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     

How to cope with negative estimates of components of variance in ecological field studies

Ecologists often need to estimate components of spatial or temporal variation. The most widely used method in ecology uses the observed and expected mean squares in an analysis of variance. A more general approach, which can be used for balanced and unbalanced designs, is based on residual maximal likelihood (REML). This method is less well known by ecologists and requires specialist software. If the design is balanced, the two methods are equivalent, except for one important respect: estimates from analysis of variance can be negative whereas REML estimates cannot. The purpose of this note is to point out a simple modification to the analyses of variance which yields the same estimates as REML for many of the designs commonly used in ecological studies. This modification has been available in the mathematical literature for over 30 years, but appears not to be well known amongst ecologists. It is useful in many cases of balanced analytical designs.     

Closed-Form Approximations to the REML Estimator of a Variance Ratio (or Heritability) in a Mixed Linear Model

In this article, we estimate heritability or intraclass correlation in a mixed linear model having two sources of variation. In most applications, the commonly used restricted maximum likelihood (REML) estimator can only be obtained via an iterative approach. In some cases, the algorithm used to compute REML estimates may be slow or may even fail to converge. We develop a set of closed-form approximations to the REML estimator, and the performance of these estimators is compared with that of the REML estimator. We provide guidelines regarding how to choose the estimator that best approximates the REML estimator. Examples presented in the article suggest that the closed-form estimators compete with and, in some cases, outperform the REML estimator.     

SNP-based heritability estimation using a Bayesian approach

Heritability is a central element in quantitative genetics. New molecular markers to assess genetic variance and heritability are continually under development. The availability of molecular single nucleotide polymorphism (SNP) markers can be applied for estimation of variance components and heritability on population, where relationship information is unknown. In this study, we evaluated the capabilities of two Bayesian genomic models to estimate heritability in simulated populations. The populations comprised different family structures of either no or a limited number of relatives, a single quantitative trait, and with one of two densities of SNP markers. All individuals were both genotyped and phenotyped. Results illustrated that the two models were capable of estimating heritability, when true heritability was 0.15 or higher and populations had a sample size of 400 or higher. For heritabilities of 0.05, all models had difficulties in estimating the true heritability. The two Bayesian models were compared with a restricted maximum likelihood (REML) approach using a genomic relationship matrix. The comparison showed that the Bayesian approaches performed equally well as the REML approach. Differences in family structure were in general not found to influence the estimation of the heritability. For the sample sizes used in this study, a 10-fold increase of SNP density did not improve precision estimates compared with set-ups with a less dense distribution of SNPs. The methods used in this study showed that it was possible to estimate heritabilities on the basis of SNPs in animals with direct measurements. This conclusion is valuable in cases when quantitative traits are either difficult or expensive to measure.     

Effects of data imbalance on estimation of heritability

Summary Effects of data imbalance on bias, sampling variance and mean square error of heritability estimated with variance components were examined using a random two-way nested classification. Four designs, ranging from zero imbalance (balanced data) to low, medium and high imbalance, were considered for each of four combinations of heritability (h²=0.2 and 0.4) and sample size (N=120 and 600). Observations were simulated for each design by drawing independent pseudo-random deviates from normal distributions with zero means, and variances determined by heritability. There were 100 replicates of each simulation; the same design matrix was used in all replications. Variance components were estimated by analysis of variance (Henderson's Method 1) and by maximum likelihood (ML). For the design and model used in this study, bias in heritability based on Method 1 and ML estimates of variance components was negligible. Effect of imbalance on variance of heritability was smaller for ML than for Method 1 estimation, and was smaller for heritability based on estimates of sire-plus-dam variance components than for heritability based on estimates of sire or dam variance components. Mean square error for heritability based on estimates of sire-plus-dam variance components appears to be less sensitive to data imbalance than heritability based on estimates of sire or dam variance components, especially when using Method 1 estimation. Estimation of heritability from sire-plus-dam components was insensitive to differences in data imbalance, especially for the larger sample size.Supported by grants from the Illinois Agricultural Experiment Station and the University of Illinois Research Board. Charles Smith, H. W. Norton and D. Gianola contributed valuable suggestions     

(Co)Variance components and genetic parameter estimates for growth traits in Moghani sheep

Genetic parameters were estimated for birth weight (BW), weaning weight (WW), yearling weight (YW), average daily gain from birth to weaning (ADG1) and average daily gain from weaning to yearling (ADG2) in Moghani sheep. Maximum number of data was 4237 at birth, but only 1389 records at yearling were investigated. The data was collected from 1995 to 2007 at the Breeding Station of Moghani sheep in Jafarabad, Moghan, Iran. (Co)Variance components and genetic parameters were estimated with different models which including direct effects, with and without maternal additive genetic effects as well as maternal permanent environmental effects using restricted maximum likelihood (REML) method. The most appropriate model for each trait was determined based on likelihood ratio tests and Akaike's Information Criterion (AIC). Maternal effects were important only for pre-weaning traits. Direct heritability estimates for BW, ADG1, WW, ADG2 and YW were 0.07, 0.08, 0.09, 0.09 and 0.17, respectively. Fractions of variance due to maternal permanent environmental effects on phenotypic variance were 0.08 for ADG1. Maternal heritability estimates for BW and WW were 0.18 and 0.06, respectively. Multivariate analysis was performed using the most appropriate models obtained in univariate analysis. Direct genetic correlations among studied traits were positive and ranged from 0.37 for BW–ADG2 to 0.85 for ADG1–YW. Maternal genetic correlation estimate between BW and WW was 0.33. Phenotypic and environmental correlation estimates were generally lower than those of genetic correlation. Low direct heritability estimates imply that mass selection for these traits results in slow genetic gain.     

Comparison of REML and Gibbs sampling estimates of multi-trait genetic parameters in Scots pine

Multi-trait (co)variance estimation is an important topic in plant and animal breeding. In this study we compare estimates obtained with restricted maximum likelihood (REML) and Bayesian Gibbs sampling of simulated data and of three traits (diameter, height and branch angle) from a 26-year-old partial diallel progeny test of Scots pine (Pinus sylvestris L.). Based on the results from the simulated data we can conclude that the REML estimates are accurate but the mode of posterior distributions from the Gibbs sampling can be overestimated depending on the level of the heritability. The mean and median of the posteriors were considerably higher than the expected values of the heritabilities. The confidence intervals calculated with the delta method were biased downwardly. The highest probablity density (HPD) interval provides a better interval estimate, but could be slightly biased at the lower level. Similar differences between REML and Gibbs sampling estimates were found for the Scots pine data. We conclude that further simulation studies are needed in order to evaluate the effect of different priors on (co)variance components in the genetic individual model.     

The effects of selection for gain in mice on the direct-maternal genetic correlation

Components of genetic variation for postweaning growth traits were estimated for both control and growth stocks of mice. The effect of phenotypic selection for gain, which genetically combines selection for additive direct and maternal effects, on additive genetic variance components, heritability, and additive genetic correlationsis discussed. Quantitative genetic theory predicts that simultaneous selection for two metric traits in the same direction will cause the genetic correlation between the two traits to become more negative. The results presented in this paper conflict with this theory. The direct-maternal additive genetic correlation was more negative in the control line (with 356 mice) than in the growth-selected line (with 320 mice) for the three traits analyzed (0.310 vs 0.999 for 21-day weight, 0.316 vs 1.000 for 42-day weight, and 0.506 vs 1.000 for gain from 21–42 days). Estimates were obtained by restricted maximum likelihood (REML) computed under a derivative free algorithm (DFREML).     

Estimation of heritability using variety trials data from incomplete blocks

An assessment of the heritability of a trait is useful in formulating a breeding strategy for crop improvement. We have considered the estimation of broad-sense heritability from a single-location trial and from multi-locational trials conducted in incomplete blocks. Using residual maximum likelihood estimates of variance components, we estimated the heritability and obtained expressions for the estimate of its bias and its standard error. The estimation procedure is illustrated for 25 barley genotypes evaluated at four locations in West Asia and North Africa during 1992.     

Genetic parameters of coat colour in golden fox (Vulpes vulpes L.)

Genetic parameters (heritability, genetic and phenotypic correlations) of chosen coat colour traits of golden fox were estimated. 1013 animals, born on the Sniaty fox farm (Poland) in 1985-1999 were evaluated. In 1993-1999 colour type was additionally assessed for 833 animals, by detailed evaluation of coat colour on the back and sides of the body, throat colour, belly colour and the amount of silver hair. The REML method was used to estimate genetic (co)variance components. Data were transformed using the probit transformation. Heritability estimates for coat traits were low (0.04 to 0.22). Values of most of the estimated genetic parameters (h2, rG, rP) were comparable to those frequently reported for other colour types of silver fox.     

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.     

How does selfing affect the genetic variance of quantitative traits? An updated meta‐analysis on empirical results in angiosperm species

Most theoretical works predict that selfing should reduce the level of additive genetic variance available for quantitative traits within natural populations. Despite a growing number of quantitative genetic studies undertaken during the last two decades, this prediction is still not well supported empirically. To resolve this issue and confirm or reject theoretical predictions, we reviewed quantitative trait heritability estimates from natural plant populations with different rates of self‐fertilization and carried out a meta‐analysis. In accordance with models of polygenic traits under stabilizing selection, we found that the fraction of additive genetic variance is negatively correlated with the selfing rate. Although the mating system explains a moderate fraction of the variance, the mean reduction of narrow‐sense heritability values between strictly allogamous and predominantly selfing populations is strong, around 60%. Because some nonadditive components of genetic variance become selectable under inbreeding, we determine whether self‐fertilization affects the relative contribution of these components to genetic variance by comparing narrow‐sense heritability estimates from outcrossing populations with broad‐sense heritability estimated in autogamous populations. Results suggest that these nonadditive components of variance may restore some genetic variance in predominantly selfing populations; it remains, however, uncertain how these nonadditive components will contribute to adaptation.     

Theory for the number of genes affecting quantitative characters

Summary A general expression for gene number estimation which encompasses the conventional formula was derived. It provides a basis for gene number estimation from the data of recurrent selection experiments that are not of sufficient duration to measure total response to selection.Gene number estimates are considerably more reliable when heritability is high. The effect of heritability on sampling variance is particularly important when gene number is large.Generally the most effective ways of decreasing the variance of a gene number estimate will be 1) to increase the number of generations in a primary selection program, 2) to increase the number of generations in the two way selection program and 3) to increase population size.From a thesis submitted by the author in partial fulfillment of the requirements for the Ph.D. degree. Received March, 1975. Work supported by Public Health Service Grant GM 16074, by the Minnesota Agricultural Experiment Station and by National Institutes of Environmental Health Sciences Grant No. 5T32 ES07011-02.Former Research Assistant, Genetics and Cell Biology, University of Minnesota; Currently Post-doctoral Fellow in Environmental Health Measurement and Statistics.     

Can dominance genetic variance be ignored in evolutionary quantitative genetic analyses of wild populations?

Accurately estimating genetic variance components is important for studying evolution in the wild. Empirical work on domesticated and wild outbred populations suggests that dominance genetic variance represents a substantial part of genetic variance, and theoretical work predicts that ignoring dominance can inflate estimates of additive genetic variance. Whether this issue is pervasive in natural systems is unknown, because we lack estimates of dominance variance in wild populations obtained in situ. Here, we estimate dominance and additive genetic variance, maternal variance, and other sources of nongenetic variance in eight traits measured in over 9000 wild nestlings linked through a genetically resolved pedigree. We find that dominance variance, when estimable, does not statistically differ from zero and represents a modest amount (2-36%) of genetic variance. Simulations show that (1) inferences of all variance components for an average trait are unbiased; (2) the power to detect dominance variance is low; (3) ignoring dominance can mildly inflate additive genetic variance and heritability estimates but such inflation becomes substantial when maternal effects are also ignored. These findings hence suggest that dominance is a small source of phenotypic variance in the wild and highlight the importance of proper model construction for accurately estimating evolutionary potential.     

Detection of significant disease risks using a spatial conditional autoregressive model

SUMMARY: The conditional autoregressive (CAR) model is widely used to describe the geographical distribution of a specific disease risk in lattice mapping. Successful developments based on frequentist and Bayesian procedures have been extensively applied to obtain two-stage disease risk predictions at the subregional level. Bayesian procedures are preferred for making inferences, as the posterior standard errors (SE) of the two-stage prediction account for the variability in the variance component estimates; however, some recent work based on frequentist procedures and the use of bootstrap adjustments for the SE has been undertaken. In this article we investigate the suitability of an analytical adjustment for disease risk inference that provides accurate interval predictions by using the penalized quasilikelihood (PQL) technique to obtain model parameter estimates. The method is a first-order approximation of the naive SE based on a Taylor expansion and is interpreted as a conditional measure of variability providing conditional calibrated prediction intervals, given the data. We conduct a simulation study to demonstrate how the method can be used to estimate the specific subregion risk by interval. We evaluate the proposed methodology by analyzing the commonly used example data set of lip cancer incidence in the 56 counties of Scotland for the period 1975-1980. This evaluation reveals a close similarity between the solutions provided by the method proposed here and those of its fully Bayesian counterpart.     

REML estimation of variance parameters in nonlinear mixed effects models using the SAEM algorithm

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken.     

Reliability of Confidence Interval Estimators under Various Nested Design Parental Sample Sizes

The coverage probabilities of several confidence limit estimators of genetic parameters, obtained from North Carolina I designs, were assessed by means of Monte Carlo simulations. The reliability of the estimators was compared under three different parental sample sizes. The coverage of confidence intervals set on the Normal distribution, and using standard errors either computed by the “delta” method or derived using an approximation for the variance of a variance component estimated by means of a linear combination of mean squares, was affected by the number of males and females included in the experiment. The “delta” method was found to provide reliable standard errors of the genetic parameters only when at least 48 males were each mated to six different females randomly selected from the reference population. Formulae are provided for obtaining “delta” method standard errors, and appropriate statistical software procedures are discussed. The error rates of confidence limits based on the Normal distribution and using standard errors obtained by an approximation for the variance of a variance component varied widely. The coverage of F-distribution confidence intervals for heritability estimates was not significantly affected by parental sample size and consistently provided a mean coverage near the stated coverage. For small parental sample sizes, confidence intervals for heritability estimates should be based on the F-distribution.     

Testing for linkage under robust genetic models.

Robust genetic models are used to assess linkage between a quantitative trait and genetic variation at a specific locus using allele-sharing data. Little is known about the relative performance of different possible significance tests under these models. Under the robust variance components model approach there are several alternatives: standard Wald and likelihood ratio tests, a quasilikelihood Wald test, and a Monte Carlo test. This paper reports on the relative performance (significance level and power) of the robust sibling pair test and the different alternatives under the robust variance components model. Simulations show that (1) for a fixed sample size of nuclear families, the variance components model approach is more powerful than the robust sibling pair approach; (2) when the number of nuclear families is at least approximately 100 and heritability at the trait locus is moderate to high (>0.20) all tests based on the variance components model are equally effective; (3) when the number of nuclear families is less than approximately 100 or heritability at the trait locus is low (<0. 20), on balance, the Monte Carlo test provides the best power and is the most valid. The different testing procedures are applied to determine which are able to detect the known association between low density lipoprotein cholesterol and the common genotypes at the locus encoding apolipoprotein E. Results from this application show that the robust sibling pair method may be more effective in practice than that indicated by simulations.     

混合家系遗传参数估计方法研究

针对混合家系遗传参数估计,本文在假定公畜方差组分和母畜方差组分相等这一理论基础上,通过对方差分析的期望均方组成分析,提出了新的遗传力估计方法,以及某些特殊情况下的近似估计方法。通过一个估测实例比较了几种遗传力估计方法,结果表明,本文方法与全同胞组分估计最为接近,而且遗传力标准误最小,本文近似估计方法的效果也较好。对各种方法而言,资料越不平衡其差异越大。本文方法可以在一定程度上弥补全同胞分析时,因实际资料的公母畜方差组分差异过大的缺陷,具有实际可行性。此外,由于本文方法是用单因方差分析解决二因方差分析问题,计算更为简便,并可免于计算混合家系平均亲缘相关系数。     

Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996