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J. Anim. Sci. 2005. 83:757-763
© 2005 American Society of Animal Science

ANIMAL GENETICS

Genetic parameters estimated with multitrait and linear spline-random regression models using Gelbvieh early growth data1

H. Iwaisaki*,2, S. Tsuruta{dagger},3, I. Misztal{dagger} and J. K. Bertrand{dagger}

* Department of Agro-biology, Niigata University, Niigata 950-2181, Japan; and and {dagger} Department of Animal and Dairy Science, University of Georgia, Athens 30602-2771


   Abstract
Top
 Abstract
Introduction
 Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 
Estimates of direct and maternal genetic parameters in beef cattle were obtained with a random regression model with a linear spline function (SFM) and were compared with those obtained by a multitrait model (MTM). Weight data of 18,900 Gelbvieh calves were used, of which 100, 75, and 17% had birth (BWT), weaning (WWT), and yearling (YWT) weights, respectively. The MTM analysis was conducted with a three-trait maternal animal model. The MTM included an overall linear partial fixed regression on age at recording for WWT and YWT, and direct-maternal genetic and maternal permanent environmental effects. The SFM included the same effects as MTM, plus a direct permanent environmental effect and heterogeneous residual variance. Three knots, or breakpoints, were set to 1, 205, and 365 d. (Co)variance components in both models were estimated with a Bayesian implementation via Gibbs sampling using flat priors. Because BWT had no variability of age at recording, there was good agreement between corresponding components of variance estimated from both models. For WWT and YWT, with the exception of the sum of direct permanent environmental and residual variances, there was a general tendency for SFM estimates of variances to be lower than MTM estimates. Direct and maternal heritability estimates with SFM tended to be lower than those estimated with MTM. For example, the direct heritability for YWT was 0.59 with MTM, and 0.48 with SFM. Estimated genetic correlations for direct and maternal effects with SFM were less negative than those with MTM. For example, the direct-maternal correlation for WWT was –0.43 with MTM and –0.33 with SFM. Estimates with SFM may be superior to MTM due to better modeling of age in both fixed and random effects.

Key Words: Beef Cattle • Early Growth • Genetic Parameters • Linear Spline Function • Random Regression


   Introduction
Top
 Abstract
 Introduction
Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 
A traditional and conventional approach for genetic evaluations of growth traits in beef cattle is to employ a multitrait model (MTM), treating the records taken at different ages as different traits. The MTM approach provides parameter estimates only for the given constant ages.

With applications in other livestock species (Schaeffer and Dekkers, 1994Go; Huisman et al., 2002Go), random regression models (RRM; Kirkpatrick et al., 1990Go) have recently attracted the interest of beef cattle breeders for modeling the changes in growth in time. For RRM application in beef cattle, Legendre polynomials of age at recording have typically been used (Albuquerque and Meyer, 2001Go; Nobre et al., 2003aGo,bGo). In contrast to MTM, RRM permits the calculation of (co)variances and EPD at any age between two growth points. However, it has been indicated that parameter estimates obtained by fitting polynomials could be affected by sparse data and extremes of trajectories, especially at later ages (Nobre et al., 2003aGo,bGo).

One alternative approach could be spline-fitting, which is expected to estimate parameters more smoothly, possibly free from sharp bends at extremes at later ages (Verbyla et al., 1999Go), while still adequately modeling the features of longitudinal data (White et al., 1999Go). However, no numerical comparison is available between genetic parameters from the application of MTM and RRM with spline functions to early growth data in beef cattle.

The objectives of this study were to estimate (co)variance components and genetic parameters for direct and maternal effects of birth, weaning, and yearling weights in Gelbvieh cattle by RRM with a linear spline function (SFM) and to numerically compare the resulting estimates with those estimated with MTM.


   Materials and Methods
Top
 Abstract
 Introduction
 Materials and Methods
Results and Discussion
 Implications
 Literature Cited
 
Data
Data used for the analyses were provided by the American Gelbvieh Association and comprised 18,900 birth weights (BWT), 14,077 weaning weights (WWT), and 3,216 yearling weights (YWT), which were collected from 1995 to 2001. Approximately half the records were female. Age means and ranges were 254.4 d and 160 to 250 d for WWT, and 363.3 d and 320 to 410 d for YWT. Contemporary groups were defined as herd-year subclasses, resulting in 2,806 groups. There were 9,277 dams, with ages ranging from 2 to 13 yr at the time of measurement. Six age-of-dam groups were constructed by treating 7-yr-old and older dams as one group. Twelve calving subclasses were defined on a monthly basis. The pedigree file included 46,923 animals from three generations, and 5,894 of them were sires. Table 1Go shows a statistical description of the data.


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  		Table 1. Description of the data
 
Spline Analysis
The SFM used in this study for records of BWT, WWT, and YWT was written with yqr representing the record at age aq of animal q with dam r, as follows:


where Fqr denotes a set of fixed effects, as stated below; dqh and pqh are the random regression coefficients for direct additive genetic and permanent environmental effects of animal q at knot h, respectively; mqh and pmrh are the random regression coefficients for maternal additive genetic and maternal permanent environmental effects of dam r at knot h, respectively; fh(aq) stands for the linear spline coefficient at knot h for age aq; and eqr is the random residual term.

Linear spline functions were used to explain fixed effects as follows:


with cgih, aodjh, mockh, and sexlh representing the fixed regression coefficients for contemporary group i, age of dam class j, month of calving k, and sex l at knot h, respectively. In this model, knots were set to 1, 205, and 365 d, which were practically averages of the three growth points, so that two line segments were joined at 205 d of age.

The SFM above could be expressed in matrix notation as:


where yS is the vector of observations for BWT, WWT, and YWT; bS is the vector of the fixed effects; dS is the random vector of direct additive genetic effects; mS is the random vector of maternal additive genetic effects; pS is the random vector of direct permanent environmental effects; pmS is the random vector of maternal permanent environmental effects; eS is the vector of random residuals; and XS, ZS, ZS2, ZS3, and ZS4 are incidence covariance matrices relating elements of yS to elements of bS, dS, mS, pS, pmS, and eS, respectively.

For the model, the covariance matrix was assumed to be:


where GSd, GSm and GSdm were covariance matrices of random regression for direct additive genetic effect, maternal additive genetic effect, and their covariances, respectively; PS, PmS, and RS were covariance matrices of random regression for direct permanent environmental, maternal permanent environmental, and residual effects, respectively; A was the additive relationship matrix among all animals in the pedigree file, In, Ic, and Ik were identity matrices, whose orders are the numbers of animals, dams, and records, respectively, and {otimes} denotes the direct product operator. Heterogeneous residual variances were considered for ages of BWT, WWT, and YWT in SFM. The (co)variance matrix for target ages was defined as B'KB, where B is a matrix containing the random effects of the spline for the target ages, and K is the estimated covariance matrix of the spline coefficients.

Multitrait Analysis
The three-trait maternal animal model used was, in matrix notation, of the following form:


where y is the vector of observations for BWT, WWT, and YWT; b is the vector of the fixed effects, or discrete variables for the fixed factors as considered in SFM, including age measurements as a linear covariate for WWT and YWT; d is the random vector of direct additive genetic effects; m is the random vector of maternal additive genetic effects; pm is the random vector of maternal permanent environmental effects; e is the vector of random residuals; and X, Z1, Z2, and Z3 are known incidence matrices relating elements of y to elements of b, d, m, and pm, respectively.

For this model, it was assumed that:


where Gd and Gm are covariance matrices of random direct additive genetic effects and random maternal additive genetic effects, respectively; Gdm is a matrix of additive genetic covariances between direct and maternal effects; Pm is a covariance matrix of random maternal permanent environmental effects; R is a covariance matrix of random residual effects; A is the additive relationship matrix among all animals in the pedigree file; and Ic and In are identity matrices, whose orders were the numbers of dams and animals, respectively. Note that in MTM, the residual effect corresponded to the sum of the direct permanent environmental and the residual effects in the SFM.

Computation of (Co)variance Components
Estimates of (co)variance components from SFM and MTM were calculated by a Bayesian implementation via Gibbs sampling using flat improper priors for nuisance parameters and flat priors for (co)variance components. A total chain length of 100,000 rounds was run by a single long-chain algorithm. After discarding the initial 50,000 samples as the burn-in, one in every 10 samples was stored to generate the Gibbs samples that were used to compute the means and standard deviations of the posterior distribution. The means and the posterior distribution were used as the estimates of the (co)variance components, and their posterior standard deviations were considered to be a measure of their accuracy. Computations were carried out using the program GIBBSF90 (Misztal, 1999Go).


   Results and Discussion
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
Implications
 Literature Cited
 
Estimates of (Co)variance Components
Estimates of (co)variance components and their posterior standard deviations for direct additive genetic, maternal additive genetic, maternal permanent environmental, and residual effects obtained using SFM and MTM are shown in Table 2Go. As expected, estimates of variances for body growth increased from birth to yearling due to a scale effect. Some of the estimates of direct genetic (co)variances for birth, weaning, and yearling obtained by MTM and SFM were found to be reasonably comparable to each other; however, there was a tendency for SFM estimates of (co)variances for WWT and YWT to be smaller than those obtained using MTM. The SFM estimate of the direct genetic variance for YWT had a noticeably smaller estimated sampling variance. For estimates concerning BWT, taking into account the magnitude of the estimated sampling variances, there was a better agreement observed between both models. This finding would largely be due to the fact that there is no variability of age at measurement for BWT.


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  		Table 2. Estimates and posterior standard deviations of (co)variance components for direct additive genetic, maternal additive genetic, maternal permanent environmental, and residual effects using a multitrait model (MTM) and a random regression model with a linear spline function (SFM)
 
Maternal genetic variances estimated with both models were generally similar to each other. There was a carryover preweaning maternal effect for YWT; however, estimates of maternal genetic variances from both models were considerably lower than estimates of direct genetic variances. Whereas estimated maternal genetic variances for SFM and MTM were quite similar for WWT, the estimate and its sampling variance for YWT were both found to be lower with SFM than with MTM. It seems that these lower values obtained by SFM might be more realistic because the linear spline function could fit the model more smoothly to this longitudinal data in spite of a relatively small number of records available for YWT due to selection on WWT. Genetic covariances for maternal effects estimated among different growth points were low, with relatively large sampling variances in some estimates for YWT.

The results for maternal permanent environmental effects were more variable than those estimates for the direct and maternal genetic effects. Sampling variances were larger for all SFM estimates compared with MTM estimates, and some of the SFM estimates were larger. Approximately half the current data were represented by progeny of first-parity cows, and the proportion of missing data was high for YWT, which would not be an exception in the case of beef field weight data. Therefore, (co)variance estimates of maternal permanent environmental effects, especially for YWT could reasonably be susceptible to larger sampling variances, which should be taken into consideration when assessing these results.

The residual term in MTM would be, for the case of no missing data, approximately equivalent to the sum of direct permanent environmental and residual effects in SFM. Actually, the sums of permanent environmental and residual (co)variances obtained with SFM corresponded well to the appropriate residual (co)variances estimated with MTM, taking into account their posterior standard deviations. For BWT, with no missing data, estimated residual variances from both models were equivalent to each other, similar to the case of estimates of other variance components for BWT.

Table 3Go shows estimates of covariances between direct genetic and maternal genetic effects for BWT, WWT, and YWT. For covariance components within the same trait, estimates were all negative and generally less negative for SFM than for MTM. Because the fraction of progeny of first-parity cows was large, and the proportion of missing data was very high for YWT, estimates of direct-maternal genetic covariances between different traits had relatively large estimated sampling variances, resulting in estimates not significantly different from zero for most trait combinations in both MTM and SFM. The only exception was the estimated covariance between the maternal genetic effect for WWT and the direct genetic effect for YWT. These estimates were 133.7 ± 40.7 and 74.6 ± 39.8 for SFM and MTM, respectively.


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  		Table 3. Estimates and posterior standard deviations of covariance components between direct and maternal additive genetic effects using a multitrait model (MTM) and a random regression model with a linear spline function (SFM)
 
Estimates of Genetic Parameters
Selected estimates of direct and maternal genetic parameters for each of the three traits obtained with MTM and SFM are presented in Table 4Go. In general, parameter estimates were within the range of values reported in the literature (Meyer, 1992Go; Mohiuddin, 1993Go; Robinson, 1996aGo). Estimates of direct heritability (h2d) for BWT were similar and high (0.51 and 0.52) for both SFM and MTM analyses. The estimates decreased for WWT and then increased for YWT with both models. A lower estimate of for WWT relative to BWT and YWT was in agreement with previous findings obtained using animal models (Meyer, 1992Go; Waldron at al., 1993Go; Koch et al., 1994Go). The estimates (0.28 and 0.48) for WWT and YWT obtained with SFM were approximately 20% lower than the corresponding MTM estimates (0.36 and 0.59). Maternal heritability (h2m) estimates for BWT, WWT, and YWT ranged from 0.08 to 0.13 with MTM and from 0.06 to 0.11 with SFM, and they were substantially lower than those for . Reviewing reported estimates, Mohiuddin (1993)Go showed that the weighted averages of estimated were around 0.10 for the three traits, whereas the averages for were higher. For WWT, depending on various factors such as breed and population structure, some previous estimates of were similar to, or larger than, those of (Meyer, 1992Go; Waldron et al., 1993Go; Koch et al., 1994Go). There was a tendency for lower estimates of with SFM than with MTM, although differences between the estimates from the two models were small. Estimates of the fraction of maternal permanent environmental variance (c2) ranged from 0.09 to 0.12 for MTM and from 0.07 to 0.14 for SFM, and they were almost of the same magnitude as ratios. However, the c2 estimate for WWT was higher than those for BWT and YWT, which was more evident in the estimates with SFM.


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  		Table 4. Estimates and posterior standard deviations of direct and maternal heritabilities, direct-maternal genetic correlations, and the proportion of maternal permanent environmental variance to the total variance using a multitrait model (MTM) and a random regression model with a linear spline function (SFM)a
 
Estimates of direct-maternal genetic correlations (rdm) with the two models were all negative and generally moderate. These results were consistent with the majority of those reported in the literature, especially for WWT (Mohiuddin, 1993Go; Meyer, 1997Go; Dodenhoff et al., 1999Go). Estimates of rdm from SFM were approximately 20% less negative than those from MTM for the three traits; however, estimates of rdm for BWT and WWT with SFM had larger estimates of sampling variances compared with MTM. In contrast, for YWT with a high percentage of missing records, estimates had rather larger sampling variances with MTM. Sampling variances with SFM were similar for all traits. With MTM, however, sampling variances for BWT and WWT were similar, but they were larger than those for YWT.

Genetic correlations among three growth points for direct genetic and maternal genetic effects are listed in Table 5Go. As expected, correlations for direct effects were moderately positive with both models (Table 5Go). Estimates of correlations between maternal genetic effects for BWT and WWT were negative with both models, indicating that prenatal and postnatal maternal effects would be genetically antagonistic. Genetic correlations of YWT with other traits were close to zero with MTM when taking into account their posterior standard deviations. With SFM, the genetic correlation between BWT and YWT was negative, and the correlation between YWT and WWT was not significantly different from zero.


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  		Table 5. Estimates and posterior standard deviations of genetic correlations among three growth points (traits) for direct and maternal genetic effects using a multitrait model (MTM) and a random regression model with a linear spline function (SFM)
 
General Discussion
A typical approach for the genetic evaluation of BWT, WWT, and YWT has been to discard data for WWT and YWT that are outside of standard age ranges (BIF, 2003Go) and then to apply MTM to the data after pread-justment to standard ages. The RRM is able to handle all the data at any age, requiring no adjustments for age at measurement, and thus, it is an attractive alternative to MTM.

For longitudinal weight data from birth to 630 d of age in Nellore beef cattle, assuming that direct-maternal correlation was unimportant, Albuquerque and Meyer (2001)Go estimated direct and maternal genetic covariance functions employing RRM with cubic Legendre polynomials and found that RRM modeled the pattern of (co)variances in the data well, with estimates similar to those obtained with the corresponding univariate analysis. Nonetheless, some estimation artifacts were observed for parameters estimated using RRM, particularly those at later ages. Moreover, using RRM with cubic Legendre polynomials, Nobre et al. (2003a)Go analyzed similar weight data from Nellore cattle, creating two data sets sampled from all herds and sampled from herds with no missing traits. They observed that parameter estimates with RRM were generally similar to those with MTM; however, estimates of variances were shown to contain more estimation artifacts for data with missing traits. It was found that EPD based on RRM using parameters estimated from the data were inaccurate relative to MTM (Nobre et al., 2003bGo). Therefore, RRM with Legendre polynomials would be susceptible to estimation artifacts due to the nature of polynomials to be poorly fit at extremes where small amounts of data are available. Following these findings, one alternative to fitting Legendre polynomials could be the use of certain functions, such as fractional polynomials (Robert-Granie et al., 2002Go), that are less susceptible to these artifacts. Another alternative is the use of spline-fitting (Verbyla et al., 1999Go; White et al., 1999Go; Huisman et al., 2002Go) that might possibly provide a better fit in the setting of a limited number of growth points that are separated enough in time to be considered different traits.

Age of calf effect in MTM was modeled by fitting an overall linear regression of WWT and YWT on age of calf at recording, and other fixed effects were included in the usual discrete fashion. Fixed effects in SFM were fitted as fixed regressions with linear splines. Results here suggest that the corresponding estimates of variances and covariances from SFM and MTM would be similar even with missing records, when there are few traits measured and the age range for each trait is narrow (90 d in the current study for WWT and YWT). Furthermore, it was found that the two models were in very close agreement for parameters associated with BWT that had no missing records and age range variation. In all cases, with the exception of the total residual variance, the (co)variance, heritability, and genetic correlation estimates provided via SFM were slightly lower and had smaller estimated sampling variances than those estimated via MTM.

Of interest was the decrease in the absolute values of the negative estimates for direct and maternal covariances and correlations from SFM compared with MTM when direct and maternal effects within the same trait were considered. Direct-maternal genetic correlations estimated with the conventional reduced-Willham model (Koerhuis and Thompson, 1997Go) may be overestimated, assuming no environmental covariance between dams and offspring (Meyer, 1997Go; Quintanilla et al., 1999Go) or not considering the effect of sire x herd-year interaction (Robinson, 1996bGo; Dodenhoff, 1999Go). In a sense, the less extreme estimate of the genetic correlation obtained with SFM compared with MTM might be more realistic, due to a smooth fit of the linear spline function to the data.

Conducting a simulation with some realistic generated data sets, Meyer (2004)Go suggested that some improvement in the accuracy of genetic evaluation of beef cattle for growth could be expected by replacing MTM with RRM for the analysis of BWT, and 200-, 400-, and 600-d weights. In the current setting of three early growth points from birth to yearling, it may generally be anticipated that the SFM analysis could be essentially equivalent to RRM with polynomials. However, the linear splines used here are computationally easier and faster to implement than polynomials. In contrast to polynomials, the behavior of splines in different intervals are not related (Wold, 1974Go). For the type of data represented in the current study, estimates of direct and maternal genetic parameters with SFM may be superior to those from MTM due to better modeling of differences due to age in both fixed and random effects, particularly for growth at yearling. A simulation study that considers various realistic data sets of early growth would be helpful to clarify the potential benefits of the SFM analysis.


   Implications
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
 Implications
Literature Cited
 
Random regression models that fit linear splines seem to provide direct and maternal genetic parameters for birth, weaning, and yearling weights that are similar to those estimated with conventional multitrait models. Random regression models with linear splines may give more reasonable estimates of genetic parameters, including direct-maternal genetic correlations, by lowering estimates in absolute value, especially for parameters dealing with yearling weight, for which a large number of records is usually missing. Random regression models with linear splines could be simpler and faster to implement than those with Legendre polynomials. For datasets with missing records, where few traits are measured and the age range for each trait is narrow, estimates from random regression models with splines may be superior to those from multitrait models, especially for growth at yearling age because of better modeling of age in both fixed and random effects. Further investigation using simulation would be helpful to clarify the advantages of random regression models with splines for the analysis of beef growth data.


   Footnotes
 
1 Appreciation is expressed to the American Gelbvieh Association for providing data. Back

2 On leave from Dept. of Agro-biology, Niigata Univ. Back

3 Correspondence—phone: 706-583-0017; fax: 706-583-0274; e-mail: shogo{at}uga.edu.

Received for publication September 13, 2004. Accepted for publication December 28, 2004.


   Literature Cited
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 


Albuquerque, L. G., and K. Meyer. 2001. Estimates of covariance functions for growth from birth to 630 days of age in Nelore cattle. J. Anim. Sci. 79:2776–2789.[Abstract/Free Full Text]

BIF. 2003. Guidelines for Uniform Beef Improvement Programs. Beef Improv. Fed., Athens, GA.

Dodenhoff, J., L. D. Van Vleck, and D. E. Wilson. 1999. Comparison of models to estimate genetic effects for weaning weight of Angus cattle. J. Anim. Sci. 77:3176–3184.[Abstract/Free Full Text]

Huisman, A. E., R. F. Veerkamp, and J. A. M. van Arendonk. 2002. Genetic parameters for various random regression models to describe the weight data of pigs. J. Anim. Sci. 80:575–582.[Abstract/Free Full Text]

Kirkpatrick, M., D. Lofsvold, and M. Bulmer. 1990. Analysis of the inheritance, selection and evolution of growth trajectories. Genetics 124:979–993.[Abstract]

Koch, R. M., L. V. Cundiff, and K. E. Gregory. 1994. Cumulative selection and genetic change for weaning or yearling weight or for yearling weight plus muscle score in Hereford cattle. J. Anim. Sci. 72:864–885.[Abstract]

Koerhuis, A. N. M., and R. Thompson. 1997. Models to estimate maternal effects for juvenile body weight in broiler chickens. Genet. Sel. Evol. 29:225–249.

Meyer, K. 1992. Variance components due to direct and maternal effects for growth traits of Australian beef cattle. Livest. Prod. Sci. 31:179–204.

Meyer, K. 1997. Estimates of genetic parameters for weaning weight of beef cattle accounting for direct-maternal environmental covariances. Livest. Prod. Sci. 52:187–199.

Meyer, K. 2004. Scope for a random regression model in genetic evaluation of beef cattle for growth. Livest. Prod. Sci. 86:68–83.

Misztal, I. 1999. Complex models, more data: Simpler programming. Proc. Inter. Workshop Comput. Cattle Breed. Tuusala, Finland. Interbull Bull. 20:33–42.

Mohiuddin, G. 1993. Estimates of genetic and phenotypic parameters of some performance traits in beef cattle. Anim. Breed. Abstr. 61:495–522.

Nobre, P. R. C., I. Misztal, S. Tsuruta, J. K. Bertrand, L. O. C. Silva, and P. S. Lopes. 2003a. Analyses of growth curves of Nellore cattle by multiple-trait and random regression models. J. Anim. Sci. 81:918–926.[Abstract/Free Full Text]

Nobre, P. R. C., I. Misztal, S. Tsuruta, J. K. Bertrand, L. O. C. Silva, and P. S. Lopes. 2003b. Genetic evaluation of growth in Nellore cattle by multiple-trait and random regression models. J. Anim. Sci. 81:927–932.[Abstract/Free Full Text]

Quintanilla, R., L. Varona, M. R. Pujol, and J. Piedrafita. 1999. Maternal animal model with correlation between maternal environmental effects of related dams. J. Anim. Sci. 77:2904–2917.[Abstract/Free Full Text]

Robert-Granie, C., E. Maza, R. Ruppi, and J. L. Foulley. 2002. Use of fractional polynomial for modeling somatic cell scores in dairy cattle. Proc. 7th World Congr. Genet. Appl. Livest. Prod., CD-ROM communication 16:05.

Robinson, D. L. 1996a. Estimation and interpretation of direct and maternal genetic parameters for weights of Australian Angus cattle. Livest. Prod. Sci. 45:1–11.

Robinson, D. L. 1996b. Models which might explain negative correlations between direct and maternal genetic effects. Livest. Prod. Sci. 45:111–122.

Schaeffer, L. R., and J. C. M. Dekkers. 1994. Random regression in animal models for test-day production in dairy cattle. Proc. 5th World Congr. Genet. Appl. Livest. Prod., Guelph, Canada 18:443–446.

Verbyla, A. P., B. R. Cullis, M. G. Kenward, and S. J. Welham. 1999. The analysis of designed experiments and longitudinal data using smoothing splines. J. Royal Stat. Soc. Ser. C Appl. Stat. 48:269–300.

Waldron, D. F., C. A. Morris, R. L. Baker, and D. L. Johnson. 1993. Maternal effects for growth traits in beef cattle. Livest. Prod. Sci. 34: 57–70.

White, I. M. S., R. Thompson, and S. Brotherstone. 1999. Genetic and environmental smoothing of lactation curves with cubic splines. J. Dairy Sci. 82:632–638.[Abstract]

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