# 28GLMM - GLM Mixedeffects Goal:...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 β + Zu + Considereachlevelofvariationseparately. Ahierarchicalormulti-levelmodel η = X β + Zu ∼ N ( X β , ZGZ ) Y | η = η + ∼ N ( η , ) Y | u = X β + Zu + ∼ N ( X β + Zu , ) Abovespecifiestheconditionaldistributionof Y given η 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 β , ZGZ + R ) ExtendtoGLMsbychangingconditionaldistributionof Y | u Logistic: f ( Y i | u ) ∼ Binomial ( m i ,π i ( u )) Poisson: f ( Y i | u ) ∼ Poisson ( λ i ( u )) c 2011Dept.Statistics(IowaStateUniversity) Stat511section28 3/17 Bigproblem :Usuallynoanalyticsolutionsto f ( Y ) Noclosedformsolutiontotheintegral Someexceptions: Y | η ∼ Binomial ( m , η ) , η ∼ β ( α,β ) Y ∼ BetaBinomial Y | η ∼ Poisson ( η ) , η ∼ Γ( α,β ) Y ∼ NegativeBinomial Okforonelevelofadditionalvariability,butdifficult(ifnot impossible)toextendtomultiplerandomeffects Normaldistributionsareveryverynice: Easytomodelmultiplerandomeffects: thesumofNormalsisNormal Easytomodelcorrelationsamongobservations Wantawaytofitamodellike: μ = g − 1 ( X β + Zu ) , u ∼ N ( , G ) Y |...
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

28GLMM - GLM Mixedeffects Goal:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online