This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 exponentialscale 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 multilevel 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 (...
View
Full Document
 Spring '08
 Staff
 Correlation, Normal Distribution, Deer, STATE UNIVERSITY

Click to edit the document details