Estimates of genetic covariance functions for growth of Angus cattle

Genotype-by-environment interactions for feed efficiency traits in Nellore cattle based on bi-trait reaction norm models

BACKGROUND

Selecting animals for feed efficiency directly impacts the profitability of the beef cattle industry, which contributes to minimizing the environmental footprint of beef production. Genetic and environmental factors influence animal feed efficiency, leading to phenotypic variability when exposed to different environmental conditions (i.e., temperature and nutritional level). Thus, our aim was to assess potential genotype-by-environment (G × E) interactions for dry matter intake (DMI) and residual feed intake (RFI) in Nellore cattle (Bos taurus indicus) based on bi-trait reaction norm models (RN) and evaluate the genetic association between RFI and DMI across different environmental gradient (EG) levels. For this, we used phenotypic information on 12,958 animals (young bulls and heifers) for DMI and RFI recorded during 158 feed efficiency trials.

RESULTS

The heritability estimates for DMI and RFI across EG ranged from 0.26 to 0.54 and from 0.07 to 0.41, respectively. The average genetic correlations (± standard deviation) across EG for DMI and RFI were 0.83 ± 0.19 and 0.81 ± 0.21, respectively, with the lowest genetic correlation estimates observed between extreme EG levels (low vs. high) i.e. 0.22 for RFI and 0.26 for DMI, indicating the presence of G × E interactions. The genetic correlation between RFI and DMI across EG levels decreased as the EG became more favorable and ranged from 0.79 (lowest EG) to 0.52 (highest EG). Based on the estimated breeding values from extreme EG levels (low vs. high), we observed a moderate Spearman correlation of 0.61 (RFI) and 0.55 (DMI) and a selection coincidence of 53.3% and 40.0% for RFI and DMI, respectively.

CONCLUSIONS

Our results show evidence of G × E interactions on feed efficiency traits in Nellore cattle, especially in feeding trials with an average daily gain (ADG) that is far from the expected of 1 kg/day, thus increasing reranking of animals.

Optimization of test day for milk yield recording and sire evaluation in Murrah buffaloes

In the present study, random regression models (RRM) were used to estimate genetic parameters for test-day milk yield in Murrah buffaloes using Legendre polynomial function (LP), with the objective to find the best combination of "minimum test-day model," which would be essential and sufficient to evaluate the trait successfully. Data included for analysis were 10,615 first lactation monthly test-day milk yield records (5th, 35th, 65th, …, 305th) from 965 Murrah buffaloes for the period 1975-2018. Cubic to octic-order orthogonal polynomials with homogeneous residual variances were used for the estimation of genetic parameters. Random regression models with sixth-order were selected based on goodness of fit criteria like lower AIC, BIC and residual variance. Heritability estimates ranged from 0.079 (TD6) to 0.21(TD10). For both ends of lactation, the additive genetic and environmental variances were higher and ranged from 0.21 ± 0.12 (TD6) to 0.85 ± 0.35 kg² (TD1) and 3.74 ± 0.36 (TD11) to 1.36 ± 0.14 kg² (TD9), respectively. Between adjacent test-day records, genetic correlation estimates ranged from 0.09 ± 0.31 (TD1 and TD2) to 0.97 ± 0.03 (TD3 and TD4; TD4 and TD5), but values gradually declined as the distance between test days increased. Negative genetic correlations were also obtained between TD1 with TD3 to TD9, TD2 with TD9 and TD10, and TD3 with TD10. On the basis of genetic correlations, models with 5 and/or 6 test-days combination were able to account for 86.1%-98.7% of variation along the lactation. Models with fourth and fifth-order LP functions were considered to account for variance with combinations of 5 and/or 6 test-day milk yields. The model with 6 test-day combinations had a higher rank correlation (0.93) with model using 11 monthly test-day milk yield records. On the basis of relative efficiency, the model with 6 monthly test day combinations with fifth-order was more efficient (maximum 99%) than the model using 11 monthly test-day milk yield records. Looking into the similar accuracy with the 11TD model, and the low resources requirement, we recommend the use of the "6 test-day combination model" for sire evaluation. These models may help in reducing the cost and time for data recording of milk yield.

Heteroscedastic Reaction Norm Models Improve the Assessment of Genotype by Environment Interaction for Growth, Reproductive, and Visual Score Traits in Nellore Cattle

The assessment of the presence of genotype by environment interaction (GxE) in beef cattle is very important in tropical countries with diverse climatic conditions and production systems. The present study aimed to assess the presence of GxE by using different reaction norm models for eleven traits related to growth, reproduction, and visual score in Nellore cattle. We studied five reaction norm models (RNM), fitting a linear model considering homoscedastic residual variance (RNM_homo), and four models considering heteroskedasticity, being linear (RNM_hete), quadratic (RNM_quad), linear spline (RNM_l-l), and quadratic spline (RNM_q-q). There was the presence of GxE for age at first calving (AFC), scrotal circumference (SC), weaning to yearling weight gain (WYG), and yearling weight (YW). The best models were RNM_l-l for YW and RNM_q-q for AFC, SC, and WYG. The heritability estimates for RNM_l-l ranged from 0.07 to 0.20, 0.42 to 0.61, 0.24 to 0.42, and 0.47 to 0.63 for AFC, SC, WYG, and YW, respectively. The heteroskedasticity in reaction norm models improves the assessment of the presence of GxE for YW, WYG, AFC, and SC. Additionally, the trajectories of reaction norms for these traits seem to be affected by a non-linear component, and selecting robust animals for these traits is an alternative to increase production and reduce environmental sensitivity.

Simultaneous Bayesian estimation of genetic parameters for curves of weight, feed intake, and residual feed intake in beef cattle

Rates of gain and feed efficiency are important traits in most breeding programs for growing farm animals. The rate of gain (GAIN) is usually expressed over a certain age period and feed efficiency is often expressed as residual feed intake (RFI), defined as observed feed intake (FI) minus expected feed intake based on live weight (WGT) and GAIN. However, the basic traits recorded are always WGT and FI and other traits are derived from these basic records. The aim of this study was to develop a procedure for simultaneous analysis of the basic records and then derive linear traits related to feed efficiency without retorting to any approximation. A bivariate longitudinal random regression model was employed on 13,791 individual longitudinal records of WGT and FI from 2,827 bulls of six different beef breeds tested for their own performance in the period from 7 to 13 mo of age. Genetic and permanent environmental covariance functions for curves of WGT and FI were estimated using Gibbs sampling. Genetic and permanent covariance functions for curves of GAIN were estimated from the first derivative of the function for WGT and finally the covariance functions were extended to curves for RFI, based on the conditional distribution of FI given WGT and GAIN. Furthermore, the covariance functions were extended to include GAIN and RFI defined over different periods of the performance test. These periods included the whole test period as normally used when predicting breeding values for GAIN and RFI for beef bulls. Based on the presented method, breeding values and genetic parameters for derived traits such as GAIN and RFI defined longitudinally or integrated over (parts of) of the test period can be obtained from a joint analysis of the basic records. The resulting covariance functions for WGT, FI, GAIN, and RFI are usually singular but the method presented here does not suffer from the estimation problems associated with defining these traits individually before the genetic analysis. All the results are thus estimated simultaneously, and the set of parameters is consistent.

Comparison and estimation of different linear and nonlinear lactation curve submodels in random regression analyses on dairy cattle

In the random regression model (RRM) for milk yield, by replacing empirical lactation curves with the five-order Legendre polynomial to fit fixed groups, the RRM can be transformed to a hierarchical model that consisted of a RRM in the first hierarchy with Legendre polynomials as individuals’ lactation curves resolved by restricted maximum likelihood (REML) software, and a multivariate animal model for phenotypic regression coefficients in the second hierarchy resolved by DMU software. Some empirical lactation functions can be embedded into the RRM at the first hierarchy to well fit phenotypic lactation curve of the average observations across all animals. The functional relationship between each parameter and time can be described by a Legendre polynomial or an empirical curve usually called submodel, and according to three commonly used criteria, the optimal submodels were picked from linear and nonlinear submodels except for polynomials. The so-called hierarchical estimation for the RRMs in dairy cattle indicated that more biologically meaningful models were available to fit the lactation curves; moreover, with the same number of parameters, the empirical lactation curves (MIL1, MIL5, and MK1 for 3, 4, and 5 parameters, respectively) performed higher goodness of fit than Legendre polynomial when modelling individuals’ phenotypic lactation curves.

Modelos de regressão aleatória para características de crescimento de bovinos da raça Nelore do estado de Mato Groso

Foram utilizados 138.976 registros de informações de pesos corporais variando de 60 a 610 dias de idade, provenientes de 27.327 animais da raça Nelore, oriundos de rebanhos do estado do Mato Grosso, com o objetivo de descrever a variabilidade genética e estimar parâmetros genéticos para o peso corporal em diferentes idades, utilizando-se modelos de regressão aleatória. O modelo empregado incluiu efeitos fixos de grupo de contemporâneos e idade da vaca ao parto como covariáveis, além de efeitos aleatórios genético aditivo direto, genético materno, ambiente permanente de animal, ambiente permanente materno e efeito de ambiente temporário. O modelo de regressão aleatória mais adequado foi o que empregou função de covariância com polinômios de quarta ordem para descrição da variabilidade de todos os efeitos e duas classes de variância residual. As estimativas de variância genética aditiva direta e de ambiente permanente de animal aumentaram com a idade dos animais. As variâncias genética materna e de ambiente permanente materno exibiram comportamento semelhante, com maiores valores na fase de aleitamento. Os coeficientes de herdabilidade estimados variam de 0,25 a 0,43, com maiores valores nas idades mais avançadas na trajetória de crescimento dos animais. Esses resultados indicaram presença de variabilidade genética suficiente para obtenção de ganho genético expressivo por meio da seleção, principalmente após desmama. Os resultados encontrados para a correlação genética aditiva direta exibiram baixas correlações entre pesos nas idades iniciais e finais, porém pesos altamente correlacionados entre idades mais próximas. As correlações genéticas estimadas entre os pesos da desmama com os pesos até 610 dias de idade foram altas e positivas e indicam que os genes responsáveis por maiores pesos nesse período, em sua maioria, são os mesmos.

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.

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.

Random regression models on Legendre polynomials to estimate genetic parameters for weights from birth to adult age in Canchim cattle

The objective of this work was to estimate covariance functions for direct and maternal genetic effects, animal and maternal permanent environmental effects, and subsequently, to derive relevant genetic parameters for growth traits in Canchim cattle. Data comprised 49,011 weight records on 2435 females from birth to adult age. The model of analysis included fixed effects of contemporary groups (year and month of birth and at weighing) and age of dam as quadratic covariable. Mean trends were taken into account by a cubic regression on orthogonal polynomials of animal age. Residual variances were allowed to vary and were modelled by a step function with 1, 4 or 11 classes based on animal's age. The model fitting four classes of residual variances was the best. A total of 12 random regression models from second to seventh order were used to model direct and maternal genetic effects, animal and maternal permanent environmental effects. The model with direct and maternal genetic effects, animal and maternal permanent environmental effects fitted by quadric, cubic, quintic and linear Legendre polynomials, respectively, was the most adequate to describe the covariance structure of the data. Estimates of direct and maternal heritability obtained by multi-trait (seven traits) and random regression models were very similar. Selection for higher weight at any age, especially after weaning, will produce an increase in mature cow weight. The possibility to modify the growth curve in Canchim cattle to obtain animals with rapid growth at early ages and moderate to low mature cow weight is limited.

Estimation of genetic parameters for milk yield in Murrah buffaloes by Bayesian inference

Random regression models were used to estimate genetic parameters for test-day milk yield in Murrah buffaloes using Bayesian inference. Data comprised 17,935 test-day milk records from 1,433 buffaloes. Twelve models were tested using different combinations of third-, fourth-, fifth-, sixth-, and seventh-order orthogonal polynomials of weeks of lactation for additive genetic and permanent environmental effects. All models included the fixed effects of contemporary group, number of daily milkings and age of cow at calving as covariate (linear and quadratic effect). In addition, residual variances were considered to be heterogeneous with 6 classes of variance. Models were selected based on the residual mean square error, weighted average of residual variance estimates, and estimates of variance components, heritabilities, correlations, eigenvalues, and eigenfunctions. Results indicated that changes in the order of fit for additive genetic and permanent environmental random effects influenced the estimation of genetic parameters. Heritability estimates ranged from 0.19 to 0.31. Genetic correlation estimates were close to unity between adjacent test-day records, but decreased gradually as the interval between test-days increased. Results from mean squared error and weighted averages of residual variance estimates suggested that a model considering sixth- and seventh-order Legendre polynomials for additive and permanent environmental effects, respectively, and 6 classes for residual variances, provided the best fit. Nevertheless, this model presented the largest degree of complexity. A more parsimonious model, with fourth- and sixth-order polynomials, respectively, for these same effects, yielded very similar genetic parameter estimates. Therefore, this last model is recommended for routine applications.

Covariance functions for body weight from birth to maturity in Nellore cows

The objective of this study was to estimate (co)variance functions using random regression models on Legendre polynomials for the analysis of repeated measures of BW from birth to adult age. A total of 82,064 records from 8,145 females were analyzed. Different models were compared. The models included additive direct and maternal effects, and animal and maternal permanent environmental effects as random terms. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of animal age (cubic regression) were considered as random covariables. Eight models with polynomials of third to sixth order were used to describe additive direct and maternal effects, and animal and maternal permanent environmental effects. Residual effects were modeled using 1 (i.e., assuming homogeneity of variances across all ages) or 5 age classes. The model with 5 classes was the best to describe the trajectory of residuals along the growth curve. The model including fourth- and sixth-order polynomials for additive direct and animal permanent environmental effects, respectively, and third-order polynomials for maternal genetic and maternal permanent environmental effects were the best. Estimates of (co)variance obtained with the multi-trait and random regression models were similar. Direct heritability estimates obtained with the random regression models followed a trend similar to that obtained with the multi-trait model. The largest estimates of maternal heritability were those of BW taken close to 240 d of age. In general, estimates of correlation between BW from birth to 8 yr of age decreased with increasing distance between ages.

Nonparametric functional mapping of quantitative trait loci

Functional mapping is a useful tool for mapping quantitative trait loci (QTL) that control dynamic traits. It incorporates mathematical aspects of biological processes into the mixture model-based likelihood setting for QTL mapping, thus increasing the power of QTL detection and the precision of parameter estimation. However, in many situations there is no obvious functional form and, in such cases, this strategy will not be optimal. Here we propose to use nonparametric function estimation, typically implemented with B-splines, to estimate the underlying functional form of phenotypic trajectories, and then construct a nonparametric test to find evidence of existing QTL. Using the representation of a nonparametric regression as a mixed model, the final test statistic is a likelihood ratio test. We consider two types of genetic maps: dense maps and general maps, and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples.

A semiparametric approach for composite functional mapping of dynamic quantitative traits

Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.

Estimation of genetic parameters for body weight of the Goettingen minipig with random regression models1

The Goettingen minipig is a laboratory animal especially developed for medical research. For easy and comfortable handling during experiments, and to minimize costs, a low BW is essential. To breed for an even smaller minipig, genetic parameters for BW were estimated using a random regression model (RRM). The RRM was calculated using random animal, common litter environment, and permanent environment effects, respectively. Regressions for the random effects in the RRM were modeled using Legendre polynomials from second to fourth order of fit in different combinations. The model was applied to a data set that focused on the time period from 30 to 400 d of age. Eight age classes were built to consider heterogeneous residual variances. The heritabilities were moderate and ranged from 0.211 (375 d of age) to 0.254 (275 d of age). The variances initially decreased and then increased toward the end of the examined time period for permanent environment and litter effects. Genetic and phenotypic correlations between BW in different age classes decreased with increasing distance between age classes. The major eigenfunction showed positive values throughout the whole trajectory (i.e., a selection for low BW had positive effects on this trait throughout the whole range of time). On the basis of the estimated genetic parameters, a breeding scheme can be created to develop genetically smaller Goettingen minipigs in the future.

Properties of random regression models using linear splines

Properties of random regression models using linear splines (RRMS) were evaluated with respect to scale of parameters, numerical properties, changes in variances and strategies to select the number and positions of knots. Parameters in RRMS are similar to those in multiple trait models with traits corresponding to points at knots. RRMS have good numerical properties because of generally superior numerical properties of splines compared with polynomials and sparser system of equations. These models also contain artefacts in terms of depression of variances and predictions in the middle of intervals between the knots, and inflation of predictions close to knots; the artefacts become smaller as correlations corresponding to adjacent knots increase. The artefacts can be greatly reduced by a simple modification to covariables. With the modification, the accuracy of RRMS increases only marginally if the correlations between the adjacent knots are > or =0.6. In practical analyses the knots for each effect in RRMS can be selected so that: (i) they cover the entire trajectory; (ii) changes in variances in intervals between the knots are approximately linear; and (iii) the correlations between the adjacent knots are at least 0.6. RRMS allow for simple and numerically stable implementations of genetic evaluations with artefacts present but transparent and easily controlled.

A joint model for nonparametric functional mapping of longitudinal trajectory and time-to-event

Background

The characterization of the relationship between a longitudinal response process and a time-to-event has been a pressing challenge in biostatistical research. This has emerged as an important issue in genetic studies when one attempts to detect the common genes or quantitative trait loci (QTL) that govern both a longitudinal trajectory and developmental event.

Results

We present a joint statistical model for functional mapping of dynamic traits in which the event times and longitudinal traits are taken to depend on a common set of genetic mechanisms. By fitting the Legendre polynomial of orthogonal properties for the time-dependent mean vector, our model does not rely on any curve, which is different from earlier parametric models of functional mapping. This newly developed nonparametric model is demonstrated and validated by an example for a forest tree in which stemwood growth and the time to first flower are jointly modelled.

Conclusion

Our model allows for the detection of specific QTL that govern both longitudinal traits and developmental processes through either pleiotropic effects or close linkage, or both. This model will have great implications for integrating longitudinal and event data to gain better insights into comprehensive biology and biomedicine.