factional factorial design.ppt

FractionalFactorialDesigns,BeltonGroup,天马行空官方博客：,2,ScreeningDesigns,Atthebeginningofinvestigations,itisnotuncommontohavealonglistoffactorsaspotentiallyimportantinfluencesontheresponseunderstudy.Asthenumberoffactorsina2kfactorialdesignincreases,thenumberofrunsrequiredforacompletereplicateofthedesignincreasesexponentially:,FactorsRunsRequired416532664712882569512101,024,3,ScreeningDesigns,Ascreeningdesignisanexperimentaldesignwhosepurposeistodistinguishinfluentialfactorsfromnon-influentialfactors,asefficientlyaspossible.Afractionalfactorialdesignisasubsetoftherunsofacompletefactorialdesign.,4,Example,Foracertainprocess,thereare4factorsofpotentialinterest.A24factorialdesign(16runs)wouldhavebeenasuitableexperimentaldesign.However,resourcesoftimeandcostpermitonly8runs.,天马行空官方博客：,5,Example,Considerthe24factorialdesign:Twopossiblesetsof8runs:a)thesetforwhichX1X2X3X4=1b)thesetforX1X2X3X4=-1,6,Example,Thesetof8runsforwhichX1X2X3X4=1isshownbelow:ItcanbeseenthatthepatternoflevelsforfactorX4isidenticaltothepatternoflevelsfortheinteractionX1X2X3.Infact,forthisdesignthepatternoflevelsforeveryindividualfactorandinteractionisidenticaltooneotherindividualfactororinteraction.,7,FractionalFactorialDesign,Thedesigninlastexampleiscalleda24-1fractionalfactorialdesign,where2=numberoflevelsforeachfactor4=numberoffactors1=degreeoffractionationofthecorrespondingfullfactorialdesign(inthiscase2-1=ofthecorresponding24factorialdesign)Thegeneralnotationfora2-levelfractionalfactorialdesignis2k-p,where2=numberoflevelsforeachfactork=numberoffactorsp=degreeoffractionation,ornumberofgenerators,天马行空官方博客：,8,Example,StatDOEFactorialCreateFactorialDesign,9,Example,FactorialDesignFractionalFactorialDesignFactors:4BaseDesign:4,8Resolution:IVRuns:8Replicates:1Fraction:1/2Blocks:noneCenterpts(total):0DesignGenerators:D=ABC,10,AliasRelationships,Asmentionedearlier,thepatternoflevelsforfactorX4infirstexampleisidenticaltothepatternoflevelsfortheinteractionX1X2X3.X4andX1X2X3arecalledaliasesofeachother.Analiasisanindividualfactororinteractionwhosepatternoflevelsinanexperimentisidenticaltothatofanotherfactororinteraction.,11,AliasRelationships,Inafractionalfactorialdesign,everyindividualfactorandinteractionhasanalias.Thecalculatedeffectofanyfactororinteractionisanestimateofthesumoftheeffectsofthealiasedvariables,e.g.thecalculatedeffectofX4(orofX1X2X3)isanestimateofthesumoftheeffectsofX4andX1X2X3.TheeffectofthefactorX4issaidtobeconfoundedwiththeeffectoftheX1X2X3interaction.,12,Example,Forthe24-1fractionalfactorialdesigninExample1,determinethealiasforeachfactorandinteraction.Note:X4=X1X2X3,i.e.X4isgeneratedfromX1X2X3,天马行空官方博客：,13,Example,FromX4=X1X2X3MultiplybothsidesbyX4,X4X4=X1X2X3X4I=X1X2X3X4(1)(Note:XiXi=I1)Multiplybothsidesof(1)byX1,X1=X1X1X2X3X4=X2X3X4Multiplybothsidesof(1)byX2,X2=X1X2X2X3X4=X1X3X4Multiplybothsidesof(1)byX3,X3=X1X2X3X3X4=X1X2X4,14,Example,FactorialDesignDesignGenerators:D=ABCAliasStructureI+ABCDA+BCDB+ACDC+ABDD+ABCAB+CDAC+BDAD+BC,15,DesignResolutions,Fractionalfactorialdesignscanbeclassifiedbytheirresolution.ResolutionIIIdesignsarethoseforwhichnoindividualfactorisaliasedwithanotherindividualfactor,butforwhichatleastonefactorisaliasedwitha2-factorinteraction.ResolutionIVdesignsarethoseforwhichnoindividualfactorisaliasedwithanotherindividualfactororwithany2-factorinteraction,butatleastone2-factorinteractionisaliasedwithanother2-factorinteraction,andatleastoneindividualfactorisaliasedwitha3-factorinteraction.ResolutionVdesignsarethoseforwhichnoindividualfactoror2-factorinteractionisaliasedwithanyotherindividualfactoror2-factorinteraction,butatleastone2-factorinteractionisaliasedwitha3-factorinteraction,andatleastoneindividualfactorisaliasedwitha4-factorinteraction.,天马行空官方博客：,16,DesignResolutions,Theresolutionofa2-levelfractionalfactorialdesignisthesmallestsumoftheordersofaliasedeffects.ResolutionSmallestSumofOrdersofAliasedEffectsIII1+2IV1+3,2+2V1+4,2+3Theresolutionofafractionalfactorialdesignisoftenincludedasasubscriptinthedesignationofthedesign,e.g.2design.Wherepossible,avoidResolution-III(RIII)designs.,17,Example,StatDOEFactorialCreateFactorialDesign,18,Plackett-BurmanDesigns,Plackett-BurmanDesignsaretwo-levelfractionalfactorialdesigns,ofResolutionIII,usedforstudyingK=N1variablesinNruns,whereNisamultipleof4.Thedesigncanbespecifiedbygivingjustthefirstcolumn.8Runs+12Runs+16Runs+20Runs+-+-+-+-+-24Runs+-+-+-+-+-+-,19,Plackett-Burman,Example:8RunPlackett-BurmanDesign:+RunABCDEFG1+2+3+4+5+6+7+8EachPlackett-BurmanDesigncanhaveuptok=(n1)factors.Ifk(n1),onlythefirstkcolumnsareused.,天马行空官方博客：,20,Example,Anengineerplanstoinvestigatethemaineffectsof5factors.Hecanonlyperform10experimentalrunsatbest.Selectanappropriateexperimentaldesign.,21,Example,StatDOEFactorialCreateFactorialDesign,22,Example,StdOrderRunOrderCenterPtBlocksABCDE12111111125111111133111111141111111157111111164111111176111111188111