Linear Models for the Prediction of Animal Breeding Values

Transcription

1 Linear Models for the Prediction of Animal Breeding Values R.A. Mrode, PhD Animal Data Centre Fox Talbot House Greenways Business Park Bellinger Close Chippenham Wilts, UK CAB INTERNATIONAL

2 Preface ix 1 Genetic Evaluation with Different Sources of Records The Basic Model Breeding-value Prediction from the Animal's Own Performance Single record Repeated records Breeding-value Prediction from Progeny Records Breeding-value Prediction from Pedigree Breeding-value Prediction for One Trait from Another Selection Index Accuracy of index Examples of selection indices using different sources of information Prediction of aggregate genotype Restricted selection index 22 2 Genetic Covariance Between Relatives The Numerator Relationship Matrix Decomposing the Relationship Matrix Computing the Inverse of the Relationship Matrix Inverse of the relationship matrix ignoring inbreeding Inverse of the relationship matrix taking inbreeding into account Inverse of the Relationship Matrix for Sires and Maternal Grandsires 33

3 vi An example of the inverse of the relationship matrix for sires and maternal grandsires 35 3 Best Linear Unbiased Prediction of Breeding Value: Univariate Models with One Random Effect Brief Theoretical Background A Model for an Animal Evaluation (Animal Model) Constructing the mixed-model equations Accuracy of evaluations A Sire Model An illustration Reduced Animal Model Defining the model An illustration An alternative approach Animal Model with Groups An illustration 61 4 Best Linear Unbiased Prediction of Breeding Value: Models with Random Environmental Effects Repeatability Model Denning the model An illustration Model with Common Environmental Effects Defining the model An illustration 74 5 Best Linear Unbiased Prediction of Breeding Value: Multivariate Animal Models Equal Design Matrices and No Missing Records Defining the model An illustration Canonical Transformation The model An illustration Equal Design Matrices with Missing Records An illustration Cholesky Transformation Calculating the transformation matrix and defining the model An illustration Unequal Design Matrices An illustration Different Traits Recorded on Relatives Defining the model 97

4 vii An illustration 98 6 Maternal-trait Models: Animal and Reduced Animal Models Animal" Model for a Maternal Trait An illustration Reduced Animal Model with Maternal Effects An illustration Multivariate Maternal Animal Model Non-additive Animal Models Dominance Relationship Matrix Animal Model with Dominance Effects Solving for animal and dominance genetic effects separately Solving for total genetic merit directly Method for Rapid Inversion of the Dominance Matrix Inverse of the relationship matrix for subclass effects Prediction of dominance effects Calculating the inverse of the relationship matrix among dominance and subclass effects for an example pedigree Epistatis Rules for the inverse of the relationship matrix for epistatic and subclass effects Calculating the inverse of the relationship matrix for epistatic and subclass effects for an example pedigree Solving Linear Equations Direct Inversion Iteration on the Mixed-model Equations Jacobi iteration Gauss-Seidel iteration Iterating on the Data Animal model without groups Animal model with groups Reduced animal model with maternal effects 145 Appendix A: Introductory Matrix Algebra 155 A.1 Matrix: a Definition 155 A.2 Special Matrices 156 A.2.1 Square matrix 156 A.2.2 Diagonal matrix 156 A.2.3 Triangular matrix 157 A.2.4 Symmetric matrix 157

5 viii A.3 Basic Matrix Operations 157 A.3.1 Transpose of a matrix 157 A.3.2 Matrix addition and subtraction 158 A.3.3 Matrix multiplication 158 A.3.4 Direct product of matrices 159 A.3.5 Matrix inversion 159 A.3.6 Rank of a matrix 161 A.3.7 Generalized inverses 161 A.3.8 Eigenvalues and eigenvectors 162 Appendix B: Recent Fast Algorithms for Calculating Inbreeding Based on the L Matrix 163 B.I Meuwissen and Luo Algorithm 163 B.I.I Illustration of the algorithm 164 B.2 Modified Meuwissen and Luo Algorithm 166 B.2.1 Illustration of the algorithm 168 Appendix C 170 C.I Outline of the Derivation of the Best Linear Unbiased Prediction (BLUP) ^ 170 C.2 Proof that b and a from Mixed-model Equations are the Generalized Least-square Solution of b and Best Linear Unbiased Prediction of a Respectively 171 Appendix D: A Method for Obtaining Approximate Reliability for Genetic Evaluations under an Animal Model 173 Appendix E 176 E.I Canonical Transformation: Procedure to Calculate the Transformation Matrix and its Inverse 176 E.2 Canonical Transformation with Missing Records and Same-incidence Matrices 177 E.2.1 Illustration 179 E.3 Cholesky Decomposition 181 References 182 Index 185