Quantitative Genetics

Preface
Contents
About the Authors
1: Introduction to Quantitative Genetics
1.1 Breeding Value and Its Application
1.1.1 Genetic Gain via Artificial Selection (Example 1)
1.1.2 Predicting Human Height (Example 2)
1.1.3 Breeding Value
1.2 Complicated Behavior Traits
1.3 Qualitative and Quantitative Traits
1.4 The Relationship Between Statistics and Genetics
1.5 Fundamental Statistical Methods in Quantitative Genetics
1.6 Statistical Software Packages
1.7 Classical Quantitative Genetics and Modern Quantitative Genetics
References
2: Review of Mendelian Genetics
2.1 Mendel´s Experiments
2.2 Mendel´s Laws of Inheritance
2.3 Vocabulary
2.4 Departure from Mendelian Ratio
2.5 Two Loci Without Environmental Effects
2.6 Multiple Loci with or Without Environmental Effects
2.7 Environmental Errors Can Lead to a Normal Distribution
References
3: Basic Concept of Population Genetics
3.1 Gene Frequencies and Genotype Frequencies
3.2 Hardy-Weinberg Equilibrium
3.2.1 Proof of the H-W Law
3.2.2 Applications of the H-W Law
3.2.3 Test of Hardy-Weinberg Equilibrium
3.3 Genetic Drift
3.4 Wright´s F Statistics
3.5 Estimation of F Statistics
References
4: Review of Elementary Statistics
4.1 Expectation
4.1.1 Definition
4.1.2 Properties of Expectation
4.1.3 Estimating the Mean
4.2 Variance
4.2.1 Definition
4.2.2 Properties of Variance
4.2.3 Normal Distribution
4.2.4 Estimating Variance from a Sample
4.2.5 An Application of the Variance Property
4.3 Covariance
4.3.1 Definition
4.3.2 Properties of Covariance
4.3.3 Estimating Covariance from a Sample
4.3.4 Conditional Expectation and Conditional Variance
4.4 Sample Estimates of Variance and Covariance
4.5 Linear Model
4.5.1 Regression
4.5.2 Correlation
4.5.3 Estimation of Regression and Correlation Coefficients
4.6 Matrix Algebra
4.6.1 Definitions
4.6.2 Matrix Addition and Subtraction
4.6.3 Matrix Multiplication
4.6.4 Matrix Transpose
4.6.5 Matrix Inverse
4.6.6 Generalized Inverse
4.6.7 Determinant of a Matrix
4.6.8 Trace of a Matrix
4.6.9 Orthogonal Matrices
4.6.10 Eigenvalues and Eigenvectors
4.7 Linear Combination, Quadratic Form, and Covariance Matrix
5: Genetic Effects of Quantitative Traits
5.1 Phenotype, Genotype, and Environmental Error
5.2 The First Genetic Model
5.3 Population Mean
5.4 Average Effect of Gene or Average Effect of Allele
5.5 Average Effect of Gene Substitution
5.6 Alternative Definition of Average Effect of Gene Substitution
5.7 Breeding Value
5.8 Dominance Deviation
5.9 Epistatic Effects Involving Two Loci
5.10 An Example of Epistatic Effects
5.11 Population Mean of Multiple Loci
References
6: Genetic Variances of Quantitative Traits
6.1 Total Genetic Variance
6.2 Additive and Dominance Variances
6.3 Epistatic Variances (General Definition)
6.4 Epistatic Variance Between Two Loci
6.5 Average Effect of Gene Substitution and Regression Coefficient
References
7: Environmental Effects and Environmental Errors
7.1 Environmental Effects
7.2 Environmental Errors
7.3 Repeatability
7.3.1 Estimation of Repeatability
7.3.2 Proof of the Intra-Class Correlation Coefficient
7.3.3 Estimation of Repeatability with Variable Numbers of Repeats
7.3.4 An Example for Estimating Repeatability
7.3.5 Application of Repeatability
7.4 Genotype by Environment (G x E) Interaction
7.4.1 Definition of G x E Interaction
7.4.2 Theoretical Evaluation of G x E Interaction
7.4.3 Significance Test of G x E Interaction
7.4.4 Partitioning of Phenotypic Variance
7.4.5 Tukey´s One Degree of Freedom G x E Interaction Test
References
8: Major Gene Detection and Segregation Analysis
8.1 Two Sample t-Test or F-Test
8.2 F-Test for Multiple Samples (ANOVA)
8.3 Regression Analysis
8.3.1 Two Genotypes
8.3.2 Three Genotypes
8.4 Major Gene Detection Involving Epistatic Effects
8.4.1 Test of Epistasis
8.4.2 Epistatic Variance Components and Significance Test for each Type of Effects
8.5 Segregation Analysis
8.5.1 Qualitative Traits
8.5.2 Quantitative Traits
References
9: Resemblance between Relatives
9.1 Genetic Covariance Between Offspring and One Parent
9.1.1 Short Derivation
9.1.2 Long Derivation
9.2 Genetic Covariance between Offspring and Mid-Parent
9.3 Genetic Covariance between Half-Sibs
9.4 Genetic Covariance between Full-Sibs
9.4.1 Short Way of Derivation
9.4.2 Long Way of Derivation
9.5 Genetic Covariance between Monozygotic Twins (Identical Twins)
9.6 Summary
9.7 Environmental Covariance
9.8 Phenotypic Resemblance
9.9 Derivation of within Family Segregation Variance
References
10: Estimation of Heritability
10.1 F2 Derived from a Cross of Two Inbred Parents
10.2 Multiple Inbred Lines or Multiple Hybrids
10.2.1 With Replications
10.2.2 Without Replications
10.3 Parent-Offspring Regression
10.3.1 Single Parent Vs. Single Offspring
10.3.2 Middle Parent Vs. Single Offspring
10.3.3 Single Parent Vs. Mean Offspring
10.3.4 Middle Parent Vs. Mean Offspring
10.3.5 Estimate Heritability Using Parent-Offspring Correlation
10.4 Sib Analysis
10.4.1 Full-Sib Analysis
10.4.2 Half-Sib Analysis
10.4.3 Nested or Hierarchical Mating Design
10.5 Standard Error of an Estimated Heritability
10.5.1 Regression Method (Parent Vs. Progeny Regression)
10.5.2 Analysis of Variances (Sib Analysis)
10.6 Examples
10.6.1 Regression Analysis
10.6.2 Analysis of Variances
10.6.3 A Nested Mating Design
References
11: Identity-by-Descent and Coancestry Coefficient
11.1 Allelic Variance
11.2 Genetic Covariance Between Relatives
11.3 Genetic Covariance of an Individual with Itself
11.4 Terminology
11.5 Computing Coancestry Coefficients
11.5.1 Path Analysis
11.5.2 Tabular Method
11.5.2.1 Example 1
11.5.2.2 Example 2
11.6 R Package to Calculate a Coancestry Matrix
11.6.1 Path Analysis
11.6.2 Tabular Method
11.7 SAS Program for Calculating a Coancestry Matrix
References
12: Mixed Model Analysis of Genetic Variances
12.1 Mixed Model
12.2 Maximum Likelihood (ML) Estimation of Parameters
12.3 Restricted Maximum Likelihood (REML) Estimation of Parameters
12.4 Likelihood Ratio Test
12.5 Examples
12.5.1 Example 1
12.5.2 Example 2
12.6 Monte Carlo Simulation
References
13: Multiple Traits and Genetic Correlation
13.1 Definition of Genetic Correlation
13.2 Causes of Genetic Correlation
13.3 Cross Covariance between Relatives
13.4 Estimation of Genetic Correlation
13.4.1 Estimate Genetic Correlation Using Parent-Offspring Correlation (Path Analysis)
13.4.2 Estimating Genetic Correlation from Sib Data
13.4.3 Estimating Genetic Correlation Using a Nested Mating Design
References
14: Concept and Theory of Selection
14.1 Evolutionary Forces
14.2 Change in Gene Frequency and Genotype Frequencies
14.3 Artificial Selection
14.3.1 Directional Selection
14.3.2 Stabilizing Selection
14.3.3 Disruptive Selection
14.4 Natural Selection
14.4.1 Directional Selection
14.4.2 Stabilizing Selection
14.4.3 Disruptive Selection
14.5 Change of Genetic Variance After Selection
14.5.1 Single Trait
14.5.2 Multiple Traits
14.5.3 The κ Values in Other Types of Selection
14.5.3.1 Directional Truncated Selection
14.5.3.2 Stabilizing Truncation Selection
14.5.3.3 Disruptive Truncation Selection
14.5.4 Derivation of Change in Covariance
14.5.4.1 Change in Covariance Within-Generation
14.5.4.2 Change in Covariance Between-Generation
References
15: Methods of Artificial Selection
15.1 Objective of Selection
15.2 Criteria of Selection
15.3 Methods of Selection
15.3.1 Individual Selection
15.3.2 Family Selection
15.3.3 Sib Selection
15.3.4 Progeny Testing
15.3.5 Within-Family Selection
15.3.6 Pedigree Selection
15.3.7 Combined Selection
15.4 Evaluation of a Selection Method
15.5 Examples
15.5.1 Family Selection
15.5.2 Within-Family Selection
15.5.3 Sib Selection
15.5.4 Pedigree Selection
15.5.5 Combined Selection
References
16: Selection Index and the Best Linear Unbiased Prediction
16.1 Selection Index
16.1.1 Derivation of the Index Weights
16.1.2 Evaluation of Index Selection
16.1.3 Comparison of Index Selection with a Simple Combined Selection
16.1.4 Index Selection Combining Candidate Phenotype with the Family Mean
16.2 Best Linear Unbiased Prediction (BLUP)
16.2.1 Relationship Between Selection Index and BLUP
16.2.2 Theory of BLUP
16.3 Examples and SAS Programs
16.3.1 Example 1
16.3.2 Example 2
References
17: Methods of Multiple Trait Selection
17.1 Common Methods of Multiple Trait Selection
17.1.1 Tandem Selection
17.1.2 Independent Culling Level Selection
17.1.3 Index Selection
17.1.4 Multistage Index Selection
17.1.5 Other Types of Selection
17.2 Index Selection
17.2.1 Variance and Covariance Matrices
17.2.2 Selection Index (Smith-Hazel Index)
17.2.3 Response to Index Selection
17.2.3.1 Gain in Aggregate Breeding Value
17.2.3.2 Gains of Individual Traits
17.2.4 Derivation of the Smith-Hazel Selection Index
17.2.5 An Example of Index Selection
17.3 Restricted Selection Index
17.4 Desired Gain Selection Index
17.5 Multistage Index Selection
17.5.1 Concept of Multistage Index Selection
17.5.2 Cunningham´s Weights of Multistage Selection Indices
17.5.3 Xu-Muir´s Weights of Multistage Selection Indices
17.5.4 An Example for Multistage Index Selection
17.5.4.1 Cunningham´s Weights of Multistage Selection Indices
17.5.4.2 Xu and Muir´s Weights of Multistage Selection Indices
17.6 Selection Gradients
References
18: Mapping Quantitative Trait Loci
18.1 Linkage Disequilibrium
18.2 Interval Mapping
18.2.1 Least Squares Method (the Haley-Knott Method)
18.2.2 Iteratively Reweighted Least Squares (IRWLS)
18.2.3 Maximum Likelihood Method
18.3 Composite Interval Mapping
18.4 Control of Polygenic Background
18.5 Ridge Regression
18.6 An Example (a Mouse Data)
18.6.1 Technical Detail
18.6.2 Results of Different Methods
18.6.3 Remarks on the Interval Mapping Procedures
18.7 Bonferroni Correction of Threshold for Multiple Tests
18.8 Permutation Test
18.9 Quantification of QTL Size
18.10 The Beavis Effect
References
19: Genome-Wide Association Studies
19.1 Introduction
19.2 Simple Regression Analysis
19.3 Mixed Model Methodology Incorporating Pedigree Information
19.4 Mixed Linear Model (MLM) Using Marker-Inferred Kinship
19.5 Efficient Mixed Model Association (EMMA)
19.6 Decontaminated Efficient Mixed Model Association (DEMMA)
19.7 Manhattan Plot and Q-Q Plot
19.8 Population Structure
19.9 Genome-Wide Association Study in Rice-A Case Study
19.10 Efficient Mixed Model Association Studies Expedited (EMMAX)
References
20: Genomic Selection
20.1 Genomic Best Linear Unbiased Prediction
20.1.1 Ridge Regression
20.1.2 Best Linear Unbiased Prediction of Random Effects
20.1.3 Predicting Genomic Values of Future Individuals
20.1.4 Estimating Variance Parameters
20.1.5 Eigenvalue Decomposition for Fast Computing
20.1.6 Kinship Matrix
20.1.6.1 Kinship Matrices for Dominance and Epistasis
20.1.6.2 Direct Method
20.1.6.3 Fast Algorithm for Epistatic Kinship Matrix Calculation
20.1.6.4 Computational Time Complexity Analysis
20.1.6.5 Prediction of New Plants
20.2 Reproducing Kernel Hilbert Spaces (RKHS) Regression
20.2.1 RKHS Prediction
20.2.2 Estimation of Variance Components
20.2.3 Prediction of Future Individuals
20.2.4 Bayesian RKHS Regression
20.2.5 Kernel Selection
20.3 Predictability
20.3.1 Data Centering
20.3.2 Maximum Likelihood Estimate of Parameters in Random Models
20.3.3 Predicted Residual Error Sum of Squares
20.3.4 Cross Validation
20.3.5 The HAT Method
20.3.6 Generalized Cross Validation (GCV)
20.4 An Example of Hybrid Prediction
20.4.1 The Hybrid Data
20.4.2 Proc Mixed
References
Index