Likelihood, Bayesian, and McMc Methods in Quantitative Genetics

by Sorensen, Daniel; Gianola, Daniel

• ISBN13: 9780387954400
• ISBN10: 0387954406
• Format: Hardcover
• Copyright: 2002-10-01
• Publisher: Springer Nature

Summary

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.

Table of Contents

Preface	v
I Review of Probability and Distribution Theory	1	(116)
Probability and Random Variables	3	(74)
Introduction	3	(1)
Univariate Discrete Distributions	4	(9)
The Bernoulli and Binomial Distributions	7	(3)
The Poisson Distribution	10	(2)
Binomial Distribution: Normal Approximation	12	(1)
Univariate Continuous Distributions	13	(16)
The Uniform, Beta, Gamma, Normal, and Student-t Distributions	18	(11)
Multivariate Probability Distributions	29	(33)
The Multinomial Distribution	37	(3)
The Dirichlet Distribution	40	(1)
The d-Dimensional Uniform Distribution	40	(1)
The Multivariate Normal Distribution	41	(12)
The Chi-square Distribution	53	(2)
The Wishart and Inverse Wishart Distributions	55	(5)
The Multivariate-t Distribution	60	(2)
Distributions with Constrained Sample Space	62	(5)
Iterated Expectations	67	(10)
Functions of Random Variables	77	(40)
Introduction	77	(1)
Functions of a Single Random Variable	78	(17)
Discrete Random Variables	78	(1)
Continuous Random Variables	79	(10)
Approximating the Mean and Variance	89	(4)
Delta Method	93	(2)
Functions of Several Random Variables	95	(22)
Linear Transformations	111	(3)
Approximating the Mean and Covariance Matrix	114	(3)
II Methods of Inference	117	(358)
An Introduction to Likelihood Inference	119	(42)
Introduction	119	(1)
The Likelihood Function	120	(2)
The Maximum Likelihood Estimator	122	(3)
Likelihood Inference in a Gaussian Model	125	(3)
Fisher's Information Measure	128	(14)
Single Parameter Case	128	(3)
Alternative Representation of Information	131	(3)
Mean and Variance of the Score Function	134	(1)
Multiparameter Case	135	(3)
Cramer-Rao Lower Bound	138	(4)
Sufficiency	142	(1)
Asymptotic Properties: Single Parameter Models	143	(9)
Probability of the Data Given the Parameter	144	(2)
Consistency	146	(1)
Asymptotic Normality and Efficiency	147	(5)
Asymptotic Properties: Multiparameter Models	152	(1)
Functional Invariance	153	(8)
Illustration of Functional Invariance	153	(4)
Invariance in a Single Parameter Model	157	(2)
Invariance in a Multiparameter Model	159	(2)
Further Topics in Likelihood Inference	161	(50)
Introduction	161	(1)
Computation of Maximum Likelihood Estimates	162	(4)
Evaluation of Hypotheses	166	(15)
Likelihood Ratio Tests	166	(11)
Confidence Regions	177	(2)
Wald's Test	179	(1)
Score Test	179	(2)
Nuisance Parameters	181	(9)
Loss of Efficiency Due to Nuisance Parameters	182	(1)
Marginal Likelihoods	182	(4)
Profile Likelihoods	186	(4)
Analysis of a Multinomial Distribution	190	(12)
Amount of Information per Observation	199	(3)
Analysis of Linear Logistic Models	202	(9)
The Logistic Distribution	204	(1)
Likelihood Function under Bernoulli Sampling	204	(4)
Mixed Effects Linear Logistic Model	208	(3)
An Introduction to Bayesian Inference	211	(76)
Introduction	211	(3)
Bayes Theorem: Discrete Case	214	(9)
Bayes Theorem: Continuous Case	223	(12)
Posterior Distributions	235	(14)
Bayesian Updating	249	(8)
Features of Posterior Distributions	257	(30)
Posterior Probabilities	258	(4)
Posterior Quantiles	262	(2)
Posterior Modes	264	(16)
Posterior Mean Vector and Covariance Matrix	280	(7)
Bayesian Analysis of Linear Models	287	(40)
Introduction	287	(1)
The Linear Regression Model	287	(26)
Inference under Uniform Improper Priors	288	(9)
Inference under Conjugate Priors	297	(10)
Orthogonal Parameterization of the Model	307	(6)
The Mixed Linear Model	313	(14)
Bayesian View of the Mixed Effects Model	313	(4)
Joint and Conditional Posterior Distributions	317	(5)
Marginal Distribution of Variance Components	322	(1)
Marginal Distribution of Location Parameters	323	(4)
The Prior Distribution and Bayesian Analysis	327	(72)
Introduction	327	(1)
An Illustration of the Effect of Priors on Inferences	328	(2)
A Rapid Tour of Bayesian Asymptotics	330	(4)
Discrete Parameter	330	(1)
Continuous Parameter	331	(3)
Statistical Information and Entropy	334	(22)
Information	334	(3)
Entropy of a Discrete Distribution	337	(3)
Entropy of a Joint and Conditional Distribution	340	(1)
Entropy of a Continuous Distribution	341	(5)
Information about a Parameter	346	(5)
Fisher's Information Revisited	351	(2)
Prior and Posterior Discrepancy	353	(3)
Priors Conveying Little Information	356	(43)
The Uniform Prior	356	(2)
Other Vague Priors	358	(9)
Maximum Entropy Prior Distributions	367	(12)
Reference Prior Distributions	379	(20)
Bayesian Assessment of Hypotheses and Models	399	(44)
Introduction	399	(1)
Bayes Factors	400	(24)
Definition	400	(2)
Interpretation	402	(1)
The Bayes Factor and Hypothesis Testing	403	(9)
Influence of the Prior Distribution	412	(2)
Nested Models	414	(4)
Approximations to the Bayes Factor	418	(4)
Partial and Intrinsic Bayes Factors	422	(2)
Estimating the Marginal Likelihood	424	(5)
Goodness of Fit and Model Complexity	429	(4)
Goodness of Fit and Predictive Ability of a Model	433	(6)
Analysis of Residuals	434	(2)
Predictive Ability and Predictive Cross-Validation	436	(3)
Bayesian Model Averaging	439	(4)
General	439	(1)
Definitions	440	(1)
Predictive Ability of BMA	441	(2)
Approximate Inference Via the EM Algorithm	443	(32)
Introduction	443	(1)
Complete and Incomplete Data	444	(1)
The EM Algorithm	445	(2)
Form of the Algorithm	445	(1)
Derivation	445	(2)
Monotonic Increase of ln p (θ|y)	447	(1)
The Missing Information Principle	448	(3)
Complete, Observed and Missing Information	448	(1)
Rate of Convergence of the EM Algorithm	449	(2)
EM Theory for Exponential Families	451	(1)
Standard Errors and Posterior Standard Deviations	452	(6)
The Method of Louis	453	(1)
Supplemented EM Algorithm (SEM)	454	(3)
The Method of Oakes	457	(1)
Examples	458	(17)
III Markov Chain Monte Carlo Methods	475	(86)
An Overview of Discrete Markov Chains	477	(20)
Introduction	477	(1)
Definitions	478	(1)
State of the System after n-Steps	479	(2)
Long-Term Behavior of the Markov Chain	481	(1)
Stationary Distribution	481	(2)
Aperiodicity and Irreducibility	483	(4)
Reversible Markov Chains	487	(5)
Limiting Behavior	492	(5)
Markov Chain Monte Carlo	497	(42)
Introduction	497	(1)
Preliminaries	498	(2)
Notation	498	(1)
Transition Kernels	499	(1)
Varying Dimensionality	499	(1)
An Overview of Markov Chain Monte Carlo	500	(2)
The Metropolis-Hastings Algorithm	502	(7)
An Informal Derivation	502	(2)
A More Formal Derivation	504	(5)
The Gibbs Sampler	509	(8)
Conditional Posterior Distributions	510	(1)
The Gibbs Sampling Algorithm	510	(7)
Langevin-Hastings Algorithm	517	(1)
Reversible Jump MCMC	517	(15)
The Invariant Distribution	518	(1)
Generating the Proposal	519	(1)
Specifying the Reversibility Condition	520	(2)
Derivation of the Acceptance Probability	522	(1)
Deterministic Proposals	523	(2)
Generating Proposals via the Identity Mapping	525	(7)
Data Augmentation	532	(7)
Implementation and Analysis of MCMC Samples	539	(22)
Introduction	539	(1)
A Single Long Chain or Several Short Chains?	540	(1)
Convergence Issues	541	(9)
Effect of Posterior Correlation on Convergence	541	(6)
Monitoring Convergence	547	(3)
Inferences from the MCMC Output	550	(6)
Estimators of Posterior Quantities	550	(3)
Monte Carlo Variance	553	(3)
Sensitivity Analysis	556	(5)
IV Applications in Quantitative Genetics	561	(140)
Gaussian and Thick-Tailed Linear Models	563	(42)
Introduction	563	(1)
The Univariate Linear Additive Genetic Model	564	(6)
A Gibbs Sampling Algorithm	566	(4)
Additive Genetic Model with Maternal Effects	570	(6)
Fully Conditional Posterior Distributions	575	(1)
The Multivariate Linear Additive Genetic Model	576	(8)
Fully Conditional Posterior Distributions	580	(4)
A Blocked Gibbs Sampler for Gaussian Linear Models	584	(4)
Linear Models with Thick-Tailed Distributions	588	(14)
Motivation	588	(7)
A Student-t Mixed Effects Model	595	(5)
Model with Clustered Random Effects	600	(2)
Parameterizations and the Gibbs Sampler	602	(3)
Threshold Models for Categorical Responses	605	(22)
Introduction	605	(2)
Analysis of a Single Polychotomous Trait	607	(8)
Sampling Model	607	(1)
Prior Distribution and Joint Posterior Density	608	(3)
Fully Conditional Posterior Distributions	611	(4)
The Gibbs Sampler	615	(1)
Analysis of a Categorical and a Gaussian Trait	615	(12)
Sampling Model	616	(1)
Prior Distribution and Joint Posterior Density	617	(2)
Fully Conditional Posterior Distributions	619	(6)
The Gibbs Sampler	625	(1)
Implementation with Binary Traits	626	(1)
Bayesian Analysis of Longitudinal Data	627	(44)
Introduction	627	(1)
Hierarchical or Multistage Models	628	(14)
First Stage	629	(5)
Second Stage	634	(5)
Third Stage	639	(2)
Joint Posterior Distribution	641	(1)
Two-Step Approximate Bayesian Analysis	642	(11)
Estimating Location Parameters	643	(7)
Estimating Dispersion Parameters	650	(2)
Special Case: Linear First Stage	652	(1)
Computation via Markov Chain Monte Carlo	653	(11)
Fully Conditional Posterior Distributions	655	(9)
Analysis with Thick-Tailed Distributions	664	(7)
First- and Second-Stage Models	665	(1)
Fully Conditional Posterior Distributions	666	(5)
Segregation and Quantitative Trait Loci Analysis	671	(30)
Introduction	671	(1)
Segregation Analysis Models	672	(7)
Notation and Model	672	(3)
Fully Conditional Posterior Distributions	675	(2)
Some Implementation Issues	677	(2)
QTL Models	679	(22)
Models with a Single QTL	680	(10)
Models with an Arbitrary Number of QTL	690	(11)
References	701	(26)
List of Citations	727	(6)
Subject Index	733