Random regression analyses using B-spline functions to model growth of Nellore cattle

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.     

Inference of Cross-Level Interaction between Genes and Contextual Factors in a Matched Case-Control Metabolic Syndrome Study: A Bayesian Approach

Genes, environment, and the interaction between them are each known to play an important role in the risk for developing complex diseases such as metabolic syndrome. For environmental factors, most studies focused on the measurements observed at the individual level, and therefore can only consider the gene-environment interaction at the same individual scale. Indeed the group-level (called contextual) environmental variables, such as community factors and the degree of local area development, may modify the genetic effect as well. To examine such cross-level interaction between genes and contextual factors, a flexible statistical model quantifying the variability of the genetic effects across different categories of the contextual variable is in need. With a Bayesian generalized linear mixed-effects model with an unconditional likelihood, we investigate whether the individual genetic effect is modified by the group-level residential environment factor in a matched case-control metabolic syndrome study. Such cross-level interaction is evaluated by examining the heterogeneity in allelic effects under various contextual categories, based on posterior samples from Markov chain Monte Carlo methods. The Bayesian analysis indicates that the effect of rs1801282 on metabolic syndrome development is modified by the contextual environmental factor. That is, even among individuals with the same genetic component of PPARG_Pro12Ala, living in a residential area with low availability of exercise facilities may result in higher risk. The modification of the group-level environment factors on the individual genetic attributes can be essential, and this Bayesian model is able to provide a quantitative assessment for such cross-level interaction. The Bayesian inference based on the full likelihood is flexible with any phenotype, and easy to implement computationally. This model has a wide applicability and may help unravel the complexity in development of complex diseases.     

Evidence of genetic and maternal effects on secondary sex ratio in cattle

There is a paucity of estimates of genetic variation for secondary sex ratio (i.e., sex ratio at birth) in dairy cattle. The objective of this study was to estimate the direct and maternal genetic variance as well as maternal permanent environmental variance for offspring sex in dairy herds. The data consisted of 77,508 births from 61,963 dams and 2,859 sires in 1,369 Irish dairy herds across the years 2003 to 2008, inclusive. Mixed models were used to estimate all parameters. Significant genetic variation in sex ratio existed, with a heritability for secondary sex ratio estimated at 0.02; the genetic standard deviation was 0.07 percentage units. No maternal genetic effects on secondary sex ratio were identified but the proportion of phenotypic variance in secondary sex ratio attributable to maternal permanent environmental effects was similar to that attributable to the additive genetic variance (i.e., 0.02). These results, therefore, suggest that the paternal (genetic) influence on secondary sex ratio is just as large as the maternal (non-genetic) influence, both of which are biologically substantial. The results from this study will be useful in generating a sample population of divergent animals for inclusion in a controlled experiment to elucidate the physiological mechanism underpinning differences in secondary sex ratio.     

Semiparametric Bayesian analysis of case-control data under conditional gene-environment independence

In case-control studies of gene-environment association with disease, when genetic and environmental exposures can be assumed to be independent in the underlying population, one may exploit the independence in order to derive more efficient estimation techniques than the traditional logistic regression analysis (Chatterjee and Carroll, 2005, Biometrika92, 399-418). However, covariates that stratify the population, such as age, ethnicity and alike, could potentially lead to nonindependence. In this article, we provide a novel semiparametric Bayesian approach to model stratification effects under the assumption of gene-environment independence in the control population. We illustrate the methods by applying them to data from a population-based case-control study on ovarian cancer conducted in Israel. A simulation study is conducted to compare our method with other popular choices. The results reflect that the semiparametric Bayesian model allows incorporation of key scientific evidence in the form of a prior and offers a flexible, robust alternative when standard parametric model assumptions do not hold.     

Familial Clustering of Abdominal Visceral Fat and Total Fat Mass: The Québec Family Study

The evidence for common familial factors underlying total fat mass (estimated from underwater weighing) and abdominal visceral fat (assessed from CT scan) was examined in families participating in phase 2 of the Québec Family Study (QFS) using a bivariate familial correlation model. Previous QFS investigations suggest that both genetic (major and polygenic) and familial environmental factors influence each phenotype, accounting for between 55% to 71% of the phenotypic variance in fat mass, and between 55% to 72% for abdominal visceral fat The current study suggests that the bivariate familial effect ranges from 29% to 50%. This pattern suggests that there may be common familial determinants for abdominal visceral fat and total fat mass, as well as additional familial factors which are specific to each. The relatively high spouse cross-trait correlations usually suggest that a large percent of the bivariate familial effect may be environmental in origin. However, if mating is not random, then the spouse resemblance may reflect either genetic or environmental causes, depending on the source [i.e., through similar genes or cohabitation (environmental) effects]. Finally, there are significant sex differences in the magnitude of the familial cross-trait correlations involving parents, but not offspring, suggesting complex generation (i.e., age) and sex effects. For example, genes may turn on or off as a function of age and sex, and/or there may be an accumulation over time of effects due to the environment which may vary by sex. Whether the common familial factors are genetic (major and/or polygenic), environmental, or some combination of both, and whether the familial expression depends on sex and/or age warrants further investigation using more complex models.     

Analysis of lambing distribution in the Ripollesa sheep breed. II. Environmental and genetic sources of variation

Seasonal reproduction patterns are typically observed in small ruminants and are a major limitation for production efficiency in most meat- and dairy-type production systems. Indeed, selection for reduced seasonality could be an appealing strategy for the small ruminant industry worldwide, although its genetic background has been poorly analyzed. One of the main limitations relied on the availability of appropriate analytical tools to cope with the circular (i.e. year-round) pattern of lambing and kidding data. The recent development of a heteroskedastic circular mixed model provided the statistical tool to go deeply into the knowledge of seasonality in small ruminants. In this study, 26 005 lambing distribution records from 4764 Ripollesa ewes collected in 20 purebred flocks were analyzed. The model accounted for systematic (lambing interval and ewe age), permanent environmental (flock-year-season and ewe) and additive genetic sources of variation influencing both mean and dispersion pattern (i.e. heteroskedasticity). Systematic effects suggested that first-lambing ewes and short lambing intervals delayed lambing date (~30 days) and increased dispersion of the lambing period. Nevertheless, this was partially compensated by ewe age, given that youngest females tended to concentrate the lambing peak. Flock-year-season, permanent ewe and additive genetic sources of variation reached moderate variance components for direct (and residual) effects on lambing distribution, they being 0.119 (0.156), 0.092 (0.132) and 0.195 (0.170) radians², respectively. Moreover, all 95% credibility intervals were placed far from the null estimate. Covariances between direct and residual effects where high and positive for additive genetic (posterior mean, 0.814) and permanent ewe effects (posterior mean, 0.917), whereas it was not relevant for flock-year-season. Selection for direct additive genetic effects should be able to advance or delay the lambing peak, whereas selection applied on residual additive genetic effects should increase or reduce seasonality (i.e. concentrate or flatten the lambing peak). Moreover, the positive and relevant genetic covariance between direct and residual effects also suggested correlated genetic responses. As example, genetic selection for earlier lambing peaks must also reduce seasonality, whereas selection for narrower lambing seasons may originate a delay in the lambing peak. These results must be viewed as the first attempt to analyze systematic, environmental and genetic sources of variation of lambing distribution within the circular paradigm, they providing a reliable characterization of these effects within the context of an heteroskedastic model.     

Generating data in models including direct and maternal dominance effects

Algorithms are presented to simulate multiple generations of animal data by a model including direct additive genetic, maternal additive genetic, direct dominance, maternal dominance and permanent environmental effects. Dominance effects were computed as parental subclasses. Testing involved five single trait models that included direct contemporary group and direct additive effects, and different combinations of maternal, permanent environmental, and dominance effects. Simulated populations included 5 generations of animals and 20 contemporary groups per generation. The base population contained 200 sires and 600 dams. Variance components were estimated by Average-Information Restricted Maximum Likelihood (AIREML). No significant bias was observed. The simulation algorithms can be used in research involving dominance models, such as evaluation of mating systems exploiting special combining abilities of prospective parents.     

Genetic parameters for litter size in sheep: natural versus hormone-induced oestrus

The litter size in Suffolk and Texel-sheep was analysed using REML and Bayesian methods. Litters born after hormonal induced oestrus and after natural oestrus were treated as different traits in order to estimate the genetic correlation between the traits. Explanatory variables were the age of the ewe at lambing, period of lambing, a year*flock-effect, a permanent environmental effect associated with the ewe, and the additive genetic effect. The heritability estimates for litter size ranged from 0.06 to 0.13 using REML in bi-variate linear models. Transformation of the estimates to the underlying scale resulted in heritability estimates from 0.12 to 0.17. Posterior means of the heritability of litter size in the Bayesian approach with bi-variate threshold models varied from 0.05 to 0.18. REML estimates of the genetic correlations between the two types of litter size ranged from 0.57 to 0.64 in the Suffolk and from 0.75 to 0.81 in the Texel. The posterior means of the genetic correlation (Bayesian analysis) were 0.40 and 0.44 for the Suffolk and 0.56 and 0.75 for the Texel in the sire and animal model respectively. A bivariate threshold model seems appropriate for the genetic evaluation of prolificacy in the breeds concerned.     

An application of Bayesian QTL mapping to early development in double haploid lines of rainbow trout including environmental effects

A Bayesian model and variable dimensional parameter estimation based on Markov chain Monte Carlo was applied to map quantitative trait loci (QTLs) in a doubled haploid mapping population of rainbow trout. To increase power, the analysis was performed using the multiple-QTL model, which simultaneously accounted for all the environmental and genetic main effects that influence the expression of early development life history traits. By doing so we obtained the posterior estimated effects for the environmental factors as well as the number, positions, and the effects for the QTLs. The analyses revealed QTLs for time at hatching, embryonic length and weight at swim-up stage. The posterior expectation of the number of QTLs in different linkage groups shows that at least four QTLs are needed to explain the observed differences in early development between the clonal lines. The Bayesian method effectively combined all the information available to accurately position these QTLs in the rainbow trout genome.     

On marker-assisted prediction of genetic value: beyond the ridge

Marked-assisted genetic improvement of agricultural species exploits statistical dependencies in the joint distribution of marker genotypes and quantitative traits. An issue is how molecular (e.g., dense marker maps) and phenotypic information (e.g., some measure of yield in plants) is to be used for predicting the genetic value of candidates for selection. Multiple regression, selection index techniques, best linear unbiased prediction, and ridge regression of phenotypes on marker genotypes have been suggested, as well as more elaborate methods. Here, phenotype-marker associations are modeled hierarchically via multilevel models including chromosomal effects, a spatial covariance of marked effects within chromosomes, background genetic variability, and family heterogeneity. Lorenz curves and Gini coefficients are suggested for assessing the inequality of the contribution of different marked effects to genetic variability. Classical and Bayesian methods are presented. The Bayesian approach includes a Markov chain Monte Carlo implementation. The generality and flexibility of the Bayesian method is illustrated when a Lorenz curve is to be inferred.     

Analysis of melanoma onset: assessing familial aggregation by using estimating equations and fitting variance components via Bayesian random effects models.

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.     

Breeding value accuracy estimates for growth traits using random regression and multi-trait models in Nelore cattle

We quantified the potential increase in accuracy of expected breeding value for weights of Nelore cattle, from birth to mature age, using multi-trait and random regression models on Legendre polynomials and B-spline functions. A total of 87,712 weight records from 8144 females were used, recorded every three months from birth to mature age from the Nelore Brazil Program. For random regression analyses, all female weight records from birth to eight years of age (data set I) were considered. From this general data set, a subset was created (data set II), which included only nine weight records: at birth, weaning, 365 and 550 days of age, and 2, 3, 4, 5, and 6 years of age. Data set II was analyzed using random regression and multi-trait models. The model of analysis included the contemporary group as fixed effects and age of dam as a linear and quadratic covariable. In the random regression analyses, average growth trends were modeled using a cubic regression on orthogonal polynomials of age. Residual variances were modeled by a step function with five classes. Legendre polynomials of fourth and sixth order were utilized to model the direct genetic and animal permanent environmental effects, respectively, while third-order Legendre polynomials were considered for maternal genetic and maternal permanent environmental effects. Quadratic polynomials were applied to model all random effects in random regression models on B-spline functions. Direct genetic and animal permanent environmental effects were modeled using three segments or five coefficients, and genetic maternal and maternal permanent environmental effects were modeled with one segment or three coefficients in the random regression models on B-spline functions. For both data sets (I and II), animals ranked differently according to expected breeding value obtained by random regression or multi-trait models. With random regression models, the highest gains in accuracy were obtained at ages with a low number of weight records. The results indicate that random regression models provide more accurate expected breeding values than the traditionally finite multi-trait models. Thus, higher genetic responses are expected for beef cattle growth traits by replacing a multi-trait model with random regression models for genetic evaluation. B-spline functions could be applied as an alternative to Legendre polynomials to model covariance functions for weights from birth to mature age.     

Genetic evaluation and selection response for growth in meat-type quail through random regression models using B-spline functions and Legendre polynomials

The objective was to estimate (co)variance functions using random regression models (RRM) with Legendre polynomials, B-spline function and multi-trait models aimed at evaluating genetic parameters of growth traits in meat-type quail. A database containing the complete pedigree information of 7000 meat-type quail was utilized. The models included the fixed effects of contemporary group and generation. Direct additive genetic and permanent environmental effects, considered as random, were modeled using B-spline functions considering quadratic and cubic polynomials for each individual segment, and Legendre polynomials for age. Residual variances were grouped in four age classes. Direct additive genetic and permanent environmental effects were modeled using 2 to 4 segments and were modeled by Legendre polynomial with orders of fit ranging from 2 to 4. The model with quadratic B-spline adjustment, using four segments for direct additive genetic and permanent environmental effects, was the most appropriate and parsimonious to describe the covariance structure of the data. The RRM using Legendre polynomials presented an underestimation of the residual variance. Lesser heritability estimates were observed for multi-trait models in comparison with RRM for the evaluated ages. In general, the genetic correlations between measures of BW from hatching to 35 days of age decreased as the range between the evaluated ages increased. Genetic trend for BW was positive and significant along the selection generations. The genetic response to selection for BW in the evaluated ages presented greater values for RRM compared with multi-trait models. In summary, RRM using B-spline functions with four residual variance classes and segments were the best fit for genetic evaluation of growth traits in meat-type quail. In conclusion, RRM should be considered in genetic evaluation of breeding programs.     

INTEGRATIVE TESTING OF HOW ENVIRONMENTS FROM THE PAST TO THE PRESENT SHAPE GENETIC STRUCTURE ACROSS LANDSCAPES

Tests of the genetic structure of empirical populations typically focus on the correlative relationships between population connectivity and geographic and/or environmental factors in landscape genetics. However, such tests may overlook or misidentify the impact of candidate factors on genetic structure, especially when connectivity patterns differ between past and present populations because of shifting environmental conditions over time. Here we account for the underlying demographic component of population connectivity associated with a temporarily dynamic landscape in tests of the factors structuring population genetic variation in an Australian lizard, Lerista lineopunctulata, from 24 nuclear loci. Correlative tests did not support significant effect from factors associated with a static contemporary landscape. However, spatially explicit demographic modeling of genetic differentiation shows that changes in environmental conditions (as estimated from paleoclimatic data) and corresponding distributional shifts from the past to present landscape significantly structures genetic variation. Results from model‐based inference (i.e., from an integrative modeling approach that generates spatially explicit expectations that are tested with approximate Bayesian computation) contrasts with those from correlative analyses, highlighting the importance of expanding the landscape genetic perspective to tests the links between pattern and process, revealing how factors shape patterns of genetic variation within species.     

Genetic Variation in a Heterogeneous Environment. II. Temporal Heterogeneity and Directional Selection

The maintenance of genetic variation is investigated in a finite population where selection at an autosomal locus with two alleles varies temporally between two environments and the heterozygote has an intermediate fitness value. When there is additive gene action and equal selection in both environments, the autocorrelation between subsequent environments must be negative for more maintenance of genetic variation than for neutrality. The maximum maintenance occurs when there is equal selection in the two environments and the autocorrelation approaches ^-1.0 (for a stochastic model), or when there is short repeating cycle such as one related to seasons. Also comparison of the effects of stochastic variation in selection in finite and infinite populations is made by using Monte Carlo simulation. One situation was found where temporal environmental variation maintains genetic variation very effectively even in a small population and that is when there is evolution of dominance, i.e., the heterozygote is closer in fitness to the favored homozygote than the other homozygote. An important conclusion is that in a finite population genetic tracing of environmental change, particularly when there is a positive autocorrelation between environments or a long environmental cycle, leads to an increased loss of genetic variation making such a response undesirable in the long term, a result different from that in infinite populations.     

Advances in Bayesian multiple quantitative trait loci mapping in experimental crosses

Many complex human diseases and traits of biological and/or economic importance are determined by interacting networks of multiple quantitative trait loci (QTL) and environmental factors. Mapping QTL is critical for understanding the genetic basis of complex traits, and for ultimate identification of genes responsible. A variety of sophisticated statistical methods for QTL mapping have been developed. Among these developments, the evolution of Bayesian approaches for multiple QTL mapping over the past decade has been remarkable. Bayesian methods can jointly infer the number of QTL, their genomic positions and their genetic effects. Here, we review recently developed and still developing Bayesian methods and associated computer software for mapping multiple QTL in experimental crosses. We compare and contrast these methods to clearly describe the relationships among different Bayesian methods. We conclude this review by highlighting some areas of future research.     

Analyzing large‐scale conservation interventions with Bayesian hierarchical models: a case study of supplementing threatened Pacific salmon

Myriad human activities increasingly threaten the existence of many species. A variety of conservation interventions such as habitat restoration, protected areas, and captive breeding have been used to prevent extinctions. Evaluating the effectiveness of these interventions requires appropriate statistical methods, given the quantity and quality of available data. Historically, analysis of variance has been used with some form of predetermined before‐after control‐impact design to estimate the effects of large‐scale experiments or conservation interventions. However, ad hoc retrospective study designs or the presence of random effects at multiple scales may preclude the use of these tools. We evaluated the effects of a large‐scale supplementation program on the density of adult Chinook salmon Oncorhynchus tshawytscha from the Snake River basin in the northwestern United States currently listed under the U.S. Endangered Species Act. We analyzed 43 years of data from 22 populations, accounting for random effects across time and space using a form of Bayesian hierarchical time‐series model common in analyses of financial markets. We found that varying degrees of supplementation over a period of 25 years increased the density of natural‐origin adults, on average, by 0–8% relative to nonsupplementation years. Thirty‐nine of the 43 year effects were at least two times larger in magnitude than the mean supplementation effect, suggesting common environmental variables play a more important role in driving interannual variability in adult density. Additional residual variation in density varied considerably across the region, but there was no systematic difference between supplemented and reference populations. Our results demonstrate the power of hierarchical Bayesian models to detect the diffuse effects of management interventions and to quantitatively describe the variability of intervention success. Nevertheless, our study could not address whether ecological factors (e.g., competition) were more important than genetic considerations (e.g., inbreeding depression) in determining the response to supplementation.     

Fine mapping of complex trait genes combining pedigree and linkage disequilibrium information: a Bayesian unified framework

We present a Bayesian method that combines linkage and linkage disequilibrium (LDL) information for quantitative trait locus (QTL) mapping. This method uses jointly all marker information (haplotypes) and all available pedigree information; i.e., it is not restricted to any specific experimental design and it is not required that phases are known. Infinitesimal genetic effects or environmental noise ("fixed") effects can equally be fitted. A diallelic QTL is assumed and both additive and dominant effects can be estimated. We have implemented a combined Gibbs/Metropolis-Hastings sampling to obtain the marginal posterior distributions of the parameters of interest. We have also implemented a Bayesian variant of usual disequilibrium measures like D' and r(2) between QTL and markers. We illustrate the method with simulated data in "simple" (two-generation full-sib families) and "complex" (four-generation) pedigrees. We compared the estimates with and without using linkage disequilibrium information. In general, using LDL resulted in estimates of QTL position that were much better than linkage-only estimates when there was complete disequilibrium between the mutant QTL allele and the marker. This advantage, however, decreased when the association was only partial. In all cases, additive and dominant effects were estimated accurately either with or without disequilibrium information.     

Has the "Equal Environments" assumption been tested in twin studies?

A recurring criticism of the twin method for quantifying genetic and environmental components of human differences is the necessity of the so-called "equal environments assumption" (EEA) (i.e., that monozygotic and dizygotic twins experience equally correlated environments). It has been proposed to test the EEA by stratifying twin correlations by indices of the amount of shared environment. However, relevant environments may also be influenced by genetic differences. We present a model for the role of genetic factors in niche selection by twins that may account for variation in indices of the shared twin environment (e.g., contact between members of twin pairs). Simulations reveal that stratification of twin correlations by amount of contact can yield spurious evidence of large shared environmental effects in some strata and even give false indications of genotype x environment interaction. The stratification approach to testing the equal environments assumption may be misleading and the results of such tests may actually be consistent with a simpler theory of the role of genetic factors in niche selection.     

Bayesian analysis for genetic architecture of dynamic traits

The dissection of the genetic architecture of quantitative traits, including the number and locations of quantitative trait loci (QTL) and their main and epistatic effects, has been an important topic in current QTL mapping. We extend the Bayesian model selection framework for mapping multiple epistatic QTL affecting continuous traits to dynamic traits in experimental crosses. The extension inherits the efficiency of Bayesian model selection and the flexibility of the Legendre polynomial model fitting to the change in genetic and environmental effects with time. We illustrate the proposed method by simultaneously detecting the main and epistatic QTLs for the growth of leaf age in a doubled-haploid population of rice. The behavior and performance of the method are also shown by computer simulation experiments. The results show that our method can more quickly identify interacting QTLs for dynamic traits in the models with many numbers of genetic effects, enhancing our understanding of genetic architecture for dynamic traits. Our proposed method can be treated as a general form of mapping QTL for continuous quantitative traits, being easier to extend to multiple traits and to a single trait with repeat records.