Prediction Theory 1 Introduction 2 General Linear Mixed Model

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5 Variances of Predictors Let then P = (X ′ V −1 X) − ˆb = PX ′ V −1 y û = GZ ′ V −1 (y − XPX ′ V −1 y) = GZ ′ V −1 Wy for W = (I − XPX ′ V −1 ). From the results on generalized inverses of X, and therefore, The variance of the predictor is, The covariance between ˆb and û is XPX ′ V −1 X = X, WX = (I − XPX ′ V −1 )X = X − XPX ′ V −1 X = X − X = 0. V ar(û) = GZ ′ V −1 W(V ar(y))W ′ V −1 ZG = GZ ′ V −1 WVW ′ V −1 ZG Therefore, the total variance of the predictor is = GZ ′ V −1 ZG − GZ ′ V −1 XPX ′ V −1 ZG. Cov(ˆb, û) = PX ′ V −1 V ar(y)W ′ V −1 ZG = PX ′ W ′ V −1 ZG = 0 because X ′ W = 0 V ar(K ′ˆb + M ′ û) = K ′ PK + M ′ GZ ′ V −1 ZGM −M ′ GZ ′ V −1 XPX ′ V −1 ZGM. 6 Variance of Prediction Error The main results are V ar(ˆb − b) = V ar(ˆb) + V ar(b) − Cov(ˆb, b) − Cov(b, ˆb) = V ar(ˆb) = P. V ar(û − u) = V ar(û) + V ar(u) − Cov(û, u) − Cov(u, û), 6
where Cov(û, u) = GZ ′ V −1 WCov(y, u) = GZ ′ V −1 WZG = GZ ′ (V −1 − V −1 XPX ′ V −1 )ZG = V ar(û) so that V ar(û − u) = V ar(û) + G − 2V ar(û) = G − V ar(û). Also, Cov(ˆb, û − u) = Cov(ˆb, û) − Cov(ˆb, u) = 0 − PX ′ V −1 ZG. 7 Mixed Model Equations The covariance matrix of y is V which is of order N. N is usually too large to allow V to be inverted. The BLUP predictor has the inverse of V in the formula, and therefore, would not be practical when N is large. Henderson(1949) developed the mixed model equations for computing BLUP of u and the GLS of b. However, Henderson did not publish a proof of these properties until 1963 with the help of S. R. Searle, which was one year after Goldberger (1962). Take the first and second partial derivatives of F, ( ) ( ) V X L X ′ = 0 θ Recall that V = V ar(y) = ZGZ ′ + R, and let ( ZGM K ) which when re-arranged gives S = G(Z ′ L − M) M = Z ′ L − G −1 S, then the previous equations can be re-written as ⎛ ⎞ ⎛ R X Z ⎜ ⎝ X ′ ⎟ ⎜ 0 0 ⎠ ⎝ Z ′ 0 −G −1 L θ S ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ 0 K M ⎞ ⎟ ⎠ . Take the first row of these equations and solve for L, then substitute the solution for L into the other two equations. L = −R −1 Xθ − R −1 ZS 7
Page 1 and 2: Prediction Theory 1 Introduction Be
Page 3 and 4: 3.1 Best Predictor The best predict
Page 5: 4.4 Function to be Minimized Becaus
Page 9 and 10: 8 Equivalence Proofs The equivalenc
Page 11 and 12: which gives the result that ( ) ( C
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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