Prediction Theory 1 Introduction 2 General Linear Mixed Model

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and ( ) ( ) X − ′ R −1 X X ′ R −1 Z θ Z ′ R −1 X Z ′ R −1 Z + G −1 S = ( K M Let a solution to these equations be obtained by computing a generalized inverse of ( X ′ R −1 X X ′ R −1 Z Z ′ R −1 X Z ′ R −1 Z + G −1 ) denoted as ( ) Cxx C xz , C zx C zz then the solutions are Therefore, the predictor is L ′ y = = ( θ S ( ( ) = − ( Cxx C xz C zx C zz ) ( K M ) ( ) ( K ′ M ′ C xx C xz X ′ R −1 y C zx C zz Z ′ R −1 y ) ( ) ˆb K ′ M ′ , û ) . ) ) . where ˆb and û are solutions to ( ) ( X ′ R −1 X X ′ R −1 Z ˆb Z ′ R −1 X Z ′ R −1 Z + G −1 û ) = ( X ′ R −1 y Z ′ R −1 y ) . The equations are known as Henderson’s <strong>Mixed</strong> <strong>Model</strong> Equations or MME. The equations are of order equal to the number of elements in b and u, which is usually much less than the number of elements in y, and therefore, are more practical to solve. Also, these equations require the inverse of R rather than V, both of which are of the same order, but R is usually diagonal or has a more simple structure than V. Also, the inverse of G is needed, which is of order equal to the number of elements in u. The ability to compute the inverse of G depends on the model and the definition of u. The MME are a useful computing algorithm for obtaining BLUP of K ′ b + M ′ u. Please keep in mind that BLUP is a statistical procedure such that if the conditions for BLUP are met, then the predictor has the smallest mean squared error of all linear, unbiased predictors. The conditions are that the model is the true model and the variance-covariance matrices of the random variables are known without error. In the strictest sense, all models approximate an unknown true model, and the variancecovariance parameters are usually guessed, so that there is never a truly BLUP analysis of data, except possibly in simulation studies. 8
8 Equivalence Proofs The equivalence of the BLUP predictor to the solution from the MME was published by Henderson in 1963. In 1961 Henderson was in New Zealand (on sabbatical leave) visiting Shayle Searle learning matrix algebra and trying to derive the proofs in this section. Henderson needed to prove that V −1 = R −1 − R −1 ZTZ ′ R −1 where and T = (Z ′ R −1 Z + G −1 ) −1 V = ZGZ ′ + R. Henderson says he took his coffee break one day and left the problem on Searle’s desk, and when he returned from his coffee break the proof was on his desk. VV −1 = (ZGZ ′ + R)(R −1 − R −1 ZTZ ′ R −1 ) = ZGZ ′ R −1 + I − ZGZ ′ R −1 ZTZ ′ R −1 −ZTZ ′ R −1 = I + (ZGT −1 − ZGZ ′ R −1 Z − Z) TZ ′ R −1 = I + (ZG(Z ′ R −1 Z + G −1 ) −ZGZ ′ R −1 Z − Z)TZ ′ R −1 = I + (ZGZ ′ R −1 Z + Z −ZGZ ′ R −1 Z − Z)TZ ′ R −1 = I + (0)TZ ′ R −1 = I. Now take the equation for û from the MME Z ′ R −1 Xˆb + (Z ′ R −1 Z + G −1 )û = Z ′ R −1 y which can be re-arranged as (Z ′ R −1 Z + G −1 )û = Z ′ R −1 (y − Xˆb) or The BLUP formula was û = TZ ′ R −1 (y − Xˆb). û = GZ ′ V −1 (y − Xˆb). 9
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: where Cov(û, u) = GZ ′ V −1 WC
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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