within block Must do some sort of incomplete block analysis If you do not certain eects confounded Confounding two eects are indistinguishable May sacrice certain eects thought to be small design makes setup simple 241 Confounding in with only 2 blo ID: 43164
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BlockingandConfoundinginFactorialDesignofExperiments-MontgomeryChapter7BlockinginFactorialDesignsForRCBD,eachcombinationrunineachblock4EUsperblock8EUsperblockRandomizerunorderwithinblockSupposeyoucannotrunallcomb.withinblockMustdosomesortofincompleteblockanalysisIfyoudonot,certaine®ectsconfoundedConfounding:twoe®ectsareindistinguishableMay\sacri¯ce"certaine®ectsthoughttobesmalldesignmakesset-upsimpleConfoundinginwithonly2blocksBlocksassumedtoallow2Firstconsider2factorial(2combsperblk)Possiblepairings1(1)andbtogetheraandabtogether2(1)andatogetherbandabtogether3(1)andabtogetheraandbtogether1E®ectofA,(ab+a-b-(1))/2,isblockdi®erence2E®ectofB,(ab-a+b-(1))/2,isblockdi®erenceBothhaveamaine®ectconfoundedwithblock3E®ectofAB,(ab-a-b+1)/2,isblockdi®erenceAllowsformaine®ectestimates(blkscancelout)ChoiceofConfoundingFactorsCommontoconfoundhighestorderinteractionCanuse+tabletodetermineblksFortwofactor,recallthefollowingtableABABSymbol --+(1)+--a-+-b+++ab Useconfoundingcolumntodetermineblks+'sinblk1and'sinblk2ConsiderthreefactorABCABACBCABCSymbol ---+++-(1)+----++a-+--+-+b++-+---ab--++--+c+-+-+--ac-++--+-bc+++++++abc Bestassignmentwouldbea,b,c,abctogetherCanestimateallbutthreefactorinteraction FactorialinFourBlocksFourblockseachcontaining2UsefulinsituationswhereMustselecttwoe®ectstoconfoundWillresultinathirdconfoundedfactorConsider6factorfactorialrunin4blocksof16EUsBlock1usesABC+andDEF+Block2usesABC+andDEF-Block3usesABC-andDEF+Block4usesABC-andDEF-Resultsin(ABC)(DEF)=ABCDEFconfoundedABCDandDEFABCEFconfoundedABandABEFEFconfoundedCanextendto8and16blksTable7-8summarizesthesedesigns(pg298)PartialConfoundingCanreplicateblockingdesignConfounddi®erente®ectseachreplicationAllowsestimationofalle®ectsConfoundede®ectsbasedonnonconfoundedreplicatesCanuseYates'Algorithmforallnonconfoundede®ectsSeeExample7-3(pg300)\*Example7-3*\datacool;inputblockfact1fact2fact3y;1-1-1-1-3111-12procglm;classfact1fact2fact3block;modely=blockfact1|fact2|fact3;ClassLevelsValuesFACT121-1FACT221-1FACT321-1BLOCK41234DependentVariable:YSourceDFSumofSquaresFValuePr�FModel1074.250000009.900.0103Error53.75000000CorrectedTotal1578.00000000SourceDFTypeISSFValuePr�FBLOCK33.500000001.560.3101FACT1136.0000000048.000.0010FACT2120.2500000027.000.0035FACT1*FACT210.500000000.670.4513FACT3112.2500000016.330.0099FACT1*FACT310.250000000.330.5887FACT2*FACT311.000000001.330.3004FACT1*FACT2*FACT310.500000000.670.4513Considera2factorialrunin4blocksEachreplicatewillresultin3confoundede®ectsConsider4replicatesfor32totalobservationsReplicate1:ConfoundBCandACReplicate2:ConfoundBCandABCReplicate3:ConfoundACandABCReplicate4:ConfoundABandABCThreereplicatestoestimateA,B,andCTworeplicatestoestimateAB,AC,andBCOnereplicatetoestimateABC Replicate1-AB,AC,BCConfoundedBlk1Blk2Blk3Blk475(1)89ab61a30b100abc73c45bc54acReplicate2-A,BC,ABCConfoundedBlk1Blk2Blk3Blk460(1)47a1b26ac34bc81abc35c52abReplicate3-B,AC,ABCConfoundedBlk1Blk2Blk3Blk458(1)48a18b68ab42ac52c82abc32bcReplicate4-C,AB,ABCConfoundedBlk1Blk2Blk3Blk447(1)34a50c37ac57ab4b80abc27bcSincenoe®ectisestimatedfromallreplications(excepterror),wewillcomputethee®ectsumsassociatedwitheachreplicateandcombinetheappropriateinformation.Thefollowingarethecolumn3sumsusingYates'Algorithmforeachofthereplicates.OnlythesumsthatarenotconfoundedwiththeparticularreparepresentedE®ectRep1Rep2Rep3Rep4 (1)5273364003368080AB-120120171616AC-BC--4040 (81+80+80) (1+0+0) (120+120) (16+16+17) (0+0) (40+40) 12 UsingSASoptionsnocenterps=50ls=80;datanew;inputreplblkabcresp;110007511111100121108912001734300150431118044101374401127procglm;classreplblkabc;modelresp=replblk(repl)a|b|c;SASOutputDependentVariable:RESPSumofMeanSourceDFSquaresSquareFValueModel2217128.718750778.57812528028.81Error90.2500000.027778CorrectedTotal3117128.968750SourceDFTypeISSMeanSquareFValueREPL33040.09375001013.364583336481.13BLK(REPL)127568.3750000630.697916722705.13A12420.04166672420.041666787121.50B10.04166670.04166671.50A*B13600.00000003600.000000099999.99C1100.0416667100.04166673601.50A*C10.00000000.00000000.00B*C1400.0000000400.000000014400.00A*B*C10.12500000.12500004.50SourceDFTypeIIISSMeanSquareFValueREPL33040.09375001013.364583336481.13BLK(REPL)12653.906250054.49218751961.72A12420.04166672420.041666787121.50B10.04166670.04166671.50A*B13600.00000003600.000000099999.99C1100.0416667100.04166673601.50A*C10.00000000.00000000.00B*C1400.0000000400.000000014400.00A*B*C10.12500000.12500004.50