Lecture 1: Basic statistical machinery

• Background Reading: Lynch and Walsh: Chapters 2, 3, A4
• Handouts
  • pdf file of lecture notes (10 pages)
  • R as a basic statistical calculator for obtaining p values and plotting probability distributions (6 page pdf file).
• Topics:
  • Probabilities, Distributions, and Expectations
    • Discrete and Continuous Random Variables
    • Joint and Conditional Probabilities
    • Bayes' Theorem
    • Expectations of Random Variables
    • The Normal Distribution
  • Covariances
    • Useful Properties of Variances and Covariances
  • Regressions
    • Properties of Least-squares Regressions
    • Regressions and Correlations
  • Maximum Likelihood
    • Large-sample Properties of MLEs
    • Likelihood ratio tests
  • Bayesian Statistics
• Advanced notes for various statistics topics
  • Functions of normal distributions -- t distributions, central/noncental chi-square and F distributions. (pdf, 6 pages)
  • Maximum Likelihood estimation, Likelihood ratio tests (pdf, 13 pages)
  • Bayesian methods (pdf, 21 pages)
  • MCMC and Gibbs Sampler (pdf, 24 pages)
  • The Metropolis-Hastings Sampler in R (pdf, 4 pages)

Lecture 2: Linear Algebra and Linear Models

• Background Reading: Lynch and Walsh: Chapter 8, Appendix A3
• Handouts
  • pdf file of lecture notes (25 pages)
  • Matrix Calculations in R (3 page pdf file).
• Topics:
  • Elementary Matrix Algebra
    • Basic Notation
    • Partitioned Matrices
    • Addition And Subtraction
    • Multiplication
    • Dimensional Properties and Matrix Multiplication
    • Transposition
    • Inverses and Solutions to Systems of Equations
  • Expectations Of Random Vectors And Matrices
  • Covariance Matrices Of Transformed Vectors
  • The Multivariate Normal Distribution
    • Properties of the MVN
    • Example: The Regression of Offspring Value on Parental Value
    • Example: Regression of Offspring Breeding Value on Parental Breeding Values
  • Overview Of Linear Models
    • Examples of GLMs
    • Ordinary Least Squares
    • Example: Partial Regression
    • Polynomial Regressions and Interaction Effects
    • Fixed vs. Random Effect
    • Example: Fixed vs. Random Effects in the Sire Model
  • Generalized Least Squares
    • Example: Weighted Least Squares
  • Model Goodness-of-fit and Hypothesis Testing
    • Chi-square and F-distributions
    • Sums of Squares
    • Hypothesis Testing
• Advanced matrix notes
  • Generalized inverses, systems of equations, advanced topics in linear models (pdf, 18 pages)
  • Geometry of matrices, matrix Eigenstructure (pdf, 21 pages)

Lecture 3: Basic Concepts in Mendelian, Population and Quantitative Genetics

• Background Reading: Lynch and Walsh: Chapters 5, 6
• Handouts
  • pdf file of lecture notes (23 pages)
• Topics:
  • Overview
    • A Tale of Two Papers: Darwin vs. Mendel
  • Basic Mendelian Genetics
    • Mendel's View of Inheritance: Single Locus
    • The Genotype to Phenotype Mapping: Dominance and Epistasis
    • Mendel's View of Inheritance: Multiple Loci
    • Mendel was Wrong: Linkage
    • Interlude: Chromosomal Theory of Inheritance
    • Linkage
    • Map Distances are Obtained from Recombination Frequencies via Mapping Functions
    • The Prior Probability of Linkage and Morton's Posterior Error Rate
    • Molecular Markers
  • Basic Population Genetics
    • Allele and Genotype Frequencies
    • Gamete Frequencies, Linkage, and Linkage Disequilibrium
    • The Effects of Population Structure
    • Forces that Change Allele Frequencies: Genetic Drift
    • Coalescence Theory
    • Forces that Change Allele Frequencies: Mutation
    • Forces that Change Allele Frequencies: Selection
    • Interaction of Selection and Drift
  • Basic Quantitative Genetics
    • Dichotomous (Binary) Traits
    • Contribution of a Locus to the Phenotypic Value of a Trait
    • Example: Apolipoprotein E and Alzheimer' s age of onset
    • Example: the Booroola (B) gene
    • Fisher's Decomposition of the Genotypic Value
    • Average Effects and Additive Genetic Values
    • Genetic Variances
    • Epistasis

Lecture 4: Resemblances Between Relatives

• Background Reading: Lynch and Walsh: Chapter 7
• Handouts
  • pdf file of lecture notes (15 pages)
• Topics:
  • Phenotypic Resemblance Between Relatives
    • Parent-offspring regressions
    • Collateral relationships: ANOVA
  • Causes of Phenotypic Covariance Among Relatives
    • Genetic Covariance Between Relatives
    • Offspring and one parent
    • Half-sibs
    • Full-Sibs
    • General degree of relationship
    • Environmental Causes of Relationship Between Relatives

Lecture 5: Basic Designs for Estimation of Genetic Parameters

• Background Reading: Lynch and Walsh: Chapters 17, 18
• Handouts
  • pdf file of lecture notes (18 pages)
• Topics:
  • Heritability
    • Why h² instead of h?
    • Heritabilities are Functions of a Population
    • Increasing the Heritability
    • Heritability and the Prediction of Breeding Values
    • Heritability Values and Population Divergence
  • Estimation: One-way ANOVA and the simple full-sib design
    • Estimating Variances and Variance Components
    • Estimating heritability
    • Worked Example of a Full-sib Design
  • Estimation: The Nested Full-sib, Half-sib Analysis
    • Worked Example of a Nested Design
  • Estimation: Parent-Offspring Regressions
  • Broad-Sense Heritability H² and Plant Breeding
  • Estimation of Var(A) and breeding values in general pedigress
    • The General Mixed Model
    • Estimating Fixed Effects and Predicting Random Effects
    • The animal model
    • ANOVA vs. REML Variance Estimates
    • ML Versus REML Variance estimation

Lecture 6: Inbreeding and Heterosis

• Background Reading:
  • Lynch and Walsh: Chapter 10
  • Walsh and Lynch WL Chapter 12 [ link to pdf file (rough draft in places)]
• Handouts
  • pdf file of lecture notes (17 pages)
• Topics:
  • Changes in the Mean and Variance Under Inbreeding
    • Inbreeding
    • Inbreeding Depression in Fitness Traits
    • Computing the Inbreeding Depression Coefficient, B
    • Why do traits associated with fitness show inbreeding depression?
    • Minimizing the Rate of Inbreeding
    • Change of Variance With Inbreeding
    • Example: Effect of Inbreeding on the Additive Genetic Variance
    • Change of Variance with Inbreeding and Mutation
  • Hybridization and Crossbreeding
    • Types of Crosses
    • Heterosis: Change in the Mean Under Crossbreeding
    • Heterosis in F2
    • Agricultural Importance of Heterosis
    • Hybrid Corn
    • Hybridization in Other Crops
    • Crossbreeding in Animals: General Concepts
    • Crossbreeding in Animals: Heterosis in { Bos indicus x Bos taurus Hybrids
    • Synthetics and Rotational Crossbreeding
    • Estimating the Amount of Heterosis in Maternal Effects

Lecture 7: Genetic Drift

• Handouts
  • pdf file of lecture notes (26 pages)
• Topics:
  • Single Locus Theory Of Random Genetic Drift
    • The Wright-Fisher Model of Genetic Drift
    • Inbreeding and Drift
    • The Coalescent Process for Drift
    • The Coalescent Process for a Sample
    • Drift Generates Between-Population Variance in Replicate Lines
  • Effective Population Size, Ne
    • Ne With Unequal Sex Ratio
    • Ne With Variable Population Size
    • Ne With Unequal Offspring Contribution
  • Mutation-Drift Equilibrium: Single Loci
    • Mutation-Drift Equilibrium: Single Loci
    • Divergence Between Populations
    • Kimura's Neutral Theory of Molecular Evolution
  • Mutation-Drift Equilibrium: Additive Variance
    Equilibrium Additive Variance within a Population
    • Divergence Between Populations
  • Diffusion Theory
    • Infinitesimal Means and Variances, m(x) and v(x)
    • Stationary Distributions
    • Probability of Fixation
  • Diffusion Applications to Quantitative Characters
    • Brownian Motion Model of Evolution
    • Drift and Divergence in the Fossil Record
    • Ornstein-Uhlenbeck Model of Evolution

Lecture 8: Tests for Molecular Signatures of Selection

• Handouts
  • pdf file of lecture notes (15 pages)
• Topics:
  • Basic Logic of Sequence-Based Selection Tests
    • Logic Behind Polymorphism-Based Tests
    • Recombination and Polymorphism
    • Logic Behind Divergence Tests
    • Logic Behind Joint Polymorphism and Divergence Tests
  • Tests Based Strictly on Within-Population Variation
    • The Infinite Alleles Model: Ewen's Sampling Formula
    • The Infinite Sites Model
    • Tajma's D Test
    • Fu and Li's D* and F* tests
    • Depaulis and Veuille's K and H tests
    • Fu's W and Fs Tests
    • Fay and Wu's Test
  • Genome-Wide Polymorphism Tests
    • The Ghost of Lewontin-Krakauer: Genome Wide Fst-based Scans
    • The Linkage Disequilibrium Decay (LLD) Test
  • Joint Polymorphism and Divergence Tests
    • McDonald-Kreitman Test
    • Hudson-Kreitman-Aquade (HKA) Test
  • Tests Based on Between-Population (Species) Divergence
    • Parsimony-Based Ancestral Reconstruction Tests
    • Likelihood-Based Codon Tests
    • Bayesian Estimator of Sites Under Positive Selection

Lecture 9: Short-term Response Selection

• Background Background Reading:
  • Walsh and Lynch : Chapter 4 , The Breeders' Equation [ link to pdf file ]
  • Walsh and Lynch : Chapter 5, Response for Special Traits [ link to pdf file ]
  • Walsh and Lynch : Chapter 6, Changes in the Variance [ link to pdf file ]
• Handouts
  • pdf file of lecture notes (21 pages)
• Topics:
  • Short Terms changes in the mean: the breeder's equation
    • The Selection Differential S and Response R
    • The Selection intensity i
    • Truncation Selection
    • Selection Intensities and Differentials Under Truncation Selection
    • Correcting the Selection Intensity for Finite Samples
    • Selection on Threshold Traits
    • Permanent Versus Transient Response
    • Response with Epistasis
    • Maternal Effects: Response Under Falconer's Model
  • Gene Frequency Changes Under Selection
  • Short-term changes in the variance
    • Bulmer's Equation for the Change in Variance
    • Change in Variance Under Truncation Selection

Lecture 10: Analysis of Short-term Selection Experiments

• Background Reading:
  • Walsh and Lynch : Chapter 6A, Least-square analysis [ link to pdf file ]
  • Walsh and Lynch : Chapter 7A, Mixed-model and Bayesian analysis [ link to pdf file ]
• Handouts
  • pdf file of lecture notes (22 pages)
• Topics:
  • Variance in response
  • Realized heritabilitites
    • Estimators for Several Generations of Selection
    • Weighted Least-Squares Estimates of Realized Heritability
    • Standard Errors for Realized Heritability Estimates
    • Experimental Evaluation of the Breeders' Equation
    • Asymmetric Selection Response
  • Control Populations and Experimental Designs
    • Basic Theory of Control Populations
    • Divergent Selection Designs
    • Variance in Response
    • Control Populations and Variance in Response
  • Optimal Experimental Design
    • Nicholas' Criterion
  • Mixed-model Estimation
    • The Relationship Matrix Accounts for Drift and Disequilibrium
    • Model Validation

Lecture 11: Long-Term Response and Selection Limits

• Background Reading:
  • Walsh. 2003. Population- and Quantitative-Genetic Models of Selection Limits: Plant breeding Reviews [ pdf file of reprint ]
  • Walsh and Lynch : Chapter 16 [ link to pdf file ]
• Handouts
  • pdf file of lecture notes (25 pages)
• Topics
  • Idealized Long-Term Response in a Large Population
    • Deterministic Single-Locus Theory
  • An Overview of Long-term Selection Experiments
    • Estimating Selection Limits and Half-Lives
    • General Features of Long-Term Selection Experiments
    • Increases In Variances And Accelerated Responses
    • Conflicts Between Natural And Artificial Selection
    • Characterizing The Nature Of Selection Limits
  • Long-Term Response in Finite Populations
    • Fixation Probabilities of Favorable QTL Alleles
    • Limits Under Drift and Selection
    • Variance In Response
    • Robertson's Theory of Selection Limits
    • Tests of Robertson's Theory
    • Weber's Selection Experiment on Drosphilia Flight Speed
  • Response from Mutational Input
    • Contributions from New Mutation
    • Mutational Response Under the Infinitesimal Model

Lecture 12: Individual Fitness and Measures of Univariate Selection

• Background Reading:
  • Walsh and Lynch : Chapter 19 [ link to pdf file ]
• Handouts
  • pdf file of lecture notes (19 pages)
• Topics
  • Episodes of selection
    • Fitness Components
    • Assigning Fitness Components
  • Variance in Individual Fitness
    • Some caveats in using opportunity of selection
  • Descriptions of Phenotypic Selection: Introductory Remarks
    • Fitness surfaces
  • Descriptions of Phenotypic Selection: Changes in Phenotypic Moments
    • The Price-Robertson Identity
    • Direction Selection: Differentials (S) and Gradients (beta)
    • Quadratic Selection: Differentials (C) and Gradients (gamma)
    • beta and gamma Measure the Geometery of the Fitness Surface
    • beta and gamma Describe the Selection Dynamics
  • Descriptions of Phenotypic Selection: Individual Fitness Surfaces
    • Linear and quadratic approximations of w(z)
    • Schluter's cublic-spline estimate
    • Complications from Unmeasured Correlated Variables
    • Strenght of Selection in Natural Populations
    • The importance of experimental manipulation

Lecture 13: Genetic Correlations and Multivariate Selection Response

• Background Reading:
  • Walsh and Lynch : Chapter 20 [ link to pdf file ]
  • Blows, M. W. and A. Hoffman. 2005. A reasessment of genetic limits to evolutionary change. Ecology 86: 1371--1384. [ pdf file of reprint ]
  • Pigliucci, M. 2006. Genetic variance-covariance matrices: a critique of the evolutionary quantitative genetics research program. Biology and Philosophy 2: 1-13. [ pdf file of reprint ]
  • Walsh and Lynch : Chapter 23, Theory of Index Selection [ link to pdf file ]
  • Walsh and Lynch : Chapter 24, Applications of Index Selection [ link to pdf file ]
  • Deckers and Hospital, 2002. The use of molecular genetics in the improvement of agricultural populations. Nature Reviews Genetics 3: 22-32 [ pdf file of reprint ]
• Handouts
  • pdf file of lecture notes (26 pages)
• Topics:
  • Phenotypic and Genetic Correlations
  • Estimating the Genetic Correlation
    • Half sibs
    • Parent-Offspring
  • Correlated Response to Selection
    • Correlated Selection Differentials
    • Indirect Selection
  • General Multivariate Selection Response
    • The Directional Selection Gradient
    • Derviation of the Multivariate Breeders' Equation
    • Realized Selection Gradients
    • The Effects of Genetic Correlations: Direct and Correlated Responses
    • Realized Genetic Correlations
    • Evolutionary Constraints Imposed by Genetic Correlations
  • A Short Diversion: The Geometry of a Matrix
    • Comparing Vectors: Lengths and Angles
    • Matrices Describe Vector Transformations
    • Eigenvalues and Eigenvectors
  • Quantifying Multivariate Constraints to Response
    • Is There Genetic Variation in the Multivariate Direction of Selection?
    • Schluter's Genetic Line of Least Resistance
    • Blow's Matrix Subspace Projection
  • General Theory of Selection on a Linear Index
    • MAS -- Marker Assisted Selection
    • Indirect Selection on Marker Score: Applications to Sex-limited Traits

Lecture 14: Measuring Multivariate Selection

• Handouts
  • pdf file of lecture notes (20 pages)
• Topics:
  • Selection On Multivariate Phenotypes: Differentials And Gradients
    • Changes in the Mean Vector: The Directional Selection Differential, S
    • The Directional Selection Gradient, Beta
    • Directional Gradients, Fitness Surface Geometry and Selection Response
    • Changes in the Covariance Matrix: The Quadratic Selection Differential, C
    • The Quadratic Selection Gradient, Gamma
    • Quadratic Gradients, Fitness Surface Geometry and Selection Response
    • Summary
  • Multidimensional Quadratic Fitness Regressions
    • Estimation, Hypothesis Testing and Confidence Intervals
    • Geometric Aspects
    • A Brief Digression: Orthonormal and Diagonalized Matrices
    • Canonical Transformation of Gamma
    • Strength of Selection: gamma_ii versus lambda_i
    • Canonical Subspaces of Gamma
  • Unmeasured Characters and Other Biological Caveats
    • The Bias due to Environmental Correlations Between Fitness and Characters

Lecture 15: Phenotypic Evolution Models

• Handouts
  • pdf file of lecture notes (16 pages)
• Topics:
  • Univariate Guassian Fitness Functions
    • Response Under Normalizing Selection
    • Application: Charlesworth's Model of the Cost of Phenotypic Selection
    • Application: Drift vs. Selection in the Fossil Record
  • Multivariate Gaussian Fitness Functions
    • Phenotypes and Breeding Values Remain MVN After Selection
    • The Selection Differential and Gradient
    • The Within-Generation Change in G
    • Changes in G Under the Infinitesimal Model
    • Long-term Response: Balance Between Directional and Stabilizing Selection
    • Long-term Response: The Infinitesimal Model with Drift and Mutation
  • Derivatives Of Vectors And Vector-Valued Functions
    • The Hessian Matrix, Local Maxima/Minima, and Multidimensional Taylor Series

Lecture 16: Major Genes, Polygenes, and QTLs

• Background Reading: Lynch and Walsh: Chapters 13
• Handouts
  • pdf file of lecture notes (17 pages)
• Topics:
  • Major and Minor Genes
    • Major Genes and Isoalleles
    • Polygenic Mutation and the Mutational Variance
  • Simple Tests for Detecting Major Genes
    • Mixture Models
  • Complex Segregation Analysis
    • Likelihood Functions Assuming a Single Major Gene
    • Common-family Effects
  • Genetic Maps and Candidate Genes
    • Map Distances vs. Recombination Frequencies
    • Linkage Disequilibrium Mapping
    • Fine-mapping Major Genes Using LD
    • The Transmission/Disequilibrium Test
    • Example: Mapping Type 1 Diabetes
    • Linkage vs. Association
  • Dense SNP Association Mapping
    • Genomic control
    • Structured Association Analysis

Lecture 17: QTL Mapping

• BackgroundReading: Lynch and Walsh: Chapters 15, 16
• Handouts
  • pdf file of lecture notes (17 pages)
• Topics:
  • Mapping Using inbred line crosses
    • Experimental designs
    • Conditional Probabilities of QTL Genotypes
    • Example: Conditional Probabilities for an F2
    • Expected Marker Means
    • Linear Models for QTL Detection
    • Maximum Likelihood Methods for QTL Mapping and Detection
    • Likelihood Maps
    • Precision of ML Estimates of QTL Position
    • Interval Mapping with Marker Cofactors
    • Power and Repeatability: The Beavis Effect
  • Mapping in Outbred Populations
    • QTL Mapping Using Sib Families
    • General Pedigree Methods
    • Haseman-Elston regression
    • Affected sib pair methods

Lecture 18: Gene Expression analysis: microarrays and eQTLs

• Background Reading:
  • Walsh and Henderson. 2004. Microarrays and beyond: What potential do current and future genomics tools have for breeders? Anim Sci. 82: E292-299E. [ pdf file of reprint ]
• Handouts
  • pdf file of lecture notes (16 pages)
• Topics:
  • Gene Regulation is a Complex Trait
    • QTLs Involved in Protein Regulation
  • Microarrays
    • A Brief Overview Of The Technology
    • Analysis of Microarray Data
    • Microarray Analysis Is Best Regarded As An EDA Approach
    • Problems (and Pitfalls) of Gene Discovery via Microarray Analysis
  • General Patterns of Transcriptional Variation
    • Gene Expression Levels are Typically Highly Heritable
    • Correlations Between Rates of Regulatory (Transcriptional) and Sequence Divergence
    • Correlations Between Regulatory Divergence and Expression Level/Pattern
    • Does Divergence in Expression Follow a Neutral Model?
  • Analysis of Pathways
    • Two-hybrid screen: Construction Protein-protein interaction maps
    • Kascer-Burns Metabolic Flux Theory
    • Regulatory Neworks and Graph Theory
    • Erdos-Renyi Random Graphs and Random Boolean Networks
    • Graphs: Small Worlds, Scale-Free, and Power Laws
• Advanced notes Sequential Bonferroni corrections and the False Discovery Rate . (pdf, 17 pages)

Lecture 19: Wrap-up: Quantitative genetics and genomics

• Background Reading:
  • Walsh, 2001. Quantitative Genetics in the Age of Genomics, in Theoretical Population Biology 59: 175--184. [ pdf file of reprint ]

• D. Falconer and T. Mackay. 1996. Introduction to Quantitative Genetics, 4th Edition. Longman The classic introduction to the field

• H. N. Turner and S. S. Y Young. 1969. Quantitative Genetics in Sheep Breeding. Cornell University Press. A great classic which is unfortunately out of print.

• G. Simm. 1998. Genetic Improvement of Cattle and Sheep. Farming Press. A nifty compact introduction

• B. Kinghorn, J. van der Werf, and M. Ryan. 2001. Animal Breeding. Use of New Technologies. Twynam Press. A nice review treatment of recent advances

• Wyman Nyquist's Notes on Statistical Genetics . On line at: http://nitro.biosci.arizona.edu/Nyquist/Nyquist.html Very detailed in terms of all the algebra. Lots of plant-specific stuff, but useful to breeders as well.

• J. I. Weller. 2001. Quantitative Trait Loci Analysis in Animals CABI Publishing. A nice compact introduction to QTL mapping

• N. D. Cameron. 1997. Selection Indices and the Prediciton of Genetic Merit in Animal Breeding CABI Publishing. Also a nice compact introduction to BLUP and selection indices

• R. A. Mrode. 1996. Linear Models for the Prediction of Animal Breeding Values CABI Publishing. Another good compact introduction to BLUP

• J. I. Weller. 1994. Economic Aspects of Animal Breeding Chapman and Hall. The best treatment of this usually nelected area

• D. Sorensen and D. Gianola. 2002. Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics Springer. An outstanding, and very detailed, treatment

• Sham, P. 1998. Statistics in Human Genetics Arnold, New York. A nice, compact, and often overlooked text.

• Ewens, W. J. 2004. Mathematical Population Genetics. I. Theoretical Introduction, 2nd Edition. Springer. A very sophisticated mathematical introduction to population and quantitative genetics.