Selection for environmental variation: a statistical analysis and power calculations to detect response

Data from uterine capacity in rabbits (litter size) were analyzed to determine whether the environmental variance was partly genetically determined. The fit of a classical homogeneous variance mixed linear (HOM) model and that of a genetically structured heterogeneous variance mixed linear (HET) model were compared. Various methods to assess the quality of fit favor the HET model. The posterior mean (95% posterior interval) of the additive genetic variance affecting the environmental variance was 0.16 (0.10; 0.25) and the corresponding number for the coefficient of correlation between genes affecting mean and variance was -0.74 (-0.90;-0.52). It is argued that stronger support for the HET model than that derived from statistical analysis of data would be provided by a successful selection experiment designed to modify the environmental variance. A simple selection criterion is suggested (average squared deviation from the mean of repeated records within individuals) and its predicted response and variance under the HET model are derived. This is used to determine the appropriate size and length of a selection experiment designed to change the environmental variance. Results from the analytical expressions are compared with those obtained using simulation. There is good agreement provided selection intensity is not intense.     

Evidence for genetic control of adult weight plasticity in the snail Helix aspersa

Phenotypic plasticity and canalization are important topics in quantitative genetics and evolution. Both concepts are related to environmental sensitivity. The latter can be modeled using a model with genetically structured environmental variance. This work reports the results of a genetic analysis of adult weight in the snail Helix aspersa. Several models of heterogeneous variance are fitted using a Bayesian, MCMC approach. Exploratory analyses using posterior predictive model checking and model comparisons based on the deviance information criterion favor a model postulating a genetically structured heterogeneous environmental variance. Our analysis provides a strong indication of a positive genetic correlation between additive genetic values affecting the mean and those affecting environmental variation of adult body weight. The possibility of manipulating environmental variance by selection is illustrated numerically using estimates of parameters derived from the snail data set.     

A comparison of strategies for Markov chain Monte Carlo computation in quantitative genetics

In quantitative genetics, Markov chain Monte Carlo (MCMC) methods are indispensable for statistical inference in non-standard models like generalized linear models with genetic random effects or models with genetically structured variance heterogeneity. A particular challenge for MCMC applications in quantitative genetics is to obtain efficient updates of the high-dimensional vectors of genetic random effects and the associated covariance parameters. We discuss various strategies to approach this problem including reparameterization, Langevin-Hastings updates, and updates based on normal approximations. The methods are compared in applications to Bayesian inference for three data sets using a model with genetically structured variance heterogeneity.     

Genetic heterogeneity of residual variance - estimation of variance components using double hierarchical generalized linear models

Background

The sensitivity to microenvironmental changes varies among animals and may be under genetic control. It is essential to take this element into account when aiming at breeding robust farm animals. Here, linear mixed models with genetic effects in the residual variance part of the model can be used. Such models have previously been fitted using EM and MCMC algorithms.

Results

We propose the use of double hierarchical generalized linear models (DHGLM), where the squared residuals are assumed to be gamma distributed and the residual variance is fitted using a generalized linear model. The algorithm iterates between two sets of mixed model equations, one on the level of observations and one on the level of variances. The method was validated using simulations and also by re-analyzing a data set on pig litter size that was previously analyzed using a Bayesian approach. The pig litter size data contained 10,060 records from 4,149 sows. The DHGLM was implemented using the ASReml software and the algorithm converged within three minutes on a Linux server. The estimates were similar to those previously obtained using Bayesian methodology, especially the variance components in the residual variance part of the model.

Conclusions

We have shown that variance components in the residual variance part of a linear mixed model can be estimated using a DHGLM approach. The method enables analyses of animal models with large numbers of observations. An important future development of the DHGLM methodology is to include the genetic correlation between the random effects in the mean and residual variance parts of the model as a parameter of the DHGLM.     

The influence of environmentally induced heterogeneity on age-specific genetic variance for mortality rates

Using parametric models that describe the increase in mortality rates with age, we demonstrate that environmentally induced heterogeneity among genetically identical individuals is sufficient to generate biased estimates of age-specific genetic variance. Although the magnitude of the bias may change with age, one general trend emerges: the true genetic variance at the oldest ages is likely to be dramatically underestimated. Our results are robust to different manifestations of heterogeneity and suggest that such a bias is a general feature of these models. We note that age-dependent estimates of genetic variance for characters that are correlated with mortality (either genetically or environmentally) can be expected to be similarly affected. The results are independent of sample size and suggest that the bias may be more widespread in the literature than is currently appreciated. Our results are discussed with reference to existing data on mortality variance in Drosophila melanogaster.     

Genetic and environmental heterogeneity of residual variance of weight traits in Nellore beef cattle

Background

Many studies have provided evidence of the existence of genetic heterogeneity of environmental variance, suggesting that it could be exploited to improve robustness and uniformity of livestock by selection. However, little is known about the perspectives of such a selection strategy in beef cattle.

Methods

A two-step approach was applied to study the genetic heterogeneity of residual variance of weight gain from birth to weaning and long-yearling weight in a Nellore beef cattle population. First, an animal model was fitted to the data and second, the influence of additive and environmental effects on the residual variance of these traits was investigated with different models, in which the log squared estimated residuals for each phenotypic record were analyzed using the restricted maximum likelihood method. Monte Carlo simulation was performed to assess the reliability of variance component estimates from the second step and the accuracy of estimated breeding values for residual variation.

Results

The results suggest that both genetic and environmental factors have an effect on the residual variance of weight gain from birth to weaning and long-yearling in Nellore beef cattle and that uniformity of these traits could be improved by selecting for lower residual variance, when considering a large amount of information to predict genetic merit for this criterion. Simulations suggested that using the two-step approach would lead to biased estimates of variance components, such that more adequate methods are needed to study the genetic heterogeneity of residual variance in beef cattle.     

Genetic heterogeneity of within-family variance of body weight in Atlantic salmon (Salmo salar)

Background

Canalization is defined as the stability of a genotype against minor variations in both environment and genetics. Genetic variation in degree of canalization causes heterogeneity of within-family variance. The aims of this study are twofold: (1) quantify genetic heterogeneity of (within-family) residual variance in Atlantic salmon and (2) test whether the observed heterogeneity of (within-family) residual variance can be explained by simple scaling effects.

Results

Analysis of body weight in Atlantic salmon using a double hierarchical generalized linear model (DHGLM) revealed substantial heterogeneity of within-family variance. The 95% prediction interval for within-family variance ranged from ~0.4 to 1.2 kg², implying that the within-family variance of the most extreme high families is expected to be approximately three times larger than the extreme low families. For cross-sectional data, DHGLM with an animal mean sub-model resulted in severe bias, while a corresponding sire-dam model was appropriate. Heterogeneity of variance was not sensitive to Box-Cox transformations of phenotypes, which implies that heterogeneity of variance exists beyond what would be expected from simple scaling effects.

Conclusions

Substantial heterogeneity of within-family variance was found for body weight in Atlantic salmon. A tendency towards higher variance with higher means (scaling effects) was observed, but heterogeneity of within-family variance existed beyond what could be explained by simple scaling effects. For cross-sectional data, using the animal mean sub-model in the DHGLM resulted in biased estimates of variance components, which differed substantially both from a standard linear mean animal model and a sire-dam DHGLM model. Although genetic differences in canalization were observed, selection for increased canalization is difficult, because there is limited individual information for the variance sub-model, especially when based on cross-sectional data. Furthermore, potential macro-environmental changes (diet, climatic region, etc.) may make genetic heterogeneity of variance a less stable trait over time and space.     

Breeding for robustness: investigating the genotype‐by‐environment interaction and micro‐environmental sensitivity of Genetically Improved Farmed Tilapia (Oreochromis niloticus)

Robustness has become a highly desirable breeding goal in the globalized agricultural market. Both genotype‐by‐environment interaction (G × E) and micro‐environmental sensitivity are important robustness components of aquaculture production, in which breeding stock is often disseminated to different environments. The objectives of this study were (i) to quantify the degree of G × E by assessing the growth performance of Genetically Improved Farmed Tilapia (GIFT) across three countries (Malaysia, India and China) and (ii) to quantify the genetic heterogeneity of environmental variance for body weight at harvest (BW) in GIFT as a measure of micro‐environmental sensitivity. Selection for BW was carried out for 13 generations in Malaysia. Subsets of 60 full‐sib families from Malaysia were sent to China and India after five and nine generations respectively. First, a multi‐trait animal model was used to analyse the BW in different countries as different traits. The results indicate a strong G × E. Second, a genetically structured environmental variance model, implemented using Bayesian inference, was used to analyse micro‐environmental sensitivity of BW in each country. The analysis revealed the presence of genetic heterogeneity of both BW and its environmental variance in all environments. The presence of genetic variation in residual variance of BW implies that the residual variance can be modified by selection. Incorporating both G × E and micro‐environmental sensitivity information may help in selecting robust genotypes with high performance across environments and resilience to environmental fluctuations.     

Comparison of statistical models to analyse the genetic effect on within-litter variance in pigs

Genetics affects not only the weight of piglets at birth but also the variability of birth weight within litter. Previous studies on this topic assigned the sample standard deviation of piglet birth weights within litter as an observation to the sow. However, the contribution of the difference in mean birth weight per sex on the within-litter variance has been neglected so far. This work deals with the genetic effect on within-litter variance when different statistical models with different distributional assumptions are used and considers the sex effect and appropriate weights per trait. Traits were formed from the pooled sample variance of male and female birth weights within litter. A linear model approach was fitted to the logarithmized within-litter variance and the sample standard deviation. A generalized linear model with gamma-distributed residuals and log-link function was applied to the untransformed sample variance. Models were compared by analysing data from 9439 litters from Landrace and Large White of a commercial breeding programme. The estimates of heritability for different traits ranged from 7% to 11%. Although the generalized linear mixed model is preferred from a mathematical view, the rank correlations between breeding values of the linear mixed models and the generalized linear mixed model were relatively high, i.e. 94% to 98%. By residual diagnostics, a linear mixed model using the weighted and pooled within-litter standard deviation was identified as most suitable.     

Logistic regression in capture-recapture models

The effect of population heterogeneity in capture-recapture, or dual registration, models is discussed. An estimator of the unknown population size based on a logistic regression model is introduced. The model allows different capture probabilities across individuals and across capture times. The probabilities are estimated from the observed data using conditional maximum likelihood. The resulting population estimator is shown to be consistent and asymptotically normal. A variance estimator under population heterogeneity is derived. The finite-sample properties of the estimators are studied via simulation. An application to Finnish occupational disease registration data is presented.     

Joint Analysis of Binomial and Continuous Traits with a Recursive Model: A Case Study Using Mortality and Litter Size of Pigs

This work presents a model for the joint analysis of a binomial and a Gaussian trait using a recursive parametrization that leads to a computationally efficient implementation. The model is illustrated in an analysis of mortality and litter size in two breeds of Danish pigs, Landrace and Yorkshire. Available evidence suggests that mortality of piglets increased partly as a result of successful selection for total number of piglets born. In recent years there has been a need to decrease the incidence of mortality in pig-breeding programs. We report estimates of genetic variation at the level of the logit of the probability of mortality and quantify how it is affected by the size of the litter. Several models for mortality are considered and the best fits are obtained by postulating linear and cubic relationships between the logit of the probability of mortality and litter size, for Landrace and Yorkshire, respectively. An interpretation of how the presence of genetic variation affects the probability of mortality in the population is provided and we discuss and quantify the prospects of selecting for reduced mortality, without affecting litter size.     

Long‐term selection for litter size in swine results in shifts in allelic frequency in regions involved in reproductive processes

High‐density genotype data were analyzed in three lines of swine that express substantial variation in sow fertility to uncover regions of the genome potentially influenced during selection for litter size traits. The experimental lines examined include the Nebraska Index Line (NIL), which has been subjected to long‐term selection for litter size; a control line derived from the same population that founded NIL; and a commercial Duroc × Hampshire (D × H) population, in which no selection for litter size was practiced. Regions of the genome potentially affected by selection for litter size traits in NIL were determined by multiple lines of evidence, including altered allelic frequency compared to the other lines, loss of heterozygosity and relative extended haplotype homozygosity. Additionally, a genome‐wide association study for litter size traits was conducted in a population based on NIL and commercial maternal line genetics. Several genomic regions identified as putative signatures of selection overlapped with QTL for litter size traits. One of these regions, located on SSC2 (13–14 Mb), includes the candidate gene P2X3R, which plays a role in implantation and sustained release of hormones associated with reproductive processes. Sequencing identified synonymous SNPs in P2X3R that are fixed in NIL but polymorphic with nearly equal frequencies in the D × H line, indicating a potential role of P2X3R in sow fertility. These results suggest that data derived from these lines can help to uncover and understand a portion of the genetic variance associated with fertility traits in swine.     

Demographic variance in heterogeneous populations: matrix models and sensitivity analysis

The demographic consequences of stochasticity in processes such as survival and reproduction are modulated by the heterogeneity within the population. Therefore, to study effects of stochasticity on population growth and extinction risk, it is critical to use structured population models in which the most important sources of heterogeneity (e.g. age, size, developmental stage) are incorporated as i‐state variables. Demographic stochasticity in heterogeneous populations has often been studied using one of two approaches: multitype branching processes and diffusion approximations. Here, we link these approaches, through the demographic stochasticity in age‐ or stage‐structured matrix population models. We derive the demographic variance, σ²_d, which measures the per capita contribution to the variance in population growth increment, and we show how it can be decomposed into contributions from transition probabilities and fertility across ages or stages. Furthermore, using matrix calculus we derive the sensitivity of σ²_d to age‐ or stage‐specific mortality and fertility. We apply the methods to an extensive set of data from age‐classified human populations (long‐term time‐series for Sweden, Japan and the Netherlands; two hunter–gatherer populations, and the high‐fertility Hutterites), and to a size‐classified population of the herbaceous plant Calathea ovandensis. For the human populations our analysis reveals substantial temporal changes in the demographic variance as well as its main components across age. These new methods provide a powerful approach for calculating the demographic variance for any structured model, and for analyzing its main components and sensitivities. This will make possible new analyses of demographic variance across different kinds of heterogeneity in different life cycles, which will in turn improve our understanding of mechanisms underpinning extinction risk and other important biological outcomes.     

Quantifying individual variation in reaction norms: Mind the residual

Phenotypic plasticity is a central topic in ecology and evolution. Individuals may differ in the degree of plasticity (individual‐by‐environment interaction (I × E)), which has implications for the capacity of populations to respond to selection. Random regression models (RRMs) are a popular tool to study I × E in behavioural or life‐history traits, yet evidence for I × E is mixed, differing between species, populations, and even between studies on the same population. One important source of discrepancies between studies is the treatment of heterogeneity in residual variance (heteroscedasticity). To date, there seems to be no collective awareness among ecologists of its influence on the estimation of I × E or a consensus on how to best model it. We performed RRMs with differing residual variance structures on simulated data with varying degrees of heteroscedasticity and plasticity, sample size and environmental variability to test how RRMs would perform under each scenario. The residual structure in the RRMs affected the precision of estimates of simulated I × E as well as statistical power, with substantial lack of precision and high false‐positive rates when sample size, environmental variability and plasticity were small. We show that model comparison using information criteria can be used to choose among residual structures and reinforce this point by analysis of real data of two study populations of great tits (Parus major). We provide guidelines that can be used by biologists studying I × E that, ultimately, should lead to a reduction in bias in the literature concerning the statistical evidence and the reported magnitude of variation in plasticity.     

Structured latent growth curves for twin data.

We describe methods to fit structured latent growth curves to data from MZ and DZ twins. The well-known Gompertz, logistic and exponential curves may be written as a function of three components - asymptote, initial value, and rate of change. These components are allowed to vary and covary within individuals in a structured latent growth model. Such models are highly economical, requiring a small number of parameters to describe covariation across many occasions of measurement. We extend these methods to analyse longitudinal data from MZ and DZ twins and focus on the estimation of genetic and environmental variation and covariation in each of the asymptote, initial and rate of growth factors. For illustration, the models are fitted to longitudinal Bayley Infant Mental Development Scale data published by McArdle (1986). In these data, all three components of growth appear strongly familial with the majority of variance associated with the shared environment; differences between the models were not great. Occasion-specific residual factors not associated with the curve components account for approximately 40% of variance of which a significant proportion is additive genetic. Though the growth curve model fit less well than some others, they make restrictive, falsifiable predictions about the mean, variance and twin covariance of other (not yet measured) occasions of measurement.     

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.     

Ecological–genetic approach in modeling the natural evolution of a population: Prospects and special aspects of verification

This study presents a complex approach for modeling the natural evolution of a population in terms of population number and dynamics of the genetic structure. A set of dynamic models that consider various types of natural selection was applied to describe possible mechanisms underlying the formation of existing genetic variations in litter sizes in coastal, inland, and farmed arctic fox populations (Alopex lagopus, family Canidae, order Carnivora). The r–K selection model for uniform population and the models with natural selection were assessed on various life cycle stages in a two-age population. The life cycle of arctic fox was fitted to the population model with two age stages. The different reproductive potentials and survivability of progeny on the early stage of life cycle were genetically determined using the model with a single diallelic gene. A monomorphism was obtained for a considered characteristic in a population of coastal arctic fox with constant food supply. Meanwhile, a polymorphism with cyclic fluctuations in population number and gene frequency was obtained in inland arctic fox populations, which could be due to cyclic fluctuations of prey. In farmed fox populations, the considered gene becomes pleiotropic (defines the survival rate of individuals on early and late stages of the life cycle) because of artificial selection performed by farmers to increase the reproductive success of breeders. The application of an appropriate model (with selection by pleiotropic gene) can be used to determine the elimination rate of low litter size alleles from the farmed populations. The possible applications of the proposed models for formulating and solving optimal control tasks in arctic fox populations are discussed too.     

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.     

Natural selection and inheritance of breeding time and clutch size in the collared flycatcher

Many characteristics of organisms in free-living populations appear to be under directional selection, possess additive genetic variance, and yet show no evolutionary response to selection. Avian breeding time and clutch size are often-cited examples of such characters. We report analyses of inheritance of, and selection on, these traits in a long-term study of a wild population of the collared flycatcher Ficedula albicollis. We used mixed model analysis with REML estimation ("animal models") to make full use of the information in complex multigenerational pedigrees. Heritability of laying date, but not clutch size, was lower than that estimated previously using parent-offspring regressions, although for both traits there was evidence of substantial additive genetic variance (h2 = 0.19 and 0.29, respectively). Laying date and clutch size were negatively genetically correlated (rA = -0.41 +/- 0.09), implying that selection on one of the traits would cause a correlated response in the other, but there was little evidence to suggest that evolution of either trait would be constrained by correlations with other phenotypic characters. Analysis of selection on these traits in females revealed consistent strong directional fecundity selection for earlier breeding at the level of the phenotype (beta = -0.28 +/- 0.03), but little evidence for stabilising selection on breeding time. We found no evidence that clutch size was independently under selection. Analysis of fecundity selection on breeding values for laying date, estimated from an animal model, indicated that selection acts directly on additive genetic variance underlying breeding time (beta = -0.20 +/- 0.04), but not on clutch size (beta = 0.03 +/- 0.05). In contrast, selection on laying date via adult female survival fluctuated in sign between years, and was opposite in sign for selection on phenotypes (negative) and breeding values (positive). Our data thus suggest that any evolutionary response to selection on laying date is partially constrained by underlying life-history trade-offs, and illustrate the difficulties in using purely phenotypic measures and incomplete fitness estimates to assess evolution of life-history trade-offs. We discuss some of the difficulties associated with understanding the evolution of laying date and clutch size in natural populations.