Advances in Statistical Methods for Genetic Improvement of Livestock

Developments in statistics and computing as well as their application to genetic improvement of livestock gained momentum over the last 20 years. This text reviews and consolidates the statistical foundations of animal breeding. This text will prove useful as a reference source to animal breeders, quantitative geneticists and statisticians working in these areas. It will also serve as a text in graduate courses in animal breeding methodology with prerequisite courses in linear models, statistical inference and quantitative genetics.

I: General.- 1 Statistical Methods in Animal Improvement: Historical Overview.- 1.1 Introduction.- 1.2 Pearson’s Pioneering Work.- 1.3 Fisher’s Work of the Late Teens and the Twenties.- 1.4 Wright’s Work of the Teens and Twenties.- 1.5 Lush and Wright ― Early Prediction Methods.- 1.6 Selection Index.- 1.7 Early Development of Linear Model Methods for Unbalanced Data.- 1.8 Derivation of Best Linear Unbiased Prediction.- 1.9 The Development of Methods for Estimation of Variances and Covariances.- 1.10 Some Recent Developments in Computing Strategies.- 1.11 Recent Work in Optimum Selection Criteria.- 2 Mixed Model Methodology and the Box-Cox Theory of Transformations: A Bayesian Approach.- 2.1 Introduction.- 2.2 Motivation: A Simple Sire Evaluation Model.- 2.3 Family of Transformations.- 2.3.1 Prior Distributions.- 2.4 Some Posterior Distributions.- 2.4.1 Joint Posterior Distribution of all Parameters.- 2.4.2 Posterior Distribution of the Variance Components and of ?.- 2.4.3 Posterior Distribution of Functions of the Variance Ratio and of ?.- 2.4.4 Posterior Distribution of ?.- 2.5 Estimation of the Transformation.- 2.5.1 From the Marginal Distribution of ?.- 2.5.2 From the Joint Distribution of ? and ?.- 2.5.3 From the Joint Distribution of ?e2, ?u2 and ?.- 2.6 Analysis of the Effects After Transformation.- 2.6.1 Analysis Conditional on ? and ?.- 2.6.2 Analysis Conditional on ?.- 2.7 Extensions and Conclusions.- 3 Models for Discrimination Between Alternative Modes of Inheritance.- 3.1 Introduction.- 3.2 Data on Inbred Lines, Their F1 and Backcrosses.- 3.2.1 One Locus.- 3.2.2 Polygenic Inheritance.- 3.2.3 Mixed Major Locus and Polygenic Inheritance.- 3.2.4 Two Loci.- 3.3 Pedigree Data from a Random Mating Population.- 3.3.1 One Locus.- 3.3.2 Polygenic Inheritance.- 3.3.3 Mixed Major Gene and Polygenic Inheritance.- 3.3.4 Regressive Models.- 3.4 Choice of Genetic Hypothesis.- Discussion Summary.- II: Design of Experiments and Breeding Programs.- 4 Considerations in the Design of Animal Breeding Experiments.- 4.1 Introduction.- 4.2 Formal Designs.- 4.2.1 Intra-Class Correlation of Sibs.- 4.2.2 Offspring-Parent Regression.- 4.2.3 Joint Sib and Offspring-Parent Analyses.- 4.2.4 Genetic Correlations.- 4.3 Selection Experiments.- 4.3.1 Single Generation Experiments.- 4.3.2 Multiple Generation Experiments.- 4.4 Field Experiments.- 4.5 Concluding Remarks.- 5 Use of Mixed Model Methodology in Analysis of Designed Experiments.- 5.1 Introduction.- 5.2 Mixed Model Methods.- 5.3 Selection of Breeding Animals.- 5.4 Estimation of Genetic Variances.- 5.5 Estimation of Selection Response.- 5.6 Design.- 5.7 Conclusions.- 6 Statistical Aspects of Design of Animal Breeding Programs: A Comparison Among Various Selection Strategies.- 6.1 Introduction.- 6.2 Full-Sib Structures.- 6.2.1 First Generation.- 6.2.2 Short-to Medium-Term Results.- 6.2.3 Long-Term Results.- 6.3 Discussion.- 7 Optimum Designs for Sire Evaluation Schemes.- 7.1 Introduction.- 7.2 Theory.- 7.3 Numerical Examples.- 7.3.1 Allocation of Progeny Testing Resources.- 7.3.2 Sampling New Candidates.- 7.3.3 Two-Stage Selection.- 7.4 Discussion.- Discussion Summary.- III: Estimation of Genetic Parameters.- 8 Computational Aspects of Likelihood-Based Inference for Variance Components.- 8.1 Introduction.- 8.2 Model.- 8.3 Analysis of Variance (ANOVA) and ANOVA-Related Notation.- 8.4 Likelihood Function.- 8.5 Extended Parameter Space.- 8.6 REML Estimation.- 8.7 Newton-Raphson Algorithms.- 8.8 Concentrated Log Likelihood Function.- 8.9 Linearization.- 8.10 Computation of Iterates.- 8.11 An Alternative Approach to the Computation of Iterates.- 8.12 Method of Scoring.- 8.13 EM Algorithm and the Method of Successive Approximations.- 8.14 Linearized Method of Successive Approximations.- 8.15 Confidence Intervals and Hypothesis Tests.- 8.16 Example.- 8.17 Extensions.- 8.17.1 More than One Set of Random Effects.- 8.17.2 Correlated or Heteroscedastic Random Effects.- 9 Parameter Estimation in Variance Component Models for Binary Response Data.- 9.1 Introduction.- 9.2 Review of the Linear Case.- 9.3 Mixed Model Analysis with Binary Response.- 9.3.1 Bayes Approach.- 9.3.2 Likelihood Approaches.- 10 Estimation of Genetic Parameters in Non-Linear Models.- 10.1 Introduction.- 10.2 Models.- 10.3 Linearization.- 10.3.1 Maximum Likelihood.- 10.3.2 Maximum a Posteriori.- 10.3.3 Foulley’s Method.- 10.3.4 The Method of Harville and Mee.- 10.3.5 Gilmour’s Method.- 10.3.6 Remarks.- 10.4 Numerical Methods.- 10.4.1 Preliminaiy Absorption.- 10.4.2 Accommodating Relationships.- 10.4.3 Tridiagonalization and the EM Algorithm.- 10.4.4 Remarks.- 10.5 A Preliminary Investigation.- 10.6 Conclusion.- Discussion Summary.- IV: Prediction and Estimation of Genetic Merit.- 11 A Framework for Prediction of Breeding Value.- 11.1 Introduction.- 11.2 The Mixed Linear Model.- 11.3 Joint Posterior Distribution.- 11.4 Known Variance Components.- 11.4.1 Posterior Distribution of ß with Known u.- 11.4.2 Posterior Distribution of u when ? is Known.- 11.5 Unknown Variance Components.- 11.5.1 Joint Inferences About Location Parameters and Variance Components.- 11.5.2 Marginal Inferences About Variance Components and Functions Thereof.- 11.5.3 Marginal Inferences About Location Parameters.- 11.6 Choosing a Predictor.- 11.7 Choosing a Model.- 11.8 Prediction of Future Records.- 12 BLUP (Best Linear Unbiased Prediction) and Beyond.- 12.1 Introduction.- 12.2 Formulation of the Prediction Problem.- 12.2.1 Mixed Model.- 12.2.2 Example.- 12.2.3 General Prediction Problem.- 12.3 State 1: Joint Distribution Known.- 12.3.1 Point Prediction.- 12.3.2 Interval Prediction.- 12.3.3 Special Case: Mixed Linear Model.- 12.4 State 2: Only First and Second Moments Known.- 12.4.1 Best Linear (Point) Prediction.- 12.4.2 Interval Prediction (Frequentist Approach).- 12.4.3 Bayesian Prediction.- 12.5 State 3: Only Variances and Covariances Known.- 12.5.1 Best Linear Unbiased (or Location-Equivariant) Prediction.- 12.5.2 Interval Prediction (Frequentist Approach).- 12.5.3 Special Case: Mixed Linear Model.- 12.5.4 Linear-Bayes Prediction.- 12.5.5 Bayesian Prediction.- 12.6 State 4: No Information.- 12.6.1 Estimation of ?.- 12.6.2 Point Prediction.- 12.6.3 MSE of Prediction.- 12.6.4 Approximating the MSE.- 12.6.5 Estimating the MSE.- 12.6.6 Interval Prediction (Frequentist Approach).- 12.6.7 Bayesian Prediction.- 13 Connectedness in Genetic Evaluation.- 13.1 Introduction.- 13.2 The Models.- 13.2.1 Classical Model.- 13.2.2 Certain Characteristics of the Males Known.- 13.3 The Unbiasedness Constraint.- 13.3.1 Models without Group Effects.- 13.3.2 Models with Group Effects.- 13.4 Minimum Mean Square Error.- 13.4.1 Models without Group Effects.- 13.4.2 Models with Group Effects.- 13.5 Other Objectives and Constraints.- 13.5.1 Relaxing the Unbiasedness Requirement for Group Effects.- 13.5.2 Maximum Genetic Progress.- 13.6 Discussion and Conclusions.- Discussion Summary.- V: Prediction and Estimation in Non-Linear Models.- 14 Generalized Linear Models and Applications to Animal Breeding.- 14.1 Introduction.- 14.2 Estimation of Heritability of Binary Traits by Offspring-Parent Regression.- 14.3 Estimation of Gene Frequencies.- 14.4 Variance Components for Normal Data.- 14.5 Variance Components with Generalized Linear Models.- 14.6 Discussion.- 15 Analysis of Linear and Non-Linear Growth Models with Random Parameters.- 15.1 Introduction.- 15.2 A Two-Stage Model for Linear Growth.- 15.3 Two-Step Methods for Linear Models.- 15.4 Methods for Non-Linear Growth Curves.- 16 Survival, Endurance and Censored Observations in Animal Breeding.- 16.1 Introduction.- 16.2 Characterization of Survival Times and Endurance Measures.- 16.2.1 Properties.- 16.2.2 Censoring.- 16.3 Models.- 16.3.1 Parametric Models.- 16.3.2 Semi-Parametric Models.- 16.4 Maximum a Posteriori.- 16.5 Numerical Methods.- 16.6 A Preliminary Investigation.- 16.7 Conclusion.- 17 Genetic Evaluation for Discrete Polygenic Traits in Animal Breeding.- 17.1 Introduction.- 17.2 Analysis of the Discontinuous Scale with Linear Models.- 17.2.1 Single Population Analysis.- 17.2.2 Multipopulation Analysis.- 17.3 Models Postulating an Underlying Scale.- 17.3.1 Binary Responses.- 17.3.2 Extension to Other Situations.- 17.4 Discussion and Conclusion.- Discussion Summary.- VI: Selection and Non-Random Mating.- 18 Accounting for Selection and Mating Biases in Genetic Evaluation.- 18.1 Introduction.- 18.2 Effect of Selection on u, e, G and R.- 18.3 Means and Covariances Conditional on Selection Functions.- 18.4 BLUE and BLUP in a Selection Model.- 18.5 Estimability and Predictability.- 18.6 Cow Culling.- 18.7 Translation Invariant Functions of Records Used in Selection Plus Other Unknown Selection Functions.- 18.8 Selection on Previous Records Not Available for Analysis.- 18.9 Mixed Model Equations to Estimate Genetic and Environmental Trends.- 18.10 The Problem of Association Between Sire Values and Herd Merits in Sire Evaluations.- 18.11 The Problem of Grouping in Sire Evaluations.- 18.12 The Problem of Differential Treatments.- 18.13 The Problem of Assortative Mating.- 18.14 Discussion.- 19 Statistical Inferences in Populations Undergoing Selection or Non-Random Mating.- 19.1 Introduction.- 19.2 Dynamics of a Breeding Population.- 19.2.1 Mathematical Representation of a Breeding Population.- 19.3 Making Inferences in a Population Undergoing Non-Random Mating and Selection.- 19.4 Making Inferences with Incomplete Information.- 19.5 Multivariate Normality.- 19.5.1 Maximum Likelihood Estimation.- 19.5.2 Best Linear Prediction.- 19.5.3 Best Linear Unbiased Prediction.- 20 Problems in the Use of the Relationship Matrix in Animal Breeding.- 20.1 Introduction.- 20.2 The Numerator Relationship Matrix.- 20.3 Additive Genetic Variance.- 20.4 Examples and Applications.- 20.4.1 Use of the NRM in a Simple Sire Evaluation.- 20.4.2 Use of the NRM when Sires of the Test Bulls are a Selected Group.- 20.5 The NRM and Unknown Parentage.- 20.5.1 Modification of the NRM to Handle Certain Kinds of Unknown Parentage.- 20.5.2 Example.- 20.5.3 Application.- 20.6 Shortcoming of the NRM.- 20.7 Conclusion.- Discussion Summary.- VII: Statistics and New Genetic Technology.- 21 Identification of Genes with Large Effects.- 21.1 Introduction and Motivation.- 21.1.1 Motivation.- 21.1.2 Prior Information - Number of Genes.- 21.2 Methods Using Population Differences.- 21.2.1 Segregation in Crosses and Backcrosses.- 21.2.2 Segregation Analysis.- 21.2.3 Repeated Backcrossing and Selection.- 21.2.4 Use of Linked Markers.- 21.2.5 Use of Physiological Markers.- 21.3 Within Population Analysis.- 21.3.1 Departures from Normality.- 21.3.2 Structured Exploratory Data Analysis.- 21.3.3 Complex Segregation Analysis.- 21.3.4 Miscellanea.- 21.4 Use of Selected Populations.- 21.5 Molecular Manipulation.- 21.5.1 Transposon Tagging.- 21.5.2 Transgenics.- 21.6 Discussion.- 22 A General Linkage Method for the Detection of Major Genes.- 22.1 Introduction.- 22.2 A Generalization of Haseman and Elston’s (1972) Method.- 22.3 Transformations to Approximate Normality.- 22.4 Dichotomous Traits and Disease Traits with Variable Age of Onset.- 22.5 Discussion.- 23 Reproductive Technology and Genetic Evaluation.- 23.1 Introduction.- 23.2 Reproductive Technology and Evaluation for Additive Genetic Merit.- 23.2.1 Embryo Transfer.- 23.2.2 Embryo Splitting (Cloning).- 23.2.3 Embryo and Semen Sexing.- 23.2.4 Androgenous Matings and Self-Fertilization.- 23.2.5 Chimeras.- 23.2.6 Polyploidy.- 23.2.7 Gene Transfer.- 23.3 Evaluation for Non-Additive Genetic Merit.- 23.3.1 Cytoplasmic Inheritance.- 23.3.2 Dominance Effects.- 23.3.3 Preferential Treatment.- 23.4 Conclusions.- Discussion Summary.