17DesignLME(1)

17DesignLME(1) - ExperimentalDesignsandLMEs

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Unformatted text preview: ExperimentalDesignsandLMEs LMEmodelsprovideonewaytomodelcorrelationsamong observations Veryusefulforexperimentaldesignswherethereismorethanone sizeofexperimentalunit Ordesignswheretheobservationunitisnotthesameasthe experimentalunit. Myphilosophy(widespreadatISUandelsewhere)isthattheway anexperimentisconductedspecifiestherandomeffectstructure fortheanalysis. Observationalstudiesareverydifferent Norandomizationscheme,sonoclearcutrandomeffectstructure Oftenusemodelselectionmethodstohelpchosetherandom effectstructure NOTSOforarandomizedexperiment. c 2011Dept.Statistics(IowaStateUniversity) Stat511section17 1/25 Oneexample: studydesignedasanRCBD. treatmentsarerandomlyassignedwithinablock analyzedata,findoutblockeffectsaresmall shouldyoudeleteblockfromthemodelandreanalyze? abovephilosophysaystough.Blockeffectsstayintheanalysis forthe next study,seriouslyconsider: blockinginadifferentway orusingaCRD FollowingpagesworkthroughdetailsofLMEsforsomecommon studydesigns c 2011Dept.Statistics(IowaStateUniversity) Stat511section17 2/25 Mousemusclestudy Micegroupedintoblocksbylitter.4miceperblock,4blocks. Fourtreatmentsrandomlyassignedwithineachblock Collecttworeplicatemusclesamplesfromeachmouse Measureresponseoneachmusclesample 16eu.,32ou. Questionsconcerndifferencesinmeanresponseamongthefour treatments Themodelusuallyusedforastudylikethisis: y ijk = + i + l j + m ij + e ijk ,where y ijk isthe k th measurementoftheresponseforthemousefrom litter j thatreceivedtreatment i , ( i = 1 , 2 , 3 , 4 ; j = 1 , 2 , 3 , 4 ; k = 1 , 2 ) . c 2011Dept.Statistics(IowaStateUniversity) Stat511section17 3/25 Thefixedeffects: = 1 2 3 4 IR 5 isanunknownvectoroffixedparameters u =[ l 1 , l 2 , l 3 , l 4 , m 11 , m 21 , m 31 , m 41 , m 12 ,..., m 34 , m 44 ] isavectorof randomeffectsdescribinglittersandmice. =[ e 111 , e 112 , e 211 , e 212 ,..., e 441 , e 442 ,..., e 441 , e 442 ] isavectorof randomerrors. with y =[ y 111 , y 112 , y 211 , y 212 ,..., y 411 , y 412 ,..., y 441 , y 442 ] ,themodel canbewrittenasalinearmixedeffectsmodel y = X + Z u + ,where c 2011Dept.Statistics(IowaStateUniversity) Stat511section17 4/25 X = 11000 11000 10100 10100 10010 10010 10001 10001 Z = 10001000 10001000 10000100 10000100 10000010 ... 10000010 10000001 10000001 . . . . 00010000 00010000 00010000 00010000 ... 00010000 00010000 1 00010000 1 c 2011Dept.Statistics(IowaStateUniversity) Stat511section17 5/25 WecanwritelessandbemorepreciseusingKroneckerproduct notation. X = 1 4 1 [ 1 8 1 , I 4 4 1 2 1 ] Z =[ I 4 4 1 8 1 , I 16 16 1 2 1 ] Inthisexperiment,wehavetworandomfactors:LitterandMouse.Inthisexperiment,wehavetworandomfactors:LitterandMouse....
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17DesignLME(1) - ExperimentalDesignsandLMEs

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