Impact of depth of pedigree and inclusion of historical data on the estimation of additive variance and breeding values in a sugarcane breeding program

Sugarcane breeders in Australia combine data across four selection programs to obtain estimates of breeding value for parents. When these data are combined with full pedigree information back to founding parents, computing limitations mean it is not possible to obtain information on all parents. Family data from one sugarcane selection program were analysed using two different genetic models to investigate how different depths of pedigree and amount of data affect the reliability of estimating breeding value of sugarcane parents. These were the parental and animal models. Additive variance components and breeding values estimated from different amounts of information were compared for both models. The accuracy of estimating additive variance components and breeding values improved as more pedigree information and historical data were included in analyses. However, adding years of data had a much larger effect on the estimation of variance components of the population, and breeding values of the parents. To accurately estimate breeding values of all sugarcane parents, a minimum of three generations of pedigree and 5 years of historical data were required, while more information (four generations of pedigree and 7 years of historical data) was required when identifying top parents to be selected for future cross pollination.     

The use of linear mixed models to estimate variance components from data on twin pairs by maximum likelihood.

It is shown that maximum likelihood estimation of variance components from twin data can be parameterized in the framework of linear mixed models. Standard statistical packages can be used to analyze univariate or multivariate data for simple models such as the ACE and CE models. Furthermore, specialized variance component estimation software that can handle pedigree data and user-defined covariance structures can be used to analyze multivariate data for simple and complex models, including those where dominance and/or QTL effects are fitted. The linear mixed model framework is particularly useful for analyzing multiple traits in extended (twin) families with a large number of random effects.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

Estimating multilevel logistic regression models when the number of clusters is low: a comparison of different statistical software procedures

Multilevel logistic regression models are increasingly being used to analyze clustered data in medical, public health, epidemiological, and educational research. Procedures for estimating the parameters of such models are available in many statistical software packages. There is currently little evidence on the minimum number of clusters necessary to reliably fit multilevel regression models. We conducted a Monte Carlo study to compare the performance of different statistical software procedures for estimating multilevel logistic regression models when the number of clusters was low. We examined procedures available in BUGS, HLM, R, SAS, and Stata. We found that there were qualitative differences in the performance of different software procedures for estimating multilevel logistic models when the number of clusters was low. Among the likelihood-based procedures, estimation methods based on adaptive Gauss-Hermite approximations to the likelihood (glmer in R and xtlogit in Stata) or adaptive Gaussian quadrature (Proc NLMIXED in SAS) tended to have superior performance for estimating variance components when the number of clusters was small, compared to software procedures based on penalized quasi-likelihood. However, only Bayesian estimation with BUGS allowed for accurate estimation of variance components when there were fewer than 10 clusters. For all statistical software procedures, estimation of variance components tended to be poor when there were only five subjects per cluster, regardless of the number of clusters.     

Estimation of additive, dominance and epistatic variance components using finite locus models implemented with a single-site Gibbs and a descent graph sampler

In a previous contribution, we implemented a finite locus model (FLM) for estimating additive and dominance genetic variances via a Bayesian method and a single-site Gibbs sampler. We observed a dependency of dominance variance estimates on locus number in the analysis FLM. Here, we extended the FLM to include two-locus epistasis, and implemented the analysis with two genotype samplers (Gibbs and descent graph) and three different priors for genetic effects (uniform and variable across loci, uniform and constant across loci, and normal). Phenotypic data were simulated for two pedigrees with 6300 and 12,300 individuals in closed populations, using several different, non-additive genetic models. Replications of these data were analysed with FLMs differing in the number of loci. Simulation results indicate that the dependency of non-additive genetic variance estimates on locus number persisted in all implementation strategies we investigated. However, this dependency was considerably diminished with normal priors for genetic effects as compared with uniform priors (constant or variable across loci). Descent graph sampling of genotypes modestly improved variance components estimation compared with Gibbs sampling. Moreover, a larger pedigree produced considerably better variance components estimation, suggesting this dependency might originate from data insufficiency. As the FLM represents an appealing alternative to the infinitesimal model for genetic parameter estimation and for inclusion of polygenic background variation in QTL mapping analyses, further improvements are warranted and might be achieved via improvement of the sampler or treatment of the number of loci as an unknown.     

Multifactorial genetic models for quantitative traits in humans.

Quantitative traits measured in human families can be analyzed to partition the total population variance into genetic and environmental components, or to elucidate the genetic mechanism involved. We review the estimation of variance components directly from human pedigree data, or in the form of path coefficients from correlations between pairs of relatives. To elucidate genetic mechanisms, a mixed model that allows for segregation at a major locus, a polygenic effect and a sibling environmental correlation is described for nuclear families. In each case appropriate likelihoods are derived as a basis, using numerical maximum likelihood methods, for parameter estimation and hypothesis testing. A general model is then described that allows for several familial sources of environmental variation, assortative mating, and both major gene and polygenic effects; and an algorithm for calculating the likelihood of a pedigree under this model is indicated. Finally, some of the remaining problems in this area of biometric analysis are pointed out.     

A Bayesian framework for comparative quantitative genetics

Bayesian approaches have been extensively used in animal breeding sciences, but similar approaches in the context of evolutionary quantitative genetics have been rare. We compared the performance of Bayesian and frequentist approaches in estimation of quantitative genetic parameters (viz. matrices of additive and dominance variances) in datasets typical of evolutionary studies and traits differing in their genetic architecture. Our results illustrate that it is difficult to disentangle the relative roles of different genetic components from small datasets, and that ignoring, e.g. dominance is likely to lead to biased estimates of additive variance. We suggest that a natural summary statistic for G-matrix comparisons can be obtained by examining how different the underlying multinormal probability distributions are, and illustrate our approach with data on the common frog (Rana temporaria). Furthermore, we derive a simple Monte Carlo method for computation of fraternity coefficients needed for the estimation of dominance variance, and use the pedigree of a natural Siberian jay (Perisoreus infaustus) population to illustrate that the commonly used approximate values can be substantially biased.     

Estimation of variance components of quantitative traits in inbred populations

Use of variance-component estimation for mapping of quantitative-trait loci in humans is a subject of great current interest. When only trait values, not genotypic information, are considered, variance-component estimation can also be used to estimate heritability of a quantitative trait. Inbred pedigrees present special challenges for variance-component estimation. First, there are more variance components to be estimated in the inbred case, even for a relatively simple model including additive, dominance, and environmental effects. Second, more identity coefficients need to be calculated from an inbred pedigree in order to perform the estimation, and these are computationally more difficult to obtain in the inbred than in the outbred case. As a result, inbreeding effects have generally been ignored in practice. We describe here the calculation of identity coefficients and estimation of variance components of quantitative traits in large inbred pedigrees, using the example of HDL in the Hutterites. We use a multivariate normal model for the genetic effects, extending the central-limit theorem of Lange to allow for both inbreeding and dominance under the assumptions of our variance-component model. We use simulated examples to give an indication of under what conditions one has the power to detect the additional variance components and to examine their impact on variance-component estimation. We discuss the implications for mapping and heritability estimation by use of variance components in inbred populations.     

Estimates of genetic parameters and breeding values from western larch open-pollinated families using marker-based relationship

The availability and affordability of genetic markers made it possible to estimate quantitative genetic parameters without mating designs' structured pedigree. Here, we compared 4-year height's heritability and individuals' breeding values for a western larch common-garden population of 1,418 offspring representing 15 open-pollinated families from a 41-clone seed orchard using (a) classical pedigree models such as half- and full-sib families and (b) a molecular marker-based pedigree-free model using four pair-wise relationship estimation methods using eight informative SSR markers. The results highlighted the commonly observed inflated estimates of genetic parameters often obtained from half-sib analyses, as well as demonstrating some of the full-sib analyses' caveats. The pedigree reconstruction permitted the identification of selfed individuals, thus allowing evaluating the impact of selfing on marker-based genetic parameter estimation. The results demonstrated the utility of marker-based methods as an alternative to the classical pedigree-based approaches. Unlike the pedigree-based methods, the marker-based approach allowed better partitioning the variance components as well as separating the non-additive and additive genetic variance. The theoretical underpinning of the marker-based approach was discussed.     

Unraveling Additive from Nonadditive Effects Using Genomic Relationship Matrices

The application of quantitative genetics in plant and animal breeding has largely focused on additive models, which may also capture dominance and epistatic effects. Partitioning genetic variance into its additive and nonadditive components using pedigree-based models (P-genomic best linear unbiased predictor) (P-BLUP) is difficult with most commonly available family structures. However, the availability of dense panels of molecular markers makes possible the use of additive- and dominance-realized genomic relationships for the estimation of variance components and the prediction of genetic values (G-BLUP). We evaluated height data from a multifamily population of the tree species Pinus taeda with a systematic series of models accounting for additive, dominance, and first-order epistatic interactions (additive by additive, dominance by dominance, and additive by dominance), using either pedigree- or marker-based information. We show that, compared with the pedigree, use of realized genomic relationships in marker-based models yields a substantially more precise separation of additive and nonadditive components of genetic variance. We conclude that the marker-based relationship matrices in a model including additive and nonadditive effects performed better, improving breeding value prediction. Moreover, our results suggest that, for tree height in this population, the additive and nonadditive components of genetic variance are similar in magnitude. This novel result improves our current understanding of the genetic control and architecture of a quantitative trait and should be considered when developing breeding strategies.     

The estimation of genetic relationships using molecular markers and their efficiency in estimating heritability in natural populations

Molecular marker data collected from natural populations allows information on genetic relationships to be established without referencing an exact pedigree. Numerous methods have been developed to exploit the marker data. These fall into two main categories: method of moment estimators and likelihood estimators. Method of moment estimators are essentially unbiased, but utilise weighting schemes that are only optimal if the analysed pair is unrelated. Thus, they differ in their efficiency at estimating parameters for different relationship categories. Likelihood estimators show smaller mean squared errors but are much more biased. Both types of estimator have been used in variance component analysis to estimate heritability. All marker-based heritability estimators require that adequate levels of the true relationship be present in the population of interest and that adequate amounts of informative marker data are available. I review the different approaches to relationship estimation, with particular attention to optimizing the use of this relationship information in subsequent variance component estimation.     

The use of marker‐based relationship information to estimate the heritability of body weight in a natural population: a cautionary tale

A number of procedures have been developed that allow the genetic parameters of natural populations to be estimated using relationship information inferred from marker data rather than known pedigrees. Three published approaches are available; the regression, pair‐wise likelihood and Markov Chain Monte Carlo (MCMC) sib‐ship reconstruction methods. These were applied to body weight and molecular data collected from the Soay sheep population of St. Kilda, which has a previously determined pedigree. The regression and pair‐wise likelihood approaches do not specify an exact pedigree and yielded unreliable heritability estimates, that were sensitive to alteration of the fixed effects. The MCMC method, which specifies a pedigree prior to heritability estimation, yielded results closer to those determined using the known pedigree. In populations of low average relationship, such as the Soay sheep population, determination of a reliable pedigree is more useful than indirect approaches that do not specify a pedigree.     

MAXIMUM-LIKELIHOOD APPROACHES APPLIED TO QUANTITATIVE GENETICS OF NATURAL POPULATIONS

Growing interest in adaptive evolution in natural populations has spurred efforts to infer genetic components of variance and covariance of quantitative characters. Here, I review difficulties inherent in the usual least-squares methods of estimation. A useful alternative approach is that of maximum likelihood (ML). Its particular advantage over least squares is that estimation and testing procedures are well defined, regardless of the design of the data. A modified version of ML, REML, eliminates the bias of ML estimates of variance components. Expressions for the expected bias and variance of estimates obtained from balanced, fully hierarchical designs are presented for ML and REML. Analyses of data simulated from balanced, hierarchical designs reveal differences in the properties of ML, REML, and F-ratio tests of significance. A second simulation study compares properties of REML estimates obtained from a balanced, fully hierarchical design (within-generation analysis) with those from a sampling design including phenotypic data on parents and multiple progeny. It also illustrates the effects of imposing nonnegativity constraints on the estimates. Finally, it reveals that predictions of the behavior of significance tests based on asymptotic theory are not accurate when sample size is small and that constraining the estimates seriously affects properties of the tests. Because of their great flexibility, likelihood methods can serve as a useful tool for estimation of quantitative-genetic parameters in natural populations. Difficulties involved in hypothesis testing remain to be solved.     

Variance component estimation techniques compared for two mating designs with forest genetic architecture through computer simulation

Computer simulation was used to compare minimum variance quadratic estimation (MIVQUE), minimum norm quadratic unbiased estimation (MINQUE), restricted maximum likelihood (REML), maximum likelihood (ML), and Henderson's Method 3 (HM3) on the basis of variance among estimates, mean square error (MSE), bias and probability of nearness for estimation of both individual variance components and three ratios of variance components. The investigation also compared three procedures for dealing with negative estimates and included the use of both individual observations and plot means as the experimental unit of the analysis. The structure of data simulated (field design, mating designs, genetic architecture and imbalance) represented typical analysis problems in quantitative forest genetics. Results of comparing the estimation techniques demonstrated that: estimates of probability of nearness did not discriminate among techniques; bias was discriminatory among procedures for dealing with negative estimates but not among estimation techniques (except ML); sampling variance among estimates was discriminatory among procedures for dealing with negative estimates, estimation techniques and unit of observation; and MSE provided no additional information to variance of the estimates. HM3 and REML were the closest competitors under these criteria; however, REML demonstrated greater robustness to imbalance. Of the three negative estimate procedures, two are of practical significance and guidelines for their application are presented. Estimates from individual observations were always preferable to those from plot means over the experimental levels of this study.This is Journal Series NO. R-03768 of the Institute of Food and Agricultural Sciences     

Comparison of genetic parameters from marker-based relationship, sibship, and combined models in Scots pine multi-site open-pollinated tests

Nine microsatellite DNA markers (simple sequence repeats, SSRs) were used to estimate pairwise relationships among 597 Scots pine (Pinus sylvestris) trees as well as to generate a sibship structure for quantitative genetic parameters’ estimation comparison. The studied trees were part of an open-pollinated progeny test of 102 first-generation parents. Three methods were used to estimate variance components and heritabilities, namely, structured pedigree (half- and full-sib), marker-based pairwise relationships (four pairwise estimators), and a combined pedigree and marker-based relationship. In each of the three methods, the same animal model was used to compute variances except when marker-based relationship was used wherein we substituted the average numerator relationship matrix (i.e., pedigree-based matrix) with that computed based on markers’ pairwise relationships. Our results showed a high correlation in estimated breeding values between the pedigree (full-sib) and the combined marker-pedigree estimates. The marker-based relationship method produced high correlations when individual site data were analyzed. In contrast, the marker-based relationship method resulted in a significant decrease in both variance estimation and their standard errors which were in concordance with earlier published results; however, no estimates were produced when across-site analyses were attempted. We concluded that the combined pedigree method is the best approach as it represents the historical (pairwise) and contemporary (pedigree) relationships among the tested individuals, a situation that cannot be attained by any of the used methods individually. This method is dependent on the number and informativeness of the markers used.     

Estimation in a multiplicative mixed model involving a genetic relationship matrix

Genetic models partitioning additive and non-additive genetic effects for populations tested in replicated multi-environment trials (METs) in a plant breeding program have recently been presented in the literature. For these data, the variance model involves the direct product of a large numerator relationship matrix A, and a complex structure for the genotype by environment interaction effects, generally of a factor analytic (FA) form. With MET data, we expect a high correlation in genotype rankings between environments, leading to non-positive definite covariance matrices. Estimation methods for reduced rank models have been derived for the FA formulation with independent genotypes, and we employ these estimation methods for the more complex case involving the numerator relationship matrix. We examine the performance of differing genetic models for MET data with an embedded pedigree structure, and consider the magnitude of the non-additive variance. The capacity of existing software packages to fit these complex models is largely due to the use of the sparse matrix methodology and the average information algorithm. Here, we present an extension to the standard formulation necessary for estimation with a factor analytic structure across multiple environments.     

MIXED MODEL APPROACHES FOR ESTIMATING GENETIC VARIANCES AND COVARIANCES

The limitations of methods for analysis of variance(ANOVA)in estimating genetic variances are discussed. Among the three methods(maximum likelihood ML, restricted maximum likelihood REML, and minimum norm quadratic unbiased estimation MINQUE)for mixed linear models, MINQUE method is presented with formulae for estimating variance components and covariances components and for predicting genetic effects. Several genetic models, which cannot be appropriately analyzed by ANOVA methods, are introduced in forms of mixed linear models. Genetic models with independent random effects can be analyzed by MINQUE(1)method whieh is a MINQUE method with all prior values setting 1. MINQUE(1)method can give unbiased estimation for variance components and covariance components, and linear unbiased prediction (LUP) for genetic effects. There are more complicate genetic models for plant seeds which involve correlated random effects. MINQUE(0/1)method, which is a MINQUE method with all prior covariances setting 0 and all prior variances setting 1, is suitable for estimating variance and covariance components in these models. Mixed model approaches have advantage over ANOVA methods for the capacity of analyzing unbalanced data and complicated models. Some problems about estimation and hypothesis test by MINQUE method are discussed.     

Impact of reconstructed pedigrees on progeny-test breeding values in red spruce

Pedigrees reconstructed through DNA marker assigned paternities in polymix (PMX) and open pollinated (OP) progeny tests were analyzed using mixed models to test the effect of unequal male reproductive success and pedigree errors on quantitative genetic parameters. The reconstructed pedigree increased heritabilities in the larger PMX test. Increased heritability resulted from adding the paternities to the pedigree per se, not by correcting the male reproductive bias by specifying the exact pedigree. Removing hypothesized pedigree errors had no effect on quantitative parameters, either because the magnitude of the errors was too small (PMX) or the progeny test was too small to detect variance components reliably (OP). Although there was no advantage in backwards selection, the increased additive variance, heritabilities and accuracy of progeny with assigned paternities in the pedigree, should permit forward selection of offspring with greater genetic gain and complete control of coancestry for future breeding decisions. Some possible breeding population structures with the new genetic information are discussed.     

Impact of reconstructed pedigrees on progeny-test breeding values in red spruce

Pedigrees reconstructed through DNA marker assigned paternities in polymix (PMX) and open pollinated (OP) progeny tests were analyzed using mixed models to test the effect of unequal male reproductive success and pedigree errors on quantitative genetic parameters. The reconstructed pedigree increased heritabilities in the larger PMX test. Increased heritability resulted from adding the paternities to the pedigree per se, not by correcting the male reproductive bias by specifying the exact pedigree. Removing hypothesized pedigree errors had no effect on quantitative parameters, either because the magnitude of the errors was too small (PMX) or the progeny test was too small to detect variance components reliably (OP). Although there was no advantage in backwards selection, the increased additive variance, heritabilities and accuracy of progeny with assigned paternities in the pedigree, should permit forward selection of offspring with greater genetic gain and complete control of coancestry for future breeding decisions. Some possible breeding population structures with the new genetic information are discussed.