Topics in Quantitative Genetics

Components of Variance

Components of Variance
Objectives
Phenotypic Components of Variance
Algebraic Description
Genetic Components of Variance
Explanation of Formula
Heritability in the Broad Sense
Broad-Sense Components
Heritability in the Narrow Sense
Deriving Variance Components
Covariance
Component Relationships
Principles to Remember
Influence of Epistasis
Epistatic Model
Example 1
Example 2
Genetic Variance Components

Gene Frequencies

Gene Frequencies
Objectives
Introduction
Allelic and Genotypic Variation
Ideal Population
Assumptions
Allelic and Genotypic Frequencies
Variance of Allele Frequency
Frequency Estimators
Expected Number of Alleles
Hardy-Weinberg Equilibrium
Assumptions
Disequilibrium
Variance
Goodness of Fit
Non-Random Mating
Factors Affecting Allele Frequency
Migration
Mutations
Fate of a Single Mutation
Probability of Loss
Recurrent Mutations
Selection
Effects of Selection
Frequency Equations
Small Population Size
Inbreeding and Small Populations
Identical by Descent and Identical by State
Coefficient of Inbreeding
Equation of Coefficient of Inbreeding
Inbreeding Coefficient
Dispersion
Changes in Frequencies
References
Population Genetics Foundations

Estimates of Variance

Estimates of Variance
Introduction
Covariance of Relatives
Linear Models for Phenotypic Values
Linear Models for Phenotypic Values
Common Types of Relatives
F2 and F3 Variances
F3 Variances
ANOVA for F3 Progenies
Estimate of Heritability
Bi-Parental Progenies
Summary
Expected Mean Squares
Using the Algorithm
Step 1
Step 2
Step 3
Step 4
Step 5
References
Genetic Covariance and Heritability

Mating Designs

Mating Designs
Introduction
Introduction
Objectives
Design Setup
Diallel Crosses
Diallel Crosses - General Mating Scheme
Number of Diallel Crosses and Entries
Common Diallel Experiment
ANOVA Table
F-Tests
Gardner and Eberhart Diallel Analysis II
Gardner and Eberhart Diallel Analysis II
Equations
Models
ANOVA Table
Equivalent Analysis
Analysis III of Gardner and Eberhart
North Carolina Design I
North Carolina Design I
ANOVA Table
Variance Estimates
North Carolina Design II
North Carolina Design II
Model
Covariance of Relatives
Variance Estimates
North Carolina Design III
North Carolina Design III
ANOVA Table
F-Tests
Mating Designs

Multi Environment Trials: Linear Mixed Models

Multi Environment Trials: Linear Mixed Models
Objectives
Henderson's Concept
Henderson's Concept - Introduction
Example Prediction 1
Summary Data
Best Linear Unbiased Prediction
Best Linear Unbiased Prediction - Introduction
BLUEs and BLUPs
Effects of BLUPs
Example
Linear Mixed Model Solution
Estimated Residual Variance
Multi Environment Trials: Linear Mixed Models
Scalar Notation
Matrix Notation
Definitions and Notation
Special Kinds of Matrices
Operations
Multiplying Vectors
Multiplying Vectors In Reverse
Matrix Multiplication
Additional Important Operations

G x E

G X E
Objectives
Introduction
Environmental Components of Variance
Many Meanings of Environment
Environmental Sources of Variation
Simple Types of GxE Interactions
Type 3 GxE Interaction
Type 4 GxE Interaction
Type 5 GxE Interaction
Type 6 GxE Interaction
Complex Types of GxE Interactions
Pattern Analysis Methods
Cluster Analyses
Euclidean Distance
Variation Flux
Partition of GxE Variances
Muir's Partition Method
GE Interaction Equation
Interaction Components
Example Data Sets 1 and 2
Example Data Set 3
Alternative Analyses
Grouping Similar Responses
Grouping Lines and Environments
Impact of Multiple Environments
Variance Component Estimation Example
Estimators
Application Notes
Multi Environment Trials: Types of GxE

Multiple Trait Selection

Multiple Trait Selection
Objectives
Multiple Trait Selection Strategies
Index Selection
Index Selection - Definition and Objective
Index Selection Theory
Assumptions
Mean and Covariance of H and I
Derivation of the Optimum Index
Derivation of the Optimum Index
Rewriting the Normal Equations
Further Calculations
Minimizing E(I-H)^2
Expected Genetic Gains
Expected Genetic Gains
Truncation Selection
Selection Relationships
Standardized Regression Coefficient
Expected Gain
Predicted Gain
Expected Genetic Gains for Each Trait
Matrix Representation
Matrix Representation of Selection Indices
Some Results
Construction of a Selection index
Optimum Index
Smith-Hazel Index
Base Index
Multiplicative Index
Desired Gain Index
Restricted Selection Index
Rank Summation Index
Selection Index Efficiency
Methods to Compare Selection Index Efficiency
Effect of Correlations on Index Weights
Derivation
Multiple Trait Selection, Heritability and Genetic Gain

Resemblance Between Relatives

Resemblance Between Relatives
Objectives
Introduction
Explanation
Background
Coefficient of Inbreeding
Coefficient of Inbreeding - Definition
General Principle
Impact on Disequilibrium
Coefficient of Parentage
Coefficient of Parentage - Definition
Calculation of Effects
Example Calculations
Example Calculations - Inbreeding Coefficient
Selfing
Panmictic Index
Full-Sib Mating (1)
Full-Sib Mating (2)
Full-Sib Mating (3)
Full-Sib Mating (4)
Selfing
Full-Sibing
Population Genetics Relationship Coefficients

Linkage

Linkage
Objectives
Introduction
Disequilibrium Definitions
Disequilibrium Example
Gametic Disequilibrium
Disequilibrium Table
Dissipation of Disequilibrium
Estimation and Testing
Chi-Square Statistic
Population Genetics Gamete and Linkage Disequilibrium

Measures of Similarity

Measures of Similarity
Objectives
Introduction
Population Structure Based on Pedigree Information
Recursive Tabular Method
Population Structure Based on Markers
Quantitative Measure for Similarity
Measures of Distance
Application of Distance and Similarity Measures
Principal Component Analysis (1)
Principal Component Analysis
Principal Component Analysis - Interpretation
PCA Example
Cluster Analysis
Cluster Analysis Example
Hierarchical Clustering
Population Genetics Population Structure

Gene Effects

Gene Effects
Introduction
Linear Models for Phenotypic Values
Single Locus
Population Mean
Degree of Dominance
Allele Frequencies and Population Mean
Additive Gene Action
Two Loci
Population Mean
Extension to More Than 2 Loci
Average Genetic Effects
Average Genetic (Allelic) Effects
Formula of Average Effect of an Allele
Allele Substitution Effect
Breeding Value
Mean Breeding Value in Random Population
Deviations for Average Genetic Effects
Dominance Deviation
Observations About Dominance
Breeding Values and Dominance Deviations
Dominance Deviations
Epistasis
Graphical View of Epistasis
Physiological and Statistical Dominance
Physiological Epistasis
Model for Physiological Epistasis
Single Locus Genotype
Non-Epistatic Genotypic Value
Epistatic Genotypic Value
Statistical Epistasis
Epistasis Effects
Quantitative Genetic Models: Theory

Supplemental Resources

Supplemental Resources
Plant Breeding Basics
Objectives
Definition
Other Definitions
Organization of Plant Breeding Activities
Cultivar Development
Decision-Making Process
A Brief History of Quantitative Genetics
Modern Synthesis Theory
Marker Technologies
Trait Measures
Trait Measures - Objectives
Categorical Scales
Quantitative Scales
Types of Models
Types of Models - Objectives
Definition And Purpose Of Models
Data Modeling
Example Of Steps For Modeling Data In A Plant Breeding Project
Organizing Data
Preserving Data
A Relational Database
Phenotypic Models
Two Linear Models
Exploratory Data Analysis (EDA)
Exploratory Data Analysis - Objectives
Statistical Inference
Exploratory Data Analyses
Estimation: Sample Average
Estimation of Means
Estimation of Means: Example
Estimation of Variances
Estimation of Covariance
Estimation of Variance Components
Questions to Consider
Mixed Models
Installation of R
Introduction and Objectives - Installation of R
Background
Exercise
Install R
RStudio Layout
Working Directory
Libraries
Getting Help in R
Reading the CSV File
Examining the Data
Viewing and Removing Datasets
EDA with R
EDA with R - Objectives
Set Working Directory
Reading the CSV File
Exploring the Data
Histogram
Boxplots
Mean and Coefficient of Variance
One-Factor ANOVA of a CRD
Interpret the Results
Hypothesis Tests
Hypothesis Tests - Objectives
Statistical Inference
Null and Alternative Hypotheses
Inferential Errors from Hypothesis Testing
Error Types
Significance Levels
Power of the Test
Analysis of Variance
Analysis of Variance - Objectives
Statistical Inference
Background
Design Principles
Completely Random Design
Fixed and Random Effects
AOV Based on Yield
Blocking Ranges
Mixed Linear Model
Regression and Prediction
Regression and Prediction - Objectives
Statistical Inference
Linear Regression
Basic Assumptions
Simple Linear Regression
Parameter Estimates
Prediction
Analysis of Covariance
Analysis of Covariance - Objective
Statistical Inference
Example
Model Equation
Computational Considerations
Computational Considerations - Key Concepts
Statistical Inference
Computational Methods
Regression, Anova, and AOC
Further Considerations
Matrix Algebra
Review Material

Selection Response

Selection Response
Objectives
Introduction
Selection Response
Selection Response - Definition
Genetic Gain
Heritability on an Entry-Mean Basis
Context of Heritability
Holland's Synthesis
Family Structure
Narrow-Sense Heritability of Half-Sibs
Covariance Estimation
Example A
Example B
Computational Considerations
Example C
Method of Moments
When to Use Method of Moments
REML
Selection, Heritability and Genetic Gain

Simulation Modeling

Simulation Modeling
Objectives
Introduction
History of Simulations
Core Elements
Excel-Based Simulation
Step 1
Step 2
Step 3
Step 4
Other Population Structures
Genetic Architecture of the Trait
Parameter Assignment
Genotypic Values
Calculations
Excel Application
Normal Distribution Interval
Simulated Phenotypes
Example Calculations
Polygenic Trait Simulation
QTL Simulations
Simulation Modeling