττ !
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• Partitioningthetotalvariability
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
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
= = = =
= =
= =
− = − + −
+ − − +
= − + −
+ − − +
= + +
∑∑ ∑∑
∑ ∑
∑∑
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Thedegreesoffreedomforthesumsofsquaresin
areasfollows:Therefore,ratiosofsumsofsquarestotheirdegreesof
freedomresultinmeansquaresandtheratioofthemeansquarefortreatmentstotheerrormeansquareisanFstatisticthatcanbeusedtotestthehypothesisofequaltreatmentmeans
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T Treatments Blocks ESS SS SS SS= + +
1 1 1 ( 1)( 1)ab a b a b− = − + − + − −
TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
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• Meansquares
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
( )
( )
( ) 2
1
2
2
1
2
2
1
1
σ
βσ
τσ
=−
+=
−+=
∑
∑
=
=
E
a
ij
Block
a
ii
treatment
MSEb
aMSE
a
bMSE
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• F-testwith(a-1),(a-1)(b-1)degreeoffreedom
• RejectthenullhypothesisifF0>Fα,a-1,(a-1)(b-1)
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
E
Treatments
MSMSF =0
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• ANOVATable
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
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Manual computing:
TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
• MeaningofF0=MSBlocks/MSE?– TherandomizationinRBCDisappliedonlytotreatmentwithinblocks
– TheBlockrepresentsarestrictiononrandomization
– Twokindsofcontroversialtheories
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TheRandomizedCompleteBlockDesign–ExtensionfromANOVA
• 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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TheRandomizedCompleteBlockDesign–Example—Minitab
Vascular-Graft.MTW
• Basicresidualplotsindicatethatnormality,constantvarianceassumptionsaresatisfied
• Noobviousproblemswithrandomization• Nopatternsintheresidualsvs.block• Canalsoplotresidualsversusthepressure(residualsbyfactor)
• Theseplotsprovidemoreinformationabouttheconstantvarianceassumption,possibleoutliers
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TheRandomizedCompleteBlockDesign–Example—ResidualAnalysis
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TheRandomizedCompleteBlockDesign–Example—Minitab
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
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TheLatinSquareDesign
• TheLatinsquaredesignsystematicallyallowsblockingintwodirections
• Ingeneral,aLatinsquareforpfactorsisasquarecontainingprowsandpcolumns.
• Eachcellcontainoneandonlyoneofplettersthatrepresentthetreatments.
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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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StatisticalAnalysisoftheLatinSquareDesign
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StatisticalAnalysisoftheLatinSquareDesign
TheStandardLatinSquareDesign• Asquarewithfirstrowandcolumninalphabeticalorder.
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OtherTopics
• Missingvaluesinblockeddesigns– RCBD– Latinsquare– Estimatedby
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)1)(1(2)( '
...'..
'..
'..
−−
−++=
ppyyyyp
y kjiijk
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OtherTopics
• ReplicationofLatinSquares– Toincreasetheerrordegreesoffreedom– Threemethods
1.Usethesamebatchesandoperatorsineachreplicate2.Usethesamebatchesbutdifferentoperatorsineachreplicate
3.Usedifferentbatchesanddifferentoperator
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OtherTopics
• ReplicationofLatinSquares– ANOVAinCase1
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OtherTopics
• ReplicationofLatinSquares– ANOVAnCase2
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OtherTopics• ReplicationofLatinSquares
– ANOVAnCase3
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OtherTopics• Crossoverdesign
– ptreatmentstobetestedinptimeperiodsusingnpexperimentunits.
– Ex:20subjectstobeassignedtotwoperiods
– Firsthalfofthesubjectsareassignedtoperiod1(inrandom)andtheotherhalfareassignedtoperiod2.
– Taketurnafterexperimentsaredone.
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OtherTopics• Crossoverdesign
– ANOVA
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Graeco-LatinSquare
• For a pxp Latin square, one can superimpose asecond pxp Latin square that treatments aredenotedbyGreekletters.
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Graeco-LatinSquare
• IfthetwosquareshavethepropertythateachGreekletterappearsonceandonlyoncewitheachLatinletter,thetwoLatinsquaresaretobeorthogonalandthisdesignisnamedasGraeco-LatinSquare.
• Itcancontrolthreesourcesofextraneousvariability.
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Graeco-LatinSquare• ANOVA
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Graeco-LatinSquare--Example
• Intherocketpropellantproblem,batchofmaterial,operators,andtestassembliesareimportant.
• If5ofthemareconsidered,aGraeco-Latnsquarecanbeused.
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Graeco-LatinSquare--Example
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Graeco-LatinSquare--Example
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