Realized Sampling Variances of Estimates of Genetic Parameters and the Difference between Genetic and Phenotypic Correlations

A data set of 1572 heritability estimates and 1015 pairs of genetic and phenotypic correlation estimates, constructed from a survey of published beef cattle genetic parameter estimates, provided a rare opportunity to study realized sampling variances of genetic parameter estimates. The distribution of both heritability estimates and genetic correlation estimates, when plotted against estimated accuracy, was consistent with random error variance being some three times the sampling variance predicted from standard formulae. This result was consistent with the observation that the variance of estimates of heritabilities and genetic correlations between populations were about four times the predicted sampling variance, suggesting few real differences in genetic parameters between populations. Except where there was a strong biological or statistical expectation of a difference, there was little evidence for differences between genetic and phenotypic correlations for most trait combinations or for differences in genetic correlations between populations. These results suggest that, even for controlled populations, estimating genetic parameters specific to a given population is less useful than commonly believed. A serendipitous discovery was that, in the standard formula for theoretical standard error of a genetic correlation estimate, the heritabilities refer to the estimated values and not, as seems generally assumed, the true population values.     

Genetic and phenotypic correlations in a natural population of song sparrows

We estimated heritabilities, and genetic and phenotypic correlations between beak and body traits in the song sparrow ( Melospiza melodia ). We compared these estimates to values for the same traits in the Galápagos finches, Geospiza (Boag, 1983; Grant, 1983). Morphological variance is low in the song sparrow, and our results show that genetic and phenotypic correlations are considerably lower than correlations in the morphologically more variable Geospiza. Comparison using a larger sample of Galapagos populations confirms the existence of an association between variance and correlation for phenotypic values. We suggest two possible explanations for this association. First, most traits studied are functionally related, and the joint evolution of variance and correlation may have resulted from stabilizing selection about a line of optimal allometry between traits. Alternatively, introgression between populations and species could have caused correlation and variance to evolve jointly. Both selection and introgression were probably influential in producing the observed pattern, but it is not possible to estimate their relative importance with current data. Genetic and phenotypic correlations were correlated in the song sparrow, but heritabilities of traits varied greatly. As a result, the genetic variance-covariance matrix for traits is not simply a constant multiple of the phenotypic matrix. Evolutionary response to natural selection cannot, therefore, be predicted from the measurement of phenotypic characteristics alone.     

Laboratory Estimates of Heritabilities and Genetic Correlations in Nature

A lower bound on heritability in a natural environment can be determined from the regression of offspring raised in the laboratory on parents raised in nature. An estimate of additive genetic variance in the laboratory is also required. The estimated lower bounds on heritabilities can sometimes be used to demonstrate a significant genetic correlation between two traits in nature, if their genetic and phenotypic correlations in nature have the same sign, and if sample sizes are large, and heritabilities and phenotypic and genetic correlations are high.     

Power of the classical twin design revisited.

Statistical power of the classical twin design was revisited. The approximate sampling variances of a least-squares estimate of the heritability in a univariate analysis and estimate of the genetic correlation coefficient in a bivariate analysis were derived analytically for the ACE model. Statistical power to detect additive genetic variation under the ACE model was derived analytically for least-squares, goodness-of-fit and maximum likelihood-based test statistics. The noncentrality parameter for the likelihood ratio test statistic is shown to be a simple function of the MZ and DZ intraclass correlation coefficients and the proportion of MZ and DZ twin pairs in the sample. All theoretical results were validated using simulation. The derived expressions can be used to calculate power of the classical twin design in a simple and rapid manner.     

On the properties of maximum likelihood estimators of familial correlations under variable sibship size

Selected distributional properties of the maximum likelihood estimator and its z-transformation of three familial correlations (parental, parent-offspring, filial) were investigated numerically for the case of nuclear families with variable sibship size. This investigation was based on six different sets of the three correlations, and four different sample sizes, defining 24 sampling conditions, which were replicated 1,000 times each. It was found that the distributional properties of the correlation estimator are affected by the magnitude of the correlations even in large samples although approximate normality is achieved locally. Fisher's z-transformation, here used only in its interclass form, achieves reduction of skewness, stabilization of variance, and approach to normality already in small samples, except for the filial correlation (where it may be deemed inappropriate) in smaller samples. For both the correlation estimator and its z-transformation, the (estimated) relative efficiency was shown to be high (better than 90% in most sampling conditions), suggesting that the estimated minimum variance bound is a satisfactory estimator of the sampling variance. It is concluded that the maximum likelihood estimation of familial correlations under variable sibship size is feasible and, when prudently applied, especially in the form of their z-transformations, provides an appropriate method in analyses of family studies.     

Estimating heritabilities and genetic correlations: comparing the 'animal model' with parent-offspring regression using data from a natural population

Quantitative genetic parameters are nowadays more frequently estimated with restricted maximum likelihood using the 'animal model' than with traditional methods such as parent-offspring regressions. These methods have however rarely been evaluated using equivalent data sets. We compare heritabilities and genetic correlations from animal model and parent-offspring analyses, respectively, using data on eight morphological traits in the great reed warbler (Acrocephalus arundinaceus). Animal models were run using either mean trait values or individual repeated measurements to be able to separate between effects of including more extended pedigree information and effects of replicated sampling from the same individuals. We show that the inclusion of more pedigree information by the use of mean traits animal models had limited effect on the standard error and magnitude of heritabilities. In contrast, the use of repeated measures animal model generally had a positive effect on the sampling accuracy and resulted in lower heritabilities; the latter due to lower additive variance and higher phenotypic variance. For most trait combinations, both animal model methods gave genetic correlations that were lower than the parent-offspring estimates, whereas the standard errors were lower only for the mean traits animal model. We conclude that differences in heritabilities between the animal model and parent-offspring regressions were mostly due to the inclusion of individual replicates to the animal model rather than the inclusion of more extended pedigree information. Genetic correlations were, on the other hand, primarily affected by the inclusion of more pedigree information. This study is to our knowledge the most comprehensive empirical evaluation of the performance of the animal model in relation to parent-offspring regressions in a wild population. Our conclusions should be valuable for reconciliation of data obtained in earlier studies as well as for future meta-analyses utilizing estimates from both traditional methods and the animal model.     

Selection of sires to reduce sampling variance in the estimates of heritability by half-sib correlation

Summary Standard methods to estimate heritability by half-sib correlation are biased if selection has operated in the parental generation. In this paper a simple method to correct for selection of animals used as sires is described. By selection of both the top and the bottom ranking sires, the sampling variances of the corrected estimates of heritability are substantially reduced. Algebraic expressions to predict the sampling variance of the estimates of heritability using selected sires are derived. Theoretical predictions were checked by Monte-Carlo simulation. The results may have application in the design of experiments to estimate heritabilities.     

Estimating genetic correlations in natural populations in the absence of pedigree information: accuracy and precision of the Lynch method

Usually, genetic correlations are estimated from breeding designs in the laboratory or greenhouse. However, estimates of the genetic correlation for natural populations are lacking, mostly because pedigrees of wild individuals are rarely known. Recently Lynch (1999) proposed a formula to estimate the genetic correlation in the absence of data on pedigree. This method has been shown to be particularly accurate provided a large sample size and a minimum (20%) proportion of relatives. Lynch (1999) proposed the use of the bootstrap to estimate standard errors associated with genetic correlations, but did not test the reliability of such a method. We tested the bootstrap and showed the jackknife can provide valid estimates of the genetic correlation calculated with the Lynch formula. The occurrence of undefined estimates, combined with the high number of replicates involved in the bootstrap, means there is a high probability of obtaining a biased upward, incomplete bootstrap, even when there is a high fraction of related pairs in a sample. It is easier to obtain complete jackknife estimates for which all the pseudovalues have been defined. We therefore recommend the use of the jackknife to estimate the genetic correlation with the Lynch formula. Provided data can be collected for more than two individuals at each location, we propose a group sampling method that produces low standard errors associated with the jackknife, even when there is a low fraction of relatives in a sample.     

Linear and threshold analysis of direct and maternal genetic effects for secondary sex ratio in Iranian buffaloes

The objective of this study was to estimate variance components and genetic parameters for secondary sex ratio (SSR) in Iranian buffaloes. Calving records from April 1995 to June 2010 comprising 15,207 calving events from the first three lactations of 1066 buffalo herds of Iran were analyzed using linear and threshold animal models to estimate variance components, heritabilities and genetic correlations between direct and maternal genetic effects for SSR. Linear and threshold animal models included direct and maternal genetic effects with covariance between them and maternal permanent environmental effects were implemented by Gibbs sampling methodology. Posterior means of direct and maternal heritabilities and repeatability for SSR obtained from linear animal model were 0.15, 0.10, and 0.17, respectively. Threshold estimates of direct and maternal heritabilities and repeatability for SSR were 0.48, 0.27, and 0.52, respectively. The results showed that the correlations between direct and maternal genetic effects of SSR were negative and high in both models. In addition, the ratios of maternal permanent environmental variance were low. Exploitable genetic variation in SSR can take advantage of sexual dimorphism for economically important traits which may facilitate greater selection intensity and thus greater response to selection, as well as reducing the replacement costs. Threshold animal model may be applied in selection programs where animals are to be genetically ranked for female rate.     

Validation of an approximate approach to compute genetic correlations between longevity and linear traits

The estimation of genetic correlations between a nonlinear trait such as longevity and linear traits is computationally difficult on large datasets. A two-step approach was proposed and was checked via simulation. First, univariate analyses were performed to get genetic variance estimates and to compute pseudo-records and their associated weights. These pseudo-records were virtual performances free of all environmental effects that can be used in a BLUP animal model, leading to the same breeding values as in the (possibly nonlinear) initial analyses. By combining these pseudo-records in a multiple trait model and fixing the genetic and residual variances to their values computed during the first step, we obtained correlation estimates by AI-REML and approximate MT-BLUP predicted breeding values that blend direct and indirect information on longevity. Mean genetic correlations and reliabilities obtained on simulated data confirmed the suitability of this approach in a wide range of situations. When nonzero residual correlations exist between traits, a sire model gave nearly unbiased estimates of genetic correlations, while the animal model estimates were biased upwards. Finally, when an incorrect genetic trend was simulated to lead to biased pseudo-records, a joint analysis including a time effect could adequately correct for this bias.     

Egg shell quality in Japanese quail: characteristics,heritabilities and genetic and phenotypic relationships

The objective of the present study was to estimate heritabilities as well as genetic and phenotypic correlations for egg weight, specific gravity, shape index, shell ratio, egg shell strength, egg length, egg width and shell weight in Japanese quail eggs. External egg quality traits were measured on 5864 eggs of 934 female quails from a dam line selected for two generations. Within the Bayesian framework, using Gibbs Sampling algorithm, a multivariate animal model was applied to estimate heritabilities and genetic correlations for external egg quality traits. The heritability estimates for external egg quality traits were moderate to high and ranged from 0.29 to 0.81. The heritability estimates for egg and shell weight of 0.81 and 0.76 were fairly high. The genetic and phenotypic correlations between egg shell strength with specific gravity, shell ratio and shell weight ranging from 0.55 to 0.79 were relatively high. It can be concluded that it is possible to determine egg shell quality using the egg specific gravity values utilizing its high heritability and fairly high positive correlation with most of the egg shell quality traits. As a result, egg specific gravity may be the choice of selection criterion rather than other external egg traits for genetic improvement of egg shell quality in Japanese quails.     

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.     

Genetic correlations of fighting ability with somatic cells and longevity in cattle

The success in competitions may be stressful for animals and costly in terms of immune functions and longevity. Focusing on Aosta Chestnut and Aosta Black Pied cattle, selected for their fighting ability in traditional competitions, this study investigated the genetic relationships of fighting ability with udder health traits (somatic cell score and two threshold traits for somatic cells), longevity (length of productive life and number of calvings) and test-day milk, fat and protein yield. Herdbook information and phenotypic records that have been routinely collected for breeding programs in 16 years were used for the abovementioned traits. Data belonged to 9328 cows and 19 283 animals in pedigree. Single-trait animal model analyses were run using a Gibbs sampling algorithm to estimate the variance components of traits, and bivariate analyses were then performed to estimate the genetic correlations. Moderate positive genetic correlations (r_a) were found for fighting ability with somatic cell score (r_a=0.255), suggesting that greater fighting ability is genetically related to a detriment in udder health, in agreement with the theory. The high positive genetic correlation between fighting ability and longevity (average r_a=0.669) suggests that the economic importance of fighting ability (the winning cows get an higher price at selling) had probably masked the true genetic covariances. The genetic correlation between milk yield traits and fighting ability showed large intervals, but the negative values (average r_a=−0.121) agreed with previous research. This study is one of the few empirical studies on genetic correlations for the competitive success v. immune functions and longevity traits. The knowledge of the genetic correlations among productive and functional traits of interest, including fighting ability, is important in animal breeding for a sustainable genetic improvement.     

Estimating genetic correlations in natural populations

Information on the genetic correlation between traits provides fundamental insight into the constraints on the evolutionary process. Estimates of such correlations are conventionally obtained by raising individuals of known relatedness in artificial environments. However, many species are not readily amenable to controlled breeding programmes, and considerable uncertainty exists over the extent to which estimates derived under benign laboratory conditions reflect the properties of populations in natural settings. Here, non-invasive methods that allow the estimation of genetic correlations from phenotypic measurements derived from individuals of unknown relatedness are introduced. Like the conventional approach, these methods demand large sample sizes in order to yield reasonably precise estimates, and special precautions need to be taken to eliminate bias from shared environmental effects. Provided the sample consists of at least 20% or so relatives, informative estimates of the genetic correlation are obtainable with sample sizes of several hundred individuals, particularly if supplemental information on relatedness is available from polymorphic molecular markers.     

Genetic and phenotypic correlations among size-related traits, and heritability variation between body parts in Drosophila buzzatii

Recent studies have shown that body size is a heritable trait phenotypically correlated with several fitness components in wild populations of the cactophilic fly Drosophila buzzatii. To obtain further information on size-related variation, heritabilities as well as genetic and phenotypic correlations among size-related traits of several body parts (head, thorax and wings) were estimated. The study was carried out on an Argentinean natural population in which size-related selection was previously detected. The genetic parameters were estimated using offspring-parent regressions (105 families) in the laboratory G₂ generation of a sample of wild flies. The traits were also scored in Wild-Caught Flies (WCF). Laboratory-Reared Flies (LRF) were larger and less variable than WCF. Although heritability estimates were significant for all traits, heritabilities were higher for thorax-wing traits than for head traits. Phenotypic and genetic correlations were all positive. The highest genetic correlations were found between traits which are both functionally and developmentally related. Genetic and phenotypic correlations estimated in the lab show similar correlation patterns (r = 0.49; TP = 0.02, Mantel's test). However, phenotypic correlations were found to be typically larger in WCF than in LRF. The genetic correlation matrix estimated in the relatively homogeneous lab environment is not simply a constant multiplicative factor of the phenotypic correlation matrix estimated in WCF. This revised version was published online in July 2006 with corrections to the Cover Date.     

QUANTITATIVE GENETICS OF SURVIVAL AND GROWTH IN IMPATIENS CAPENSIS

When variation in life-history characters is caused by many genes of small effect, then quantitative-genetic parameters may quantify constraints on rate and direction of microevolutionary change. I estimated heritabilities and genetic correlations for 16 life-history and morphological characters in two populations of Impatiens capensis, a partially self-pollinating herbaceous annual. The Madison population had little or no additive genetic variance for any of these characters, while the Milwaukee population had significant narrowsense heritabilities and genetic correlations for several traits, including adult size, which is highly correlated with fitness. All genetic correlations among fitness components were positive, hence there is no evidence for antagonistic pleiotropy among these traits. Dissimilarity of heritabilities in the two populations supports theoretical predictions that long-term changes in genetic variance-covariance patterns may occur when population sizes are small and selection is strong, as may occur in many plant species.     

Multivariate selection under adverse genetic correlations: impacts of population sizes and selection strategies on gains and coancestry in forest tree breeding

We examined gene models for two traits with and without antagonistic pleiotropy using a locus-based simulation model to investigate the effects of different population sizes, heritabilities and economic weights, using index selection, and index selection with optimum selection (OS), over 10 generations. Gene models included additive and dominance gene action, with equal and varying initial allele frequencies with and without pleiotropy for a fixed level of resources (i.e. founder sizes each generation of 40, 80 and 160 with progeny arrays that totaled 800 per generation). Pleiotropy (with an initial r _g of −0.5), reduced gain by ~8–10% when heritabilities for both traits were the same (0.2), relative to non-pleiotropic cases. When traits had different heritabilities (i.e. 0.2 and 0.4), gains in the lower heritability trait were substantially lower, especially with pleiotropy present. In general, OS with slightly larger population sizes could offset losses in gain, but rarely overrode the large effects of different heritabilities or economic weights. Pleiotropy increased response variance among traits, which was magnified when heritabilities were different. Identifying an appropriate weight on relatedness in the OS process is important to manage coancestry expectations around the loss of alleles (or fixation of recessive alleles) and to minimise response variance. The dynamics of selection intensity, drift, rate of coancestry build-up, response variance, etc. are complex for multi-trait selection; however, a few economically viable strategies could reduce the adverse effects of selecting against genetic correlations without drastically impairing gain.     

Likelihood calculations to evaluate experimental designs to estimate genetic variances

Mixed model analyses via restricted maximum likelihood, fitting the so-called animal model, have become standard methodology for the estimation of genetic variances. Models involving multiple genetic variance components, due to different modes of gene action, are readily fitted. It is shown that likelihood-based calculations may provide insight into the quality of the resulting parameter estimates, and are directly applicable to the validation of experimental designs. This is illustrated for the example of a design suggested recently to estimate X-linked genetic variances. In particular, large sample variances and sampling correlations are demonstrated to provide an indication of 'problem' scenarios. Using simulation, it is shown that the profile likelihood function provides more appropriate estimates of confidence intervals than large sample variances. Examination of the likelihood function and its derivatives are recommended as part of the design stage of quantitative genetic experiments.     

Quantifying intraclass correlations for count and time‐to‐event data

The intraclass correlation is commonly used with clustered data. It is often estimated based on fitting a model to hierarchical data and it leads, in turn, to several concepts such as reliability, heritability, inter‐rater agreement, etc. For data where linear models can be used, such measures can be defined as ratios of variance components. Matters are more difficult for non‐Gaussian outcomes. The focus here is on count and time‐to‐event outcomes where so‐called combined models are used, extending generalized linear mixed models, to describe the data. These models combine normal and gamma random effects to allow for both correlation due to data hierarchies as well as for overdispersion. Furthermore, because the models admit closed‐form expressions for the means, variances, higher moments, and even the joint marginal distribution, it is demonstrated that closed forms of intraclass correlations exist. The proposed methodology is illustrated using data from agricultural and livestock studies.     

A COMPARISON OF GENETIC AND PHENOTYPIC CORRELATIONS

Genetic variances and correlations lie at the center of quantitative evolutionary theory. They are often difficult to estimate, however, due to the large samples of related individuals that are required. I investigated the relationship of genetic- and phenotypic-correlation magnitudes and patterns in 41 pairs of matrices drawn from the literature in order to determine their degree of similarity and whether phenotypic parameters could be used in place of their genetic counterparts in situations where genetic variances and correlations cannot be precisely estimated. The analysis indicates that squared genetic correlations were on average much higher than squared phenotypic correlations and that genetic and phenotypic correlations had only broadly similar patterns. These results could be due either to biological causes or to imprecision of genetic-correlation estimates due to sampling error. When only those studies based on the largest sample sizes (effective sample size of 40 or more) were included, squared genetic-correlation estimates were only slightly greater than their phenotypic counterparts and the patterns of correlation were strikingly similar. Thus, much of the dissimilarity between phenotypic- and genetic-correlation estimates seems to be due to imprecise estimates of genetic correlations. Phenotypic correlations are likely to be fair estimates of their genetic counterparts in many situations. These further results also indicate that genetic and environmental causes of phenotypic variation tend to act on growth and development in a similar manner.