Estimating maternal genetic effects in livestock

This study investigates the estimation of direct and maternal genetic (co)variances, accounting for environmental covariances between direct and maternal effects. Estimated genetic correlations between direct and maternal effects presented in the literature have often been strongly negative, and their validity has been questioned. Explanations of extreme estimates have focused on the existence of environmental covariances between dam and offspring. As a solution, models including a regression on dam-phenotype have been proposed, but have yielded biased estimates. The performance of models that implement the variance structure arising from the classical model of Willham, however, has not been evaluated. This study investigated the covariance structure of the parts of the residual term that arise from Willham's model. Results show that a correlation between the residual of the record of an individual and that of its dam is a direct consequence of combining Willham's model with the usual assumption that phenotypic covariances between different traits are the sum of additive genetic and environmental covariances. Stochastic simulations show that fitting this structure yields unbiased estimates of the genetic (co)variances. When correlated residuals were ignored in the cases investigated, the bias in the estimated genetic correlations was approximately equal to the value of the environmental correlation. In contrast to models including a regression on dam-phenotype, there were no difficulties with interpretation of results, and the approach was consistent with standard quantitative genetic theory. The use of Willham's model while accounting for correlated residuals is conceptually appealing and yields unbiased results, with no need for regression on dam phenotype. Inclusion of the ability to fit the residual variance structure required for maternal effects into existing software packages would be helpful to animal breeders.     

Bayesian conjugate analysis using a generalized inverted Wishart distribution accounts for differential uncertainty among the genetic parameters--an application to the maternal animal model

Consider the estimation of genetic (co)variance components from a maternal animal model (MAM) using a conjugated Bayesian approach. Usually, more uncertainty is expected a priori on the value of the maternal additive variance than on the value of the direct additive variance. However, it is not possible to model such differential uncertainty when assuming an inverted Wishart (IW) distribution for the genetic covariance matrix. Instead, consider the use of a generalized inverted Wishart (GIW) distribution. The GIW is essentially an extension of the IW distribution with a larger set of distinct parameters. In this study, the GIW distribution in its full generality is introduced and theoretical results regarding its use as the prior distribution for the genetic covariance matrix of the MAM are derived. In particular, we prove that the conditional conjugacy property holds so that parameter estimation can be accomplished via the Gibbs sampler. A sampling algorithm is also sketched. Furthermore, we describe how to specify the hyperparameters to account for differential prior opinion on the (co)variance components. A recursive strategy to elicit these parameters is then presented and tested using field records and simulated data. The procedure returned accurate estimates and reduced standard errors when compared with non-informative prior settings while improving the convergence rates. In general, faster convergence was always observed when a stronger weight was placed on the prior distributions. However, analyses based on the IW distribution have also produced biased estimates when the prior means were set to over-dispersed values.     

Genetic parameters for a maternal breeding goal in beef production

New maternal breeding values have been developed for use in UK beef evaluations. To undertake multitrait BLUP evaluations, it is necessary to have a full covariance matrix. This study outlines the approach taken to construct the full covariance matrices for the four beef breeds that most widely contribute to suckler beef cows in the United Kingdom. The maternal traits investigated were age at first calving, calving interval, lifespan, mature cow weight, 200-d weight, and calving difficulty. Three terminal sire traits (weight at 400 d, ultrasonic fat depth, and muscle score) were included to estimate covariances between the new and existing traits. A sire-maternal-grandsire model was used for the estimation procedure in a series of bivariate and multivariate models. A weighted bending procedure was employed to construct positive definite covariance matrices. Parameter estimates broadly agreed with literature values, although for some traits, literature information was very scarce. Some differences between parameters for different breeds were evident.     

Estimates of genetic and phenotypic covariance functions for postweaning growth and mature weight of beef cows

Annual weights of cows from 19 to 119 months of age in two herds were analysed fitting a random regression model, regressing on orthogonal polynomials of age in months. Estimates of covariances between random regression coefficients were obtained by restricted maximum likelihood, and the resulting estimates of covariance functions were used to construct covariance matrices for all ages in the data. Analyses were carried out fitting regression coefficients corresponding to overall animal effects only and fitting regressions for animals' additive genetic and permanent, environmental effects. Different definitions of fixed effects subclasses were examined. Models were compared using likelihood ratio tests and estimated standard deviations for the ages in the data. Cubic regressions were sufficient to model both population trajectories and individual growth curves. Random regression coefficients were highly correlated, so that estimation forcing their covariance matrices to have reduced rank (2 or 3) did not reduce likelihoods significantly, allowing parsimonious modelling. Results showed that records were clearly not repeated measurements of a single trait with constant variances. As cows grew up to about 5 years of age, variances. As cows grew up to about 5 years of age, variances increased. Estimates of genetic correlations between 3-year-old and older cows were close to unity in one herd but more erratic in the other. For both herds, genetic correlations between weights on 2-year-old cows and older animals were clearly less than unity.     

Semiparametric animal models via penalized splines as alternatives to models with contemporary groups

Contemporary groups (CG) are used in genetic evaluation to account for systematic environmental effects of management, nutritional level, or any other differentially expressed group effect; however, because the functional form of the distribution of those effects is unknown, CG serve as an approximation to a time-varying mean. Conversely, in semiparametric models, there is no need to assume any functional form for the time-varying effects. In this research, we present a semiparametric animal model (AMS) using the covariate day of birth (DOB) by means of penalized splines (P-splines), as an alternative to fitting CG. In the AMS, the functionality of the data on DOB is expressed by means of a Basic segmented polynomial line (B-spline) basis, and proper covariance matrices are used to reflect the serial correlation among the points of support (or knots) at different times. Three different covariance matrices that reflect either short- or long-range dependences among knots are discussed. Different models were fitted to birth weight data from Polled Hereford calves. Models compared were an animal model with CG, an animal model with CG and the covariate DOB nested within CG (CG + DOB), and P-splines with the first difference penalty matrix and three different AMS with 20, 40, 60, 80, or 120 knots. Models were compared using a modified Akaike information criterion (AICC), which was calculated as a byproduct of the estimation of variance components by REML using the expectation maximization algorithm. All three AMS had smaller (better) values of AICC than the regular model with CG, while producing almost the same ranking of predicted breeding values and similar average predicted error variance. In all AMS, the inference and all measures of comparison were similar when the number of knots was equal > or = 40. The model CG + DOB had analogous performance to the AMS, but at the expense of using more parameters. It is concluded that the use of penalized regression splines using a B-spline basis with proper covariance matrices is a competitive method to the fitting of CG into animal models for genetic evaluation, without having to assume any functional form for the covariate DOB.     

Simple preconditioners for the conjugate gradient method: experience with test day models

Preconditioned conjugate gradient method can be used to solve large mixed model equations quickly. Convergence of the method depends on the quality of the preconditioner. Here, the effect of simple preconditioners on the number of iterations until convergence was studied by solving breeding values for several test day models. The test day records were from a field data set, and several simulated data sets with low and high correlations among regression coefficients. The preconditioner matrices had diagonal or block diagonal parts. Transformation of the mixed model equations by diagonalization of the genetic covariance matrix was studied as well. Preconditioner having the whole block of the fixed effects was found to be advantageous. A block diagonal preconditioner for the animal effects reduced the number of iterations the higher the correlations among animal effects, but increased memory usage of the preconditioner. Diagonalization of the animal genetic covariance matrix often reduced the number of iterations considerably without increased memory usage.     

A method for the prediction of multitrait breeding values for use in stochastic simulation to compare progeny-testing schemes, with large progeny groups for proven sires

A method of approximating estimated breeding values (EBV) from a multivariate distribution of true breeding values (TBV) and EBV is proposed for use in large-scale stochastic simulation of alternative breeding schemes with a complex breeding goal. The covariance matrix of the multivariate distributions includes the additive genetic (co)variances and approximated prediction error (co)variances at different selection stages in the life of the animal. The prediction error (co)variance matrix is set up for one animal at a time, utilizing information on the selection candidate and its offspring, the parents, as well as paternal and maternal half- sibs. The EBV are a regression on TBV taking individual uncertainty into account, but with additional 'free' variation drawn at random. With the current information included in the calculation of the prediction error variance of a selection candidate, it is concluded that the method can be used to optimize progeny-testing schemes, where the progeny-tested sires are utilized with large progeny groups, e.g. through artificial insemination.     

Covariances among sire by breed group of dam interaction effects in multibreed sire evaluation procedures

In multibreed populations, bulls need to be evaluated for additive and nonadditive genetic effects. When the nonadditive genetic effects associated with a bull are defined as sire x breed-group-of-dam interactions, they can be expressed as linear combinations of interactions between alleles of one or more breeds at one or more loci. If these specific allelic interactions are assumed to be independent, then variances and covariances between sire x breed-group-of-dam interaction subclasses can be shown to be linear combinations of variances and covariances of specific intra- and interlocus intra- and interbreed allelic interactions. Furthermore, covariances between sire x breed-group-of-dam interactions due to specific interactions at one, two, or more loci are zero. If dams are assumed to be unrelated to bulls and among themselves, except through their sires and maternal grandsires, efficient procedures to compute the inverse of the covariance matrices of nonadditive genetic effects can be devised, both in subclass and in regression models. Recursive procedures developed make possible the evaluation of large numbers of bulls for nonadditive genetic effects using mixed-model methodology. For completeness, recursive procedures to compute nonadditive covariance matrices in subclass and in regression models also were developed. The prediction of nonadditive genetic values for bulls, in addition to their additive genetic values, will help plan matings, make selection decisions more accurate and, possibly, make economic projections better.     

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.     

Genetic parameters for various random regression models to describe total sperm cells per ejaculate over the reproductive lifetime of boars

The objective of this study was to model the variances and covariances of total sperm cells per ejaculate (TSC) over the reproductive lifetime of AI boars. Data from boars (n = 834) selected for AI were provided by Smithfield Premium Genetics. The total numbers of records and animals were 19,629 and 1,736, respectively. Parameters were estimated for TSC by age of boar classification with a random regression model using the Simplex method and DxMRR procedures. The model included breed, collector, and year-season as fixed effects. Random effects were additive genetic, permanent environmental effect of boar, and residual. Observations were removed when the number of data at a given age of boar classification was < 10 records. Preliminary evaluations showed the best fit with fifth-order polynomials, indicating that the best model would have fifth-order fixed regression and fifth-order random regressions for animal and permanent environmental effects. Random regression models were fitted to evaluate all combinations of first- through seventh-order polynomial covariance functions. Goodness of fit for the models was tested using Akaike's Information Criterion and the Schwarz Criterion. The maximum log likelihood value was observed for sixth-, fifth-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. However, the best fit as determined by Akaike's Information Criterion and the Schwarz Criterion was by fitting sixth-, fourth-, and seventh-order polynomials; and fourth-, second-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. Heritability estimates for TSC ranged from 0.27 to 0.48 across age of boar classifications. In addition, heritability for TSC tended to increase with age of boar classification.     

Effects of nonrandom parental selection on estimation of variance components

Bayesian estimation via Gibbs sampling, REML, and Method R were compared for their empirical sampling properties in estimating genetic parameters from data subject to parental selection using an infinitesimal animal model. Models with and without contemporary groups, random or nonrandom parental selection, two levels of heritability, and none or 15% randomly missing pedigree information were considered. Nonrandom parental selection caused similar effects on estimates of variance components from all three methods. When pedigree information was complete, REML and Bayesian estimation were not biased by nonrandom parental selection for models with or without contemporary groups. Method R estimates, however, were strongly biased by nonrandom parental selection when contemporary groups were in the model. The bias was empirically shown to be a consequence of not fully accounting for gametic phase disequilibrium in the subsamples. The joint effects of nonrandom parental selection and missing pedigree information caused estimates from all methods to be highly biased. Missing pedigree information did not cause biased estimates in random mating populations. Method R estimates usually had greater mean square errors than did REML and Bayesian estimates.     

Random regression analyses using B-splines functions to model growth from birth to adult age in Canchim cattle

The objective of this work was to estimate covariance functions using random regression models on B-splines functions of animal age, for weights from birth to adult age in Canchim cattle. Data comprised 49,011 records on 2435 females. The model of analysis included fixed effects of contemporary groups, age of dam as quadratic covariable and the population mean trend taken into account by a cubic regression on orthogonal polynomials of animal age. Residual variances were modelled through a step function with four classes. The direct and maternal additive genetic effects, and animal and maternal permanent environmental effects were included as random effects in the model. A total of seventeen analyses, considering linear, quadratic and cubic B-splines functions and up to seven knots, were carried out. B-spline functions of the same order were considered for all random effects. Random regression models on B-splines functions were compared to a random regression model on Legendre polynomials and with a multitrait model. Results from different models of analyses were compared using the REML form of the Akaike Information criterion and Schwarz' Bayesian Information criterion. In addition, the variance components and genetic parameters estimated for each random regression model were also used as criteria to choose the most adequate model to describe the covariance structure of the data. 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 adequate to describe the covariance structure of the data. Random regression models using B-spline functions as base functions fitted the data better than Legendre polynomials, especially at mature ages, but higher number of parameters need to be estimated with B-splines functions.     

考力代羊生长性状的协方差函数估计

试验旨在模拟考力代羊生长过程中的遗传和表型变异,为其选种提供一定的依据.用1 269只考力代羊的3 332个生长发育记录,配合自变量为年龄的勒让德多项式的随机回归模型(RRM),其中个体加性遗传效应配合5阶,母体永久环境效应配合2~5阶,对数约束最大似然值和信息标准用来检测最佳配合阶数.用平均信息约束最大似然(AIREML)法来估计体重的遗传、环境和表型协方差函数,结果表明,母体永久环境效应配合4阶时,协方差图更符合考力代羊的生长变化情况,涉及的参数也较少,且羔羊在6月龄前后表现出较大的遗传和表型变异.这些结果提示随机回归模型在有限的数据下能够充分描述考力代羊体重在连续时间轴上的增长变化情况.     

Covariance functions and random regression models for cow weight in beef cattle

Data from the first four cycles of the Germplasm Evaluation program at the U.S. Meat Animal Research Center were used to evaluate weights of Angus, Hereford, and F1 cows produced by crosses of 22 sire and 2 dam (Angus and Hereford) breeds. Four weights per year were available for cows from 2 through 8 yr of age (AY) with age in months (AM). Weights (n = 61,798) were analyzed with REML using covariance function-random regression models (CF-RRM), with regression on orthogonal (Legendre) polynomials of AM. Models included fixed regression on AM and effects of cow line, age in years, season of measurement, and their interactions; year of birth; and pregnancy-lactation codes. Random parts of the models fitted RRM coefficients for additive (a) and permanent environmental (c) effects. Estimates of CF were used to estimate covariances among all ages. Temporary environmental effects were modeled to account for heterogeneity of variance by AY. Quadratic fixed regression was sufficient to model population trajectory and was fitted in all analyses. Other models varied order of fit and rank of coefficients for a and c. A parsimonious model included linear and quartic regression coefficients for a and c, respectively. A reduced cubic order sufficed for c. Estimates of all variances increased with age. Estimates for older ages disagreed with estimates using traditional bivariate models. Plots of covariances for c were smooth for intermediate, but erratic for extreme ages. Heritability estimates ranged from 0.38 (36 mo) to 0.78 (94 mo), with fluctuations especially for extreme ages. Estimates of genetic correlations were high for most pairs of ages, with the lowest estimate (0.70) between extreme ages (19 and 103 mo). Results suggest that although cow weights do not fit a repeatability model with constant variances as well as CF-RRM, a repeatability model might be an acceptable approximation for prediction of additive genetic effects.     

Impact of variation in length of individual testing periods on estimation of (co)variance components of a random regression model for feed intake of growing pigs

A simulation study was conducted to assess the influence of differences in the length of individual testing periods on estimates of (co)variance components of a random regression model for daily feed intake of growing pigs performance tested between 30 and 100 kg live weight. A quadratic polynomial in days on test with fixed regressions for sex, random regressions for additive genetic and permanent environmental effects and a constant residual variance was used for a bivariate simulation of feed intake and daily gain. (Co)variance components were estimated for feed intake only by means of a Bayesian analysis using Gibbs sampling and restricted maximum likelihood (REML). A single trait random regression model analogous to the one used for data simulation was used to analyse two versions of the data: full data sets with 18 weekly means of feed intake per animal and reduced data sets with the individual length of testing periods determined when tested animals reached 100 kg live weight. Only one significant difference between estimates from full and reduced data (REML estimate of genetic covariance between linear and quadratic regression parameters) and two significant differences from expected values (Gibbs estimates of permanent environmental variance of quadratic regression parameters) occurred. These differences are believed to be negligible, as the number lies within the expected range of type I error when testing at the 5% level. The course of test day variances calculated from estimates of additive genetic and permanent environmental covariance matrices also supports the conclusion that no bias in estimates of (co)variance components occurs due to the individual length of testing periods of performance‐tested growing pigs. A lower number of records per tested animal only results in more variation among estimates of (co)variance components from reduced compared with full data sets. Compared with the full data, the effective sample size of Gibbs samples from the reduced data decreased to 18% for residual variance and increased up to five times for other (co)variances. The data structure seems to influence the mixing of Gibbs chains.     

Variance components and breeding values for growth traits from different statistical models.

Estimates of genetic parameters resulting from various analytical models for birth weight (BWT, n = 4,155), 205-d weight (WWT, n = 3,884), and 365-d weight (YWT, n = 3,476) were compared. Data consisted of records for Line 1 Hereford cattle selected for postweaning growth from 1934 to 1989 at ARS-USDA, Miles City, MT. Twelve models were compared. Model 1 included fixed effects of year, sex, age of dam; covariates for birth day and inbreeding coefficients of animal and of dam; and random animal genetic and residual effects. Model 2 was the same as Model 1 but ignored inbreeding coefficients. Model 3 was the same as Model 1 and included random maternal genetic effects with covariance between direct and maternal genetic effects, and maternal permanent environmental effects. Model 4 was the same as Model 3 but ignored inbreeding. Model 5 was the same as Model 1 but with a random sire effect instead of animal genetic effect. Model 6 was the same as Model 5 but ignored inbreeding. Model 7 was a sire model that considered relationships among males. Model 8 was a sire model, assuming sires to be unrelated, but with dam effects as uncorrelated random effects to account for maternal effects. Model 9 was a sire and dam model but with relationships to account for direct and maternal genetic effects; dams also were included as uncorrelated random effects to account for maternal permanent environmental effects. Model 10 was a sire model with maternal grandsire and dam effects all as uncorrelated random effects. Model 11 was a sire and maternal grandsire model, with dams as uncorrelated random effects but with sires and maternal grandsires assumed to be related using male relationships. Model 12 was the same as Model 11 but with all pedigree relationships from the full animal model for sires and maternal grandsires. Rankings on predictions of breeding values were the same regardless of whether inbreeding coefficients for animal and dam were included in the models. Heritability estimates were similar regardless of whether inbreeding effects were in the model. Models 3 and 9 best fit the data for estimation of variances and covariances for direct, maternal genetic, and permanent environmental effects. Other models resulted in changes in ranking for predicted breeding values and for estimates of direct and maternal heritability. Heritability estimates of direct effects were smallest with sire and sire-maternal grandsire models.     

藏北高寒草地样带物种多样性沿降水梯度的分布格局

 在藏北高原高寒草地样带上对４０个围栏内草地群落物种多样性和地上生物量进行测定,探讨了生长季降水对高寒草地生物量和物种多样性分布格局的影响以及地上生物量－物种多样性之间的关系模式。结果表明,降水格局显著地影响藏北高原内部高寒草地群落物种丰富度、多样性和均匀度,群落结构特征与初级生产力关系密切;藏北地区高寒草地地上生物量、物种丰富度、Ｓｈａｎｎｏｎ－Ｗｉｅｎｅｒ指数和Ｐｉｅｌｏｕ均匀度指数随生长季累积降水呈指数增加趋势;在高寒草地群落物种丰富度－生产力关系研究中单峰模式的判别系数R^２（０．７５４）略高于线性回归模型（０．７４３）。沿藏北高原样带高寒草地物种丰富度随地上生物量单调递增,单峰模式的单调递减区间并未出现;然而单峰模型预示着在地上生物量高于１２１．１７ｇ／ｍ^２ 的高寒草地群落物种丰富度可能随生物量单调递减,从而使物种丰富度－地上生物量表现为较为标准的单峰模式;Ｓｈａｎｎｏｎ－Ｗｉｅｎｅｒ多样性指数和Ｐｉｅｌｏｕ均匀度指数与地上生物量也均呈单峰模式,但其单调递减区间窄于单调递增区间,峰值分别对应的草地群落地上生物量为７１．９０和６０．９０ｇ／ｍ^２。     

Efficient approximation of reliabilities for single-step genomic best linear unbiased predictor models with the Algorithm for Proven and Young

The objectives of this study were to develop an efficient algorithm for calculating prediction error variances (PEVs) for genomic best linear unbiased prediction (GBLUP) models using the Algorithm for Proven and Young (APY), extend it to single-step GBLUP (ssGBLUP), and apply this algorithm for approximating the theoretical reliabilities for single- and multiple-trait models in ssGBLUP. The PEV with APY was calculated by block sparse inversion, efficiently exploiting the sparse structure of the inverse of the genomic relationship matrix with APY. Single-step GBLUP reliabilities were approximated by combining reliabilities with and without genomic information in terms of effective record contributions. Multi-trait reliabilities relied on single-trait results adjusted using the genetic and residual covariance matrices among traits. Tests involved two datasets provided by the American Angus Association. A small dataset (Data1) was used for comparing the approximated reliabilities with the reliabilities obtained by the inversion of the left-hand side of the mixed model equations. A large dataset (Data2) was used for evaluating the computational performance of the algorithm. Analyses with both datasets used single-trait and three-trait models. The number of animals in the pedigree ranged from 167,951 in Data1 to 10,213,401 in Data2, with 50,000 and 20,000 genotyped animals for single-trait and multiple-trait analysis, respectively, in Data1 and 335,325 in Data2. Correlations between estimated and exact reliabilities obtained by inversion ranged from 0.97 to 0.99, whereas the intercept and slope of the regression of the exact on the approximated reliabilities ranged from 0.00 to 0.04 and from 0.93 to 1.05, respectively. For the three-trait model with the largest dataset (Data2), the elapsed time for the reliability estimation was 11 min. The computational complexity of the proposed algorithm increased linearly with the number of genotyped animals and with the number of traits in the model. This algorithm can efficiently approximate the theoretical reliability of genomic estimated breeding values in ssGBLUP with APY for large numbers of genotyped animals at a low cost.     

Approximating prediction error covariances among additive genetic effects within animals in multiple-trait and random regression models

A method for approximating prediction error variances and covariances among estimates of individual animals’ genetic effects for multiple‐trait and random regression models is described. These approximations are used to calculate the prediction error variances of linear functions of the terms in the model. In the multiple‐trait case these are indexes of estimated breeding values, and for random regression models these are estimated breeding values at individual points on the longitudinal scale. Approximate reliabilities for terms in the model and linear functions thereof are compared with corresponding reliabilities obtained from the inverse of the coefficient matrix in the mixed model equations. Results show good agreement between approximate and ‘true’ values.     

Genetic evaluation using random regression models with different covariance functions for test‐day milk yield in an admixture population of Thailand goats

The objectives of this study were to compare covariance functions (CF) and estimate the heritability of milk yield from test‐day records among exotic (Saanen, Anglo‐Nubian, Toggenburg and Alpine) and crossbred goats (Thai native and exotic breed), using a random regression model. A total of 1472 records of test‐day milk yield were used, collected from 112 does between 2003 and 2006. CF of the study were Wilmink function, second‐ and third‐order Legendre polynomials, and linear splines 4 knots located at 5, 25, 90 and 155 days in milk (SP25–90) and 5, 35, 95 and 155 of days in milk (SP35–95). Variance components were estimated by restricted maximum likelihood method (REML). Goodness of fit, Akaike information criterion (AIC), percentage of squared bias (PSB), mean square error (MSE), and empirical correlation (RHO) between the observed and predicted values were used to compare models. The results showed that CF had an impact on (co)variance estimation in random regression models (RRM). The RRM with splines 4 knots located at 5, 25, 90 and 155 of days in milk had the lowest AIC, PSB and MSE, and the highest RHO. The heritability estimated throughout lactation obtained with this model ranged from 0.13 to 0.23.