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J. Anim. Sci. 2003. 81:933-938
© 2003 American Society of Animal Science

Evaluation models and genetic parameters for calving difficulty in beef cattle

F. Phocas1 and D. Laloë

Institut National de la Recherche Agronomique, Station de Génétique Quantitative et Appliquée, 78 352 Jouy-en-Josas Cedex, France

1 Correspondence:
INRA-CRJ, 78352 Jouy-en-Josas Cedex, France (phone: 33-1-34652199; fax: 33-1-34652210; E-mail:
phocas{at}dga.jouy.inra.fr).


   Abstract
Top
 Abstract
Introduction
 Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 
Calving difficulty was analyzed under threshold and linear models considering either a fixed or random herd-year effect. The aim of the study was to compare models for predicting breeding values according to the size of herd-year groups. When simulating data sets with small herds, in order to obtain an unbiased evaluation under a nonrandom and negative association of sire and herd effects, the best model for a practical evaluation was the fixed linear model. Field data included 246,576 records of the largest Charolais herds in France. Models were compared using the correlations of estimated breeding values between the different models. Although the best model from a theoretical point of view was a threshold model with a fixed herd-year effect, a linear model with a fixed herd-year effect was the best choice from a practical point of view for predicting direct effects for calving difficulty in beef cattle and was a sufficient choice for predicting the associated maternal effects for data set with large herds. Correlations between direct estimated breeding values under the reference model and the fixed linear model and the random threshold model were 0.94 and 0.91, respectively. Correlations between the corresponding maternal estimated breeding values were 0.94 and 0.98. Heritabilities of direct effects were 0.27 and 0.14 under fixed threshold and fixed linear models, respectively. The corresponding heritabilities of maternal effects were 0.18 and 0.13, and the genetic correlation between direct and maternal effects were -0.36 and -0.34, respectively.

Key Words: Contemporary Comparisons • Dystocia • Linear Models • Threshold Models


   Introduction
Top
 Abstract
 Introduction
Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 
Calving difficulty needs to be reduced in beef cattle because it greatly affects the animal welfare and profitability of herds via increased labor and veterinary costs, mortality of calf, and time to rebreeding of the cow. Reductions need to be made mainly in the breeds with the highest incidence of calving difficulty, such as the Charolais breed, whose 8% of the calvings in France of this breed require some mechanical assistance or caesarian section. Calving difficulties are scored by farmers from 1 (no assistance) to 4 (caesarian). Theoretically, the discrete nature of the performance should be taken into account for the genetic evaluation by using a threshold model (Gianola and Foulley, 1983). This fact was confirmed for calving difficulty in beef cattle by the studies of Varona et al. (1999) and Ramirez-Valverde et al. (2001). However, the advantage of the threshold model was very slight. Moreover, there might be some statistical problems with the estimation of fixed herd-year effects in the threshold model when dealing with small herds or with the absence of some scores within some herd-years. Consequently, the two studies mentioned above assumed a random herd-year-season effect in the models. Such an assumption can lead to biased estimates of breeding values (Henderson, 1975; Visscher and Goddard, 1993). Objectives of this study were to choose the most appropriate statistical model for the prediction of breeding values for calving difficulty and to estimate the genetic direct and maternal parameters for calving difficulty in the Charolais breed.


   Materials and Methods
Top
 Abstract
 Introduction
 Materials and Methods
Results and Discussion
 Implications
 Literature Cited
 
Field Data
Calving difficulty records were analysed from 246,576 Charolais calves born from 1985 to 1999, bred by 3,787 sires and 81,408 dams. The total number of sires, including ancestors, was 14,179. Most calvings (56.0%) were unassisted (score = 1), 37.2% had minor difficulty (score = 2), 3.2% were mechanically assisted (score = 3), and 3.6% were caesarian births (score = 4). The characteristics of the data set are given in Table 1Go. Records were selected from the largest 225 herds in order to ensure a minimal herd-year group size of 40 calves and were then selected from sires with at least 20 calves. Records from herds where all the calvings were scored as unassisted for a few years were also removed because it is impossible to correctly estimate a fixed herd-year effect due to the well-known extreme category problem (Mistzal et al., 1989).


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  		Table 1. Summary of the Charolais data set
 
Records were selected because it is impossible to estimate variance components on the entire Charolais data set (approximately 3 million records) and to get adequate genetic parameters under a threshold model with small herds (Moreno et al. 1997). This last point appears more crucial than the possible bias due to differences in management that may exist between large and small herds because the estimation of genetic parameters will be influenced mainly by the number of progeny per family.

Analysis Models
Threshold Model.
This model assumed an underlying normal distribution of calving difficulty controlling the observed value Yi for animal i through a set of j-1 thresholds {tau}j (for j categories; here, j = 4). The thresholds on the underlying distribution were derived according to the probabilities of observing the different scores. Analyses were performed using a sire-maternal grandsire model due to the inadequacy of the threshold model to estimate genetic parameters under an animal model (Moreno et al., 1997) and also because computational time was reduced. Therefore, a linear model was written after a probit transformation of the data Y. In practice, let Pj(Yi) be the probability of observing Yi in category j:



where {Phi} denotes the cumulative normal density function, {eta}i is the mean of the underlying variable yi, ß is the vector of fixed effects (birth season, sex of calf, class of age of dam), h is the vector of random or fixed herd-year effects, s is the vector of random sire effects, and t is the vector of random grandsire effects, and X, H, Z and W are the corresponding incidence matrices.

From a theoretical point of view, h was considered to be a fixed effect in order to take into account the potential nonrandom distribution of sires across herds and years (Henderson, 1975; Visscher and Goddard, 1993). On the other hand, h was also considered to be a random effect from a practical point of view in order to be able to estimate herd-year effects for herd-years of small size or with some scores not registered. However, the reader must keep in mind that considering the vector of herd-year effects as random is completely uncorrect for statistical theory.

The software used for the estimation of variance components and the prediction of breeding values under a threshold model was developed by Ducrocq (2000) for the calving ease evaluation of French dairy cattle. The method used is the GFCAT method based on marginal maximum likelihood derived simultaneously by Gianola and Foulley (1983) and by Harville and Mee (1984).

Linear Model.
The same model as decribed in the previous section was considered, but it was directly applied on the expectation of the observed data. The software ASREML developed by Gilmour et al. (2000) was used for such models. Because ASREML can only deal with binary data, it could not be used for the threshold models because of the polychotomous nature of calving difficulty.

Estimation of Genetic Parameters
The relationships between the (co)variances estimated under a sire-maternal grandsire model and the (co)variances of direct and maternal effects were as follows: the direct genetic variance was 4{sigma}2s, the maternal genetic variance was {sigma}2s + 4{sigma}2t - 4{sigma}st, the direct-maternal covariance was -2{sigma}2s + 4{sigma}st, the environmental variance was {sigma}2r - 2{sigma}2s - 3{sigma}2t, and the phenotypic variance was {sigma}2s + {sigma}2t + {sigma}2r + {sigma}2h, where {sigma}2s, {sigma}2t, {sigma}st and {sigma}2h denote the sire variance, the maternal grandsire variance, the sire-maternal grandsire covariance, and the herd-year variance, respectively, and {sigma}2a, {sigma}2m, {sigma}am, {sigma}2e, {sigma}2p are the derived estimates. The residual variance ({sigma}2r ) is fixed to the value 1 under a threshold model.

In a preliminary analysis, the random effect of the dam within the maternal grandsire was considered in the model in order to estimate the permanent environmental effect due to the dam. This variance component was estimated to be very close to zero. Consequently, the dam effect was ignored in the subsequent analyses. Moreover, it was verified that calving difficulty for primiparous cows was genetically the same trait as calving difficulty for multiparous cows. Dealing with a bivariate model, the genetic correlation between calving difficulty for primiparous cows and for multiparous cows was indeed estimated close to 1.

Criteria for Comparing Models
The fixed threshold model (FTM) is the reference model since it is theoretically the best statistical model to deal with a discrete trait and to provide an unbiased genetic evaluation. Consequently, the other models were compared to the FTM in order to determine which models predict similar breeding values. The motivation of this study is that the FTM cannot be applied in French breeding evaluations due to small herd-year sizes. The criterion for the choice of the most appropriate model was the correlation between estimated breeding values or so-called indices. Two kinds of correlation were derived: a correlation between direct indices (i.e., the sire effects) and a correlation between maternal indices (i.e., twice the maternal grandsire effect minus its sire effect).

Monte-Carlo Simulations
These simulations were motivated by the feeling that the results on a data set with large herd-year group sizes may present the fixed linear model (FLM) in the best light. Monte-Carlo simulations were necessary to give some idea of how the models compare in small herds since the reference model FTM cannot be used in that case. Monte-Carlo simulations were used to compare the true breeding values (under a threshold model) and the breeding values predicted under random threshold model (RTM) and FLM model for populations with 225 small herds. A unique year of records was considered, so the "herd-year" effect considered previously became a herd effect. It was assumed that 225 natural service sires had progeny in a single herd, and one AI sire sired 30% of the calves born in each of the 225 herds. These figures mimicked the real degree of connectedness across Charolais herds. Levels of the herd effects and levels of the sire effects within herds were simulated with different values in order to evaluate the corresponding behavior of the models. In order to simplify the simulations, the maternal grandsires were ignored and the dam population was assumed to be unselected and unrelated to the sires.

The genetic evaluation was given under a sire model:


where y is the observed performance for FLM or the underlying performance for the RTM, b is the vector of the fixed effects of the herds under the FLM or the vector of random effects of the herds for the RTM, s the vector of sire random effects, which is supposed to be distributed as N(0, I {sigma}a2/4), and the vector of residuals e* was distributed as N(0, I {sigma}*e2 ), with {sigma}*e2 = 3/4{sigma}a2 + {sigma}e2.

The phenotypic variance ({sigma}p2 = {sigma}a2 + {sigma}e2) was equal to 100 for the underlying performance and the true heritability (h2 = {sigma}a2/{sigma}p2) was h = 0.25 for the underlying variable. To derive the thresholds, it was assumed that 56% of calves were unassisted, 36% had minor difficulty, 4% were mechanically assisted, and the last 4% were caesarian births.

Either 20 calves or 40 calves were recorded per herd, corresponding to 14 or 28 calves bred by a single sire within herd and 6 or 12 calves bred by the AI sire used in the 225 herds. The phenotypes of progeny were simulated by adding their genotype (sire effect + sampling component N(0, 3/4{sigma}a2) due to the dam effect and the Mendelian sampling) to an environmental random residual sampled from N(mh, {sigma}e2), where mh = 0 or mh = mh-1 + 0.01 {sigma}P (with h varying from 1 to 225, m1 = -112 x 0.01 {sigma}P and m225 = +112 x 0.01 {sigma}P). Considering the addition of 0.01 {sigma}P from one herd to the next led to the estimation of a herd variance around 0.46 under a random threshold model. The breeding values of natural service sires (equal to 2s) were sampled from a distribution N (mah,{sigma}a2), where mah was equal to 0, +0.5mh, or -0.5mh to consider a random, positive, or negative association between sires and herds, respectively. Expectation of the AI sire effect was equal to 0 for all simulations. Variance components estimation and sire evaluation were done simultaneously with ASREML (Gilmour et al., 2000) for linear model and with a program developed by Ducrocq (2000) for the discrete model. Average correlations and their standard deviations between true breeding values and RTM and FLM indices of sires were derived from 25 replicates per Monte Carlo simulation.


   Results and Discussion
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
Implications
 Literature Cited
 
Estimation of Genetic and Phenotypic Parameters
Table 2Go presents the estimates of variance components according to the statistical model considered. The highest estimates of heritabilities were obtained under a FTM: 0.27 for direct effects and 0.18 for maternal effects. The direct-maternal genetic correlation was estimated at -0.36. For a RTM, the heritability of direct effects was 0.24 and the heritability of maternal effects was 0.12. The corresponding direct-maternal genetic correlation was -0.33. The heritability of direct effects was 0.14 under a FLM and 0.16 for a random linear model (RLM). The heritability of maternal effects was 0.13 under a FLM and 0.11 under a RLM. Under linear models, the direct-maternal genetic correlation was more sensitive than heritabilities to the treatment of the herd-year effect as fixed or random: the correlation was -0.34 and -0.19 under a FLM and a RLM, respectively.


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  		Table 2. Variance components for calving difficulty under a random threshold model (RTM), fixed threshold model (FTM), random linear model (RLM), or fixed linear model (FLM)
 
These estimates were in the range of the estimates obtained for other beef cattle breeds either with linear models (Gregory et al., 1995; Varona et al., 1999; Carnier et al., 2000; Bennett and Gregory, 2001) or with threshold models (Dong et al., 1991; Varona et al., 1999).

Comparison of Models for the Field Data Set of Large Herds
Previous comparisons of models applied to discrete traits, such as calving ease concentrated on comparing either threshold vs. linear models or sire vs. animal models (Varona et al., 1999; Ramirez-Valverde et al., 2001), assuming a random contemporary group effect. In this study, the comparison of threshold vs. linear models includes the statistical treatment of the herd-year effect, either as a fixed or a random effect. Results are given in Table 3Go. Genetic parameters used to predict the breeding values were those presented in Table 2Go.


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  		Table 3. Index correlations between models in the pedigree population of field dataa
 
Correlations between direct indices ranked from 0.95 for the comparison RTM-RLM to 0.85 for either the FTM-RLM or the RTM-FLM comparisons. As far as direct indices are concerned, the reference model is more correlated to the FLM (r = 0.94) than to the RTM (r = 0.91). Correlations between maternal indices are higher than those between direct indices and ranked from 0.98 for the comparison FTM-RTM to 0.92 for the comparison FTM-RLM. Concerning maternal indices, the reference model is more correlated to the RTM (r = 0.98) than to the FLM (r = 0.94).

Table 4Go presents correlations between direct indices for two extreme populations of sires selected for their calving ease breeding values predicted under the reference model: the top 1,000 sires and the bottom 1,000 sires. The correlations were higher for the top population than for the bottom population. In the two populations, the FLM was the model that gave indices closer to those of the reference model; the correlations were 0.93 for the top 1,000 sires and 0.80 for the bottom 1,000 sires. The ranking of sires was more sensitive to the model for high incidence of calving difficulty than for low incidence. When selecting sires, the bottom sires for calving ease evaluation will be eliminated and, consequently, there will be little impact of the statistical model in the population of selected sires. This will be true if the worst sires are consistently detected.


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  		Table 4. Direct index correlations between models in two extreme sire populations: the top 1,000 bulls (above diagonal) and the worst 1,000 bulls (below diagonal) for evaluation under a fixed threshold model
 
Table 5Go presents correlations between maternal indices for the two extreme populations of sires. As for the correlations between direct indices, the correlations between maternal indices were higher for the top population than for the bottom population. In contrast to the correlations between direct indices, the FLM was not the model that gave the closest indices to those of the reference model; the correlations between maternal indices were 0.96 and 0.95 for the bottom and top populations, respectively, whereas the correlations between maternal FTM and maternal RTM were estimated at 0.99 and 0.98 for the bottom and top sires, respectively. However, in any case, the correlations between maternal indices are larger than those between direct indices; the minimum value is 0.94 between maternal indices derived from FTM and FLM for the overall population of sires, whereas the minimum value is 0.74 for the correlation between direct indices for FTM and RTM in the bottom population of sires. Therefore, the FLM is the statistical model that gives globally the closest results to the FTM for large herd-year groups.


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  		Table 5. Maternal index correlations between models in two extreme sire populations: the top 1,000 bulls (above diagonal) and the worst 1,000 bulls (below diagonal) for sire evaluation under a fixed threshold model
 
Comparison of Models for Simulated Data Sets of Small Herds
Table 6Go presents correlations between true breeding values and estimated breeding values predicted from RTM and FLM in two simulated sire populations from 225 small herds, the first one with 20 records and the second one with 40 records. Correlations between RTM and FLM indices or between true breeding values and estimated breeding values increased with the size of the herds. When there was no genetic difference across herds, the best model to predict the true breeding values was the RTM. When different genetic levels across herds were considered, models compared differently according to the sign of the nonrandom association between sire and herd effects. As already described in the literature (Visscher and Goddard, 1993), higher correlations between true and predicted breeding values when treating herd effects as random were found when a positive association existed, whereas smaller correlations corresponded to a negative association (i.e., the best sires in the worst herds).


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  		Table 6. Index correlations between random threshold (RTM) and fixed linear (FLM) models in simulated populations of 225 herds with either 20 or 40 records (N) per herd (standard deviations in brackets)
 

   Implications
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
 Implications
Literature Cited
 
Because the genetic evaluation of calving difficulty in French beef cattle cannot be based on a fixed threshold model due to the small herd-year sizes, an alternative is to use a fixed linear model. With regard to large herds, the direct estimated breeding values under the reference model were more correlated to those estimated under the fixed linear model than to those estimated under a random threshold model. The fixed linear model should also be preferred for an unbiased evaluation across small herds when a negative association exists between sire and herd-year effects. Consequently, a fixed linear model was chosen for the genetic evaluation for calving ease in French beef cattle. Genetic parameters in the Charolais breed were then estimated at 0.14, 0.13, and -0.34 for direct heritability, maternal heritability, and direct-maternal genetic correlation, respectively.

Received for publication August 2, 2002. Accepted for publication January 8, 2003.


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


Bennett, G. L., and K. E. Gregory. 2001. Genetic (co)variances for calving difficulty score in composite and parental populations of beef cattle: I. Calving difficulty score, birth weight, weaning weight, and postweaning gain. J. Anim. Sci. 79:45–51.[Abstract/Free Full Text]

Carnier, P., A. Albera, R. Dal Zotto, A. F. Groen, M. Bona, and G. Bittante. 2000. Genetic parameters for direct and maternal calving ability over parities in Piedmontese cattle. J. Anim. Sci. 78:2532–2539.[Abstract/Free Full Text]

Dong, M. C., R. L. Quaas, and E. J. Pollak. 1991. Estimation of genetic parameters of calving ease and birth weight by a threshold model. J. Anim. Sci. 69(Suppl 1):204. (Abstr.)

Ducrocq, V. 2000. Calving ease evaluation of French dairy bulls with a heteroskedastic threshold model with direct and maternal effects. Proc. Interbull Open Mtg., Bled, Slovenia 25:123–130.

Gianola, D., and J. L. Foulley. 1983. Sire evaluation for ordered categorical data with a threshold model. Genet. Sel. Evol. 15:201–224.

Gilmour, A. R., B. R. Cullis, and S. J. Welham. 2000. ASREML Reference Manual. NSW Agriculture, Orange, Australia.

Gregory, K. E., L. V. Cundiff, and R. M. Koch. 1995. Genetic and phenotypic (co)variances for production traits of female populations of purebred and composite beef cattle. J. Anim. Sci. 73:2235–2242.[Abstract]

Harville, D. A., and R. W. Mee. 1984. A mixed-model procedure for analysed ordered categorical data. Biometrics 40:393–408.

Henderson, C. R. 1975. Best linear unbiased estimation and prediction under a selection model. Biometrics 31:343–360.

Hoeschele I., and B. Tier. 1995. Estimation of variance components of threshold characters by marginal poseterior modes and means via Gibbs sampling. Genet. Sel. Evol. 27:519–540.

Misztal, I., D. Gianola, and J. L. Foulley. 1989. Computing aspects of a nonlinear sire evaluation for categorical data. J. Dairy Sci. 72:1557–1568.[Abstract/Free Full Text]

Moreno, C., D. Sorensen, L. A. Garcia-Cortés, L. Varona, and J. Altarriba. 1997. On biased inferences about variance components in the binary threshold model. Genet. Sel. Evol. 29:145–160.

Ramirez-Valverde, R., I. Misztal, and J. K. Bertrand. 2001. Comparison of threshold vs linear and animal vs sire models for predicting direct and maternal genetic effects on calving difficulty in beef cattle. J. Anim. Sci. 79:333–338.[Abstract/Free Full Text]

Varona, L., I. Misztal, and J. K. Bertrand. 1999. Threshold-linear versus linear-linear analysis of birth weight and calving ease using an animal model: I. variance component estimation. J. Anim. Sci. 77:1994–2002.[Abstract/Free Full Text]

Visscher, P. M., and M. E. Goddard. 1993. Fixed and random contemporary groups. J. Dairy Sci. 76:1444–1454.[Abstract]


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