The Prediction of Random Effects in Generalized Linear Mixed Model Wenxue Ge Department of Mathematics and Statistics University of Victoria, Canada Nov 20, 2017 @University of Victoria
Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
Outline Background GLM and GLMM 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
GLM and GLMM Exponential Family f(y;θ) = s(y) t(θ) exp[a(y)b(θ)] f(y;θ) = exp[ a(y)b(θ) + c(θ) + d(y) ] -where s(y) = exp[ d(y) ]; t(θ) = exp[ c(θ) ] Canonical if a(y) = y; b(θ) is natural parameter
GLM and GLMM Generalized Linear Model - Y 1,...,Y N from the exponential family 1. Y i has the canonical form f(y i ;θ i ) = exp [ y i b i (θ i ) + c i (θ i ) + d i (y i )] 2. Y i s same distribution form - Link function g(µ i ) = x T i β Monotone, Differentiable
GLM and GLMM Generalized Linear Mixed Model - Random Effects Incorporating correlation, broader inference - General Model
Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
, or BLUP - "Best": miminum MSE (not within all predictors) - Model Var [ ] u = e [ ] G 0 σ 2 0 R E(u) = 0, E(e) = 0
Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
Henderson s Justification, u N q (0,Gσ 2 ) f (y, u) = g(y u)h(u) (1) = (2πσ 2 ) 1 2 n 1 2 q (det G 0 ) 1 1 2 exp{ 0 R 2σ 2 [u T G 1 u + (y Xβ Z u) T R 1 (y Xβ Z u)]} Bayesian Derivation, u N q (0,Gσ 2 ) - β: uniform, improper prior - Posterior density is proportional to (1)
Minimize u T G 1 u + (y Xβ Z u) T R 1 (y Xβ Z u) Mixed model equations X T R 1 X ˆβ + X T R 1 Z û = X T R 1 y (2) Z T R 1 X ˆβ + (Z T R 1 Z + G 1 )û = Z T R 1 y (3) - Eliminate u û = (Z T R 1 Z + G 1 ) 1 Z T R 1 (y X ˆβ) (4) gives X T R 1 X ˆβ X T R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 X ˆβ = X T R 1 y X T R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 y
Simplify X T (R 1 R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 )X ˆβ = X T (R 1 R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 )y and (R+ZGZ T ) 1 = R 1 R 1 Z (Z T R 1 Z +G 1 ) 1 Z T R 1 gives X T (R + ZGZ T ) 1 X ˆβ = X T (R + ZGZ T ) 1 y (5) Solution (Note:û = (Z T R 1 Z +G 1 ) 1 Z T R 1 (y X ˆβ)) ˆβ = (X T (R + ZGZ T ) 1 X) 1 X T (R + ZGZ T ) 1 y (6) û = (Z T R 1 Z + G 1 ) 1 {[Z T R 1 Z T R 1 X] [X T (R + ZGZ T ) 1 X] 1 X T (R + ZGZ T ) 1 }y (7)
Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
BLUP(b T β + c T u) - b T and c T are given - Seek λ : λ T y unbiase for b T β + c T u and minimize var(λ T y (b T β + c T u)) + 2m T (X T λ X T b) = λ T var(y)λ + c T var(u)c-2λ T cov(u, y T ) +2 m T (X T λ X T b) 2m Lagrange multipliers Yields λ T soly as BLUP(b T β + c T u) = b T ˆβ + c T (Z T R 1 Z + G 1 ) 1 Z T R 1 (y X ˆβ)
Harville and Robinson - More intuitive - Linear Combination: b T β + c T u - Any Linear Unbiased Estimator: b T ˆβ + c T û + a T y - Where X T a = 0 b T and c T are given
Robinson Continued E[yy T ] = Xββ T X T + ZGZ T σ 2 + Rσ 2 E[uy T ] = GZ T σ 2 E[(ˆβ β)y T a] =[X T (R + ZGZ T ) 1 X] 1 X T (ZGZ T + R) 1 E[yy T ]a βe[y T ]a =ββ T X T a + [X T (R + ZGZ T ) 1 X] 1 X T σ 2 a ββ T X T a =0 Same steps E[(û u)y T a] = 0
"Best" E{[b T (ˆβ β) + c T (û u)a T y] [b T (ˆβ β) + c T (û u)a T y] T } = E{[b T (ˆβ β) + c T (û u)][b T (ˆβ β) + c T (û u)] T } + E[a T yy T a] + E{[b T (ˆβ β) + c T (û u)]y T a} + E{a T y[b T (ˆβ β) + c T (û u)]} = E{[b T (ˆβ β) + c T (û u)][b T (ˆβ β) + c T (û u)] T } + E[a T yy T a]
Outline Background First Lactation Yields 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
First Lactation Yields Model: y 9x1 = X 9x3 β 3x1 + Z 9x4 u 4x1 + e 9x1 y : Yield β : Herd u : Sire
First Lactation Yields Mixed model Equations X T R 1 X ˆβ + X T R 1 Z û = X T R 1 y (2) Z T R 1 X ˆβ + (Z T R 1 Z + G 1 )û = Z T R 1 y (3) R 9x9 = I 9x9 G 4x4 = 0.1I 4x4
First Lactation Yields Mixed model Equations X T R 1 X ˆβ + X T R 1 Z û = X T R 1 y (2) Z T R 1 X ˆβ + (Z T R 1 Z + G 1 )û = Z T R 1 y (3)
First Lactation Yields Solutions
Outline Background Expansions 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
Expansions Why have random effects Expansion - Goldberger s Derivition Not require normal - Restricted Maximum Likelihood Variance parameters
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