28GLMM(1) - Generalized Linear Mixed Models I GLM Mixed effects I Goal Add random effects or correlations among observations to a model where

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Unformatted text preview: Generalized Linear Mixed Models I GLM + Mixed effects I Goal: Add random effects or correlations among observations to a model where observations arise from a distribution in the exponential-scale family (other than the normal) I Why: I More than one source of variation (e.g. farm and animal within farm) I Account for temporal correlation I Provides another way to deal with overdispersion I Take home message: Can be done, but a lot harder than a linear mixed effect model I Because: both computation and interpretation issues c 2011 Dept. Statistics (Iowa State University) Stat 511 section 28 1 / 17 I Another look at the canonical LME: Y = X β + Zu + I Consider each level of variation separately. A hierarchical or multi-level model η = X β + Zu ∼ N ( X β , ZGZ ) Y | η = η + ∼ N ( η , ) Y | u = X β + Zu + ∼ N ( X β + Zu , ) I Above specifies the conditional distribution of Y given η or equivalently u c 2011 Dept. Statistics (Iowa State University) Stat 511 section 28 2 / 17 I To write down a likelihood, need the marginal pdf of 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 I When u ∼ N () and ∼ N () , that integral has a closed form solution Y ∼ N ( X β , ZGZ + R ) I Extend to GLMs by changing conditional distribution of Y | u I Logistic: f ( Y i | u ) ∼ Binomial ( m i , π i ( u )) I Poisson: f ( Y i | u ) ∼ Poisson ( λ i ( u )) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 28 3 / 17 I Big problem : Usually no analytic solutions to f ( Y ) No closed form solution to the integral I Some exceptions: I Y | η ∼ Binomial ( m , η ) , η ∼ β ( α, β ) Y ∼ BetaBinomial I Y | η ∼ Poisson ( η ) , η ∼ Γ( α, β ) Y ∼ NegativeBinomial I Ok for one level of additional variability, but difficult (if not impossible) to extend to multiple random effects I Normal distributions are very very nice: I Easy to model multiple random effects: the sum of Normals is Normal I Easy to model correlations among observations I Want a way to fit a model like: μ = g- 1 ( X β + Zu ) , u ∼ N ( , G ) Y | μ = f (...
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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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28GLMM(1) - Generalized Linear Mixed Models I GLM Mixed effects I Goal Add random effects or correlations among observations to a model where

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