16LinMixedEffects(1)

16LinMixedEffects(1) - LINEARMIXED-EFFECTMODELS...

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Unformatted text preview: LINEARMIXED-EFFECTMODELS Studies/data/modelsseenpreviouslyin511assumedasingle sourceof“error”variation y = X β + . β arefixedconstants(inthefrequentistapproachtoinference) istheonlyrandomeffect Whatiftherearemultiplesourcesof“error”variation? c 2011Dept.Statistics(IowaStateUniversity) Stat511section16 1/21 Examples:Subsampling Seedlingweightin2genotypestudyfromAitkenmodelsection. Seedlingweightmeasuredoneachseedling. Two(potential)sourcesofvariation: amongflatsandamongseedlingswithinaflat. Y ijk = μ + γ i + T ij + ijk T ij ∼ N ( ,σ 2 F ) ijk ∼ N ( ,σ 2 e ) where i indexesgenotype, j indexesflatwithingenotype,and k indexesseedlingwithinflat σ 2 F quantifiesvariationamongflats, ifhaveperfectknowledgeoftheseedlingsinthem σ 2 e quantifiesvariationamongseedlings c 2011Dept.Statistics(IowaStateUniversity) Stat511section16 2/21 Examples:Splitplotexperimentaldesign Influenceoftwofactors:temperatureandacatalyston fermentationofdrydistillersgrain(byproductofEtOHproduction fromcorn) ResponseisCO 2 production,measuredinatube Catalyst(+/-)randomlyassignedtotubes Temperaturerandomlyassignedtogrowthchambers 6tubespergrowthchamber,3+catalyst,3-catalyst all6atsametemperature Use6growthchambers,2ateachtemperature Theo.u.isatube.Whatistheexperimentalunit? c 2011Dept.Statistics(IowaStateUniversity) Stat511section16 3/21 Examples:Splitplotexperimentaldesign-2 Twosizesofexperimentalunit:tubeandtemperature Y ijkl = μ + α i + γ ik + β j + αβ ij + ijkl γ ik ∼ N ( ,σ 2 G ) Varamonggrowthchambers ijkl ∼ N ( ,σ 2 ) Varamongtubes where i indexestemperature, j indexesg.c.withintemp., k indexes catalyst,and l indexestubewithintemp.,g.c.,andcatalyst σ 2 G quantifiesvariationamongg.c., ifhaveperfectknowledgeofthetubesinthem σ 2 quantifiesvariationamongtubes c 2011Dept.Statistics(IowaStateUniversity) Stat511section16 4/21 Examples:GaugeR&Rstudy Wanttoquantifyrepeatabilityandreproducibilityofam easurementprocess Repeatability:variationinmeas.takenbyasinglepersonor instrumentonthesameitem. Reproducibility:variationinmeas.madebydifferentpeople(or differentlabs),ifperfectrepeatability Onepossibledesign: 10parts,eachmeasuredtwiceby10operators.200obs. Y ijk = μ + α i + β j + αβ ij + ijk α i ∼ N ( ,σ 2 P ) Varamongparts β j ∼ N ( ,σ 2 R ) Reproducibilityvariance αβ ij ∼ N ( ,σ 2 I ) Interactionvariance...
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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16LinMixedEffects(1) - LINEARMIXED-EFFECTMODELS...

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