Linear models for the prediction of animal breeding values / R.A. Mrode ; with a chapter contributed by R. Thompson.

Format:

Book

Author/Creator:

Mrode, R. A.

Contributor:

Thompson, R.

Clarence J. Marshall Memorial Library Fund.

Language:

English

Subjects (All):

Livestock--Breeding--Mathematical models.

Livestock.

Livestock--Breeding--Statistical methods.

Livestock--Genetics--Mathematical models.

Livestock--Genetics--Statistical methods.

Livestock--Genetics.

Mathematical models.

Livestock--Breeding.

Physical Description:

xiii, 344 pages ; 25 cm

Edition:

Second edition.

Place of Publication:

Wallingford, UK ; Cambridge, MA : CABI Pub., [2005]

Summary:

Best Linear Unbiased Prediction (BLUP) has become the most widely accepted method for genetic evaluation of domestic livestock. Since its introduction, the method has evolved and, despite this, until the publication of the first edition of this book in 1996 there has been no simple text on the application of linear models to the prediction of breeding values. This second edition has been fully updated and has been written with a good balance of theory and application to fill this gap. Equations for partitioning breeding values into contributions from various sources of information are derived under the various models. Recent developments in the analysis of longitudinal data with random regression models and the inclusion of genetic marker information in the evaluation of animals are also incorporated. Threshold models have been introduced in addition to basic concepts and methods for variance component estimation, including the use of the Gibbs sampler. This book is aimed at advanced students of animal breeding and genetics, as well as practitioners in these subjects.

Contents:

1 Genetic Evaluation with Different Sources of Records 1

1.1 The Basic Model 1

1.2 Breeding Value Prediction from the Animal's Own Performance 3

1.2.1 Single record 3

1.2.2 Repeated records 4

1.3 Breeding Value Prediction from Progeny Records 7

1.4 Breeding Value Prediction from Pedigree 10

1.5 Breeding Value Prediction for One Trait from Another 11

1.6 Selection Index 12

1.6.1 Accuracy of index 14

1.6.2 Examples of selection indices using different sources of information 15

1.6.3 Prediction of aggregate genotype 18

1.6.4 Overall economic indices using predicted genetic merit 20

1.6.5 Restricted selection index 21

1.6.6 Index combining breeding values from phenotype and genetic marker information 23

2 Genetic Covariance Between Relatives 25

2.1 The Numerator Relationship Matrix 25

2.2 Decomposing the Relationship Matrix 27

2.3 Computing the Inverse of the Relationship Matrix 28

2.3.1 Inverse of the numerator relationship matrix, ignoring inbreeding 29

2.3.2 Inverse of the numerator relationship matrix accounting for inbreeding 32

2.4 Inverse of the Relationship Matrix for Sires and Maternal Grandsires 34

2.4.1 An example of the inverse of the relationship matrix for sires and maternal grandsires 36

3 Best Linear Unbiased Prediction of Breeding Value: Univariate Models with One Random Effect 39

3.1 Brief Theoretical Background 40

3.2 A Model for an Animal Evaluation (Animal Model) 42

3.2.1 Constructing the mixed model equations 43

3.2.2 Progeny (daughter) yield deviation 48

3.2.3 Accuracy of evaluations 50

3.3 A Sire Model 52

3.3.1 An illustration 53

3.4 Reduced Animal Model 55

3.4.1 Defining the model 56

3.4.2 An illustration 58

3.4.3 An alternative approach 61

3.5 Animal Model with Groups 62

3.5.1 An illustration 64

4 Best Linear Unbiased Prediction of Breeding Value: Models with Random Environmental Effects 71

4.1 Repeatability Model 71

4.1.1 Defining the model 72

4.1.2 An illustration 73

4.1.3 Calculating daughter yield deviations 77

4.2 Model with Common Environmental Effects 77

4.2.1 Defining the model 78

4.2.2 An illustration 79

5 Best Linear Unbiased Prediction of Breeding Value: Multivariate Models 83

5.1 Equal Design Matrices and No Missing Records 84

5.1.1 Defining the model 84

5.1.2 An illustration 85

5.1.3 Partitioning animal evaluations from multivariate analysis 88

5.1.4 Accuracy of multivariate evaluations 90

5.1.5 Calculating daughter yield deviations in multivariate models 91

5.2 Canonical Transformation 92

5.2.1 The model 93

5.2.2 An illustration 93

5.3 Equal Design Matrices with Missing Records 95

5.3.1 An illustration 96

5.4 Cholesky Transformation 98

5.4.1 Calculating the transformation matrix and defining the model 98

5.4.2 An illustration 99

5.5 Unequal Design Matrices 101

5.5.1 Numerical example 102

5.5.2 Illustrating the computation of DYD from a multivariate model 104

5.6 Multivariate Models with No Environmental Covariance 105

5.6.1 Different traits recorded on relatives 106

5.6.2 The multi-trait across-country evaluations (MACE) 109

6 Maternal Trait Models: Animal and Reduced Animal Models 121

6.1 Animal Model for a Maternal Trait 122

6.1.1 An illustration 123

6.2 Reduced Animal Model with Maternal Effects 127

6.2.1 An illustration 129

6.3 Multivariate Maternal Animal Model 133

7 Analysis of Longitudinal Data 135

7.1 Fixed Regression Model 136

7.2 Random Regression Model 143

7.2.1 Numerical application 144

7.2.2 Partitioning animal solutions from the random regression model 148

7.2.3 Calculating daughter yield deviations 152

7.2.4 Reliability of breeding values 152

7.2.5 Random regression model for maternal traits 154

7.3 Covariance Functions 154

7.3.1 Fitting a reduced-order covariance function 157

7.4 Equivalence of the Random Regression Model to the Covariance Function 161

8 Use of Genetic Markers in Prediction of Breeding Values 163

8.1 Defining a Model with Marker Information 163

8.2 Calculating the Covariance Matrix (G[subscript v]) for MQTL Effects 164

8.2.1 Numerical application 166

8.3 An Alternative Approach for Calculating G[subscript v] 167

8.4 Calculating the Inverse of G[subscript v] 169

8.5 Prediction of Breeding Values with Marker Information 173

8.5.1 An illustration 173

8.6 Reduced Animal Model with Marker Information 174

8.6.1 Numerical example 176

8.6.2 Back-solving for solutions of non-parents 178

8.7 Directly Predicting the Additive Genetic Merit at the MQTL 179

8.7.1 An illustration 181

8.8 Predicting Total Additive Genetic Merit 182

8.8.1 Numerical application 183

8.9 Analysis of Data with QTL Bracketed by Two Markers 184

8.9.1 Basic model 184

8.9.2 Calculating the covariance matrix, G 185

8.9.3 An illustration 188

8.10 Reduced Animal Model 191

9 Non-additive Animal Models 193

9.1 Dominance Relationship Matrix 193

9.2 Animal Model with Dominance Effect 194

9.2.1 Solving for animal and dominance genetic effects separately 195

9.2.2 Solving for total genetic merit directly 198

9.3 Method for Rapid Inversion of the Dominance Matrix 198

9.3.1 Inverse of the relationship matrix of subclass effects 200

9.3.2 Prediction of dominance effects 202

9.3.3 Calculating the inverse of the relationship matrix among dominance and subclass effects for example data 202

9.4 Epistasis 206

9.4.1 Rules for the inverse of the relationship matrix for epistatic and subclass effects 206

9.4.2 Calculating the inverse relationship matrix for epistasis and the subclass matrix for an example pedigree 207

10 Analysis of Ordered Categorical Traits 212

10.1 The Threshold Model 212

10.1.1 Defining some functions of the normal distribution 212

10.1.2 Data organization and the threshold model 213

10.1.3 Numerical example 215

10.2 Joint Analysis of Quantitative and Binary Traits 224

10.2.1 Data and model definition 224

10.2.2 Numerical application 228

11 Estimation of Genetic Parameters / Robin Thompson 235

11.1 Univariate Sire Model 235

11.2 Numerical Example of Sire Model 236

11.3 Extended Model 237

11.4 Numerical Example 239

11.5 Animal Model 240

11.6 Numerical Example 242

12 Application of Gibbs Sampling in Variance Component Estimation and Prediction of Breeding Value 247

12.2 Univariate Animal Model 248

12.2.1 Prior distributions 248

12.2.2 Joint and full conditional distributions 249

12.2.3 Inferences from the Gibbs sampling output 251

12.2.4 Numerical application 253

12.3 Multivariate Animal Model 254

12.3.1 Prior distributions 254

12.3.2 Conditional distributions 255

12.3.3 Numerical illustration 257

13 Solving Linear Equations 259

13.1 Direct Inversion 259

13.2 Iteration on the Mixed Model Equations 260

13.2.1 Jacobi iteration 260

13.2.2 Gauss-Seidel iteration 263

13.3 Iterating on the Data 264

13.3.1 Animal model without groups 267

13.3.2 Animal model with groups 271

13.3.3 Reduced animal model with maternal effects 274

13.4 Preconditioned Conjugate Gradient Algorithm 283

13.4.1 Computation strategy 284

13.4.2 Numerical application 285

Appendix A Introductory Matrix Algebra 289

A.1 Matrix: a Definition 289

A.2 Special Matrices 290

A.2.1 Square matrix 290

A.2.2 Diagonal matrix 290

A.2.3 Triangular matrix 290

A.2.4 Symmetric matrix 291

A.3 Basic Matrix Operations 291

A.3.1 Transpose of a matrix 291

A.3.2 Matrix addition and subtraction 292

A.3.3 Matrix multiplication 292

A.3.4 Direct product of matrices 293

A.3.5 Matrix inversion 293

A.3.6 Rank of a matrix 294

A.3.7 Generalized inverses 295

A.3.8 Eigenvalues and eigenvectors 295

Appendix B Fast Algorithms for Calculating Inbreeding Based on the L Matrix 297

B.1 Meuwissen and Luo Algorithm 297

B.1.1 Illustration of the algorithm 298

B.2 Modified Meuwissen and Luo Algorithm 300

B.2.1 Illustration of the algorithm 301

C.1 Outline of the Derivation of the Best Linear Unbiased Prediction (BLUP) 303

C.2 Proof that b and a from the Mixed Model Equations are the Generalized Least-square Solution of b and the Best Linear Unbiased Prediction of a, Respectively 304

C.3 Deriving the Equation for Progeny Contribution (PC) 305

Appendix D Methods for Obtaining Approximate Reliability for Genetic Evaluations 307

D.1 Computing Approximate Reliabilities for an Animal Model 307

D.2 Computing Approximate Reliabilities for Random Regression Models 309

E.1 Canonical Transformation: Procedure to Calculate the Transformation Matrix and its Inverse 311

E.2 Canonical Transformation with Missing Records and Same Incidence Matrices 312

E.3 Cholesky Decomposition 316

Appendix F Procedure for Computing De-regressed Breeding Values 317

Appendix G Calculting [Phi], a Matrix of Legendre Polynomials Evaluated at Different Ages or Time Periods 321

Appendix H Computing the Covariance Matrix of Additive Genetic Effect of Marked QTL (MQTL) when Paternal or Maternal Origin of Marker Alleles cannot be Determined and Marker Information is Incomplete 325

H.1 Covariance Between Individuals 325

H.1.1 Computing PDMs 327

H.2 Covariance Within Individuals 327

H.3 Constructing the Matrix G[subscript v] 328

H.3.1 An illustration 328

H.4 Computing Inverse of G[subscript v] 330

H.5 Incomplete Marker Data 331.

Notes:

Includes bibliographical references (pages 333-340) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Clarence J. Marshall Memorial Library Fund.

ISBN:

0851990002

OCLC:

56509471