Repeatability for Gaussian and non‐Gaussian data: a practical guide for biologists

Repeatability (more precisely the common measure of repeatability, the intra‐class correlation coefficient, ICC) is an important index for quantifying the accuracy of measurements and the constancy of phenotypes. It is the proportion of phenotypic variation that can be attributed to between‐subject (or between‐group) variation. As a consequence, the non‐repeatable fraction of phenotypic variation is the sum of measurement error and phenotypic flexibility. There are several ways to estimate repeatability for Gaussian data, but there are no formal agreements on how repeatability should be calculated for non‐Gaussian data (e.g. binary, proportion and count data). In addition to point estimates, appropriate uncertainty estimates (standard errors and confidence intervals) and statistical significance for repeatability estimates are required regardless of the types of data. We review the methods for calculating repeatability and the associated statistics for Gaussian and non‐Gaussian data. For Gaussian data, we present three common approaches for estimating repeatability: correlation‐based, analysis of variance (ANOVA)‐based and linear mixed‐effects model (LMM)‐based methods, while for non‐Gaussian data, we focus on generalised linear mixed‐effects models (GLMM) that allow the estimation of repeatability on the original and on the underlying latent scale. We also address a number of methods for calculating standard errors, confidence intervals and statistical significance; the most accurate and recommended methods are parametric bootstrapping, randomisation tests and Bayesian approaches. We advocate the use of LMM‐ and GLMM‐based approaches mainly because of the ease with which confounding variables can be controlled for. Furthermore, we compare two types of repeatability (ordinary repeatability and extrapolated repeatability) in relation to narrow‐sense heritability. This review serves as a collection of guidelines and recommendations for biologists to calculate repeatability and heritability from both Gaussian and non‐Gaussian data.     

Inclusion of Dominance Effects in the Multivariate GBLUP Model

New proposals for models and applications of prediction processes with data on molecular markers may help reduce the financial costs of and identify superior genotypes in maize breeding programs. Studies evaluating Genomic Best Linear Unbiased Prediction (GBLUP) models including dominance effects have not been performed in the univariate and multivariate context in the data analysis of this crop. A single cross hybrid construction procedure was performed in this study using phenotypic data and actual molecular markers of 4,091 maize lines from the public database Panzea. A total of 400 simple hybrids resulting from this process were analyzed using the univariate and multivariate GBLUP model considering only additive effects additive plus dominance effects. Historic heritability scenarios of five traits and other genetic architecture settings were used to compare models, evaluating the predictive ability and estimation of variance components. Marginal differences were detected between the multivariate and univariate models. The main explanation for the small discrepancy between models is the low- to moderate-magnitude correlations between the traits studied and moderate heritabilities. These conditions do not favor the advantages of multivariate analysis. The inclusion of dominance effects in the models was an efficient strategy to improve the predictive ability and estimation quality of variance components.     

Social cues affect quantitative genetic variation and covariation in animal personality traits

The social environment is expected to have substantial effects on behavior, and as a consequence, its heritability and evolvability. We investigated these effects by exposing Australian field crickets (Teleogryllus oceanicus) to either silence or recordings of male acoustic sexual signals. We used a combined pedigree and full‐sib/half‐sib breeding design to estimate the repeatability, heritability, and evolvability of behaviors related to boldness, exploration, and activity. All behaviors measured were significantly repeatable in both social environments. Additionally, most behaviors showed significant heritabilities in the two environments. We found no difference in repeatabilities between the silent and the acoustic environment but did find significant differences in the heritabilities and evolvabilities between these environments. There was a high degree of similarity between the phenotypic covariance matrices across the two environments, while the genotypic covariance matrices were highly dissimilar. Reflecting this, we found significant genotype‐by‐environment interactions for most of the behaviors. Lastly, we found that the repeatable aspect of behavior (“personality”) was significantly heritable for most behaviors, but that these heritabilities were higher in the acoustic than in the silent environment. We conclude that the social environment can have a significant impact on the heritability and evolvability of behavior, and argue that evolutionary inferences from phenotypic studies should be made with caution.     

Sex‐specific heritability of cell‐mediated immune response in the blue tit nestlings (Cyanistes caeruleus)

Here, we aimed at estimating sex‐specific heritabilities of cell‐mediated immune response (CMI) in the blue tit nestlings (Cyanistes caeruleus). To separate genetic and environmental components of the phenotypic variance in CMI (measured using phytohaemagglutinin assay), we performed a cross‐fostering experiment. Additionally, controlled environmental variation was introduced by enlarging some broods. Our analyses revealed a significant genetic component (as approximated by the nest‐of‐origin term) of the phenotypic variance in immune response. More importantly, these genetic effects differed between sexes and experimentally manipulated brood sizes, as indicated by significant genotype‐by‐sex and genotype‐by‐environment interactions. We discuss possible causes of such sexual dimorphism in gene expression and suggest that sex‐ and environment‐specific genetic interactions may contribute to the maintenance of genetic variability in traits related to immune functions.     

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.     

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.     

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.     

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.     

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.     

Direct and indirect genetic effects on reproductive investment in a grasshopper

A fundamental part of the quantitative genetic theory deals with the partitioning of the phenotypic variance into additive genetic and environmental components. During interaction with conspecifics, the interaction partner becomes a part of the environment from the perspective of the focal individual. If the interaction effects have a genetic basis, they are called indirect genetic effects (IGEs) and can evolve along with direct genetic effects. Sexual reproduction is a classic context where potential conflict between males and females can arise from trade‐offs between current and future investments. We studied five female fecundity traits, egg length and number, egg pod length and number and latency to first egg pod, and estimated the direct and IGEs using a half‐sib breeding design in the grasshopper Chorthippus biguttulus. We found that the male IGEs were an order of magnitude lower than the direct genetic effects and were not significantly different from zero. However, there was some indication that IGEs were larger shortly after mating, consistent with the idea that IGEs fade with time after interaction. Female direct heritabilities were moderate to low. Simulation shows that the variance component estimates can appear larger with less data, calling for care when interpreting variance components estimated with low power. Our results illustrate that the contribution of male IGEs is overall low on the phenotypic variance of female fecundity traits. Thus, even in the relevant context of sexual conflict, the influence of male IGEs on the evolutionary trajectory of female reproductive traits is likely to be small.     

Mixed effects models with censored data with application to HIV RNA levels

Mixed effects models are often used for estimating fixed effects and variance components in longitudinal studies of continuous data. When the outcome being modelled is a laboratory measurement, however, it may be subject to lower and upper detection limits (i.e., censoring). In this paper, the usual EM estimation procedure for mixed effects models is modified to account for left and/or right censoring.     

Age and sex affect quantitative genetic parameters for dominance rank and aggression in free‐living greylag geese

Knowledge of the genetic and environmental influences on a character is pivotal for understanding evolutionary changes in quantitative traits in natural populations. Dominance and aggression are ubiquitous traits that are selectively advantageous in many animal societies and have the potential to impact the evolutionary trajectory of animal populations. Here we provide age‐ and sex‐specific estimates of additive genetic and environmental components of variance for dominance rank and aggression rate in a free‐living, human‐habituated bird population subject to natural selection. We use a long‐term data set on individually marked greylag geese (Anser anser) and show that phenotypic variation in dominance‐related behaviours contains significant additive genetic variance, parental effects and permanent environment effects. The relative importance of these variance components varied between age and sex classes, whereby the most pronounced differences concerned nongenetic components. In particular, parental effects were larger in juveniles of both sexes than in adults. In paired adults, the partner's identity had a larger influence on male dominance rank and aggression rate than in females. In sex‐ and age‐specific estimates, heritabilities did not differ significantly between age and sex classes. Adult dominance rank was only weakly genetically correlated between the sexes, leading to considerably higher heritabilities in sex‐specific estimates than across sexes. We discuss these patterns in relation to selection acting on dominance rank and aggression in different life history stages and sexes and suggest that different adaptive optima could be a mechanism for maintaining genetic variation in dominance‐related traits in free‐living animal populations.     

Quantitative Genetics of Drosophila Melanogaster. II. Heritabilities and Genetic Correlations between Sexes for Head and Thorax Traits

A quantitative genetic analysis is reported for traits on the head and thorax of adult fruit flies, Drosophila melanogaster. Females are larger than males, and the magnitude of sexual dimorphism is similar for traits derived from the same imaginal disc, but the level of sexual dimorphism varies widely across discs. The greatest difference between males and females occurs for the dimensions of the sclerotized mouthparts of the proboscis. Most of the traits studied are highly heritable with heritabilities ranging from 0.26 to 0.84 for males and 0.27 to 0.81 for females. In general, heritabilities are slightly higher for males, possibly reflecting the effect of dosage compensation on X-linked variance. The X chromosome contributes substantially to variance for many of these traits, and including results reported elsewhere, the variance for over two-thirds of the traits studied includes X-linked variance. The genetic correlations between sexes for the same trait are generally high and close to unity. Coupled with the small differences in the traits between sexes for heritabilities and phenotypic variances, these results suggest that selection would be very slow to change the level of sexual dimorphism in size of various body parts.     

Heritability of phenotypic udder traits to improve resilience to mastitis in Texel ewes

There are no estimates of the heritability of phenotypic udder traits in suckler sheep, which produce meat lambs, and whether these are associated with resilience to mastitis. Mastitis is a common disease which damages the mammary gland and reduces productivity. The aims of this study were to investigate the feasibility of collecting udder phenotypes, their heritability and their association with mastitis in suckler ewes. Udder and teat conformation, teat lesions, intramammary masses (IMM) and litter size were recorded from 10 Texel flocks in Great Britain between 2012 and 2014; 968 records were collected. Pedigree data were obtained from an online pedigree recording system. Univariate quantitative genetic parameters were estimated using animal and sire models. Linear mixed models were used to analyse continuous traits and generalised linear mixed models were used to analyse binary traits. Continuous traits had higher heritabilities than binary with teat placement and teat length heritability (h²) highest at 0.35 (SD 0.04) and 0.42 (SD 0.04), respectively. Udder width, drop and separation heritabilities were lower and varied with udder volume. The heritabilities of IMM and teat lesions (sire model) were 0.18 (SD 0.12) and 0.17 (SD 0.11), respectively. All heritabilities were sufficiently high to be in a selection programme to increase resilience to mastitis in the population of Texel sheep. Further studies are required to investigate genetic relationships between traits and to determine whether udder traits predict IMM, and the potential benefits from including traits in a selection programme to increase resilience to chronic mastitis.     

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.     

Maternal genetic effects set the potential for evolution in a free-living vertebrate population

Heritable maternal effects have important consequences for the evolutionary dynamics of phenotypic traits under selection, but have only rarely been tested for or quantified in evolutionary studies. Here we estimate maternal effects on early-life traits in a feral population of Soay sheep (Ovis aries) from St Kilda, Scotland. We then partition the maternal effects into genetic and environmental components to obtain the first direct estimates of maternal genetic effects in a free-living population, and furthermore test for covariance between direct and maternal genetic effects. Using an animal model approach, direct heritabilities (h2) were low but maternal genetic effects (m2) represented a relatively large proportion of the total phenotypic variance for each trait (birth weight m2=0.119, birth date m2=0.197, natal litter size m2=0.211). A negative correlation between direct and maternal genetic effects was estimated for each trait, but was only statistically significant for natal litter size (ram= -0.714). Total heritabilities (incorporating variance from heritable maternal effects and the direct-maternal genetic covariance) were significant for birth weight and birth date but not for natal litter size. Inadequately specified models greatly overestimated additive genetic variance and hence direct h2 (by a factor of up to 6.45 in the case of birth date). We conclude that failure to model heritable maternal variance can result in over- or under-estimation of the potential for traits to respond to selection, and advocate an increased effort to explicitly measure maternal genetic effects in evolutionary studies.     

Genetic variability of body weight in two goose strains under long-term selection

Body weight is one of the most important traits in any genetic improvement program in geese for at least 2 reasons. First, measurements of the trait are very easy. Second, body weight is correlated with a number of other meat performance traits. However, the genetic background of body weight shows considerable complexity. Three genetic models (with direct, maternal genetic and permanent maternal environmental effects) were employed in this study. Records of 3076 individuals of maternal strain W11 and 2656 individuals of paternal strain W33 over 6 consecutive generations, kept in the pedigree farm of Ko?uda Wielka, were analysed. Body weight (in kilograms) was measured in weeks 8 (BW8) and 11 (BW11). The inbreeding levels in both populations were relatively low (0.14% and 0.02% for W11 and W33, respectively), therefore these effects were not included in the linear models to estimate genetic parameters. Three fixed effects (hatch period, sex and year) were included in each linear model. Two criteria (AIC, BIC) were used to check the goodness of fit of the models. The computations were performed by WOMBAT software. In general, the genetic parameter estimates varied across the traits, models and strains studied. Direct additive heritability estimates ranged from 0.0001 (for BW11 of W33) to 0.55 (for BW11 of W33). Maternal and total heritabilities were also variable. Estimates of ratios of direct-maternal effect covariance in phenotypic variance were both positive and negative, but they were negligible, whereas ratios of the permanent environmental maternal variance to phenotypic variance were close to zero. Both of the applied criteria of model adequacy indicate that the model with maternal genetic and environmental effects should be considered as optimal. Genetic trends were close to zero. It seems that they were influenced by long-term selection. Similar tendencies have been observed for phenotypic trends, as well.     

Estimates of direct and maternal covariance functions for growth of Australian beef calves from birth to weaning

Records for birth and subsequent, monthly weights until weaning on beef calves of two breeds in a selection experiment were analysed fitting random regression models. Independent variables were orthogonal (Legendre) polynomials of age at weighing in days. Orders of polynomial fit up to 6 were considered. Analyses were carried out fitting sets of random regression coefficients due to animals'' direct and maternal, additive genetic and permanent environmental effects, with changes in variances due to temporary environmental effects modelled through a variance function, estimating up to 67 parameters. Results identified similar patterns of variation for both breeds, with maternal effects considerably more important in purebred Polled Herefords than a four-breed synthetic, the so-called Wokalups. Conversely, repeatabilities were higher for the latter. For both breeds, heritabilities decreased after birth, being lowest when maternal effects were most important around 100 days of age. Estimates at birth and weaning were consistent with previous, univariate results.     

Genetic relationship between milk urea nitrogen and reproductive performance in Holstein dairy cows

The objective of this study was to describe the genetic and phenotypic relationship between milk urea nitrogen (MUN) and reproductive traits in Iranian Holstein dairy cows. Test-day MUN data obtained from 57 301 dairy cows on 20 large dairy herds in Iran between January 2005 and June 2009. Genetic parameters for MUN and reproductive traits were estimated with a five-trait model using ASREML program. Random regression test-day models were used to estimate heritabilities separately for MUN from first, second and third lactations. Regression curves were modeled using Legendre polynomials of order 3. Herd-year-season along with age at calving was included as fixed effects in all models for reproductive traits. Heritabilities for MUN and reproductive traits were estimated separately for first lactation, second lactation and third lactation. The estimated heritabilities for MUN varied from 0.18 to 0.22. The heritability estimate was low for reproductive traits, which ranged from 0.02 to 0.06 for different traits and across parities. Except for days open, phenotypic and genetic correlations of MUN with reproductive performance traits were close to zero. Genetic correlations between MUN and days open were 0.23, 0.35 and 0.45 in first, second and third lactation, respectively. However, the phenotypic correlation between MUN at different parities was moderate (0.28 to 0.35), but the genetic correlation between MUN at different parities was high and ranged from 0.84 to 0.97. This study shows a limited application of MUN for use in selection programs to improve reproductive performance.