The prediction of producing desirable traits in offspring--such as increased growth rate or superior meat, milk and wool production--is a vital economic tool to the animal scientist. Summarizing the latest developments in genomics relating to animal breeding values and design of breeding programs, this new edition includes models of survival analysis, social interaction and sire and dam models, as well as advancements in the use of SNPs in the computation of genomic breeding values.

Preface XIII

Abbreviations xv

1. Genetic Evaluation with Different Sources of Records 1

1.1. Introduction 1

1.2. The Basic Model 1

1.3. Breeding Val ue Prediction from the Animal's Own Performance 2

      1.3.1. Single record 2

      1.3.2. Repeated records 3

1.4. Breeding Value Prediction from Progeny Records 6

1.5. Breeding Value Prediction from Pedigree 9

1.6. Breeding Value Prediction for One Trait from Another 10

1.7. Selection lndex 11

      1.7.1. Accuracy of index 12

      1.7.2. Examples of selection indicesusing different sources of in formation 14

      1.7.3. Prediction of aggregate genotype 16

      1.7.4. Overa ll economic indices using predicted genetic merit 18

      1.7.5. Restricted selection ind ex 19

      1.7.6. Index combining breeding values from phenotype and genetic marker information 21

2. Genetic Covariance Between Relatives 22

2.1. Introduction 22

2.2. The Numerator Relationship Matrix 22

2.3. Decomposing the Relationship Matrix 23

2.4. Computing the Inverse of the Relationship Matrix 25

      2.4.1. Jnverse of the numerator relationship matrixignoring inbreeding 26

      2.4.2. Inverse of the numerator relationship matrix accou n ti ng for i n breeding 28

2.5. Inverse of the Relationship Matrix for Sires and Maternal Grandsires 30

2.6. An Example of the Inverse of a Sire and Maternal Gra ndsire Relationship Matrix 32

3. Best Linear Unbiased Prediction of Breeding Value: Univariate Models with One Random Effect 34

3.1. Introduction 34

3.2. Brief Theoretical Background 35

3.3. A Model for an Animal Evaluation ( Animallvlodel ) 37

      3.3.1. Constructing the mixed model equations 38

      3.3.2. Progeny (daughter) yield deviation 42

      3.3.3. Accuracy of evaluations 44

3.4. A Sire Model 46

      3.4.1. An illustration 46

3.5. Reduced Animal Model 49

      3.5.1. Defining the model 49

      3.5.2. An illustration 51

      3.5.3. An alternative approach 54

3.6. Animal Model with Groups 54

      3.6.1. An illustration 56

4. Best Linear Unbiased Prediction of Breeding Value: Models with Random Environmental Effects 61

4.1. Introduction 61

4.2. Repeata bility Model 61

      4.2.1. Defining the model 62

      4.2.2. An illustration 62

      4.2.3. Calculating daughter yield deviations 66

4.3. Model with Common Environmental Effects 66

      4.3.1. Defining the model 67

      4.3.2. An illustration 67

5. Best Linear Unbiased Prediction of Breeding Value: Multivariate Animal Models 70

5.1. Introduction 70

5.2. Equal Design Matrices and No Missing Records 71

      5.2.1. Defining the model 71

      5.2.2. An illustration 72

      5.2.3. Partitioning animal evaluations from multivariate analysis 74

      5.2.4. Accuracy of multivariate evaluations 76

      5.2.5. Calculating daughter yield deviations in multivariate models 77

5.3. Equal Design Matrices with Missing Records 78

      5.3.1. An illustration 78

5.4. Unequal Design Matrices 80

      5.4.1. Numerical example 80

      5.4.2. Illustrating the computation of DYD from a m ultivariate model 82

5.5. Multivariate Models with No Environmental Covariance 84

      5.5.1. Different traits recorded on relatives 84

      5.5.2. The multi-trait across-country evaluations (MACE) 86

6. Methods to Reduce the Dimension of Multivariate Models 95

6.1. Introduction 95

6.2. Canonical Transformation 95

      6.2.1. The model 96

      6.2.2. An illustration 97

6.3. Cholesk y Transformation 98

      6.3.1. Calculati ng the transformation matrix and defining the model 98

      6.3.2. An illustration 99

6.4. Factor and Pri ncipal Com ponent Analysis 101

      6.4.1. Factor analysis 102

      6.4.2. Pri ncipal com ponent analysis 105

      6.4.3. Analysis with reduced rank PC model 106

7. Maternal Trait Models: Animal and Reduced Animal Models 109

7.1. Introduction 109

7.2. Animal Model for a Maternal Trait 110

      7.2.1. Anillustration 110

7.3. Reduced Animal Model with Maternal Effects 115

      7.3.1. Anillustration 116

      7.4. Sire and Maternal Grandsire Model 119

8. Social [nteraction Models 121

8.1. Introduction 121

8.2. A nimal Model with Social Interaction Effects 123

      8.2.1. Illustration of a model with social interaction 125

8.3. Partitioning Evaluations from Associative Models 127

8.4. Analysis Using Correlated Error Structure 128

9. Analysis of Longitudinal Data 130

9.1. lntroduction 130

9.2. Fixed Regression Model 131

      9.2.1. Anillustration 132

9.3. Random Regression Model 136

      9.3.1. Nu merical application 138

      9.3.2. Partitioning animal solutions from random regression model 142

      9.3.3. Calculating daughter y ield deviations 145

      9.3.4. Reliabil ity of breeding va l ues 146

      9.3.5. Rand om regression models with spl ine function 147

      9.3.6. Random regression model for maternal tra its 148

9.4. Covaria nce Fu nctions 149

      9.4.1. Fitting a reduced order cova riance function 151

9.5. Equiva lence of the Random Regression Model to the Cova riance Function 155

10. Use of Genetic Markers in Breeding Value Prediction 156

10.1. Introduction 156

10.2. Defining a Model with Marker Information 156

10.3. Calculating the Covariance Matrix (G,,) for MQTL Effects 157

      10.3.1. Numerical application 158

10.4. An Alternative A pproach for Ca lculating G., 160

10.5. Calcu la ti ng rhe Inverse of Gv 161

10.6. Pred iction of Breeding Values with Marker Information 160

      10.6.1. Anillustration 165

10.7. Directly Predicting the Additive Genetic Merit at the MQTL 167

      10.7.1. An illustration 168

10.8. Predicting Total Additive Genetic Merit 169

      10.8.1. Numerical application 169

10.9. Analysis of Data with QTL Bracketed by Two Markers 171

      10.9.1. Basic model 171

      10.9.2. Calculating the covariance matrix, G 172

      10.9.3. An illustration 174

11. Computation of Genomic Breeding Values and Genomic Selection 177

11.1. Introduction 177

11.2. General Linea r Model 178

11.3. Coding and Scaling Genotypes 178

11.4. Fixed Effect Model for SNP Effects 179

11.5. Mixed Linear Model for Computing SNP Effects 182

      11.5.1. SNP-BLUP model 183

      11.5.2. Equivalent models: GBLUP 184

      11.5.3. Equivalent models: selection index approach 187

11.6. Mixed Linear Models with Polygenic Effects 188

11.7. Single-step Approach 190

11.8. Bayesian Methods for Computing SNP Effects 193

      11.8.1. Ba yesA 194

      11.8.2. BayesB 197

      11.8.3. BayesC 199

      11.8.4. BayesC1t 201

11.9. Cross-validation and Genomic Relia bilities 202

11.10. Understanding SNP Solutions from the Various Models 202

12. Non-additive Animal Models 204

12.1. Introd uction 204

12.2. Dominance Relationship Matrix 204

12.3. Ani mal Model with Dominance Effect 205

      12.3.1. Solving for animal and dominance genetic effects separately 206

      12.3.2. Solving for total genetic merit directly 208

12.4. Method for Rapid Inversion of the Dominance Matrix 209

      12.4.1. Inverse of the relationship matrix of subclass effects 210

      12.4.2. Prediction of domi nance effects 211

      12.4.3. Calculati ng the inverse of the relationship matrix among dominance and subclass effects for exam ple data 212

12.5. Epistasis 215

      12.5.1. Rules for the inverse of the relationship matrix for epistatic and subclass effects 216

      12.5.2. Calculating the inverse relationship matrix for epistasis and the subclass matrix for an example pedigree 217

13. Analysis of Ordered Categorical Traits 219

13.1. Introduction 219

13.2. The Threshold Model 220

      13.2.1. Defi ning some functions of the normal distribution 220

      13.2.2. Dara organization and the threshold model 221

      13.2.3. Numerical example 223

13.3. Joint Analysis of Quantitative and Binary Traits 230

      13.3.1 Dara and model definition 230

      13.3.2. Numerical application 234

14. Survival Analysis 240

14.1. Introduction 240

14.2. Functional Survival 240

14.3. Censoring 240

14.4. Model for Analysis of Survival 241

      14.4.1. Linear models 241

      14.4.2. Random regression models for survival 241

      14.4.3. Proportional hazard mod els 243

      14.4.4. on-para metric estimation of the survival function 245

      14.4.5. Regression u rv i va l models 246

      14.4.6. Mi xed su rv iva l models 247

      14.4.7. Group data su rvi va l model 250

15. Estimation of Genetic Parameters 251

Robin Thompson

15.1. Introduction 251

15.2. Univari ate Sire Model 251

15.3. Numerical Example of Sire Model 252

15.4. Extended lode! 253

15.5. umerical Exa m ple 2 -4

15.6. An imal Model 255

15.7. Nu merical Exa m ple 257

16. Use of Gibbs Sampling in Variance Component Esti mation and Breeding Value Prediction 260

16.1. Introduction 260

16.2. Uni va riate Animal Model 261

      16.2.l. Prior distri butions 261

      16.2.2. Joint and full conditional distribution 262

      16.2.3. In ferences from the Gibb sampling output 264

      16.2.4. Numerical application 265

16.3. Multivariate Animal Model 266

      16.3.1. Prior distribu tions 267

      16.3.2. Conditiona l probabilities 267

      16.3.3. Numerical illustration 269

17. Solving Linear Equations 271

17.1. Introduction 271

17.2. Direct Inversion 271

17.3. Iteration on the Mixed Model Equations 271

      17.3.1. Jacobi iteration 272

      17.3.2. Gauss-Seidel iteration 275

17.4. Iterating on the Data 276

      17.4.1. Animal model without groups 278

      17.4.2. Animal model with groups 282

      17.4.3. Reduced animal model with maternal effects 284

17.5. Precondi tioned Conjugate Gradient Algorithm 292

      17.5.1. Computation strategy 293

      17.5.2. Numerical applica tion 294

Appendix A: Introduction to Matrix Algebra 299

A.1 Matrix: A Definition 299

A.2 Special Matrices 300

Appendix B: Fast Algorithms for Calculating Inbreeding Based on the L Matrix 306

B.1 Meuwissen and Luo Algorithm 306

      B.1.1 Illustration of the algorithm 307

B.2 Modified Meuwissen and Luo Algorithm 308

      B.2.1 Illustration of the algorithm 309

Appendix C 311

C.1 Outline of the Derivation of the Best Linear Un biased Prediction (BLUP)

311

C.2 Proof that b and a from MME are the GLS of band BLUP of a, Respectively 312

C.3 Deriving the Equation for Progeny Contribution (PC) 313

Appendix D: Methods for Obtaining Approximate Reliability for Genetic Evaluations 314

D.1 Computing Approximate Reliabilities for an Ani mal Model D.2 Computing Approximate Reliabilities for Random Regression Models 314

      D.2.1 Determine value of observation for an animal 316

      D.2.2 Val ue of record on descendants 316

      D.2.3 Val ue of records on ancestors 317

Appendix E 318

E.1 Canonical Transformation: Procedure to Calculate the Transformation Matrix and its Inverse 318

E.2 Canonical Transformation wi th Missing Records and Same Incidence Matrices 319

      E.2.1 Illustration 320

E.3 Cholesky Decom position 322

Appendix F: Procedure for Computing Deregressed Breeding Values 323

Appendix G: Calculating, a Matrix of Legendre Polynomials Evaluated at Different Ages or Time Periods 325

References 327

Index 337

Raphael Mrode is Professor of Quantitative Genetics and Genomics at Scotland's Rural College. He has been lecturing on Edinburgh University's Masters course on quantitative genetics and genome analysis since 2005, and has given lectures on mixed linear models and the use of various BLUP models for genetic prediction. His research interests include data modelling and analysis, the incorporation of molecular information in genetic evaluation procedures, the application of innovative approaches for data capture, analysis and feedback and investigating methods for generating alternative and novel phenotypes in small dairy systems in developing countries.

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