Linear Models for the Prediction of the Genetic Merit of Animals, 4th Edition

Fundamental to any livestock improvement program by animal scientists is the prediction of genetic merit in the offspring generation for desirable production traits such as increased growth rate, superior meat, milk, and wool production. Covering the foundational principles on the application of linear models for the prediction of genetic merit in livestock, this new edition is fully updated to incorporate recent advances in genomic prediction approaches, genomic models for multi-breed and crossbred performance, dominance, and epistasis. It provides models for the analysis of main production traits as well as functional traits and includes numerous worked examples. For the first time, R codes for key examples in the textbook are provided online. The book covers: The relationship between the genome and the phenotype. BLUP models for various livestock data and structure. Incorporation of related ancestral parents and metafounders in prediction models. Models for survival analysis and social interaction. Advancements in genomic prediction approaches and selection. Genomic models for multi-breed and crossbred performance. Models for non-additive genetic effects including dominance and epistasis. Estimation of genetic parameters including Gibbs sampling approaches. Computation methods for solving linear mixed model equations. Suitable for graduate and postgraduate students, researchers and lecturers of animal breeding, genetics and genomics, this established textbook provides a thorough grounding in both the basics and in new developments of linear models and animal genetics. Cover Linear Models for the Prediction of the Genetic Merit of Animals Copyright Contents Preface Abbreviations 1 The Genome and Phenotypes 1.1 Introduction 1.2 Variation in DNA 1.3 Variation in Phenotype Values 1.4 DNA Lottery 1.5 Additional Points 2 Genetic Evaluation with Different Sources of Records 2.1 Introduction 2.2 The Basic Model 2.3 Breeding Value Prediction from the Animal’s Own Performance 2.3.1 Single record 2.3.2 Repeated records 2.4 Breeding Value Prediction from Progeny Records 2.5 Breeding Value Prediction from Pedigree 2.6 Breeding Value Prediction for One Trait from Another 2.7 Selection Index 2.7.1 Accuracy of index 2.7.2 Examples of selection indices using different sources of information Data available on correlated traits Using single records on individual and relatives Using means of records from animal and relatives 2.7.3 Prediction of aggregate genotype 2.7.4 Overall economic indices using predicted genetic merit 2.7.5 Restricted selection index 2.7.6 Index combining breeding values from phenotype and genetic marker information 3 Genetic Covariance Between Relatives 3.1 Introduction 3.2 Identity-by-State Versus Identity-by-Descent 3.3 The Numerator Relationship Matrix 3.4 Decomposing the Relationship Matrix 3.5 Computing the Inverse of the Relationship Matrix 3.5.1 Inverse of the numerator relationship matrix ignoring inbreeding 3.5.2 Inverse of the numerator relationship matrix accounting for inbreeding 3.6 Inverse of the Relationship Matrix for Sires and Maternal Grandsires 3.7 An Example of the Inverse of a Sire and Maternal Grandsire Relationship Matrix 3.8 Inferring Ancestral Relationships and Metafounders 3.8.1 Ancestral relationship within and across individuals Pedigree relationship with related base population 3.8.2 Metafounders Computing additive relationship matrix and inbreeding with one metafounder Computing inverse of (Ag)-1 with one metafounder 3.8.3 Computing relationships and inbreeding with several metafounders 3.8.4 Estimation of metafounders’ ancestral relationships from genomic data 4  Best Linear Unbiased Prediction of Breeding Value: Univariate Models with One Random Effect 4.1 Introduction 4.2 Brief Theoretical Background 4.3 A Model for an Animal Evaluation (Animal Model) 4.3.1 Constructing the mixed-model equations 4.3.2 Progeny (daughter) yield deviation Illustrating the calculation of PYD or DYD 4.3.3 Accuracy of evaluations 4.4 A Sire Model 4.4.1 An illustration Setting up the design of matrices and MME 4.5 Reduced Animal Model 4.5.1 Defining the model 4.5.2 An illustration Constructing the MME Solutions for non-parents 4.5.3 An alternative approach 4.6 Animal Model with Groups 4.6.1 An illustration Setting up the design matrices and MME 5 Best Linear Unbiased Prediction of Breeding Value: Models with Random Environmental Effects 5.1 Introduction 5.2 Repeatability Model 5.2.1 Defining the model 5.2.2 An illustration Setting up the design matrices 5.2.3 Calculating daughter yield deviations 5.3 Model with Common Environmental Effects 5.3.1 Defining the model 5.3.2 An illustration Setting up the design matrices 6 Best Linear Unbiased Prediction of Breeding Value: Multivariate Models 6.1 Introduction 6.2 Equal Design Matrices and No Missing Records 6.2.1 Defining the model 6.2.2 An illustration Setting up the design matrices 6.2.3 Partitioning animal evaluations from multivariate analysis 6.2.4 Accuracy of multivariate evaluations 6.2.5 Calculating daughter yield deviations in multivariate models 6.3 Equal Design Matrices with Missing Records 6.3.1 An illustration Setting up the design matrices 6.4 Unequal Design Matrices 6.4.1 Numerical example Setting up the design matrices and MME 6.4.2 Illustrating the computation of DYD from a multivariate model 6.5 Multivariate Models with No Environmental Covariance 6.5.1 Different traits recorded on relatives Defining the model An illustration 6.5.2 The multi-trait across-country evaluations (MACE) Computing effective daughter contribution An example of MACE for two countries Computing sire breeding values Equations for partitioning bull evaluations from MACE 7 Methods to Reduce the Dimension of Multivariate Models 7.1 Introduction 7.2 Canonical Transformation 7.2.1 The model 7.2.2 An illustration Setting up the design matrices 7.3 Cholesky Transformation 7.3.1 Calculating the transformation matrix and defining the model 7.3.2 An illustration 7.4 Factor and Principal Component Analysis 7.4.1 Factor analysis Mixed-model equations An illustration Analysis with FA model 7.4.2 Principal component analysis Analysis with full PC model 7.4.3 Analysis with reduced-rank PC model 8 Maternal Trait Models: Animal and Reduced Animal Models 8.1 Introduction 8.2 Animal Model for Maternal Traits 8.2.1 An illustration Setting up the design matrices 8.3 Reduced Animal Model with Maternal Effects 8.3.1 An illustration Setting up the design matrices Back-solving for non-parents Back-solving for direct effects Back-solving for maternal effects 8.4 Sire and Maternal Grandsire Model 9 Social Interaction Models 9.1 Introduction 9.2 Animal Model with Social Interaction Effects 9.2.1 Illustration of a model with social interaction 9.3 Partitioning Evaluations from Associative Models 9.4 Analysis Using Correlated Error Structure 10 Analysis of Longitudinal Data 10.1 Introduction 10.2 Fixed Regression Model 10.2.1 An illustration Setting up the incidence matrices for the MME Partitioning breeding values and solutions for permanent environmental effects 10.3 Random Regression Model 10.3.1 Numerical application Setting up the matrices for the MME 10.3.2 Partitioning animal solutions from random regression model 10.3.3 Calculating daughter yield deviations 10.3.4 Reliability of breeding values 10.3.5 Random regression models with spline function 10.3.6 Random regression models for maternal traits 10.4 Covariance Functions 10.4.1 Fitting a reduced-order covariance function 10.5 Equivalence of the Random Regression Model to the Covariance Function 11 Genomic Prediction and Selection 11.1 Introduction 11.2 Coding and Scaling Genotypes 11.3 Fixed-effect Model for SNP Effects 11.4 Mixed Linear Model for Computing SNP Effects 11.4.1 SNP-BLUP model Computing the required matrices and a 11.4.2 Equivalent models: GBLUP 11.4.3 Females in reference 11.4.4 Computing SNP solutions from GBLUP 11.4.5 Equivalent models: selection index approach 11.4.6 Computing base population allele frequencies 11.5 Mixed Linear Models with Polygenic Effects 11.6 Haplotype Models 11.7 Bayesian Methods for Computing SNP Effects 11.7.1 BayesA Prior distributions 11.7.2 BayesB 11.7.3 BayesC 11.7.4 BayesCp 11.8 Multivariate Genomic Models 11.9 Cross-validation and Genomic Reliabilities 12 Single-step Approaches to Genomics 12.1 Basic Principle 12.2 Alternative Approaches 12.3 Groups and Metafounders in Single-step Procedures 12.4 APY Approach 13 Non-additive Animal Models 13.1 Introduction 13.2 Dominance Relationship Matrix Using Pedigree Information 13.3 Animal Model with Dominance Effect 13.3.1 Solving for additive and dominance genetic effects separately Setting up the MME 13.3.2 Solving for total genetic merit directly Setting up the MME 13.4 Genomic Models for Dominance Effects 13.4.1. Estimating breeding values and dominance deviation effects Equivalent GBLUP dominance model 14 Genetic and Genomic Models for Multibreed and Crossbred Analyses 14.1 Introduction 14.2 Multibreed Analysis by Splitting the Breeding Values 15 Analysis of Ordered Categorical Traits 15.1 Introduction 15.2 The Threshold Model 15.2.1 Defining some functions of the normal distribution 15.2.2 Data organization and the threshold model 15.2.3 Numerical example 15.3 Joint Analysis of Quantitative and Binary Traits 15.3.1 Data and model definition Calculating m and the residual regression coefficient 15.3.2 Numerical application 16 Survival Analysis 16.1 Introduction 16.2 Functional Survival 16.3 Censoring 16.4 Models for Analysis of Survival 16.4.1 Linear models 16.4.2 Random regression models for survival 16.4.3 Proportional hazard models Defining some distributions Exponential distribution Weibull distribution 16.4.4 Non-parametric estimation of the survival function 16.4.5 Regression survival models Stratified proportional hazard model Accelerated failure time model Time-dependent risk factors 16.4.6 Mixed survival models 16.4.7 Group data survival model 17 Estimation of Genetic Parameters 17.1 Introduction 17.2 Univariate Sire Model 17.3 Numerical Example of Sire Model 17.4 Extended Model 17.5 Numerical Example 17.6 Animal Model 17.7 Numerical Example 18 Use of Gibbs Sampling in Variance Component Estimation and Breeding Value Prediction 18.1 Introduction 18.2 Univariate Animal Model 18.2.1 Prior distributions 18.2.2 Joint and full conditional distributions 18.2.3 Inferences from the Gibbs sampling output 18.2.4 Numerical application 18.3 Multivariate Animal Model 18.3.1 Prior distributions 18.3.2 Conditional probabilities 18.3.3 Numerical illustration 19 Solving Linear Equations 19.1 Introduction 19.2 Absorption 19.3 Direct Inversion 19.4 Iteration on the Mixed-model Equations 19.4.1 Jacobi iteration 19.4.2 Gauss–Seidel iteration 19.5 Iterating on the Data 19.5.1 Animal model without groups Data arrangement Iteration stage 19.5.2 Animal model with groups Data preparation Iterative stage 19.5.3 Reduced animal model with maternal effects Data arrangement Iteration stage 19.6 Preconditioned Conjugate Gradient Algorithm 19.6.1 Computation strategy 19.6.2 Numerical application Computing starting values Iterative stage Appendix A: Introduction to Matrix Algebra A.1 Matrix: A Definition A.2 Special Matrices A.2.1 Square matrix A.2.2 Diagonal matrix A.2.3 Triangular matrix A.2.4 Symmetric matrix A.3 Basic Matrix Operations A.3.1 Transpose of a matrix A.3.2 Matrix addition and subtraction A.3.3 Matrix multiplication A.3.4 Direct product of matrices A.3.5 Matrix inversion A.3.6 Rank of a matrix A.3.7 Generalized inverses A.3.8 Eigenvalues and eigenvectors Appendix B: Fast Algorithms for Calculating Inbreeding Based on the L Matrix B.1 Meuwissen and Luo Algorithm B.1.1 Illustration of the algorithm B.2 Modified Meuwissen and Luo Algorithm B.2.1 Illustration of the algorithm Appendix C Appendix D: Methods for Obtaining Approximate Reliability for Genetic Evaluations D.1 Computing Approximate Reliabilities for an Animal Model D.2 Computing Approximate Reliabilities for Random Regression Models D.2.1 Determining value of observation for an animal D.2.2 Value of records on descendants D.2.3 Value of records on ancestors Appendix F: Procedure for Computing De-regressed Breeding Values Appendix G:  Calculating Φ, a Matrix of Legendre Polynomials Evaluated at Different Ages or Time Periods References Index Back Cover Untitled Untitled