28GLMM - GeneralizedLinearMixedModels GLM+Mixedeffects

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Unformatted text preview: GeneralizedLinearMixedModels GLM+Mixedeffects Goal:Addrandomeffectsorcorrelationsamongobservationstoa modelwhereobservationsarisefromadistributioninthe exponential-scalefamily(otherthanthenormal) Why: Morethanonesourceofvariation(e.g.farmandanimalwithin farm) Accountfortemporalcorrelation Providesanotherwaytodealwithoverdispersion Takehomemessage:Canbedone,buta lot harderthanalinear mixedeffectmodel Because:bothcomputationandinterpretationissues c 2011Dept.Statistics(IowaStateUniversity) Stat511section28 1/17 AnotherlookatthecanonicalLME: Y = X + Zu + Considereachlevelofvariationseparately. Ahierarchicalormulti-levelmodel = X + Zu N ( X , ZGZ ) Y | = + N ( , ) Y | u = X + Zu + N ( X + Zu , ) Abovespecifiestheconditionaldistributionof Y given or equivalently u c 2011Dept.Statistics(IowaStateUniversity) Stat511section28 2/17 Towritedownalikelihood,needthemarginalpdfof Y f ( Y , u )= f ( Y | u ) f ( u ) f ( Y )= Z u f ( Y , u ) d u = Z u f ( Y | u ) f ( u ) d u When u N () and N () ,thatintegralhasaclosedformsolution Y N ( X , ZGZ + R ) ExtendtoGLMsbychangingconditionaldistributionof Y | u Logistic: f ( Y i | u ) Binomial ( m i , i ( u )) Poisson: f ( Y i | u ) Poisson ( i ( u )) c 2011Dept.Statistics(IowaStateUniversity) Stat511section28 3/17 Bigproblem :Usuallynoanalyticsolutionsto f ( Y ) Noclosedformsolutiontotheintegral Someexceptions: Y | Binomial ( m , ) , ( , ) Y BetaBinomial Y | Poisson ( ) , ( , ) Y NegativeBinomial Okforonelevelofadditionalvariability,butdifficult(ifnot impossible)toextendtomultiplerandomeffects Normaldistributionsareveryverynice: Easytomodelmultiplerandomeffects: thesumofNormalsisNormal Easytomodelcorrelationsamongobservations Wantawaytofitamodellike: = g 1 ( X + Zu ) , u N ( , G ) Y |...
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28GLMM - GeneralizedLinearMixedModels GLM+Mixedeffects

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