- Research article
- Open Access

# Development of a structural growth curve model that considers the causal effect of initial phenotypes

- Akio Onogi
^{1}Email authorView ORCID ID profile, - Atsushi Ogino
^{2}, - Ayako Sato
^{3}, - Kazuhito Kurogi
^{2}, - Takanori Yasumori
^{3}and - Kenji Togashi
^{2}

**Received:**12 October 2018**Accepted:**18 April 2019**Published:**2 May 2019

## Abstract

### Background

Growth curves have been widely used in genetic analyses to gain insights into the growth characteristics of both animals and plants. However, several questions remain unanswered, including how the initial phenotypes affect growth and what is the duration of any such impact. For beef cattle production in Japan, calves are procured from farms that specialize in reproduction and then moved to other farms where they are fattened to achieve their market/purchase value. However, the causal effect of growth, while calves are on the reproductive farms, on their growth during fattening remains unclear. To investigate this, we developed a model that combines a structural equation with a growth curve model. The causal effect was modeled with B-splines, which allows inference of the effect as a curve. We fitted the proposed structural growth curve model to repeated measures of body weight from a Japanese beef cattle population (n = 3831) to estimate the curve of the causal effect of the calves’ initial weight on their trajectory of growth when they are on fattening farms.

### Results

Maternal and reproduction farm effects explained 26% of the phenotypic variance of initial weight at fattening farms. The structural growth curve model was fitted to remove the effects of these factors in growth curve analysis at fattening farms. The estimated curve of causal effects remained at approximately 0.8 for 200 d after the calves entered the fattening farms, which means that 64% of the phenotypic variance was explained by the initial weight. Then, the effect decreased linearly and disappeared approximately 620 d after entering the fattening farms, which corresponded to an average age of 871.5 d.

### Conclusions

The proposed model is expected to provide more accurate estimates of genetic values for growth patterns because the confounding causal factors such as maternal and reproduction farm effects are removed. Moreover, examination of the inferred curve of the causal effect enabled us to estimate the effect of a calf’s initial weight at arbitrary times during growth, which could provide suitable information for decision-making when shifting the time of slaughter, building models for genetic evaluation, and selecting calves for market.

## Background

Fitting curves to longitudinal phenotypic data is a common methodology that is used in animal and plant genetics to gain insights into individual growth patterns. Research on growth curves has a long history, and to date a number of curves have been used to model growth, including logistic [1], Richards [2], Gompertz [3], von Bertalanffy [4], and Brody [5] curves. These curves include three to four parameters that are often regarded as new traits and are subjected to various genetic analyses, such as the estimation of genetic parameters or mapping of quantitative trait loci [6–10], to better understand the genetic architecture of growth patterns.

Regarding growth curve analysis, it is important to understand the impact of initial measures on subsequent growth and the duration of this impact. For example, birth weight is affected by various factors such as maternal effects and the environment, which can have causal effects on subsequent growth. For beef cattle production in Japan, calves are usually born on farms that specialize in reproduction, and then at about 9 months of age they are moved to other farms for fattening to achieve their market value at around 30 months of age. However, although the phenotypes of the calves when they enter the fattening farms are known to influence their growth patterns during the fattening phase, the duration of this impact is unclear. The causal effect of a phenotype such as initial weight will interfere with growth curve analyses if it is affected by factors that are not considered in the growth curves, such as maternal effects.

Therefore, to address this issue, we developed a growth curve model that considers the causal effect of initial weight by combining a structural equation for causality inference with growth curves in a Bayesian framework. In quantitative genetics, structural equations are often used to infer causal relationships between phenotypes in multivariate mixed models [11–13]. In the current study, we applied structural equation modeling to longitudinal data and inferred the causal effect of the initial phenotypic value as a curve over time by using B-splines. We fitted this structural growth curve (SGC) model to real data on weight from a beef cattle population in Japan to reveal the causal effect of calf weight at entry to the fattening farms.

## Methods

### Data

Data summary

Characteristic | Value |
---|---|

Number of animals with weight records | 3831 (1600 heifers and 2231 steers) |

Number of animals in the numerator relationship matrix | 24,284 |

Mean number of weight records per animal | 4.4 ± 0.7 |

Number of reproduction farms | 1845 |

Number of experimental stations (fattening farms) | 3 |

Average age at entry to the stations (d) | 251.5 ± 20.4 |

Average age at slaughter (d) | 886.8 ± 46.7 |

Average weight on the day of entry to the stations (kg) | 236.0 ± 39.0 (heifers: 221.3 ± 35.4; steers: 246.6 ± 38.0) |

Average weight at slaughter (kg) | 729.8 ± 81.5 (heifers: 699.2 ± 76.5; steers: 751.8 ± 77.7) |

### Structural growth curve model

*T*

_{L}is the time point of the last measurement. These knots were repeated to constrain the span of B-splines [15]. The remaining knots were set at \(t = T_{L} /4,\) \(T_{L} /2,3T_{L} /4\), and \(T_{L}\), respectively.

### Comparison of models

When fitting the SGC model to the data, the day of entry to the experimental stations was set as the initial day (\(t = 0\)). When fitting the ordinal growth curve model to the data, either the day of entry or the day of birth was set as the initial day [referred to as the growth curve model fitted to the entry day data (GC_A) and birth day data (GC_B), respectively]. The unit of time was d in each model. We compared these models using the mean log-likelihood, the deviance information criterion (DIC) [20], and the widely applicable information criterion (WAIC) [21].

In the final analyses, \({\mathbf{S}}_{{\mathbf{u}}}^{\varvec{*}}\), and \({\mathbf{S}}_{{\mathbf{e}}}\) were set following the posterior means of the preliminary analysis. In addition, we fitted a linear mixed-effect model, which was the same as Eq. 1 in the SGC model, to the entry day weight using airemlf90 ver. 1.103 [22], with the default value of 1e−10 as the convergence criterion. \({\mathbf{S}}_{{\mathbf{u}}}\) was determined from the estimates provided by these two preliminary analyses (the GC_B and linear mixed models for the weight on entry day). The off-diagonal elements in \({\mathbf{S}}_{{\mathbf{u}}}\) that corresponded to the covariances between \({\mathbf{u}}_{\mathbf0}\) and \({\mathbf{u}}_{{\mathbf{A}}}\), \({\mathbf{u}}_{{\mathbf{B}}}\), and \({\mathbf{u}}_{{\mathbf{K}}}\) were determined as the empirical covariances of these random effects obtained from the two preliminary analyses.

### Estimation of parameters

The parameters in the SGC, GC_A, and GC_B models were estimated using the Markov chain Monte Carlo (MCMC) method. Gibbs sampling could be applied to all of the parameters except \(A_{i}\), \(B_{i}\), \(K_{i}\), and \(P_{j}\). \({\varvec{\upbeta}}_{\mathbf0}\), \({\varvec{\upbeta}}_{{\mathbf{A}}}\), \({\varvec{\upbeta}}_{{\mathbf{B}}}\), and \({\varvec{\upbeta}}_{{\mathbf{K}}}\) had normal posterior distributions, whereas \({\mathbf{v}}_{{\mathbf{j}}}\), and \({\mathbf{u}}_{{\mathbf{A}}}\), \({\mathbf{u}}_{{\mathbf{B}}}\), \({\mathbf{u}}_{{\mathbf{K}}}\), and \({\mathbf{u}}_{0}\) had multivariate normal posterior distributions. The posterior distributions of \({\varvec{\upbeta}}_{{\mathbf{A}}}\), \({\varvec{\upbeta}}_{{\mathbf{B}}}\), \({\varvec{\upbeta}}_{{\mathbf{K}}}\), \({\mathbf{u}}_{{\mathbf{A}}}\), \({\mathbf{u}}_{{\mathbf{B}}}\), and \({\mathbf{u}}_{{\mathbf{K}}}\) could be derived by considering the growth curve parameters (\(A_{i}\), \(B_{i}\) and \(K_{i}\)) as response variables. \(\upsigma_{\text{vj}}^{2}\), \(\upsigma_{{{\text{e}}0}}^{2}\), \(\upsigma_{\text{e}}^{2}\), and \(\upsigma_{\text{p}}^{2}\) had scaled inverse Chi squared posterior distributions, while \({\varvec{\Sigma}}_{{\mathbf{u}}}\), \({\varvec{\Sigma}}_{{\mathbf{u}}}^{\varvec{*}}\), and \({\varvec{\Sigma}}_{{\mathbf{e}}}\) had inverse Wishart posterior distributions, all of which were derived following a previous study [23]. Since the posterior distributions of \(A_{i}\), \(B_{i}\), \(K_{i}\), and \(P_{j}\) were not closed form expressions, Metropolis–Hastings sampling was applied by adopting a random-walk algorithm.

The number of iterations for the SGC model was 2.5 million with the first 2 million being discarded. By contrast, there were 1 million iterations for both the GC_A and GC_B models, with the first 0.6 million being discarded. The sampling interval was 10 for each model. We ran three chains with different initial values and checked the convergence of MCMC, as described previously [24].

### Parametric bootstrapping

To evaluate the accuracy of the parameter estimation of the SGC model, parametric bootstrapping was conducted. The weight at slaughter of each animal was simulated using the estimates of the variance components, the fixed effects, and the causal effect generated by the SGC model. A single-trait animal model that included sex and experimental station as fixed effects and age as a covariate was then fitted to the simulated weights to estimate the heritability at slaughter. This procedure was repeated 1000 times and the heritability that was estimated from the simulated data was compared with that estimated from the real data.

## Results and discussion

### Convergence of the MCMC chains

The \(\hat{R}\) statistics of the convergence diagnosis [24] were calculated for the log-likelihood values and the parameters for each model. The \(\hat{R}\) statistics decreased to 1.0 as the MCMC chains converged. The statistics for the log-likelihood values were 1.047, 1.060, and 1.029 for the SGC, GC_A, and GC_B models, respectively; these values were lower than 1.1, which was previously suggested to be a rough threshold [24]. Most \(\hat{R}\) statistics for the parameters were also lower than the threshold, with the exception of a few parameters in the SGC model, including \(P_{5}\) (1.153), \(P_{6}\) (1.185), and \(P_{7}\) (1.148), which are the weights of splines, and the additive genetic variances for parameters \(A\) (1.122) and \(Y_{i,0}\) (1.127). However, the statistics for these parameters are close to 1.1, indicating that the MCMC chains for each model converged to stationary distributions. The SGC model took more than three times as many iterations as the other models to reach convergence (2 million vs. 0.6 million), which may be due to the model complexity.

### Comparison of models using information criteria

Mean log-likelihood and information criteria for each model

Model | Mean log-likelihood | DIC | WAIC |
---|---|---|---|

SGC | − 5,523,358.6 | 7,871,834.7 | 12,466,275.7 |

GC_A | − 5,574,092.6 | 7,872,045.8 | 12,567,909.0 |

GC_B | − 7,335,939.9 | 10,341,550.1 | 16,537,364.6 |

### Causal effect of the initial weight

Variance components of the initial weight estimated by the structural growth curve model

Additive genetic effect | Maternal effect | Reproduction farm effect | Residual | |
---|---|---|---|---|

Variance | 499.4 (432.3, 568.7) | 49.6 (10.3, 90.5) | 219.0 (180.8, 259.1) | 265.0 (218.0, 310.7) |

Proportion to phenotypic variance | 0.48 (0.43, 0.54) | 0.05 (0.01, 0.09) | 0.21 (0.18, 0.25) | 0.26 (0.21, 0.31) |

Several animals showed a relatively high causal effect of the initial weight at the time of slaughter because of their (probably unintended) early slaughter. For example, 14 animals had coefficients of causal effect that were higher than 0.316, which indicates that about 10% of the phenotypic variance at slaughter was explained by the variance of initial weight. Consequently, since approximately 21% of the variance of initial weight could be explained by the reproduction farm effect (Table 3), about 2.1% of the variance of slaughter weight of these 14 animals could be explained by the effect of reproduction farm, which is not considered in the current model for genetic evaluation. Consequently, since these animals may affect the results, they should be eliminated from the genetic evaluation.

The estimated curve suggests that if slaughter of calves is planned at approximately 620 d after purchase, their growth can be recovered on the fattening farm if it was less than optimal on the reproduction farm. However, if animals are slaughtered earlier than 620 d, e.g., to reduce feeding costs, the causal effect of the initial weight will linearly increase as the slaughter age decreases (Fig. 2). For example, if the animal is slaughtered 440 d after purchase (approximately half a year earlier than 620 d), the effect of the initial phenotype increases to 0.405, suggesting that 16.4% of the phenotypic variance at slaughter is explained by the variance of initial weight. Thus, growth on reproduction farms should be considered with caution when selecting calves at market.

This is the first report on the causal effect of an initial phenotype on subsequent growth. However, rough estimates of this effect have been reported using multivariate analyses. For example, Meyer et al. [25] used four-variate analyses consisting of the birth, weaning (120–300 d after birth), yearling (301–500 d), and final (501–700 d) weights to estimate maternal genetic correlations in Hereford cattle and in a synthetic breed, and found that the correlations between weaning and yearling weights were 0.97 and 0.99 in Hereford and the synthetic breed, respectively, while those between weaning and final weights were 0.92 and 0.88, respectively. Similarly, Eler et al. [26] found that the maternal genetic correlation between weaning and yearling weights was 0.84 in Nelore cattle. In these studies, the maternal effects during the post-weaning periods were considered as a carry-over of those on the weaning weight. However, if the weaning weight is regarded as the initial phenotype, these correlations can be regarded as estimates of the causal effect. The causal effect that was estimated by the SGC model (Fig. 2) was lower than these estimates, which is likely due to differences in breeds and management conditions such as the type of feed.

### Genetic parameters

Heritability (diagonal), genetic correlation (lower triangular), and residual correlation (upper triangular) of the growth curve parameters *A*, *B*, and *K* estimated by each model

\(A\) | \(B\) | \(K\) | ||
---|---|---|---|---|

SGC | \(A\) | 0.97 (0.90, 1.00) | 0.34 (− 0.20, 0.83) | − 0.57 (− 0.96, − 0.05) |

\(B\) | 0.63 (0.57, 0.69) | 0.91 (0.84, 0.97) | − 0.30 (− 0.79, 0.22) | |

\(K\) | − 0.69 (− 0.73, − 0.64) | − 0.67 (− 0.76, − 0.58) | 0.91 (0.78, 0.99) | |

GC_A | \(A\) | 0.96 (0.89, 0.99) | 0.35 (− 0.25, 0.83) | − 0.42 (− 0.92, 0.20) |

\(B\) | 0.69 (0.59, 0.77) | 0.66 (0.53, 0.78) | − 0.26 (− 0.76, 0.34) | |

\(K\) | − 0.77 (− 0.81, − 0.73) | − 0.67 (− 0.75, − 0.58) | 0.96 (0.91, 0.99) | |

GC_B | \(A\) | 0.94 (0.89, 0.99) | 0.61 (0.21, 0.91) | − 0.51 (− 0.93, 0.03) |

\(B\) | − 0.30 (− 0.44, − 0.17) | 0.69 (0.55, 0.84) | − 0.28 (− 0.78, 0.30) | |

\(K\) | − 0.75 (− 0.79, − 0.71) | 0.65 (0.55, 0.74) | 0.95 (0.91, 0.99) |

The genetic correlations between growth curve parameters differed according to the model that was used. The correlations between \(A\) and \(B\) were positive in the SGC and GC_A models but negative in the GC_B model, whereas the correlations between \(B\) and \(K\) were negative in the SGC and GC_A models but positive in the GC_B model. These opposite tendencies may be due to the day of entry in the stations being set as the initial day in the SGC and GC_A models, although the actual age of the calves’ at entry differed (251.5 ± 20.4 d). Because parameter \(B\) shifts the growth curve back and forth, setting the entry day as the initial day would affect the estimates of \(B\) and its correlation with the other parameters. Similar contrasting results in genetic correlations were also found in two independent pig studies: Koivula et al. [8] reported strong negative genetic correlations between \(A\) and \(K\) (− 0.80) and between \(B\) and \(K\) (−0.80) but a positive correlation between \(A\) and \(B\) (0.88), whereas Coyne et al. [9] reported negative correlations between \(A\) and \(B\) (− 0.69) and between \(A\) and \(K\) (− 0.78) but a positive correlation between \(B\) and \(K\) (0.76). Although the estimates in the latter study [9] differed depending on the method used for estimation, a negative correlation between \(A\) and \(B\) was consistently observed. Interestingly, Koivula et al. [8] used test age, starting from when the body weight was approximately 30 kg, which may have affected the estimation of the genetic correlation, as observed in our study. It will be difficult to determine which day (birth or entry) is most valid as the initial day, and this may depend on the data. However, these findings indicate that genetic correlations between the parameters must be interpreted with caution.

The interpretation of genetic correlations is also complicated by compensation between the parameters. For example, \(K\) which increases the maximum growth speed, also increases the mature weight controlled by \(A\). Therefore, when fitting a curve to a measured value of mature weight, \(K\) should decrease as \(A\) increases, and vice versa, resulting in a negative correlation between the two. This may explain why a negative correlation was consistently observed between these parameters in previous studies [8, 9] and the present study. Thus, the biological interpretation of the genetic correlation between growth curve parameters may be controversial.

## Conclusions

By fitting our newly developed SGC model to data on weight of beef cattle, we were able to estimate the causal effect of the initial weight (weight on the day of entry to the stations) on growth. Because all of the evaluated criteria supported the proposed model, we suggest that the SGC model can provide more accurate estimates of the genetic effects on growth, particularly for the cattle cohort that was assessed in this study. Moreover, our data suggest that the inferred curve of the causal effect can provide valuable information for planning the time of slaughter, building models for genetic evaluation, and selecting calves at markets.

## Declarations

### Authors’ contributions

AkO designed this study, developed the statistical models, analyzed the data, and drafted the manuscript. AtO, AS, KK, TY, and KT also analyzed the data and assisted in drafting the manuscript. All authors have read and approved the final manuscript.

### Acknowledgements

The authors thank the staff at the Livestock Improvement Association of Japan, Inc. for their assistance in the progeny-testing program and for providing high-quality data.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

The datasets supporting the conclusions of this article are included within the article and its additional files [Additional files: 1–8]. The programs are available upon request.

### Funding

This study was supported by Japan Racing Association Livestock Promotion Funds.

### Implementation

The SGC, GC_A, and GC_B models were programed using C. All other statistical analyses were performed using R [31].

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## Authors’ Affiliations

## References

- Verhulst PF. Notice sur la loi que la population suit dans son accroissement. Curr Math Phys. 1838;10:113–20.Google Scholar
- Richards FJ. A flexible growth function for empirical use. J Exp Bot. 1959;10:290–301.View ArticleGoogle Scholar
- Winsor CP. The Gompertz curve as a growth curve. Proc Natl Acad Sci USA. 1932;18:1–8.View ArticlePubMedGoogle Scholar
- von Bertalanffy L. Quantitative laws in metabolism and growth. Q Rev Biol. 1957;32:217–31.View ArticleGoogle Scholar
- Brody S. Bioenergetics and growth. New York: Reinhold Publishing Corporation; 1945.Google Scholar
- DeNise RS, Brinks JS. Genetic and environmental aspects of the growth curve parameters in beef cows. J Anim Sci. 1985;61:1431–40.View ArticlePubMedGoogle Scholar
- Koenen EP, Groen AF. Genetic analysis of growth patterns of black and white dairy heifers. J Dairy Sci. 1996;79:495–501.View ArticlePubMedGoogle Scholar
- Koivula M, Sevon-Aimonen ML, Stranden I, Matilainen K, Serenius T, Stalder KJ, et al. Genetic (co)variances and breeding value estimation of Gompertz growth curve parameters in Finnish Yorkshire boars, gilts and barrows. J Anim Breed Genet. 2008;125:168–75.View ArticlePubMedGoogle Scholar
- Coyne JM, Matilainen K, Berry DP, Sevon-Aimonen ML, Mantysaari EA, Juga J, et al. Estimation of genetic (co)variances of Gompertz growth function parameters in pigs. J Anim Breed Genet. 2017;134:136–43.View ArticlePubMedGoogle Scholar
- Crispim AC, Kelly MJ, Guimaraes SE, e Silva FF, Fortes MR, Wenceslau RR, et al. Multi-trait GWAS and new candidate genes annotation for growth curve parameters in Brahman cattle. PLoS One. 2015;10:e0139906.View ArticlePubMedPubMed CentralGoogle Scholar
- Wu XL, Heringstad B, Chang YM, de Los Campos G, Gianola D. Inferring relationships between somatic cell score and milk yield using simultaneous and recursive models. J Dairy Sci. 2007;90:3508–21.View ArticlePubMedGoogle Scholar
- Varona L, Sorensen D, Thompson R. Analysis of litter size and average litter weight in pigs using a recursive model. Genetics. 2007;177:1791–9.View ArticlePubMedPubMed CentralGoogle Scholar
- Valente BD, Rosa GJ, de Los Campos G, Gianola D, Silva MA. Searching for recursive causal structures in multivariate quantitative genetics mixed models. Genetics. 2010;185:633–44.View ArticlePubMedPubMed CentralGoogle Scholar
- Onogi A, Ideta O, Yoshioka T, Ebana K, Yamasaki M, Iwata H. Uncovering a nuisance influence of a phenological trait of plants using a nonlinear structural equation: application to days to heading and culm length in Asian cultivated rice (
*Oryza sativa*L.). PLoS One. 2016;11:e0148609.View ArticlePubMedPubMed CentralGoogle Scholar - Hastie T, Tibshirani R, Friedman J. The elements of statistical learning. 2nd ed. New York: Chapman and Hall; 2009.View ArticleGoogle Scholar
- Gianola D, Sorensen D. Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes. Genetics. 2004;167:1407–24.View ArticlePubMedPubMed CentralGoogle Scholar
- Varona L, Moreno C, García Cortés LA, Altarriba J. Multiple trait genetic analysis of underlying biological variables of production functions. Livest Prod Sci. 1997;47:201–9.View ArticleGoogle Scholar
- Blasco A, Piles M, Varona L. A Bayesian analysis of the effect of selection for growth rate on growth curves in rabbits. Genet Sel Evol. 2003;35:21–41.View ArticlePubMedPubMed CentralGoogle Scholar
- Lázaro SF, Ibáñez-Escriche N, Varona L, e Silva FF, Brito LC, Guimarães SEF, et al. Bayesian analysis of pig growth curves combining pedigree and genomic information. Livest Sci. 2017;201:34–40.View ArticleGoogle Scholar
- Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A. Bayesian measures of model complexity and fit. J R Stat Soc B. 2002;64:583–639.View ArticleGoogle Scholar
- Watanabe S. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J Mach Learn Res. 2010;11:3571–94.Google Scholar
- Misztal I, Tsuruta S, Strabel T, Auvray B, Druet T, Lee DH. BLUPF90 and related programs. In: Proceedings of the 7th world congress on genetics applied to livestock production: 18–23 August 2002, Montpellier; 2002.Google Scholar
- Sorensen D, Gianola D. Likelihood, Bayesian, and MCMC methods in quantitative genetics. New York: Springer; 2002.View ArticleGoogle Scholar
- Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian data analysis. 3rd ed. London: CRC Press; 2014.Google Scholar
- Meyer K, Carrick MJ, Donnelly BJP. Genetic parameters for growth traits of Australian beef cattle from a multibreed selection experiment. J Anim Sci. 1993;71:2614–22.View ArticlePubMedGoogle Scholar
- Eler JP, Van Vleck LD, Ferraz JBS, Lobo RB. Estimation of variances due to direct and maternal effects for growth traits of Nelore cattle. J Anim Sci. 1995;73:3253–8.View ArticlePubMedGoogle Scholar
- Takeda M, Uemoto Y, Inoue K, Ogino A, Nozaki T, Kurogi K, et al. Evaluation of feed efficiency traits for genetic improvement in Japanese Black cattle. J Anim Sci. 2018;96:797–805.View ArticlePubMedPubMed CentralGoogle Scholar
- Varona L, Moreno C, García Cortés LA, Yagüe G, Altarriba J. Two-step versus joint analysis of Von Bertalanffy function. J Anim Breed Genet. 1999;116:331–8.View ArticleGoogle Scholar
- Nogi T, Honda T, Mukai F, Okagaki T, Oyama K. Heritabilities and genetic correlations of fatty acid compositions in longissimus muscle lipid with carcass traits in Japanese Black cattle. J Anim Sci. 2011;89:615–21.View ArticlePubMedGoogle Scholar
- Onogi A, Ogino A, Komatsu T, Shoji N, Simizu K, Kurogi K, et al. Genomic prediction in Japanese Black cattle: application of a single-step approach to beef cattle. J Anim Sci. 2014;92:1931–8.View ArticlePubMedGoogle Scholar
- R Development Core Team. R. A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. ISBN 3-900051-07-0. http://www.R-project.org/.2011. Accessed 26 Apr 2019.