Analysis of cytoplasmic and maternal effects. II. Genetic models for triploid endosperms

Genetic models for quantitative traits of triploid endosperms are proposed for the analysis of direct gene effects, cytoplasmic effects, and maternal gene effects. The maternal effect is partitioned into maternal additive and dominance components. In the full genetic model, the direct effect is partitioned into direct additive and dominance components and high-order dominance component, which are the cumulative effects of three-allele interactions. If the high-order dominance effects are of no importance, a reduced genetic model can be used. Monte Carlo simulations were conducted in this study for demonstrating unbiasedness of estimated variance and covariance components from the MINQUE (0/1) procedure, which is a minimum norm quadratic unbiased estimation (MINQUE) method setting 0 for all the prior covariances and 1 for all the prior variances. Robustness of estimating variance and covariance components for the genetic models was tested by simulations. Both full and reduced genetic models are shown to be robust for estimating variance and covariance components under several situations of no specific effects. Efficiency of predicting random genetic effects for the genetic models by the MINQUE (0/1) procedure was compared with the best linear unbiased prediction (BLUP). A worked example is given to illustrate the use of the reduced genetic model for kernel growth characteristics in corn (Zea mays L.).     

Analysis of cytoplasmic and maternal effects I. A genetic model for diploid plant seeds and animals

A genetic model for modified diallel crosses is proposed for estimating variance and covariance components of cytoplasmic, maternal additive and dominance effects, as well as direct additive and dominance effects. Monte Carlo simulations were conducted to compare the efficiencies of minimum norm quadratic unbiased estimation (MINQUE) methods. For both balanced and unbalanced mating designs, MINQUE (0/1), which has 0 for all the prior covariances and 1 for all the prior variances, has similar efficiency to MINQUE(), which has parameter values for the prior values. Unbiased estimates of variance and covariance components and their sampling variances could be obtained with MINQUE(0/1) and jackknifing. A t-test following jackknifing is applicable to test hypotheses for zero variance and covariance components. The genetic model is robust for estimating variance and covariance components under several situations of no specific effects. A MINQUE(0/1) procedure is suggested for unbiased estimation of covariance components between two traits with equal design matrices. Methods of unbiased prediction for random genetic effects are discussed. A linear unbiased prediction (LUP) method is shown to be efficient for the genetic model. An example is given for a demonstration of estimating variance and covariance components and predicting genetic effects.     

Diallel analysis for sex-linked and maternal effects

Genetic models including sex-linked and maternal effects as well as autosomal gene effects are described. Monte Carlo simulations were conducted to compare efficiencies of estimation by minimum norm quadratic unbiased estimation (MINQUE) and restricted maximum likelihood (REML) methods. MINQUE(1), which has 1 for all prior values, has a similar efficiency to MINQUE(), which requires prior estimates of parameter values. MINQUE(1) has the advantage over REML of unbiased estimation and convenient computation. An adjusted unbiased prediction (AUP) method is developed for predicting random genetic effects. AUP is desirable for its easy computation and unbiasedness of both mean and variance of predictors. The jackknife procedure is appropriate for estimating the sampling variances of estimated variances (or covariances) and of predicted genetic effects. A t-test based on jackknife variances is applicable for detecting significance of variation. Worked examples from mice and silkworm data are given in order to demonstrate variance and covariance estimation and genetic effect prediction.     

Estimating effects of a single gene and polygenes on quantitative traits from a diallel design

A genetic model is developed with additive and dominance effects of a single gene and polygenes as well as general and specific reciprocal effects for the progeny from a diallel mating design. The methods of ANOVA, minimum norm quadratic unbiased estimation (MINQUE), restricted maximum likelihood estimation (REML), and maximum likelihood estimation (ML) are suggested for estimating variance components, and the methods of generalized least squares (GLS) and ordinary least squares (OLS) for fixed effects, while best linear unbiased prediction, linear unbiased prediction (LUP), and adjusted unbiased prediction are suggested for analyzing random effects. Monte Carlo simulations were conducted to evaluate the unbiasedness and efficiency of statistical methods involving two diallel designs with commonly used sample sizes, 6 and 8 parents, with no and missing crosses, respectively. Simulation results show that GLS and OLS are almost equally efficient for estimation of fixed effects, while MINQUE (1) and REML are better estimators of the variance components and LUP is most practical method for prediction of random effects. Data from a Drosophila melanogaster experiment (Gilbert 1985a, Theor appl Genet 69:625–629) were used as a working example to demonstrate the statistical analysis. The new methodology is also applicable to screening candidate gene(s) and to other mating designs with multiple parents, such as nested (NC Design I) and factorial (NC Design II) designs. Moreover, this methodology can serve as a guide to develop new methods for detecting indiscernible major genes and mapping quantitative trait loci based on mixture distribution theory. The computer program for the methods suggested in this article is freely available from the authors.     

Analysis of genetic effects of major genes and polygenes on quantitative traits. II. Genetic models for seed traits of crops

Genetic models for quantitative seed traits with effects of several major genes and polygenes, as well as their GE interaction, were proposed. Mixed linear model approaches were suggested for analyzing the genetic models. Monte Carlo simulations were conducted to evaluate unbiasedness and efficiency for estimating fixed effects and variance components of the embryo and the endosperm models, including effects of a major gene from an unbalanced modified diallel mating design with nine parents, respectively. Simulation results showed that estimates of generalized least squares (GLS) were unbiased and efficient, while those of ordinary least squares (OLS) were almost as good as GLS. Minimum norm quadratic unbiased estimation (MINQUE) could obtain unbiased estimates of the variance components. It was also suggested that precision of MINQUE estimation would be improved with augmentation of experimental size. Data from a modified diallel design in upland cotton ( Gossypium hirsutum L.) were used as a worked example to illustrate the parameter estimation.     

On selection criteria and estimation of parameters when the variance is heterogeneous

Summary Procedures for ranking candidates for selection and for estimating genetic and environmental parameters when variances are heterogeneous are discussed. The best linear unbiased predictor (BLUP) accounts automatically for heterogeneous variance provided that the covariance structure is known and that the assumptions of the model hold. Under multivariate normality BLUP allowing for heterogeneous variance maximizes expected genetic progress. Examples of application of BLUP to selection when residual or genetic variances are heterogeneous are given. Restricted maximum likelihood estimation of heterogeneous variances and covariances via the expectation-maximization algorithm is presented.     

Analysis of genetic effects of major genes and polygenes on quantitative traits I. Genetic model for diploid plants and animals

A genetic model was proposed to simultaneously investigate genetic effects of both polygenes and several single genes for quantitative traits of diploid plants and animals. Mixed linear model approaches were employed for statistical analysis. Based on two mating designs, a full diallel cross and a modified diallel cross including F₂, Monte Carlo simulations were conducted to evaluate the unbiasedness and efficiency of the estimation of generalized least squares (GLS) and ordinary least squares (OLS) for fixed effects and of minimum norm quadratic unbiased estimation (MINQUE) and Henderson III for variance components. Estimates of MINQUE (1) were unbiased and efficient in both reduced and full genetic models. Henderson III could have a large bias when used to analyze the full genetic model. Simulation results also showed that GLS and OLS were good methods to estimate fixed effects in the genetic models. Data on Drosophila melanogaster from Gilbert were used as a worked example to demonstrate the parameter estimation. Received: 11 November 2000 / Accepted: 2 May 2001     

Accurate genetic and environmental covariance estimation with composite likelihood in genome-wide association studies

Genetic and environmental covariances between pairs of complex traits are important quantitative measurements that characterize their shared genetic and environmental architectures. Accurate estimation of genetic and environmental covariances in genome-wide association studies (GWASs) can help us identify common genetic and environmental factors associated with both traits and facilitate the investigation of their causal relationship. Genetic and environmental covariances are often modeled through multivariate linear mixed models. Existing algorithms for covariance estimation include the traditional restricted maximum likelihood (REML) method and the recent method of moments (MoM). Compared to REML, MoM approaches are computationally efficient and require only GWAS summary statistics. However, MoM approaches can be statistically inefficient, often yielding inaccurate covariance estimates. In addition, existing MoM approaches have so far focused on estimating genetic covariance and have largely ignored environmental covariance estimation. Here we introduce a new computational method, GECKO, for estimating both genetic and environmental covariances, that improves the estimation accuracy of MoM while keeping computation in check. GECKO is based on composite likelihood, relies on only summary statistics for scalable computation, provides accurate genetic and environmental covariance estimates across a range of scenarios, and can accommodate SNP annotation stratified covariance estimation. We illustrate the benefits of GECKO through simulations and applications on analyzing 22 traits from five large-scale GWASs. In the real data applications, GECKO identified 50 significant genetic covariances among analyzed trait pairs, resulting in a twofold power gain compared to the previous MoM method LDSC. In addition, GECKO identified 20 significant environmental covariances. The ability of GECKO to estimate environmental covariance in addition to genetic covariance helps us reveal strong positive correlation between the genetic and environmental covariance estimates across trait pairs, suggesting that common pathways may underlie the shared genetic and environmental architectures between traits.     

Estimators of repeatability

Summary Analysis of variance and principal components methods have been suggested for estimating repeatability. In this study, six estimation procedures are compared: ANOVA, principal components based on the sample covariance matrix and also on the sample correlation matrix, a related multivariate method (structural analysis) based on the sample covariance matrix and also on the sample correlation matrix, and maximum likelihood estimation. A simulation study indicates that when the standard linear model assumptions are met, the estimators are quite similar except when the repeatability is small. Overall, maximum likelihood appears the preferred method. If the assumption of equal variance is relaxed, the methods based on the sample correlation matrix perform better although others are surprisingly robust. The structural analysis method (with sample correlation matrix) appears to be best.Paper number 776 from the Department of Meat and Animal Science, University of Wisconsin-Madison.     

Analysis of Conditional Genetic Effects and Variance Components in Developmental Genetics

A genetic model with additive-dominance effects and genotype X environment interactions is presented for quantitative traits with time-dependent measures. The genetic model for phenotypic means at time t conditional on phenotypic means measured at previous time (t - 1) is defined. Statistical methods are proposed for analyzing conditional genetic effects and conditional genetic variance components. Conditional variances can be estimated by minimum norm quadratic unbiased estimation (MINQUE) method. An adjusted unbiased prediction (AUP) procedure is suggested for predicting conditional genetic effects. A worked example from cotton fruiting data is given for comparison of unconditional and conditional genetic variances and additive effects.     

Bayesian adaptive Markov chain Monte Carlo estimation of genetic parameters

Accurate and fast estimation of genetic parameters that underlie quantitative traits using mixed linear models with additive and dominance effects is of great importance in both natural and breeding populations. Here, we propose a new fast adaptive Markov chain Monte Carlo (MCMC) sampling algorithm for the estimation of genetic parameters in the linear mixed model with several random effects. In the learning phase of our algorithm, we use the hybrid Gibbs sampler to learn the covariance structure of the variance components. In the second phase of the algorithm, we use this covariance structure to formulate an effective proposal distribution for a Metropolis-Hastings algorithm, which uses a likelihood function in which the random effects have been integrated out. Compared with the hybrid Gibbs sampler, the new algorithm had better mixing properties and was approximately twice as fast to run. Our new algorithm was able to detect different modes in the posterior distribution. In addition, the posterior mode estimates from the adaptive MCMC method were close to the REML (residual maximum likelihood) estimates. Moreover, our exponential prior for inverse variance components was vague and enabled the estimated mode of the posterior variance to be practically zero, which was in agreement with the support from the likelihood (in the case of no dominance). The method performance is illustrated using simulated data sets with replicates and field data in barley.     

Complex genetic effects in quantitative trait locus identification: a computationally tractable random model for use in F(2) populations

Methodology for mapping quantitative trait loci (QTL) has focused primarily on treating the QTL as a fixed effect. These methods differ from the usual models of genetic variation that treat genetic effects as random. Computationally expensive methods that allow QTL to be treated as random have been explicitly developed for additive genetic and dominance effects. By extending these methods with a variance component method (VCM), multiple QTL can be mapped. We focused on an F(2) crossbred population derived from inbred lines and estimated effects for each individual and their corresponding marker-derived genetic covariances. We present extensions to pairwise epistatic effects, which are computationally intensive because a great many individual effects must be estimated. But by replacing individual genetic effects with average genetic effects for each marker class, genetic covariances are approximated. This substantially reduces the computational burden by reducing the dimensions of covariance matrices of genetic effects, resulting in a remarkable gain in the speed of estimating the variance components and evaluating the residual log-likelihood. Preliminary results from simulations indicate competitiveness of the reduced model with multiple-interval mapping, regression interval mapping, and VCM with individual genetic effects in its estimated QTL positions and experimental power.     

Covariance between relatives in multibreed populations: additive model

Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (RLF) and 35-0367 (MG)     

THE MIXED-MODEL ANALYSIS OF VARIANCE APPLIED TO QUANTITATIVE GENETICS: BIOLOGICAL MEANING OF THE PARAMETERS

The mixed-model factorial analysis of variance has been used in many recent studies in evolutionary quantitative genetics. Two competing formulations of the mixed-model ANOVA are commonly used, the “Scheffe” model and the “SAS” model; these models differ in both their assumptions and in the way in which variance components due to the main effect of random factors are defined. The biological meanings of the two variance component definitions have often been unappreciated, however. A full understanding of these meanings leads to the conclusion that the mixed-model ANOVA could have been used to much greater effect by many recent authors. The variance component due to the random main effect under the two-way SAS model is the covariance in true means associated with a level of the random factor (e.g., families) across levels of the fixed factor (e.g., environments). Therefore the SAS model has a natural application for estimating the genetic correlation between a character expressed in different environments and testing whether it differs from zero. The variance component due to the random main effect under the two-way Scheffe model is the variance in marginal means (i.e., means over levels of the fixed factor) among levels of the random factor. Therefore the Scheffe model has a natural application for estimating genetic variances and heritabilities in populations using a defined mixture of environments. Procedures and assumptions necessary for these applications of the models are discussed. While exact significance tests under the SAS model require balanced data and the assumptions that family effects are normally distributed with equal variances in the different environments, the model can be useful even when these conditions are not met (e.g., for providing an unbiased estimate of the across-environment genetic covariance). Contrary to statements in a recent paper, exact significance tests regarding the variance in marginal means as well as unbiased estimates can be readily obtained from unbalanced designs with no restrictive assumptions about the distributions or variance-covariance structure of family effects.     

A Fast Method that Uses Polygenic Scores to Estimate the Variance Explained by Genome-wide Marker Panels and the Proportion of Variants Affecting a Trait

Several methods have been proposed to estimate the variance in disease liability explained by large sets of genetic markers. However, current methods do not scale up well to large sample sizes. Linear mixed models require solving high-dimensional matrix equations, and methods that use polygenic scores are very computationally intensive. Here we propose a fast analytic method that uses polygenic scores, based on the formula for the non-centrality parameter of the association test of the score. We estimate model parameters from the results of multiple polygenic score tests based on markers with p values in different intervals. We estimate parameters by maximum likelihood and use profile likelihood to compute confidence intervals. We compare various options for constructing polygenic scores, based on nested or disjoint intervals of p values, weighted or unweighted effect sizes, and different numbers of intervals, in estimating the variance explained by a set of markers, the proportion of markers with effects, and the genetic covariance between a pair of traits. Our method provides nearly unbiased estimates and confidence intervals with good coverage, although estimation of the variance is less reliable when jointly estimated with the covariance. We find that disjoint p value intervals perform better than nested intervals, but the weighting did not affect our results. A particular advantage of our method is that it can be applied to summary statistics from single markers, and so can be quickly applied to large consortium datasets. Our method, named AVENGEME (Additive Variance Explained and Number of Genetic Effects Method of Estimation), is implemented in R software.     

An alternative derivation of Harville's restricted log likelihood function for variance component estimation

Estimation of variance components in linear mixed models is important in clinical trial and longitudinal data analysis. It is also important in animal and plant breeding for accurately partitioning total phenotypic variance into genetic and environmental variances. Restricted maximum likelihood (REML) method is often preferred over the maximum likelihood (ML) method for variance component estimation because REML takes into account the lost degree of freedom resulting from estimating the fixed effects. The original restricted likelihood function involves a linear transformation of the original response variable (a collection of error contrasts). Harville's final form of the restricted likelihood function does not involve the transformation and thus is much easier to manipulate than the original restricted likelihood function. There are several different ways to show that the two forms of the restricted likelihood are equivalent. In this study, I present a much simpler way to derive Harville's restricted likelihood function. I first treat the fixed effects as random effects and call such a mixed model a pseudo random model (PDRM). I then construct a likelihood function for the PDRM. Finally, I let the variance of the pseudo random effects be infinity and show that the limit of the likelihood function of the PDRM is the restricted likelihood function.     

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.     

A mixed model method to predict QTL-cluster effects using trait and marker information in a multi-group population.

In this study, a mixed model method using trait phenotype and marker information was developed for genetic evaluation of animals in a crossbred population originated from several founder genetic groups. The situation in which a cluster of QTLs is located in a particular chromosome region and is marked by two flanking markers is considered. With this method, the conditional expectation of the identity-by-descent proportion for the QTL-cluster marked and the genetic variances and covariances, given genetic group and marker information, are properly taken into account. The structure of segregation variance used in this method is different from that in the case of a single QTL marked. The current method provides best linear unbiased estimation of the relevant fixed effects and best linear unbiased prediction of the additive effects for the QTL-cluster marked and of the additive effects of the remaining polygenes. A small numerical example is given to illustrate the current prediction procedure.     

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     

非等试验设计胚乳品质性状的遗传分析方法

根据朱军(1996)提出的包括基因型×环境互作的胚乳品质性状三倍体遗传模型,运用蒙特卡罗模拟证明,以混合线性模型统计分析的MINQUE法,对非等试验设计获得的实验数据进行数量遗传分析是可行的.蒙特卡罗模拟结果表明在样本群体大小基本一致的条件下,采用相等试验设计或非等试验设计所估算的遗传参数的偏差(Blas)和功效值(Power)没有明显差异,表明以非等试验设计获得的非平衡数据也可用来进行遗传分析,估算上述遗传模型中的各项遗传方差分量和协方差分量,并且可以采用朱军(1993)提出的AUP法来预测遗传模型中的各项遗传效应值.