2006 Postgraduate course in Evolutionary Quantitative Genetics, Roenbjerg field station, University of Aarhus
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 since 17 April 2006

Instructor: Bruce Walsh, University of Arizona
We will be using the R statistical software
R is free. You can download it (for almost all computers) at
Course Schedule, Readings and Lecture Notes
Walsh and Lynch refers to online draft chapters from our (yet to be finished) volume, Evolution and Selection on Quantitative Traits.
Day 1: (Tuesday, 6 June)
As the first day of class, we will start a little later (10am) to allow students a chance to arrive and settle in at the Field Station
Lecture 1: Basic statistical machinery
 Background Reading: Lynch and Walsh: Chapters 2, 3, A4
 Handouts
 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 Leastsquares Regressions
 Regressions and Correlations
 Maximum Likelihood
 Largesample Properties of MLEs
 Likelihood ratio tests
 Bayesian Statistics

Advanced notes for various statistics topics
Lecture 2: Linear Algebra and Linear Models
 Background Reading: Lynch and Walsh: Chapter 8, Appendix A3
 Handouts
 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 Goodnessoffit and Hypothesis Testing
 Chisquare and Fdistributions
 Sums of Squares
 Hypothesis Testing

Advanced matrix notes
Day 2: (Wednesday 7 June)
Lecture 3: Basic Concepts in Mendelian, Population and Quantitative Genetics
 Background Reading: Lynch and Walsh: Chapters 5, 6
 Handouts
 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
 Topics:
 Phenotypic Resemblance Between Relatives
 Parentoffspring regressions
 Collateral relationships: ANOVA
 Causes of Phenotypic Covariance Among Relatives
 Genetic Covariance Between Relatives
 Offspring and one parent
 Halfsibs
 FullSibs
 General degree of relationship
 Environmental Causes of Relationship Between Relatives
Day 3: (Thursday, 8 June)
Lecture 5: Basic Designs for Estimation of Genetic Parameters
 Background Reading: Lynch and Walsh: Chapters 17, 18
 Handouts
 Topics:
 Heritability
 Why h^{2} 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: Oneway ANOVA and the simple fullsib design
 Estimating Variances and Variance Components
 Estimating heritability
 Worked Example of a Fullsib Design
 Estimation: The Nested Fullsib, Halfsib Analysis
 Worked Example of a Nested Design
 Estimation: ParentOffspring Regressions
 BroadSense Heritability H^{2} 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
 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
Day 4: (Friday, 9 June)
Lecture 7: Genetic Drift
 Handouts
 Topics:
 Single Locus Theory Of Random Genetic Drift
 The WrightFisher Model of Genetic Drift
 Inbreeding and Drift
 The Coalescent Process for Drift
 The Coalescent Process for a Sample
 Drift Generates BetweenPopulation Variance in Replicate Lines
 Effective Population Size, Ne
 Ne With Unequal Sex Ratio
 Ne With Variable Population Size
 Ne With Unequal Offspring Contribution
 MutationDrift Equilibrium: Single Loci
 MutationDrift Equilibrium: Single Loci
 Divergence Between Populations
 Kimura's Neutral Theory of Molecular Evolution
 MutationDrift 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
 OrnsteinUhlenbeck Model of Evolution
Lecture 8: Tests for Molecular Signatures of Selection
 Handouts
 Topics:
 Basic Logic of SequenceBased Selection Tests
 Logic Behind PolymorphismBased Tests
 Recombination and Polymorphism
 Logic Behind Divergence Tests
 Logic Behind Joint Polymorphism and Divergence
Tests
 Tests Based Strictly on WithinPopulation 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
 GenomeWide Polymorphism Tests
 The Ghost of LewontinKrakauer: Genome Wide Fstbased Scans
 The Linkage Disequilibrium Decay (LLD) Test
 Joint Polymorphism and Divergence Tests
 McDonaldKreitman Test
 HudsonKreitmanAquade (HKA) Test
 Tests Based on BetweenPopulation (Species) Divergence
 ParsimonyBased Ancestral Reconstruction Tests
 LikelihoodBased Codon Tests
 Bayesian Estimator of Sites Under Positive Selection
Day 5: (Saturday, 10 June)
Lecture 9: Shortterm Response Selection
 Background Background Reading:
 Handouts
 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
 Shortterm changes in the variance
 Bulmer's Equation for the Change in Variance
 Change in Variance Under Truncation Selection
Lecture 10: Analysis of Shortterm Selection Experiments
 Background Reading:
 Handouts
 Topics:
 Variance in response
 Realized heritabilitites
 Estimators for Several Generations of Selection
 Weighted LeastSquares 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
 Mixedmodel Estimation
 The Relationship Matrix Accounts for Drift and Disequilibrium
 Model Validation
Day 6: (Monday, 12 June)
Lecture 11: LongTerm Response and Selection Limits
 Background Reading:

Walsh. 2003. Population and QuantitativeGenetic Models of Selection Limits: Plant breeding Reviews [ pdf file of reprint ]
 Walsh and Lynch : Chapter 16 [ link to pdf file ]
 Handouts
 Topics
 Idealized LongTerm Response in a Large Population
 Deterministic SingleLocus Theory
 An Overview of Longterm Selection Experiments
 Estimating Selection Limits and HalfLives
 General Features of LongTerm Selection Experiments
 Increases In Variances And Accelerated Responses
 Conflicts Between Natural And Artificial Selection
 Characterizing The Nature Of Selection Limits
 LongTerm 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:
 Handouts
 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
 Descriptions of Phenotypic Selection: Changes in
Phenotypic Moments
 The PriceRobertson 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 cublicspline estimate
 Complications from Unmeasured Correlated Variables
 Strenght of Selection in Natural Populations
 The importance of experimental manipulation
Day 7: (Tuesday, 13 June)
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: 13711384.
[ pdf file of reprint ]

Pigliucci, M. 2006. Genetic variancecovariance matrices: a critique of the evolutionary quantitative genetics research program.
Biology and Philosophy 2: 113. [ 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: 2232 [ pdf file of reprint ]
 Handouts
 Topics:
 Phenotypic and Genetic Correlations
 Estimating the Genetic Correlation
 Half sibs
 ParentOffspring
 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 Sexlimited Traits
Lecture 14: Measuring Multivariate Selection
 Handouts
 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
Day 8: (Wednesday, 14 June)
Lecture 15: Phenotypic Evolution Models
 Handouts
 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 WithinGeneration Change in G
 Changes in G Under the Infinitesimal Model
 Longterm Response: Balance Between Directional and Stabilizing
Selection
 Longterm Response: The Infinitesimal Model with Drift and Mutation
 Derivatives Of Vectors And VectorValued 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
 Topics:
 Major and Minor Genes
 Major Genes and Isoalleles
 Polygenic Mutation and the Mutational Variance
 Simple Tests for Detecting Major Genes
 Complex Segregation Analysis
 Likelihood Functions Assuming a Single Major Gene
 Commonfamily Effects
 Genetic Maps and Candidate Genes
 Map Distances vs. Recombination Frequencies
 Linkage Disequilibrium Mapping
 Finemapping 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
Day 9: (Thursday, 15 June)
Lecture 17: QTL Mapping
 BackgroundReading: Lynch and Walsh: Chapters 15, 16
 Handouts
 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
 HasemanElston 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: E292299E. [ pdf file of reprint ]
 Handouts
 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
 Twohybrid screen: Construction Proteinprotein interaction maps
 KascerBurns Metabolic Flux Theory
 Regulatory Neworks and Graph Theory
 ErdosRenyi Random Graphs and Random Boolean Networks
 Graphs: Small Worlds, ScaleFree, and Power Laws
 Advanced notes
Sequential Bonferroni corrections and the False Discovery Rate . (pdf, 17 pages)
Day 10 (Friday, 16 June)
Lecture 19: Wrapup: Quantitative genetics and genomics
 Background Reading:

Walsh, 2001. Quantitative Genetics in the Age of Genomics, in Theoretical Population Biology 59: 175184. [ 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 plantspecific 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.