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Bayesian Analysis for Estimation of Genetic Parameters of Calving Ease and Stillbirth for Canadian Holsteins1
Bayesian Analysis for Estimation of Genetic Parameters of Calving Ease and Stillbirth for Canadian Holsteins M. F. LUO,* P. J. BOETTCHER,* J.C.M. DEKKERS,+ and L. R. SCHAEFFER* *Center for Genetic Improvement of Livestock, Department of Animal and Poultry Science, University of Guelph, Guelph, ON, Canada N1G 2W1 + Department of Animal Science, Iowa State University, Ames 50011
ABSTRACT A total of 129,765 calving ease and stillbirth records of Canadian Holsteins were analyzed. Bayesian analysis was applied to a mixed linear sire and maternal grandsire model, and Gibbs Sampling was used to obtain posterior densities of variance components. Both traits were recorded in ordered categories. For calving ease, scores were transformed to Snell scores, for which higher numbers indicated easier calving. Live births were scored 1, and stillbirths were scored 2. Low heritability (combining both direct and maternal effects) estimates were obtained for calving ease (0.07) and stillbirth (0.09). The specific heritabilities for direct and maternal calving ease and for direct and maternal stillbirth were 0.05, 0.03, 0.04, and 0.06, respectively. Negative correlations were obtained between direct and maternal components for calving ease (0.16) and stillbirth (-0.24). The genetic correlation between direct calving ease and direct stillbirth was high (-0.59), but the genetic correlation between maternal calving ease and maternal stillbirth was slightly lower (-0.34). The genetic correlations between direct calving ease and maternal stillbirth (0.06) and between maternal calving ease and direct stillbirth (0.04) were positive but very low. The genetic (maternal plus direct), phenotypic, and residual correlations between calving ease and stillbirth were -0.47, -0.19, and -0.16, respectively. (Key words: calving ease, stillbirth, genetic parameter estimation, Bayesian analysis) Abbreviation Key: CE = calving ease, MGS = maternal grandsire, SB = stillbirth. ____________________ Received December 2, 1998. Accepted June 29, 1999. 1999 J. Dairy Sci. (Aug)
INTRODUCTION
Calving ease (CE) and stillbirth (SB) in cattle are reproductive traits of economic importance in the dairy industry (3), especially for first-calf females. The direct losses caused by dystocia and SB include loss of calf, death of dam, veterinary fees, and extra labor requirements. Dystocia also contributes to some long-term indirect losses, such as increased risk of subsequent health and fertility problems, increased culling, and reduced production (8). These economic concerns and international competition for semen sales demand the genetic evaluation of sires for CE and SB. The Canadian genetic evaluation for CE is carried out by the Canadian Dairy Network (Guelph, Canada) and is based on a linear animal model with maternal effects. The genetic model assumes that CE is the same trait for heifers and cows. Because of the perceived high correlation of CE and SB, additive genetic effects on SB have never been considered important enough in Canada to warrant a separate genetic evaluation for SB. However, concerns about the increasing rates of SB have arisen in Scandinavia recently, coinciding with the importation of semen from North American sires (10). Data on SB has been collected in Canada. A preliminary study showed that the SB rate of offspring of Canadian Holstein sires ranged from 0 to 40% (Luo 1998, unpublished data). A genetic evaluation for SB may be initiated soon in Canada to meet the demand of the market. However, no estimates from Canadian data of genetic parameters are available for the genetic evaluation of SB. Furthermore, the parameters being employed for the current genetic evaluation of CE are based on estimates made more than 10 yr ago by Dwyer et al. (5). Therefore, more recent estimates of the genetic parameters of CE and SB are urgently needed for precise genetic evaluation of the current Holstein population. The CE and SB are influenced by both direct genetic (the effect due to genes of the calf on its ease of birth) and maternal genetic effects (the effects of the genes of the calf's dam on CE through the environment provided by the dam). Studies (2, 5, 8, 14, 17) have shown an antagonistic correlation between these two components. Jensen et al. (6) proposed a model to estimate genetic variances for a trait influenced by maternal and direct genetic effects using a Bayesian analysis. Wang et al. (16) presented a similar model for two traits, of which one was influenced by maternal effects. A Bayesian analysis for estimation of genetic parameters for two traits with maternal effects has not yet been investigated. Although application of linear models to categorical traits is theoretically inadequate, many studies have used a linear model to predict genetic merit (1, 4, 12) and to estimate genetic parameters (2, 5, 14, 17) of traits scored as ordered categories. Some studies (9, 17) also showed that the sire solutions from BLUP and a nonlinear model were highly correlated (r = 0.99) for dystocia and SB. Weller (17) used a sire-maternal grandsire (MGS) model that assumed sires were unrelated to compare genetic parameters estimated by linear versus nonlinear models. The purpose of this study was to apply Bayesian analysis to the estimation of the genetic parameters of CE and SB for Canadian Holsteins, using a bivariate linear model that included direct and maternal genetic effects for each trait. The Gibbs sampler was applied to obtain the posterior densities. A linear model was used in this study to remain consistent with the current genetic evaluation methodology for CE in Canada and for comparisons with threshold models under a later study.
MATERIALS AND METHODS Data A total of 312,281 CE and SB records of first lactation Canadian Holsteins from the national database were used. These data were routinely screened by the Canadian Dairy Network to help ensure complete recording of all births within a herd. Heifers ranged from 18 to 36 mo of age. Two calving seasons (May to October and November to April) were defined. Scores for CE and SB were recorded by farmers at or around the time of calving. The CE scores ranged from 1 to 4 (1 = unassisted, 2 = easy pull, 3 = hard pull, and 4 = surgery needed). These scores were transformed according to the Snell scoring system (13) to a range from 0 to 100. On this transformed scale, higher scores indicated easier calving. The SB record was scored as 1 if the calf was alive 24 h after birth or 2 if the calf was dead. The incidences of CE and SB rate of Canadian Holsteins from three regions with different dairy records processing centers (Alberta, Ontario, and Quebec) in Canada are in Table 1. The Alberta center also processes records from Manitoba and Saskatchewan. Quebec processes data from Nova Scotia, New Brunswick, Newfoundland, and Prince Edward Island. Approximately 19.4% of the male calves were born with difficulty (hard pull plus surgery), but only 13.2% of female calves were born with difficulty. The SB rate of male calves was almost 2% higher than that of female calves. Among the three regions in Canada, heifers from Alberta had the highest calving difficulty scores and SB rate, and heifers from Quebec showed the lowest calving difficulty scores and SB rate. The data consisted of 129,765 records of daughters of 6734 sires after editing for incomplete records, missing identification numbers, birth dates, and unreasonable or duplicate records. Sires and MGS were required to have records in at least 15 herd-year-seasons to ensure the connectedness of the data.
Model Luo et al. (7) analyzed the nongenetic factors influencing CE and SB of Canadian Holsteins and found that incidence of dystocia and SB rate of male calf births were much greater than that of female calves. Incidence of dystocia and SB rate were significantly (P < 0.01) different among herd-year-seasons. The effect of age of cow at first calving was significant (P < 0.01), but no discernable pattern was observed, except that very young (less than 22 mo) and very old (over 36 mo) cows gave birth with decreased CE and greater SB. Based on these results, sex of calf, herdyear-season and age of cows as a covariate fitted to a quadratic curve were included as fixed effects in the mixed linear model that was used for estimation of genetic parameters: [1] where yijklm = CE or SB score for a calf, a = age of cow at first calving, β 1 and β2 = regression coefficients, hi = fixed effect of HYS i, xj = fixed effect of sex j of calf (j = 1, 2 for male,
female), sk = random genetic effect of sire k of calf, ml = random genetic effect of MGS l of calf, and eijklm = residual effect. In matrix notation, the model can be expressed as [2] where b = vector of fixed effects; s and m = vectors of sire and MGS effects, respectively; X, Z1, and Z2 = known design matrices associated with b, s, and m. Let Z = (Z1 Z2) and u' = (s' m'). The expectations and variance structure of the model [2] were
, [3] and
[4] where A = additive genetic relationship matrix, G0 = sire-MGS (co)variance matrix, R0 = residual variance matrix , and n = the number of records. More assumptions and limitations of the sire-MGS model were that 1) sires were mated randomly to dams, and 2) dams were mated to only one sire and had just one progeny. Assumption 1 may be violated in some instances if breeders choose bulls with favorable CE as mates for first-calf heifers, but inclusion of the age of the dam effects in the model was expected to account for some of those effects. Some of the dams may have had twins, which would have violated assumption 2. Only first lactation data were used so that all animals had only one record. In addition to these assumptions, sires must have had both progeny and grand progeny for correct estimation of maternal variance and covariance between direct and maternal. In general, most of the sires represented in the data had both progeny and grand progeny, although a few of the youngest sires had only progeny.
Method
A Bayesian analysis was applied to the sire and MGS model (6, 15, 16). We assumed that the joint distribution of these two traits was bivariate normal: . [5] The distribution of breeding value u was multivariate normal: . [6] We also assumed a uniform distribution for the prior for the fixed effects b and inverted Wishart distributions, which are the multivariate form of the inverted gamma and appropriate for covariances, for genetic variance-covariance matrix G0, and for the residual variance-covariance matrix R0: p(b) ∝ constant [7] [8] [9] where Sg = E (G0 | Sg, νg ), Se = E (R0 | Se, νe), and ν g and νe are called shape parameters or degrees of freedom in the priors. The subscripts 4 and 2 indicate the size of matrices G0 and R0.
So, ν g and νe have to be greater than 5 and 3, respectively, to have a proper prior. Based on these priors and the given data, by applying the Bayes theorem we have the following fully conditional posterior densities (6, 16): [10] [11] [12] [13] where SSg = sum of squares and cross products of sire and MGS effects, and SSe = sum of
squares and cross products of residuals (i.e., SSg = u' A-1 u, and SSe = e' e), Cii = inverse of the coefficient matrix of the mixed model equation, q = number of sires, and n = number of records.
Gibbs Sampler The Gibbs sampler was applied to obtain posterior densities of (co)variance components. The program MTGSAM from C. P. Van Tassell and L. D. Van Vleck was used (15). Visual inspection of plots of variances against sample number and statistics of Raftery and Lewis (11) were applied to determine the burn-in period (the number of initial samples discarded to ensure sampling from the proper distributions) and the length of Gibbs sampler. The lag correlation (autocorrelation) between samples after burn-in was used to determine the number of rounds between independent samples. Based on the lag correlations, samples taken every 50 rounds after burn-in were sufficient to estimate the posterior densities for all variance and covariance components. From the sire (s) and MGS (m) (co)variances, the direct (D) and maternal (M) genetic (co) variances for given trait were obtained by
[14] and the covariances between traits 1 and 2 by
. [15] The phenotypic (p) variance was . [16] The direct and maternal heritability were calculated by . [17] Total (t) heritabilities for CE and SB and the total genetic correlation between CE (trait 1) and SB (trait 2) were calculated (18) by , and [18]
. [19] The residual and phenotypic correlations between two traits (1 and 2) were
. [20]
RESULTS AND DISCUSSION Estimated posterior means, modes, medians, maxima, minima, coefficients of variation, and the first and third quartiles of heritabilities and genetic correlation coefficients are presented in Table 2. Posterior means of genetic correlation coefficients had greater variation than did posterior means for heritabilities. Ranges and coefficients of variation of genetic correlation coefficients were very large compared with heritabilities. The posterior means, modes, and medians were very similar for heritabilities but were different for correlation coefficients. These results indicated that the distributions of heritabilities were nearly symmetric, but the distributions of correlation coefficients were usually skewed. Means of heritabilities for direct and maternal effects for CE and SB were low. The direct heritability was greater than was the maternal heritability for CE but was lower for SB. Heritabilities estimated from this study were consistent with most previous reports on dairy cattle (2, 4, 14, 17) but were lower than the estimates by Dwyer et al. (5) for Canadian Holsteins. The Canadian Dairy Network has used the estimates of Dwyer et al. (5) for genetic evaluation for CE since 1992. Their estimate was 0.11 for the direct heritability and 0.12 for the maternal heritability of CE of Canadian Holsteins. The heritabilities are two times greater than the average of other studies, including this paper, for Holsteins. Several factors might be involved in greater estimates of heritabilities in their study. First was that all parity data were used in the study by Dwyer et al. (5). Second, perhaps accuracy and diligence of records has declined over time in Canada since the time when the data were collected for the study by Dwyer et al. (5). A negative correlation was observed between direct and maternal genetic components for both CE and SB. The correlation was -0.16 between direct CE and maternal CE and was -0.25 between direct SB and maternal SB for CE. The genetic correlation between direct and maternal genetic effects was close to those results from Djemali et al. (4) (-0.47), and Cue and Hayes (2) (-0.40). Because of the negative correlation between direct and maternal genetic components, total heritability was less than the sum of the direct and maternal heritabilities. Total heritabilities of CE and SB were nearly the same (0.07 and 0.09, respectively). This negative relationship between direct and maternal genetic components also makes selection for these two traits more
difficult in practice. Obviously, both direct and maternal genetic effects should be accounted for in the selection for CE and SB. Meijering (8) and Dekkers (3) suggested some basic breeding strategies for simultaneous selection on CE and SB. The genetic correlations between CE and SB were moderately high and negative (-0.59 for direct effects and -0.34 for maternal effects). The total genetic correlation between CE and SB was also negative (-0.47). This result indicated that increased calving difficulty was associated with increased probability of SB. Therefore, improving the CE genetically could decrease the rate of SB at the same time but direct selection for both traits would be more effective. The residual correlation between CE and SB was -0.16. The phenotypic correlation between CE and SB was -0.19. Moderate negative genetic correlation but low phenotypic correlation between CE and SB imply that phenotypic selection on one trait may not effectively change the other one on the phenotypic scale.
CONCLUSIONS The heritabilities of CE and SB were lower than those from previous analysis based on Canadian Holstein data. Heritabilities used in the current CE evaluation in Canada may be too high for a linear model. Using heritability 0.05 in a genetic evaluation for CE will yield low accuracy for predicting genetic merits of animals. Therefore, alternatives to the linear model that are more appropriate for categorical data or measures of CE that could be recorded on a more continuous scale should be studied for the genetic evaluation of calving traits. Moderate negative genetic correlation between CE and SB provides an opportunity to improve simultaneously the CE and decrease the SB rate. The negative correlation between direct and maternal components restricts the efficiency of genetic improvement of both direct and maternal effects on the two traits. A selection index that includes both direct and maternal effects, as suggested by Dekkers (3), might be the most efficient approach to increase CE.
ACKNOWLEDGMENTS Financial support was provided by Ontario Ministry of Agriculture, Food and Rural Affairs (Guelph, Canada), and Cattle Breeding Research Council (Guelph, Canada). We thank C. P. Van Tassell and L. D. Van Vleck for use of their software.
REFERENCES 1 Berger, P. J., and A. E. Freeman. 1978. Prediction of sire merit for calving difficulty. J. Dairy Sci. 61:1146-1150. 2 Cue, R. I., and J. F. Hayes. 1985. Correlations of various direct and maternal effects for calving ease. J. Dairy Sci. 68:374-381.
3 Dekkers, J.C.M. 1994. Optimal breeding strategies for calving ease. J. Dairy Sci. 77:3441-3453. 4 Djemali, M., P. J. Berger, and A. E. Freeman. 1987. Ordered categorical sire evaluation for dystocia in Holsteins. J. Dairy Sci. 70:2374-2384. 5 Dwyer, D. J., L. R. Schaeffer, and B. W. Kennedy. 1986. Bias due to corrective matings in sire evaluations for calving ease. J. Dairy Sci. 69:794-799. 6 Jensen, J., C. S. Wang, D. A. Sorensen, and D. Gianola. 1994. Bayesian inference on variance and covariance components for traits influenced by maternal and direct genetic effects, using the Gibbs sampler. Acta. Agric. Scand., Sect. A, Anim. Sci. 44:193-201. 7 Luo, M. F., P. J. Boettcher, and J.C.M. Dekkers. 1998. Factors influencing calving difficulty and stillbirth of Canadian Holsteins. J. Dairy Sci. 76(Suppl.1):84.(Abstr.) 8 Meijering, A. 1984. Dystocia and stillbirth in cattle a review of causes, relations and implications. Livest. Prod. Sci. 11:143-177. 9 Meijering, A. 1985. Sire evaluation for calving traits by Best Linear Unbiased Prediction and nonlinear methodology. Z. Tierzüchtg. Züchtgsbiol. 102:95-105. 10 Philipsson, J. 1998. Considering stillbirths in the breeding program? Pages 25-27 in Proc. Int. Workshop Genet. Improvement of Functional Traits in Cattle; Fertility and Reproduction. Bull. No. 18. Int. Committee Anim. Recording, Uppsala, Sweden. 11 Raftery, A. E., and Lewis S. M. 1992. Pages 765-776 in Bayesian Statistics 4. J. M. Bernardo, J. O. Berger, A. P. David, and A.F.M. Smith, ed. Oxford, Oxford Univ. Press, London, United Kingdom. 12 Schaeffer, L. R., and J. W. Wilton. 1976. Methods of sire evaluation for calving ease. J. Dairy Sci. 59: 544-551. 13 Snell, E. J. 1964. A scaling procedure for ordered categorical data. Biometrics 20:592607. 14 Thompson, J. R., A. E. Freeman, and P. J. Berger. 1981. Age of dam and maternal effects for dystocia in Holsteins. J. Dairy Sci. 64:1603-1609. 15 Van Tassel, C. P., and L. D. Van Vleck. 1996. Multiple-trait Gibbs sampler for animal models: flexible program for Bayesian and likelihood-based (co)variance component inference. J. Anim. Sci. 74:2586-2597. 16 Wang, C. S., R. L. Quaas, and E. J. Pollak. 1997. Bayesian analysis of calving ease scores and birth weights. Genet. Sel. Evol. 29:117-143. 17 Weller, J. I., I. Misztal, and D. Gianola. 1988. Genetic analysis of dystocia and calf
mortality in Israeli-Holsteins by threshold and linear models. J. Dairy Sci. 71:2491-2501. 18 Willham, R. L. 1972. The role of maternal effects in animal breeding: III. Biometrical aspects of maternal effects in animal. J. Anim. Sci. 35:1288-1292.
TABLE 1. Distribution of calving ease scores and stillbirth rate of Canadian Holsteins across sexes from three centers for data processing in Canada.
Calving ease scores
Data center
Sex of calf
1
2
3
4
Stillbirth rate
(%) Alberta
Ontario
Quebec
Average
Male
44.5
33.4
21.2
1.0
11.8
Female
50.1
34.8
14.5
0.6
9.9
Male
37.8
38.7
22.8
0.8
7.7
Female
42.0
40.1
17.4
0.5
6.6
Male
57.2
30.4
11.7
0.6
7.8
Female
65.7
27.3
6.3
0.3
5.6
Male
46.5
34.2
18.6
0.8
9.1
Female
52.6
34.1
12.7
0.5
7.4
TABLE 2. Statistics of marginal posterior distributions of direct and maternal heritabilities (h2) and correlation coefficients (r) between direct and maternal genetic effects.
Parameter1
Mean
Mode
CV
Min
Q1
Med
Q3
Max
h2DCE
0.052
0.051
11.8
0.034
0.048
0.052
0.056
0.076
rDCE, DSB
0.593
-0.671
-11.6
-0.755
-0.641
-0.601
-0.553
-0.270
rDCE, MCE
0.160
-0.194
-77.7
-0.561
-0.247
-0.167
-0.067
0.232
rDCE, MSB
0.063
0.139
138.8
-0.273
0.003
0.064
0.124
0.313
h2DSB
0.033
0.032
13.6
0.022
0.030
0.033
0.036
0.050
rDSB MCE
0.044
-0.210
260.0
-0.374
-0.038
0.046
0.133
0.332
rDSB MSB
0.242
-0.343
-31.7
-0.505
-0.296
-0.247
0.193
0.106
h2MCE
0.035
0.036
18.2
0.020
0.031
0.035
0.040
0.055
rMCE, MSB
0.344
-0.311
-21.8
-0.557
-0.397
-0.347
-0.294
-0.083
h2MSB
0.060
0.058
11.0
0.039
0.055
0.059
0.064
0.085
1DCE
= Direct calving ease, DSB = direct stillbirth, MCE = maternal calving ease, MSB = maternal stillbirth, Q1 = quartile 1, and Q3 = quartile 3.
ABSTRACT A total of 129,765 calving ease and stillbirth records of Canadian Holsteins were analyzed. Bayesian analysis was applied to a mixed linear sire and maternal grandsire model, and Gibbs Sampling was used to obtain posterior densities of variance components. Both traits were recorded in ordered categories. For calving ease, scores were transformed to Snell scores, for which higher numbers indicated easier calving. Live births were scored 1, and stillbirths were scored 2. Low heritability (combining both direct and maternal effects) estimates were obtained for calving ease (0.07) and stillbirth (0.09). The specific heritabilities for direct and maternal calving ease and for direct and maternal stillbirth were 0.05, 0.03, 0.04, and 0.06, respectively. Negative correlations were obtained between direct and maternal components for calving ease (0.16) and stillbirth (-0.24). The genetic correlation between direct calving ease and direct stillbirth was high (-0.59), but the genetic correlation between maternal calving ease and maternal stillbirth was slightly lower (-0.34). The genetic correlations between direct calving ease and maternal stillbirth (0.06) and between maternal calving ease and direct stillbirth (0.04) were positive but very low. The genetic (maternal plus direct), phenotypic, and residual correlations between calving ease and stillbirth were -0.47, -0.19, and -0.16, respectively. (Key words: calving ease, stillbirth, genetic parameter estimation, Bayesian analysis) Abbreviation Key: CE = calving ease, MGS = maternal grandsire, SB = stillbirth. ____________________ Received December 2, 1998. Accepted June 29, 1999. 1999 J. Dairy Sci. (Aug)
INTRODUCTION
Calving ease (CE) and stillbirth (SB) in cattle are reproductive traits of economic importance in the dairy industry (3), especially for first-calf females. The direct losses caused by dystocia and SB include loss of calf, death of dam, veterinary fees, and extra labor requirements. Dystocia also contributes to some long-term indirect losses, such as increased risk of subsequent health and fertility problems, increased culling, and reduced production (8). These economic concerns and international competition for semen sales demand the genetic evaluation of sires for CE and SB. The Canadian genetic evaluation for CE is carried out by the Canadian Dairy Network (Guelph, Canada) and is based on a linear animal model with maternal effects. The genetic model assumes that CE is the same trait for heifers and cows. Because of the perceived high correlation of CE and SB, additive genetic effects on SB have never been considered important enough in Canada to warrant a separate genetic evaluation for SB. However, concerns about the increasing rates of SB have arisen in Scandinavia recently, coinciding with the importation of semen from North American sires (10). Data on SB has been collected in Canada. A preliminary study showed that the SB rate of offspring of Canadian Holstein sires ranged from 0 to 40% (Luo 1998, unpublished data). A genetic evaluation for SB may be initiated soon in Canada to meet the demand of the market. However, no estimates from Canadian data of genetic parameters are available for the genetic evaluation of SB. Furthermore, the parameters being employed for the current genetic evaluation of CE are based on estimates made more than 10 yr ago by Dwyer et al. (5). Therefore, more recent estimates of the genetic parameters of CE and SB are urgently needed for precise genetic evaluation of the current Holstein population. The CE and SB are influenced by both direct genetic (the effect due to genes of the calf on its ease of birth) and maternal genetic effects (the effects of the genes of the calf's dam on CE through the environment provided by the dam). Studies (2, 5, 8, 14, 17) have shown an antagonistic correlation between these two components. Jensen et al. (6) proposed a model to estimate genetic variances for a trait influenced by maternal and direct genetic effects using a Bayesian analysis. Wang et al. (16) presented a similar model for two traits, of which one was influenced by maternal effects. A Bayesian analysis for estimation of genetic parameters for two traits with maternal effects has not yet been investigated. Although application of linear models to categorical traits is theoretically inadequate, many studies have used a linear model to predict genetic merit (1, 4, 12) and to estimate genetic parameters (2, 5, 14, 17) of traits scored as ordered categories. Some studies (9, 17) also showed that the sire solutions from BLUP and a nonlinear model were highly correlated (r = 0.99) for dystocia and SB. Weller (17) used a sire-maternal grandsire (MGS) model that assumed sires were unrelated to compare genetic parameters estimated by linear versus nonlinear models. The purpose of this study was to apply Bayesian analysis to the estimation of the genetic parameters of CE and SB for Canadian Holsteins, using a bivariate linear model that included direct and maternal genetic effects for each trait. The Gibbs sampler was applied to obtain the posterior densities. A linear model was used in this study to remain consistent with the current genetic evaluation methodology for CE in Canada and for comparisons with threshold models under a later study.
MATERIALS AND METHODS Data A total of 312,281 CE and SB records of first lactation Canadian Holsteins from the national database were used. These data were routinely screened by the Canadian Dairy Network to help ensure complete recording of all births within a herd. Heifers ranged from 18 to 36 mo of age. Two calving seasons (May to October and November to April) were defined. Scores for CE and SB were recorded by farmers at or around the time of calving. The CE scores ranged from 1 to 4 (1 = unassisted, 2 = easy pull, 3 = hard pull, and 4 = surgery needed). These scores were transformed according to the Snell scoring system (13) to a range from 0 to 100. On this transformed scale, higher scores indicated easier calving. The SB record was scored as 1 if the calf was alive 24 h after birth or 2 if the calf was dead. The incidences of CE and SB rate of Canadian Holsteins from three regions with different dairy records processing centers (Alberta, Ontario, and Quebec) in Canada are in Table 1. The Alberta center also processes records from Manitoba and Saskatchewan. Quebec processes data from Nova Scotia, New Brunswick, Newfoundland, and Prince Edward Island. Approximately 19.4% of the male calves were born with difficulty (hard pull plus surgery), but only 13.2% of female calves were born with difficulty. The SB rate of male calves was almost 2% higher than that of female calves. Among the three regions in Canada, heifers from Alberta had the highest calving difficulty scores and SB rate, and heifers from Quebec showed the lowest calving difficulty scores and SB rate. The data consisted of 129,765 records of daughters of 6734 sires after editing for incomplete records, missing identification numbers, birth dates, and unreasonable or duplicate records. Sires and MGS were required to have records in at least 15 herd-year-seasons to ensure the connectedness of the data.
Model Luo et al. (7) analyzed the nongenetic factors influencing CE and SB of Canadian Holsteins and found that incidence of dystocia and SB rate of male calf births were much greater than that of female calves. Incidence of dystocia and SB rate were significantly (P < 0.01) different among herd-year-seasons. The effect of age of cow at first calving was significant (P < 0.01), but no discernable pattern was observed, except that very young (less than 22 mo) and very old (over 36 mo) cows gave birth with decreased CE and greater SB. Based on these results, sex of calf, herdyear-season and age of cows as a covariate fitted to a quadratic curve were included as fixed effects in the mixed linear model that was used for estimation of genetic parameters: [1] where yijklm = CE or SB score for a calf, a = age of cow at first calving, β 1 and β2 = regression coefficients, hi = fixed effect of HYS i, xj = fixed effect of sex j of calf (j = 1, 2 for male,
female), sk = random genetic effect of sire k of calf, ml = random genetic effect of MGS l of calf, and eijklm = residual effect. In matrix notation, the model can be expressed as [2] where b = vector of fixed effects; s and m = vectors of sire and MGS effects, respectively; X, Z1, and Z2 = known design matrices associated with b, s, and m. Let Z = (Z1 Z2) and u' = (s' m'). The expectations and variance structure of the model [2] were
, [3] and
[4] where A = additive genetic relationship matrix, G0 = sire-MGS (co)variance matrix, R0 = residual variance matrix , and n = the number of records. More assumptions and limitations of the sire-MGS model were that 1) sires were mated randomly to dams, and 2) dams were mated to only one sire and had just one progeny. Assumption 1 may be violated in some instances if breeders choose bulls with favorable CE as mates for first-calf heifers, but inclusion of the age of the dam effects in the model was expected to account for some of those effects. Some of the dams may have had twins, which would have violated assumption 2. Only first lactation data were used so that all animals had only one record. In addition to these assumptions, sires must have had both progeny and grand progeny for correct estimation of maternal variance and covariance between direct and maternal. In general, most of the sires represented in the data had both progeny and grand progeny, although a few of the youngest sires had only progeny.
Method
A Bayesian analysis was applied to the sire and MGS model (6, 15, 16). We assumed that the joint distribution of these two traits was bivariate normal: . [5] The distribution of breeding value u was multivariate normal: . [6] We also assumed a uniform distribution for the prior for the fixed effects b and inverted Wishart distributions, which are the multivariate form of the inverted gamma and appropriate for covariances, for genetic variance-covariance matrix G0, and for the residual variance-covariance matrix R0: p(b) ∝ constant [7] [8] [9] where Sg = E (G0 | Sg, νg ), Se = E (R0 | Se, νe), and ν g and νe are called shape parameters or degrees of freedom in the priors. The subscripts 4 and 2 indicate the size of matrices G0 and R0.
So, ν g and νe have to be greater than 5 and 3, respectively, to have a proper prior. Based on these priors and the given data, by applying the Bayes theorem we have the following fully conditional posterior densities (6, 16): [10] [11] [12] [13] where SSg = sum of squares and cross products of sire and MGS effects, and SSe = sum of
squares and cross products of residuals (i.e., SSg = u' A-1 u, and SSe = e' e), Cii = inverse of the coefficient matrix of the mixed model equation, q = number of sires, and n = number of records.
Gibbs Sampler The Gibbs sampler was applied to obtain posterior densities of (co)variance components. The program MTGSAM from C. P. Van Tassell and L. D. Van Vleck was used (15). Visual inspection of plots of variances against sample number and statistics of Raftery and Lewis (11) were applied to determine the burn-in period (the number of initial samples discarded to ensure sampling from the proper distributions) and the length of Gibbs sampler. The lag correlation (autocorrelation) between samples after burn-in was used to determine the number of rounds between independent samples. Based on the lag correlations, samples taken every 50 rounds after burn-in were sufficient to estimate the posterior densities for all variance and covariance components. From the sire (s) and MGS (m) (co)variances, the direct (D) and maternal (M) genetic (co) variances for given trait were obtained by
[14] and the covariances between traits 1 and 2 by
. [15] The phenotypic (p) variance was . [16] The direct and maternal heritability were calculated by . [17] Total (t) heritabilities for CE and SB and the total genetic correlation between CE (trait 1) and SB (trait 2) were calculated (18) by , and [18]
. [19] The residual and phenotypic correlations between two traits (1 and 2) were
. [20]
RESULTS AND DISCUSSION Estimated posterior means, modes, medians, maxima, minima, coefficients of variation, and the first and third quartiles of heritabilities and genetic correlation coefficients are presented in Table 2. Posterior means of genetic correlation coefficients had greater variation than did posterior means for heritabilities. Ranges and coefficients of variation of genetic correlation coefficients were very large compared with heritabilities. The posterior means, modes, and medians were very similar for heritabilities but were different for correlation coefficients. These results indicated that the distributions of heritabilities were nearly symmetric, but the distributions of correlation coefficients were usually skewed. Means of heritabilities for direct and maternal effects for CE and SB were low. The direct heritability was greater than was the maternal heritability for CE but was lower for SB. Heritabilities estimated from this study were consistent with most previous reports on dairy cattle (2, 4, 14, 17) but were lower than the estimates by Dwyer et al. (5) for Canadian Holsteins. The Canadian Dairy Network has used the estimates of Dwyer et al. (5) for genetic evaluation for CE since 1992. Their estimate was 0.11 for the direct heritability and 0.12 for the maternal heritability of CE of Canadian Holsteins. The heritabilities are two times greater than the average of other studies, including this paper, for Holsteins. Several factors might be involved in greater estimates of heritabilities in their study. First was that all parity data were used in the study by Dwyer et al. (5). Second, perhaps accuracy and diligence of records has declined over time in Canada since the time when the data were collected for the study by Dwyer et al. (5). A negative correlation was observed between direct and maternal genetic components for both CE and SB. The correlation was -0.16 between direct CE and maternal CE and was -0.25 between direct SB and maternal SB for CE. The genetic correlation between direct and maternal genetic effects was close to those results from Djemali et al. (4) (-0.47), and Cue and Hayes (2) (-0.40). Because of the negative correlation between direct and maternal genetic components, total heritability was less than the sum of the direct and maternal heritabilities. Total heritabilities of CE and SB were nearly the same (0.07 and 0.09, respectively). This negative relationship between direct and maternal genetic components also makes selection for these two traits more
difficult in practice. Obviously, both direct and maternal genetic effects should be accounted for in the selection for CE and SB. Meijering (8) and Dekkers (3) suggested some basic breeding strategies for simultaneous selection on CE and SB. The genetic correlations between CE and SB were moderately high and negative (-0.59 for direct effects and -0.34 for maternal effects). The total genetic correlation between CE and SB was also negative (-0.47). This result indicated that increased calving difficulty was associated with increased probability of SB. Therefore, improving the CE genetically could decrease the rate of SB at the same time but direct selection for both traits would be more effective. The residual correlation between CE and SB was -0.16. The phenotypic correlation between CE and SB was -0.19. Moderate negative genetic correlation but low phenotypic correlation between CE and SB imply that phenotypic selection on one trait may not effectively change the other one on the phenotypic scale.
CONCLUSIONS The heritabilities of CE and SB were lower than those from previous analysis based on Canadian Holstein data. Heritabilities used in the current CE evaluation in Canada may be too high for a linear model. Using heritability 0.05 in a genetic evaluation for CE will yield low accuracy for predicting genetic merits of animals. Therefore, alternatives to the linear model that are more appropriate for categorical data or measures of CE that could be recorded on a more continuous scale should be studied for the genetic evaluation of calving traits. Moderate negative genetic correlation between CE and SB provides an opportunity to improve simultaneously the CE and decrease the SB rate. The negative correlation between direct and maternal components restricts the efficiency of genetic improvement of both direct and maternal effects on the two traits. A selection index that includes both direct and maternal effects, as suggested by Dekkers (3), might be the most efficient approach to increase CE.
ACKNOWLEDGMENTS Financial support was provided by Ontario Ministry of Agriculture, Food and Rural Affairs (Guelph, Canada), and Cattle Breeding Research Council (Guelph, Canada). We thank C. P. Van Tassell and L. D. Van Vleck for use of their software.
REFERENCES 1 Berger, P. J., and A. E. Freeman. 1978. Prediction of sire merit for calving difficulty. J. Dairy Sci. 61:1146-1150. 2 Cue, R. I., and J. F. Hayes. 1985. Correlations of various direct and maternal effects for calving ease. J. Dairy Sci. 68:374-381.
3 Dekkers, J.C.M. 1994. Optimal breeding strategies for calving ease. J. Dairy Sci. 77:3441-3453. 4 Djemali, M., P. J. Berger, and A. E. Freeman. 1987. Ordered categorical sire evaluation for dystocia in Holsteins. J. Dairy Sci. 70:2374-2384. 5 Dwyer, D. J., L. R. Schaeffer, and B. W. Kennedy. 1986. Bias due to corrective matings in sire evaluations for calving ease. J. Dairy Sci. 69:794-799. 6 Jensen, J., C. S. Wang, D. A. Sorensen, and D. Gianola. 1994. Bayesian inference on variance and covariance components for traits influenced by maternal and direct genetic effects, using the Gibbs sampler. Acta. Agric. Scand., Sect. A, Anim. Sci. 44:193-201. 7 Luo, M. F., P. J. Boettcher, and J.C.M. Dekkers. 1998. Factors influencing calving difficulty and stillbirth of Canadian Holsteins. J. Dairy Sci. 76(Suppl.1):84.(Abstr.) 8 Meijering, A. 1984. Dystocia and stillbirth in cattle a review of causes, relations and implications. Livest. Prod. Sci. 11:143-177. 9 Meijering, A. 1985. Sire evaluation for calving traits by Best Linear Unbiased Prediction and nonlinear methodology. Z. Tierzüchtg. Züchtgsbiol. 102:95-105. 10 Philipsson, J. 1998. Considering stillbirths in the breeding program? Pages 25-27 in Proc. Int. Workshop Genet. Improvement of Functional Traits in Cattle; Fertility and Reproduction. Bull. No. 18. Int. Committee Anim. Recording, Uppsala, Sweden. 11 Raftery, A. E., and Lewis S. M. 1992. Pages 765-776 in Bayesian Statistics 4. J. M. Bernardo, J. O. Berger, A. P. David, and A.F.M. Smith, ed. Oxford, Oxford Univ. Press, London, United Kingdom. 12 Schaeffer, L. R., and J. W. Wilton. 1976. Methods of sire evaluation for calving ease. J. Dairy Sci. 59: 544-551. 13 Snell, E. J. 1964. A scaling procedure for ordered categorical data. Biometrics 20:592607. 14 Thompson, J. R., A. E. Freeman, and P. J. Berger. 1981. Age of dam and maternal effects for dystocia in Holsteins. J. Dairy Sci. 64:1603-1609. 15 Van Tassel, C. P., and L. D. Van Vleck. 1996. Multiple-trait Gibbs sampler for animal models: flexible program for Bayesian and likelihood-based (co)variance component inference. J. Anim. Sci. 74:2586-2597. 16 Wang, C. S., R. L. Quaas, and E. J. Pollak. 1997. Bayesian analysis of calving ease scores and birth weights. Genet. Sel. Evol. 29:117-143. 17 Weller, J. I., I. Misztal, and D. Gianola. 1988. Genetic analysis of dystocia and calf
mortality in Israeli-Holsteins by threshold and linear models. J. Dairy Sci. 71:2491-2501. 18 Willham, R. L. 1972. The role of maternal effects in animal breeding: III. Biometrical aspects of maternal effects in animal. J. Anim. Sci. 35:1288-1292.
TABLE 1. Distribution of calving ease scores and stillbirth rate of Canadian Holsteins across sexes from three centers for data processing in Canada.
Calving ease scores
Data center
Sex of calf
1
2
3
4
Stillbirth rate
(%) Alberta
Ontario
Quebec
Average
Male
44.5
33.4
21.2
1.0
11.8
Female
50.1
34.8
14.5
0.6
9.9
Male
37.8
38.7
22.8
0.8
7.7
Female
42.0
40.1
17.4
0.5
6.6
Male
57.2
30.4
11.7
0.6
7.8
Female
65.7
27.3
6.3
0.3
5.6
Male
46.5
34.2
18.6
0.8
9.1
Female
52.6
34.1
12.7
0.5
7.4
TABLE 2. Statistics of marginal posterior distributions of direct and maternal heritabilities (h2) and correlation coefficients (r) between direct and maternal genetic effects.
Parameter1
Mean
Mode
CV
Min
Q1
Med
Q3
Max
h2DCE
0.052
0.051
11.8
0.034
0.048
0.052
0.056
0.076
rDCE, DSB
0.593
-0.671
-11.6
-0.755
-0.641
-0.601
-0.553
-0.270
rDCE, MCE
0.160
-0.194
-77.7
-0.561
-0.247
-0.167
-0.067
0.232
rDCE, MSB
0.063
0.139
138.8
-0.273
0.003
0.064
0.124
0.313
h2DSB
0.033
0.032
13.6
0.022
0.030
0.033
0.036
0.050
rDSB MCE
0.044
-0.210
260.0
-0.374
-0.038
0.046
0.133
0.332
rDSB MSB
0.242
-0.343
-31.7
-0.505
-0.296
-0.247
0.193
0.106
h2MCE
0.035
0.036
18.2
0.020
0.031
0.035
0.040
0.055
rMCE, MSB
0.344
-0.311
-21.8
-0.557
-0.397
-0.347
-0.294
-0.083
h2MSB
0.060
0.058
11.0
0.039
0.055
0.059
0.064
0.085
1DCE
= Direct calving ease, DSB = direct stillbirth, MCE = maternal calving ease, MSB = maternal stillbirth, Q1 = quartile 1, and Q3 = quartile 3.