Daniel Gianola

Likelihood, Bayesian, and MCMC methods in quantitative genetics.

Preface I Review of Probability and Distribution Theory 1 Probability and Random Variables 1.1 Introduction 1.2 Univariate Discrete Distributions 1.2.1 The Bernoulli and Binomial Distributions 1.2.2 The Poisson Distribution 1.2.3 Binomial Distribution: Normal Approximation 1.3 Univariate Continuous Distributions 1.3.1 The Uniform, Beta, Gamma, Normal, and Student-t Distributions 1.4 Multivariate Probability Distributions 1.4.1 The Multinomial Distribution 1.4.2 The Dirichlet Distribution 1.4.3 The d-Dimensional Uniform Distribution 1.4.4 The Multivariate Normal Distribution 1.4.5 The Chi-square Distribution 1.4.6 The Wishart and Inverse Wishart Distributions 1.4.7 The Multivariate-t Distribution 1.5 Distributions with Constrained Sample Space 1.6 Iterated Expectations 2 Functions of Random Variables 2.1 Introduction 2.2 Functions of a Single Random Variable 2.2.1 Discrete Random Variables 2.2.2 Continuous Random Variables 2.2.3 Approximating the Mean and Variance 2.2.4 Delta Method 2.3 Functions of Several Random Variables 2.3.1 Linear Transformations 2.3.2 Approximating the Mean and Covariance Matrix II Methods of Inference 3 An Introduction to Likelihood Inference 3.1 Introduction 3.2 The Likelihood Function 3.3 The Maximum Likelihood Estimator 3.4 Likelihood Inference in a Gaussian Model 3.5 Fishers Information Measure 3.5.1 Single Parameter Case 3.5.2 Alternative Representation of Information 3.5.3 Mean and Variance of the Score Function 3.5.4 Multiparameter Case 3.5.5 Cramer-Rao Lower Bound 3.6 Sufficiency 3.7 Asymptotic Properties: Single Parameter Models 3.7.1 Probability of the Data Given the Parameter 3.7.2 Consistency 3.7.3 Asymptotic Normality and Effciency 3.8 Asymptotic Properties: Multiparameter Models 3.9 Functional Invariance 3.9.1 Illustration of FunctionalInvariance 3.9.2 Invariance in a Single Parameter Model 3.9.3 Invariance in a Multiparameter Model 4 Further Topics in Likelihood Inference 4.1 Introduction 4.2 Computation of Maximum Likelihood Estimates 4.3 Evaluation of Hypotheses 4.3.1 Likelihood Ratio Tests 4.3.2 Con.dence Regions 4.3.3 Walds Test 4.3.4 Score Test 4.4 Nuisance Parameters 4.4.1 Loss of Efficiency Due to Nuisance Parameters 4.4.2 Marginal Likelihoods 4.4.3 Profile Likelihoods 4.5 Analysis of a Multinomial Distribution 4.5.1 Amount of Information per Observation 4.6 Analysis of Linear Logistic Models 4.6.1 The Logistic Distribution 4.6.2 Likelihood Function under Bernoulli Sampling 4.6.3 Mixed Effects Linear Logistic Model 5 An Introduction to Bayesian Inference 5.1 Introduction 5.2 Bayes Theorem: Discrete Case 5.3 Bayes Theorem: Continuous Case 5.4 Posterior Distributions 5.5 Bayesian Updating 5.6 Features of Posterior Distributions 5.6.1 Posterior Probabilities 5.6.2 Posterior Quantiles 5.6.3 Posterior Modes 5.6.4 Posterior Mean Vector and Covariance Matrix 6 Bayesian Analysis of Linear Models 6.1 Introduction 6.2 The Linear Regression Model 6.2.1 Inference under Uniform Improper Priors 6.2.2 Inference under Conjugate Priors 6.2.3 Orthogonal Parameterization of the Model 6.3 The Mixed Linear Model 6.3.1 Bayesian View of the Mixed Effects Model 6.3.2 Joint and Conditional Posterior Distributions 6.3.3 Marginal Distribution of Variance Components 6.3.4 Marginal Distribution of Location Parameters 7 The Prior Distribution and Bayesian Analysis 7.1 Introduction 7.2 An Illustration of the Effect of Priors on Inferences 7.3 A Rapid Tour of Bayesian Asymptotics 7.3.1 Discrete Parameter 7.3.2 Continuous Parameter 7.4 Statistical Information and Entropy 7.4.1 Information 7.4.2 Entropy of a Discrete

Prediction of genetic values of quantitative traits in plant breeding using pedigree and molecular markers

The availability of dense molecular markers has made possible the use of genomic selection (GS) for plant breeding. However, the evaluation of models for GS in real plant populations is very limited. This article evaluates the performance of parametric and semiparametric models for GS using wheat (Triticum aestivum L.) and maize (Zea mays) data in which different traits were measured in several environmental conditions. The findings, based on extensive cross-validations, indicate that models including marker information had higher predictive ability than pedigree-based models. In the wheat data set, and relative to a pedigree model, gains in predictive ability due to inclusion of markers ranged from 7.7 to 35.7%. Correlation between observed and predictive values in the maize data set achieved values up to 0.79. Estimates of marker effects were different across environmental conditions, indicating that genotype × environment interaction is an important component of genetic variability. These results indicate that GS in plant breeding can be an effective strategy for selecting among lines whose phenotypes have yet to be observed.

Predicting Quantitative Traits With Regression Models for Dense Molecular Markers and Pedigree

The availability of genomewide dense markers brings opportunities and challenges to breeding programs. An important question concerns the ways in which dense markers and pedigrees, together with phenotypic records, should be used to arrive at predictions of genetic values for complex traits. If a large number of markers are included in a regression model, marker-specific shrinkage of regression coefficients may be needed. For this reason, the Bayesian least absolute shrinkage and selection operator (LASSO) (BL) appears to be an interesting approach for fitting marker effects in a regression model. This article adapts the BL to arrive at a regression model where markers, pedigrees, and covariates other than markers are considered jointly. Connections between BL and other marker-based regression models are discussed, and the sensitivity of BL with respect to the choice of prior distributions assigned to key parameters is evaluated using simulation. The proposed model was fitted to two data sets from wheat and mouse populations, and evaluated using cross-validation methods. Results indicate that inclusion of markers in the regression further improved the predictive ability of models. An R program that implements the proposed model is freely available.

Additive Genetic Variability and the Bayesian Alphabet

The use of all available molecular markers in statistical models for prediction of quantitative traits has led to what could be termed a genomic-assisted selection paradigm in animal and plant breeding. This article provides a critical review of some theoretical and statistical concepts in the context of genomic-assisted genetic evaluation of animals and crops. First, relationships between the (Bayesian) variance of marker effects in some regression models and additive genetic variance are examined under standard assumptions. Second, the connection between marker genotypes and resemblance between relatives is explored, and linkages between a marker-based model and the infinitesimal model are reviewed. Third, issues associated with the use of Bayesian models for marker-assisted selection, with a focus on the role of the priors, are examined from a theoretical angle. The sensitivity of a Bayesian specification that has been proposed (called “Bayes A”) with respect to priors is illustrated with a simulation. Methods that can solve potential shortcomings of some of these Bayesian regression procedures are discussed briefly.

Genomic-Assisted Prediction of Genetic Value With Semiparametric Procedures

Semiparametric procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are presented. The methods focus on the treatment of massive information provided by, e.g., single-nucleotide polymorphisms. It is argued that standard parametric methods for quantitative genetic analysis cannot handle the multiplicity of potential interactions arising in models with, e.g., hundreds of thousands of markers, and that most of the assumptions required for an orthogonal decomposition of variance are violated in artificial and natural populations. This makes nonparametric procedures attractive. Kernel regression and reproducing kernel Hilbert spaces regression procedures are embedded into standard mixed-effects linear models, retaining additive genetic effects under multivariate normality for operational reasons. Inferential procedures are presented, and some extensions are suggested. An example is presented, illustrating the potential of the methodology. Implementations can be carried out after modification of standard software developed by animal breeders for likelihood-based or Bayesian analysis.

Reproducing Kernel Hilbert Spaces Regression Methods for Genomic Assisted Prediction of Quantitative Traits

Reproducing kernel Hilbert spaces regression procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are discussed from a theoretical perspective. It is argued that a nonparametric treatment may be needed for capturing the multiple and complex interactions potentially arising in whole-genome models, i.e., those based on thousands of single-nucleotide polymorphism (SNP) markers. After a review of reproducing kernel Hilbert spaces regression, it is shown that the statistical specification admits a standard mixed-effects linear model representation, with smoothing parameters treated as variance components. Models for capturing different forms of interaction, e.g., chromosome-specific, are presented. Implementations can be carried out using software for likelihood-based or Bayesian inference.

Bayesian analysis of mixed linear models via Gibbs sampling with an application to litter size in Iberian pigs

Summary - The Gibbs sampling is a Monte-Carlo procedure for generating random samples from joint distributions through sampling from and updating conditional distributions. Inferences about unknown parameters are made by: 1) computing directly summary statistics from the samples; or 2) estimating the marginal density of an unknown, and then obtaining summary statistics from the density. All conditional distributions needed to implement the Gibbs sampling in a univariate Gaussian mixed linear model are presented in scalar algebra, so no matrix inversion is needed in the computations. For location parameters, all conditional distributions are univariate normal, whereas those for variance components are scaled inverted chi-squares. The procedure was applied to solve a Gaussian animal model for litter size in the Gamito strain of Iberian pigs. Data were 1 213 records from 426 dams. The model had farrowing season (72 levels) and parity (4) as fixed effects; breeding values (597), permanent environmental effects (426) and residuals were random. In CASE I, variances were assumed known, with REML (restricted maximum likelihood) estimates used as true parameter values. Here, means and variances of the posterior distributions of all effects were obtained, by inversion, from the mixed model equations. These exact solutions were used to check the Monte-Carlo estimates given by Gibbs, using 120 000 samples. Linear regression slopes of true posterior means on Gibbs means were almost exactly 1 for fixed, additive genetic and permanent environmental effects. Regression slopes of true posterior variances on Gibbs variances were 1.00, 1.01 and 0.96, respectively. In CASE II, variances were treated as unknown, with a flat prior assigned to these. Posterior densities of selected location parameters, variance components, heritability and repeatability were estimated. Marginal posterior distributions of dispersion parameters were skewed, save the residual variance; the means, modes and medians of these distributions differed from the REML estimates, as expected from theory. The conclusions are: 1) the Gibbs sampler converged to the true posterior distributions, as suggested by CASE I; 2) it provides a richer description of uncertainty about genetic

Predicting genetic predisposition in humans: the promise of whole-genome markers.

Although genome-wide association studies have identified markers that are associated with various human traits and diseases, our ability to predict such phenotypes remains limited. A perhaps overlooked explanation lies in the limitations of the genetic models and statistical techniques commonly used in association studies. We propose that alternative approaches, which are largely borrowed from animal breeding, provide potential for advances. We review selected methods and discuss the challenges and opportunities ahead.

Marginal inferences about variance components in a mixed linear model using Gibbs sampling

Summary - Arguing from a Bayesian viewpoint, Gianola and Foulley (1990) derived a new method for estimation of variance components in a mixed linear model: variance estimation from integrated likelihoods (VEIL). Inference is based on the marginal posterior distribution of each of the variance components. Exact analysis requires numerical integration. In this paper, the Gibbs sampler, a numerical procedure for generating marginal distributions from conditional distributions, is employed to obtain marginal inferences about variance components in a general univariate mixed linear model. All needed conditional posterior distributions are derived. Examples based on simulated data sets containing varying amounts of information are presented for a one-way sire model. Estimates of the marginal densities of the variance components and of functions thereof are obtained, and the corresponding distributions are plotted. Numerical results with a balanced sire model suggest that convergence to the marginal posterior distributions is achieved with a Gibbs sequence length of 20, and that Gibbs sample sizes ranging from 300 - 3 000 may be needed to appropriately characterize the marginal distributions. variance components / linear models / Bayesian methods / marginalization / Gibbs sampler

Priors in whole-genome regression: the bayesian alphabet returns.

Whole-genome enabled prediction of complex traits has received enormous attention in animal and plant breeding and is making inroads into human and even Drosophila genetics. The term “Bayesian alphabet” denotes a growing number of letters of the alphabet used to denote various Bayesian linear regressions that differ in the priors adopted, while sharing the same sampling model. We explore the role of the prior distribution in whole-genome regression models for dissecting complex traits in what is now a standard situation with genomic data where the number of unknown parameters (p) typically exceeds sample size (n). Members of the alphabet aim to confront this overparameterization in various manners, but it is shown here that the prior is always influential, unless n ≫ p. This happens because parameters are not likelihood identified, so Bayesian learning is imperfect. Since inferences are not devoid of the influence of the prior, claims about genetic architecture from these methods should be taken with caution. However, all such procedures may deliver reasonable predictions of complex traits, provided that some parameters (“tuning knobs”) are assessed via a properly conducted cross-validation. It is concluded that members of the alphabet have a room in whole-genome prediction of phenotypes, but have somewhat doubtful inferential value, at least when sample size is such that n ≪ p.