Likelihood, Bayesian and MCMC Methods in Quantitative Genetics

By: Daniel Sorensen, Daniel Gianola

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Hardcover | 1 January 2002

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764 Pages

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Over the last ten years the introduction of computer intensive statistical methods has opened new horizons concerning the probability models that can be fitted to genetic data, the scale of the problems that can be tackled and the nature of the questions that can be posed. In particular, the application of Bayesian and likelihood methods to statistical genetics has been facilitated enormously by these methods. Techniques generally referred to as Markov chain Monte Carlo (MCMC) have played a major role in this process, stimulating synergies among scientists in different fields, such as mathematicians, probabilists, statisticians, computer scientists and statistical geneticists. Specifically, the MCMC "revolution" has made a deep impact in quantitative genetics. This can be seen, for example, in the vast number of papers dealing with complex hierarchical models and models for detection of genes affecting quantitative or meristic traits in plants, animals and humans that have been published recently. This book, suitable for numerate biologists and for applied statisticians, provides the foundations of likelihood, Bayesian and MCMC methods in the context of genetic analysis of quantitative traits. Most students in biology and agriculture lack the formal background needed to learn these modern biometrical techniques. Although a number of excellent texts in these areas have become available in recent years, the basic ideas and tools are typically described in a technically demanding style, and have been written by and addressed to professional statisticians. For this reason, considerable more detail is offered than what may be warranted for a more mathematically apt audience. The book is divided into four parts. Part I gives a review of probability and distribution theory. Parts II and III present methods of inference and MCMC methods. Part IV discusses several models that can be applied in quantitative genetics, primarily from a Bayesian perspective. An effort has been made to relate biological to statistical parameters throughout, and examples are used profusely to motivate the developments. Daniel Sorensen is a Research Professor in Statistical Genetics, at the Department of Animal Breeding and Genetics in the Danish Institute of Agricultural Sciences. Daniel Gianola is Professor in the Animal Sciences, Biostatistics and Medical Informatics, and Dairy Science Departments of the University of Wisconsin-Madison. Gianola and Sorensen pioneered the introduction of Bayesian and MCMC methods in animal breeding. The authors have published and lectured extensively in applications of statistics to quantitative genetics.

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From the reviews:

BIOINFORMATICS

"I found the coverage of material to be excellent: well chosen and well written, and I didn't spot a single typographical error...It can serve as a resource book for masters-level taught courses, but will be most useful for PhD students and other researchers who need to fill in the gaps in their knowledge, grasp the intuition behind statistical techniques, models, and algorithms, and find pointers to more extensive treatments. Overall, I find that the authors have succeeded admirably in their goals. I highly recommend this excellent book to any researcher seeking a graduate-level introduction to the modern statistical methods applied in quantitative genetics."

"Just one personal sentence as an Introduction: I like the book so much that I have decided to include several parts of it in my own lectures. ... it may be understood more easily by students and researchers that lack a strong background in statistics and mathematics. ... most examples are nicely explained. ... Summing up, I am convinced that this excellent book should be a standard book for researchers and students with a background in genetics who are interested in Bayesian and MCMC methods." (Andreas Ziegler, Metrika, February, 2004)

"Both authors ... have made significant contributions to development of statistical methods in quantitative genetics and in particular have been at the forefront of the adoption of MCMC methods for Bayesian analysis, which can be applied to an enormous range of problems ... . their coverage of likelihood methods is both extensive and fair. ... this is a valuable book, in that it presents so much background essential for the subsequent application and merits a much broader market that it is likely to get." (William G. Hill, Genetical Research, Vol. 81, 2003)

"The coverage of Bayesian theory is extensive, and includes a discussion of information and entropy, and of the notion 'uninformative' priors,as well as model assessment and model averaging. ... I found the coverage of material to be excellent: well chosen and well written, and I didn't spot a single typographical error. ... the authors have succeeded admirably in their goals. I highly recommend this excellent book to any researcher seeking a graduate-level introduction to the modern statistical methods applied in quantitative genetics." (David Balding, Bioinformatics, July, 2003)

"The book is aimed at students and researchers in agriculture, biology and medicine. ... Statisticians will appreciate the attempt to relate biological to statistical parameters. In conclusion the book shows that the authors have a lot of experience with applications of statistics to quantitative genetics. Much more details are given in this book than usual, so it can be considered and recommended for classroom use." (Prof. Dr. W. Urfer, Statistical Papers, Vol. 46 (4), 2005)

" [T]he book is worth owning for anyone interested in applying likelihood or Bayesian models, especially realistic models that may require MCMC for implementation." (Journal of the American Statistical Associaton)

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Non-Fiction Science Biology, Life Sciences Life Sciences in General Genetics excluding Medical Mathematics Applied Mathematics Stochastics Probability & Statistics Evolution Education Science in General

Maths for Scientists Medicine Medicine in General Public Health & Preventive Medicine Epidemiology & Medical Statistics Biochemistry

Preface	p. v
Review of Probability and Distribution Theory	p. 1
Probability and Random Variables	p. 3
Introduction	p. 3
Univariate Discrete Distributions	p. 4
The Bernoulli and Binomial Distributions	p. 7
The Poisson Distribution	p. 10
Binomial Distribution: Normal Approximation	p. 12
Univariate Continuous Distributions	p. 13
The Uniform, Beta, Gamma, Normal, and Student-t Distributions	p. 18
Multivariate Probability Distributions	p. 29
The Multinomial Distribution	p. 37
The Dirichlet Distribution	p. 40
The d-Dimensional Uniform Distribution	p. 40
The Multivariate Normal Distribution	p. 41
The Chi-square Distribution	p. 53
The Wishart and Inverse Wishart Distributions	p. 55
The Multivariate-t Distribution	p. 60
Distributions with Constrained Sample Space	p. 62
Iterated Expectations	p. 67
Functions of Random Variables	p. 77
Introduction	p. 77
Functions of a Single Random Variable	p. 78
Discrete Random Variables	p. 78
Continuous Random Variables	p. 79
Approximating the Mean and Variance	p. 89
Delta Method	p. 93
Functions of Several Random Variables	p. 95
Linear Transformations	p. 111
Approximating the Mean and Covariance Matrix	p. 114
Methods of Inference	p. 117
An Introduction to Likelihood Inference	p. 119
Introduction	p. 119
The Likelihood Function	p. 120
The Maximum Likelihood Estimator	p. 122
Likelihood Inference in a Gaussian Model	p. 125
Fisher's Information Measure	p. 128
Single Parameter Case	p. 128
Alternative Representation of Information	p. 131
Mean and Variance of the Score Function	p. 134
Multiparameter Case	p. 135
Cramér-Rao Lower Bound	p. 138
Sufficiency	p. 142
Asymptotic Properties: Single Parameter Models	p. 143
Probability of the Data Given the Parameter	p. 144
Consistency	p. 146
Asymptotic Normality and Efficiency	p. 147
Asymptotic Properties: Multiparameter Models	p. 152
Functional Invariance	p. 153
Illustration of Functional Invariance	p. 153
Invariance in a Single Parameter Model	p. 157
Invariance in a Multiparameter Model	p. 159
Further Topics in Likelihood Inference	p. 161
Introduction	p. 161
Computation of Maximum Likelihood Estimates	p. 162
Evaluation of Hypotheses	p. 166
Likelihood Ratio Tests	p. 166
Confidence Regions	p. 177
Wald's Test	p. 179
Score Test	p. 179
Nuisance Parameters	p. 181
Loss of Efficiency Due to Nuisance Parameters	p. 182
Marginal Likelihoods	p. 182
Profile Likelihoods	p. 186
Analysis of a Multinomial Distribution	p. 190
Amount of Information per Observation	p. 199
Analysis of Linear Logistic Models	p. 202
The Logistic Distribution	p. 204
Likelihood Function under Bernoulli Sampling	p. 205
Mixed Effects Linear Logistic Model	p. 208
An Introduction to Bayesian Inference	p. 211
Introduction	p. 211
Bayes Theorem: Discrete Case	p. 214
Bayes Theorem: Continuous Case	p. 223
Posterior Distributions	p. 235
Bayesian Updating	p. 249
Features of Posterior Distributions	p. 257
Posterior Probabilities	p. 258
Posterior Quantiles	p. 262
Posterior Modes	p. 264
Posterior Mean Vector and Covariance Matrix	p. 280
Bayesian Analysis of Linear Models	p. 287
Introduction	p. 287
The Linear Regression Model	p. 287
Inference under Uniform Improper Priors	p. 288
Inference under Conjugate Priors	p. 297
Orthogonal Parameterization of the Model	p. 307
The Mixed Linear Model	p. 313
Bayesian View of the Mixed Effects Model	p. 313
Joint and Conditional Posterior Distributions	p. 317
Marginal Distribution of Variance Components	p. 322
Marginal Distribution of Location Parameters	p. 323
The Prior Distribution and Bayesian Analysis	p. 327
Introduction	p. 327
An Illustration of the Effect of Priors on Inferences	p. 328
A Rapid Tour of Bayesian Asymptotics	p. 330
Discrete Parameter	p. 330
Continuous Parameter	p. 331
Statistical Information and Entropy	p. 334
Information	p. 334
Entropy of a Discrete Distribution	p. 337
Entropy of a Joint and Conditional Distribution	p. 340
Entropy of a Continuous Distribution	p. 341
Information about a Parameter	p. 346
Fisher's Information Revisited	p. 351
Prior and Posterior Discrepancy	p. 353
Priors Conveying Little Information	p. 356
The Uniform Prior	p. 356
Other Vague Priors	p. 358
Maximum Entropy Prior Distributions	p. 367
Reference Prior Distributions	p. 379
Bayesian Assessment of Hypotheses and Models	p. 399
Introduction	p. 399
Bayes Factors	p. 400
Definition	p. 400
Interpretation	p. 402
The Bayes Factor and Hypothesis Testing	p. 403
Influence of the Prior Distribution	p. 412
Nested Models	p. 414
Approximations to the Bayes Factor	p. 418
Partial and Intrinsic Bayes Factors	p. 422
Estimating the Marginal Likelihood	p. 424
Goodness of Fit and Model Complexity	p. 429
Goodness of Fit and Predictive Ability of a Model	p. 433
Analysis of Residuals	p. 434
Predictive Ability and Predictive Cross-Validation	p. 436
Bayesian Model Averaging	p. 439
General	p. 439
Definitions	p. 440
Predictive Ability of BMA	p. 441
Approximate Inference Via the EM Algorithm	p. 443
Introduction	p. 443
Complete and Incomplete Data	p. 444
The EM Algorithm	p. 445
Form of the Algorithm	p. 445
Derivation	p. 445
Monotonic Increase of ln p (¿ y)	p. 447
The Missing Information Principle	p. 448
Complete, Observed and Missing Information	p. 448
Rate of Convergence of the EM Algorithm	p. 449
EM Theory for Exponential Families	p. 451
Standard Errors and Posterior Standard Deviations	p. 452
The Method of Louis	p. 453
Supplemented EM Algorithm (SEM)	p. 454
The Method of Oakes	p. 457
Examples	p. 458
Markov Chain Monte Carlo Methods	p. 475
An Overview of Discrete Markov Chains	p. 477
Introduction	p. 477
Definitions	p. 478
State of the System after n-Steps	p. 479
Long-Term Behavior of the Markov Chain	p. 481
Stationary Distribution	p. 481
Aperiodicity and Irreducibility	p. 483
Reversible Markov Chains	p. 487
Limiting Behavior	p. 492
Markov Chain Monte Carlo	p. 497
Introduction	p. 497
Preliminaries	p. 498
Notation	p. 498
Transition Kernels	p. 499
Varying Dimensionality	p. 499
An Overview of Markov Chain Monte Carlo	p. 500
The Metropolis-Hastings Algorithm	p. 502
An Informal Derivation	p. 502
A More Formal Derivation	p. 504
The Gibbs Sampler	p. 509
Fully Conditional Posterior Distributions	p. 510
The Gibbs Sampling Algorithm	p. 510
Langevin-Hastings Algorithm	p. 517
Reversible Jump MCMC	p. 517
The Invariant Distribution	p. 518
Generating the Proposal	p. 519
Specifying the Reversibility Condition	p. 520
Derivation of the Acceptance Probability	p. 522
Deterministic Proposals	p. 523
Generating Proposals via the Identity Mapping	p. 525
Data Augmentation	p. 532
Implementation and Analysis of MCMC Samples	p. 539
Introduction	p. 539
A Single Long Chain or Several Short Chains?	p. 540
Convergence Issues	p. 541
Effect of Posterior Correlation on Convergence	p. 541
Monitoring Convergence	p. 547
Inferences from the MCMC Output	p. 550
Estimators of Posterior Quantities	p. 550
Monte Carlo Variance	p. 553
Sensitivity Analysis	p. 556
Applications in Quantitative Genetics	p. 561
Gaussian and Thick-Tailed Linear Models	p. 563
Introduction	p. 563
The Univariate Linear Additive Genetic Model	p. 564
A Gibbs Sampling Algorithm	p. 566
Additive Genetic Model with Maternal Effects	p. 570
Fully Conditional Posterior Distributions	p. 575
The Multivariate Linear Additive Genetic Model	p. 576
Fully Conditional Posterior Distributions	p. 580
A Blocked Gibbs Sampler for Gaussian Linear Models	p. 584
Linear Models with Thick-Tailed Distributions	p. 588
Motivation	p. 588
A Student-t Mixed Effects Model	p. 595
Model with Clustered Random Effects	p. 600
Parameterizations and the Gibbs Sampler	p. 602
Threshold Models for Categorical Responses	p. 605
Introduction	p. 605
Analysis of a Single Polychotomous Trait	p. 607
Sampling Model	p. 607
Prior Distribution and Joint Posterior Density	p. 608
Fully Conditional Posterior Distributions	p. 611
The Gibbs Sampler	p. 615
Analysis of a Categorical and a Gaussian Trait	p. 615
Sampling Model	p. 616
Prior Distribution and Joint Posterior Density	p. 617
Fully Conditional Posterior Distributions	p. 619
The Gibbs Sampler	p. 625
Implementation with Binary Traits	p. 626
Bayesian Analysis of Longitudinal Data	p. 627
Introduction	p. 627
Hierarchical or Multistage Models	p. 628
First Stage	p. 629
Second Stage	p. 634
Third Stage	p. 639
Joint Posterior Distribution	p. 641
Two-Step Approximate Bayesian Analysis	p. 642
Estimating Location Parameters	p. 643
Estimating Dispersion Parameters	p. 650
Special Case: Linear First Stage	p. 652
Computation via Markov Chain Monte Carlo	p. 653
Fully Conditional Posterior Distributions	p. 655
Analysis with Thick-Tailed Distributions	p. 664
First- and Second-Stage Models	p. 665
Fully Conditional Posterior Distributions	p. 666
Segregation and Quantitative Trait Loci Analysis	p. 671
Introduction	p. 671
Segregation Analysis Models	p. 672
Notation and Model	p. 672
Fully Conditional Posterior Distributions	p. 675
Some Implementation Issues	p. 677
QTL Models	p. 679
Models with a Single QTL	p. 680
Models with an Arbitrary Number of QTL	p. 690
References	p. 701
List of Citations	p. 727
Subject Index	p. 733
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Likelihood, Bayesian and MCMC Methods in Quantitative Genetics

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