Genetic heterogeneity of residual variance in broiler chickens

Aims were to estimate the extent of genetic heterogeneity in environmental variance. Data comprised 99 535 records of 35-day body weights from broiler chickens reared in a controlled environment. Residual variance within dam families was estimated using ASREML, after fitting fixed effects such as genetic groups and hatches, for each of 377 genetically contemporary sires with a large number of progeny (> 100 males or females each). Residual variance was computed separately for male and female offspring, and after correction for sampling, strong evidence for heterogeneity was found, the standard deviation between sires in within variance amounting to 15–18% of its mean. Reanalysis using log-transformed data gave similar results, and elimination of 2–3% of outlier data reduced the heterogeneity but it was still over 10%. The correlation between estimates for males and females was low, however. The correlation between sire effects on progeny mean and residual variance for body weight was small and negative (-0.1). Using a data set bigger than any yet presented and on a trait measurable in both sexes, this study has shown evidence for heterogeneity in the residual variance, which could not be explained by segregation of major genes unless very few determined the trait.     

Responses of alloplasmic (cytoplasm=Triticum timopheevii) and euplasmic wheats (Triticum aestivum) to photoperiod and vernalization

Summary Heritability estimated from sire family variance components, ignoring dams, pools conventional paternal and maternal half sib estimates, in a way which is biased upward, and sub-optimal for minimizing the sampling variance. Standard error of a sire family estimate will be smaller than that of the equivalent paternal half sib estimate, but not as small as that of an estimate obtained by optimal pooling of paternal and maternal half sib estimates. If only additive genetic variance components are significant, the bias may be removed by use of a computed average genetic relationship for sire families, in place of a nominal R = 0.25. Average genetic relationship may be computed from mean and variance of dam family size within sire families. If dominance, epistatic, or maternal components are significant, this simple correction is not appropriate. In situations likely to be encountered in large domestic species such as sheep and cattle (dam family size small and uniform) bias will be negligible. The method could be useful where cost of dam identification is a limiting factor.     

Estimation of average heterozygosity and genetic distance from a small number of individuals

The magnitudes of the systematic biases involved in sample heterozygosity and sample genetic distances are evaluated, and formulae for obtaining unbiased estimates of average heterozygosity and genetic distance are developed. It is also shown that the number of individuals to be used for estimating average heterozygosity can be very small if a large number of loci are studied and the average heterozygosity is low. The number of individuals to be used for estimating genetic distance can also be very small if the genetic distance is large and the average heterozygosity of the two species compared is low.     

Maximum likelihood estimation of human craniometric heritabilities

This study presents univariate narrow-sense heritability estimates for 33 common craniometric dimensions, calculated using the maximum likelihood variance components method on a skeletal sample of 298 pedigreed individuals from Hallstatt, Austria. Quantitative genetic studies that use skeletal cranial measurements as a basis for inferring microevolutionary processes in human populations usually employ heritability estimates to represent the genetic variance of the population. The heritabilities used are often problematic: most come from studies of living humans, and/or they were calculated using statistical techniques or assumptions violated by human groups. Most bilateral breadth measures in the current study show low heritability estimates, while cranial length and height measures have heritability values ranging between 0.102-0.729. There appear to be differences between the heritabilities calculated from crania and those from anthropometric studies of living humans, suggesting that the use of the latter in quantitative genetic models of skeletal data may be inappropriate. The univariate skeletal heritability estimates seem to group into distinct regions of the cranium, based on their relative values. The most salient group of measurements is for the midfacial/orbital region, with a number of measures showing heritabilities less than 0.30. Several possible reasons behind this pattern are examined. Given the fact that heritabilities calculated on one population should not be applied to others, suggestions are made for the use of the data presented.     

Competition can maintain genetic but not environmental variance in the presence of stabilizing selection

A population in which there is stabilizing selection acting on quantitative traits toward an intermediate optimum becomes monomorphic in the absence of mutation. Further, genotypes that show least environmental variation are also favored, such that selection is likely to reduce both genetic and environmental components of phenotypic variance. In contrast, intraspecific competition for resources is more severe between phenotypically similar individuals, such that those deviating from prevailing phenotypes have a selective advantage. It has been shown previously that polymorphism and phenotypic variance can be maintained if competition between individuals is "effectively" stronger than stabilizing selection. Environmental variance is generally observed in quantitative traits, so mechanisms to explain its maintenance are sought, but the impact of competition on its magnitude has not previously been studied. Here we assume that a quantitative trait is subject to selection for an optimal value and to selection due to competition. Further, we assume that both the mean and variance of the phenotypic value depend on genotype, such that both may be affected by selection. Theoretical analysis and numerical simulations reveal that environmental variance can be maintained only when the genetic variance (in mean phenotypic value) is constrained to a very low level. Environmental variance will be replaced entirely by genotypic variance if a range of genotypes that vary widely in mean phenotype are present or become so by mutation. The distribution of mean phenotypic values is discrete when competition is strong relative to stabilizing selection; but more genotypes segregate and the distribution can approach continuity as competition becomes extremely strong. If the magnitude of the environmental variance is not under genetic control, there is a complementary relationship between the levels of environmental and genetic variance such that the level of phenotypic variance is little affected.     

On the sampling variance of intraclass correlations and genetic correlations.

Widely used standard expressions for the sampling variance of intraclass correlations and genetic correlation coefficients were reviewed for small and large sample sizes. For the sampling variance of the intraclass correlation, it was shown by simulation that the commonly used expression, derived using a first-order Taylor series performs better than alternative expressions found in the literature, when the between-sire degrees of freedom were small. The expressions for the sampling variance of the genetic correlation are significantly biased for small sample sizes, in particular when the population values, or their estimates, are close to zero. It was shown, both analytically and by simulation, that this is because the estimate of the sampling variance becomes very large in these cases due to very small values of the denominator of the expressions. It was concluded, therefore, that for small samples, estimates of the heritabilities and genetic correlations should not be used in the expressions for the sampling variance of the genetic correlation. It was shown analytically that in cases where the population values of the heritabilities are known, using the estimated heritabilities rather than their true values to estimate the genetic correlation results in a lower sampling variance for the genetic correlation. Therefore, for large samples, estimates of heritabilities, and not their true values, should be used.     

Heritability and evolvability of fitness and nonfitness traits: Lessons from livestock

Data from natural populations have suggested a disconnection between trait heritability (variance standardized additive genetic variance, V_A) and evolvability (mean standardized V_A) and emphasized the importance of environmental variation as a determinant of trait heritability but not evolvability. However, these inferences are based on heterogeneous and often small datasets across species from different environments. We surveyed the relationship between evolvability and heritability in >100 traits in farmed cattle, taking advantage of large sample sizes and consistent genetic approaches. Heritability and evolvability estimates were positively correlated (r = 0.37/0.54 on untransformed/log scales) reflecting a substantial impact of V_A on both measures. Furthermore, heritabilities and residual variances were uncorrelated. The differences between this and previously described patterns may reflect lower environmental variation experienced in farmed systems, but also low and heterogeneous quality of data from natural populations. Similar to studies on wild populations, heritabilities for life‐history and behavioral traits were lower than for other traits. Traits having extremely low heritabilities and evolvabilities (17% of the studied traits) were almost exclusively life‐history or behavioral traits, suggesting that evolutionary constraints stemming from lack of genetic variability are likely to be most common for classical “fitness” (cf. life‐history) rather than for “nonfitness” (cf. morphological) traits.     

A note on the effect of observations with missing data on genetic correlation estimates

Summary Various studies have estimated covariance components as half the difference between the variance component of the sum of the variable values, for each observation, and the sum of the corresponding variable variance components. Although the variance components for the separate variables can be computed using all available data, the variance components of the sum can be computed only from those observations with records for both variables. Previous studies have suggested eliminating observations with missing data, because of possible selection bias. The effect of missing data on estimates of covariance components and genetic correlations was tested on sample beef cattle data and simulated data by randomly deleting differing proportions of records of one variable for each pair of variables analyzed. Estimates of genetic correlations computed with observations with missing data eliminated, were more accurate than estimates computed using all available data. Furthermore, when observations with missing data were included, estimates of genetic correlation far outside the parameter space were common. Therefore, this method should be used only if observations with missing data have been eliminated.     

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.     

A comparison of methods to estimate cross-environment genetic correlations

Advanced techniques for quantitative genetic parameter estimation may not always be necessary to answer broad genetic questions. However, simpler methods are often biased, and the extent of this determines their usefulness. In this study we compare family mean correlations to least squares and restricted error maximum likelihood (REML) variance component approaches to estimating cross-environment genetic correlations. We analysed empirical data from studies where both types of estimates were made, and from studies in our own laboratories. We found that the agreement between estimates was better when full-sib rather than half-sib estimates of cross-environment genetic correlations were used and when mean family size increased. We also note biases in REML estimation that may be especially important when testing to see if correlations differ from 0 or 1. We conclude that correlations calculated from family means can be used to test for the presence of genetic correlations across environments, which is sufficient for some research questions. Variance component approaches should be used when parameter estimation is the objective, or if the goal is anything other than determining broad patterns.     

Lack of nonadditive genetic effects on early fecundity in Drosophila melanogaster

Fecundity is usually considered as a trait closely connected to fitness and is expected to exhibit substantial nonadditive genetic variation and inbreeding depression. However, two independent experiments, using populations of different geographical origin, indicate that early fecundity in Drosophila melanogaster behaves as a typical additive trait of low heritability. The first experiment involved artificial selection in inbred and non-inbred lines, all of them started from a common base population previously maintained in the laboratory for about 35 generations. The realized heritability estimate was 0.151 +/- 0.075 and the inbreeding depression was very small and nonsignificant (0.09 +/- 0.09% of the non-inbred mean per 1% increase in inbreeding coefficient). With inbreeding, the observed decrease in the within-line additive genetic variance and the corresponding increase of the between-line variance were very close to their expected values for pure additive gene action. This result is at odds with previous studies showing inbreeding depression and, therefore, directional dominance for the same trait and species. All experiments, however, used laboratory populations, and it is possible that the original genetic architecture of the trait in nature was subsequently altered by the joint action of random drift and adaptation to captivity. Thus, we carried out a second experiment, involving inbreeding without artificial selection in a population recently collected from the wild. In this case we obtained, again, a maximum-likelihood heritability estimate of 0.210 +/- 0.027 and very little nonsignificant inbreeding depression (0.06 +/- 0.12%). The results suggest that, for fitness-component traits, low levels of additive genetic variance are not necessarily associated with large inbreeding depression or high levels of nonadditive genetic variance.     

Estimation of genetic variation in residual variance in female and male broiler chickens

In breeding programs, robustness of animals and uniformity of end product can be improved by exploiting genetic variation in residual variance. Residual variance can be defined as environmental variance after accounting for all identifiable effects. The aims of this study were to estimate genetic variance in residual variance of body weight, and to estimate genetic correlations between body weight itself and its residual variance and between female and male residual variance for broilers. The data sets comprised 26 972 female and 24 407 male body weight records. Variance components were estimated with ASREML. Estimates of the heritability of residual variance were in the range 0.029 (s.e. = 0.003) to 0.047 (s.e. = 0.004). The genetic coefficients of variation were high, between 0.35 and 0.57. Heritabilities were higher in females than in males. Accounting for heterogeneous residual variance increased the heritabilities for body weight as well. Genetic correlations between body weight and its residual variance were -0.41 (s.e. = 0.032) and -0.45 (s.e. = 0.040), respectively, in females and males. The genetic correlation between female and male residual variance was 0.11 (s.e. = 0.089), indicating that female and male residual variance are different traits. Results indicate good opportunities to simultaneously increase the mean and improve uniformity of body weight of broilers by selection.     

Sex differences in the genetic architecture of aggressiveness in a sexually dimorphic spider

Sex differences in the genetic architecture of behavioral traits can offer critical insight into the processes of sex‐specific selection and sexual conflict dynamics. Here, we assess genetic variances and cross‐sex genetic correlations of two personality traits, aggression and activity, in a sexually size‐dimorphic spider, Nuctenea umbratica. Using a quantitative genetic approach, we show that both traits are heritable. Males have higher heritability estimates for aggressiveness compared to females, whereas the coefficient of additive genetic variation and evolvability did not differ between the sexes. Furthermore, we found sex differences in the coefficient of residual variance in aggressiveness with females exhibiting higher estimates. In contrast, the quantitative genetic estimates for activity suggest no significant differentiation between males and females. We interpret these results with caution as the estimates of additive genetic variances may be inflated by nonadditive genetic effects. The mean cross‐sex genetic correlations for aggression and activity were 0.5 and 0.6, respectively. Nonetheless, credible intervals of both estimates were broad, implying high uncertainty for these estimates. Future work using larger sample sizes would be needed to draw firmer conclusions on how sexual selection shapes sex differences in the genetic architecture of behavioral traits.     

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.     

Non-equivalent loci and mutation-selection balance

We consider the implications of mutationally non-equivalent loci for large populations of randomly mating diploid organisms under mutation-selection balance. Variation, across loci, of parameters such as the allelic mutational variance and the mutation rate, is shown to reduce the equilibrium genetic variance. This is proved to follow from the genetic variance contributed by a single locus having an underlying convexity. We give approximate results indicating the way small deviations of the mutational parameters, from their mean values, reduce the genetic variance. Numerical estimates of the size of the effect are given for more general variations of the parameters. Variation in the mutation rates has a significantly smaller effect than variation in the mutational variances. Under accepted parameter values, the reduction in genetic variance can be substantial.     

Persistence of changes in the genetic covariance matrix after a bottleneck

Abstract.— Genetic variance, phenotypic variance, and the genetic covariance matrix ( G ) can change as a result of genetic drift. These changes will persist over time to some extent and will continue if population size remains relatively small. Nine populations founded by a single pair of Drosophila melanogaster were measured for a series of six morphological characteristics for a large number of parent-offspring families at both the third generation after the bottlenecks and after 20 generations. From these data, the phenotypic variance, additive genetic variance, and G were estimated for each line at each generation. Phenotypic and genetic variances were highly correlated over time, so that the measurements made at the third generation were predictive of the state of the population 17 generations later. Genetic covariances were also somewhat stable over time; however, the G matrices of some lines changed significantly over the intervening generations. This change did not return the populations toward their original state before the population bottlenecks. We conclude that the genetic covariance matrix can change as a result of mild genetic drift over a short span of time.     

On the Probabilities of Obtaining Negative Estimates of Genetic Variance Components

This paper derives the probabilities of obtaining negative estimates of additive and dominance genetic variances when one uses the traditional weighted least square method for estimating genetic variances as given in MATHER and JINKS (1971). The model considered involves P₁, P₂, F₂, B₁ (Backcross to P₁) and B₂ (Backcross to P₂). The results are derived under the ordinary assumptions as made in the genetic literatures. It is shown that unless the genetic effects are very large and environmental effects small, the probabilities of obtaining negative estimates of additive and dominance variances are in general quite large.     

用R法估计方差组分的原理、方法和应用

综述了R法估计方差组分的原理、方法和应用,目的是使该方法能够得到合理应用。R法是通过计算全数据集对亚数据集随机效应的回归因子（R）来估计方差组分的。利用一种基于一个变换矩阵的多变量迭代算法,结合先决条件的共扼梯度法求解混合模型方程组使R法的计算效率大为改善。R法的主要优点是计算成本低,同时可以得到方差组分估值的抽样误差和近似置信区间。其缺点是对于同样的数据,R法较其他方法的抽样误差大,而且在小样本中估计值往往有偏。做为一种可选方法,R法可以应用到大数据集的方差组分估计中,同时应进一步研究其理论特性,拓宽其应用范围。Abstract: Theory, method and application of Method R on estimation of (co)variance components were reviewed in order to make the method be reasonably used. Estimation requires R values,which are regressions of predicted random effects that are calculated using complete dataset on predicted random effects that are calculated using random subsets of the same data. By using multivariate iteration algorithm based on a transformation matrix,and combining with the preconditioned conjugate gradient to solve the mixed model equations, the computation efficiency of Method R is much improved. Method R is computationally inexpensive,and the sampling errors and approximate credible intervals of estimates can be obtained. Disadvantages of Method R include a larger sampling variance than other methods for the same data,and biased estimates in small datasets. As an alternative method, Method R can be used in larger datasets. It is necessary to study its theoretical properties and broaden its application range further.     

Estimating variance components and predicting breeding values for eventing disciplines and grades in sport horses

Eventing competitions in Great Britain (GB) comprise three disciplines, each split into four grades, yielding 12 discipline-grade traits. As there is a demand for tools to estimate (co)variance matrices with a large number of traits, the aim of this work was to investigate different methods to produce large (co)variance matrices using GB eventing data. Data from 1999 to 2008 were used and penalty points were converted to normal scores. A sire model was utilised to estimate fixed effects of gender, age and class, and random effects of sire, horse and rider. Three methods were used to estimate (co)variance matrices. Method 1 used a method based on Gibbs sampling and data augmentation and imputation. Methods 2a and 2b combined sub-matrices from bivariate analyses; one took samples from a multivariate Normal distribution defined by the covariance matrix from each bivariate analysis, then analysed these data in a 12-trait multivariate analysis; the other replaced negative eigenvalues in the matrix with positive values to obtain a positive definite (co)variance matrix. A formal comparison of models could not be conducted; however, estimates from all methods, particularly Methods 2a/2b, were in reasonable agreement. The computational requirements of Method 1 were much less compared with Methods 2a or 2b. Method 2a heritability estimates were as follows: for dressage 7.2% to 9.0%, for show jumping 8.9% to 16.2% and for cross-country 1.3% to 1.4%. Method 1 heritability estimates were higher for the advanced grades, particularly for dressage (17.1%) and show jumping (22.6%). Irrespective of the model, genetic correlations between grades, for dressage and show jumping, were positive, high and significant, ranging from 0.59 to 0.99 for Method 2a and 0.78 to 0.95 for Method 1. For cross-country, using Method 2a, genetic correlations were only significant between novice and pre-novice (0.75); however, using Method 1 estimates were all significant and low to moderate (0.36 to 0.70). Between-discipline correlations were all low and of mixed sign. All methods produced positive definite 12 × 12 (co)variance matrices, suitable for the prediction of breeding values. Method 1 benefits from much reduced computational requirements, and by performing a true multivariate analysis.     

A Comparative Statistical Analysis of Genetic Distances. I - Estimation of Distances and of their Variances: Distribution of the Estimators

We present a Monte-Carlo simulation analysis of the statistical properties of absolute genetic distance and of Nei's minimum and standard genetic distances. The estimation of distances (bias) and of their variances is analysed as well as the distributions of distance and variance estimators, taking into account both gamete and locus samplings. Both of Nei's statistics are non-linear when distances are small and consequently the distributions of their estimators are extremely asymmetrical. It is difficult to find theoretical laws that fit such asymmetrical distributions. Absolute genetic distance is linear and its distributions are better fit by a normal distribution. When distances are medium or large, minimum distance and absolute distance distributions are close to a normal distribution, but those of the standard distance can never be considered as normal. For large distances the jack-knife estimator of the standard distance variance is bad; another standard distance estimator is suggested. Absolute distance, which has the best mathematical properties, is particularly interesting for small distances if the gamete sample size is large, even when the number of loci is small. When both distance and gamete sample size are small, this statistic is biased.