Prediction Theory 1 Introduction 2 General Linear Mixed Model

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Then GZ ′ V −1 = GZ ′ (R −1 − R −1 ZTZ ′ R −1 ) = (GZ ′ R −1 − GZ ′ R −1 ZTZ ′ R −1 ) = (GT −1 − GZ ′ R −1 Z)TZ ′ R −1 = (G(Z ′ R −1 Z + G −1 ) − GZ ′ R −1 Z)TZ ′ R −1 = TZ ′ R −1 . Similarly, the MME solution for û and substituting it into the first equation in the MME gives X ′ R −1 Xˆb + X ′ R −1 Z(TZ ′ R −1 (y − Xˆb)) = X ′ R −1 y. Combine the terms in ˆb and y to give X ′ (R −1 − R −1 ZTZ ′ R −1 )Xˆb = X ′ (R −1 − R −1 ZTZ ′ R −1 )y, which are the same as the GLS equations, X ′ V −1 Xˆb = X ′ V −1 y. Goldberger (1962) published these results before Henderson (1963), but Henderson knew of these equivalences back in 1949 through numerical examples. After he discovered Goldberger’s paper (sometime after his retirement) Henderson insisted on citing it along with his work. Most people in animal breeding, however, refer to Henderson as the originator of this work and its primary proponent. 9 Variances of Predictors and <strong>Prediction</strong> Errors From MME The covariance matrices of the predictors and prediction errors can be expressed in terms of the generalized inverse of the coefficient matrix of the MME, C. Recall that ( ) ( ) ( ) ˆb Cxx C = xz X ′ R −1 û C zx C zz Z ′ R −1 y, or as ˆb = C ′ by, and û = C ′ uy. If the coefficient matrix of the MME is full rank (or a full rank subset) (to simplify the presentation of results), then ( ) ( ) ( ) Cxx C xz X ′ R −1 X X ′ R −1 Z I 0 C zx C zz Z ′ R −1 X Z ′ R −1 Z + G −1 = , 0 I 10
which gives the result that ( ) ( Cxx C xz X ′ R −1 X X ′ R −1 Z C zx C zz Z ′ R −1 X Z ′ R −1 Z = ( ) I −Cxz G −1 0 I − C zz G −1 . This last result is used over and over in deriving the remaining results. Now, V ar(ˆb) = V ar(C ′ by) = C ′ bV ar(y)C b = C ′ b(ZGZ ′ + R)C b ( ) ( ) X = C xx C ′ R −1 xz Z ′ R −1 (ZGZ ′ + R)C b ( ) ( ) X = C xx C ′ R −1 Z xz Z ′ R −1 GZ ′ C Z b ( + C xx C xz ) ( X ′ Z ′ ) C b ) = −C xz G −1 GZ ′ C b ( ) ( X + C xx C ′ R −1 X X ′ R −1 Z xz Z ′ R −1 X Z ′ R −1 Z ) ( ) Cxx C zx = C xz G −1 C zx ( ) ( ) + I −C xz G −1 C xx C zx = C xz G −1 C zx + C xx − C xz G −1 C zx = C xx . The remaining results are derived in a similar manner. These give V ar(û) = C u ′ V ar(y)C u = G − C zz Cov(ˆb, û) = 0 V ar(û − u) = V ar(û) + V ar(u) − Cov(û, u) − Cov(u, û) = V ar(u) − V ar(û) = G − (G − C zz ) = C zz 11
Page 1 and 2: Prediction Theory 1 Introduction Be
Page 3 and 4: 3.1 Best Predictor The best predict
Page 5 and 6: 4.4 Function to be Minimized Becaus
Page 7 and 8: where Cov(û, u) = GZ ′ V −1 WC
Page 9: 8 Equivalence Proofs The equivalenc
Page 13 and 14: The degrees of freedom for F are r(
Page 15 and 16: epresenting the vectors of elements
Page 17 and 18: Another alternative might be to par
Page 19 and 20: 17.1 Operational Models Let y ijk =
Page 21 and 22: 17.2.3 G and R The covariance matri
Page 23 and 24: 17.2.5 Solutions and Variance of Pr
Page 25 and 26: MME = function(X,Z,GI,RI,y) { XX =

Prediction Theory 1 Introduction 2 General Linear Mixed Model

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