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* Neiker, Basque Institute for Agricultural Research and Development, PO Box 46, 01080 Vitoria-Gasteiz, Spain
Station d’amélioration génétique des animaux, Institut National de la Recherche Agronomique, Auzeville, BP 52627, 31326 Castanet Tolosan Cedex, France
Area de ProduccióAnimal, Centro UdL-IRTA.25198, Lleida, Spain
1 Corresponding author: elmaturana{at}neiker.net
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Key Words: recursive model • genetic parameter • fertility • dystocia
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Calving ease is one of the most economically significant secondary traits (Dekkers, 1994; Dematawewa and Berger, 1997), especially for first-calf heifers; it measures the presence or absence of dystocia and its intensity. Dystocia, defined as a prolonged or difficult parturition, affects the profitability of herds, animal welfare, and acceptability of the production system by the consumer (Carnier et al., 2000). Difficult births increase direct costs of the herd (veterinary fees, calf or cow death or both, and extra farmer labor), as well as indirect costs, such as an increase in the risk of subsequent unfavorable health events, an increase in culling rate, and a reduction in yield (Dekkers, 1994; Dematawewa and Berger, 1997). Moreover, dystocia can negatively affect reproductive traits, such as days open or number of services per pregnancy (Dematawewa and Berger, 1997).
In Holstein cattle, CE records have been collected systematically in the Basque Country Autonomous Community (Spain) by trained technicians since 1992. Trained technicians are needed because calving performance is scored subjectively and might require a special recording routine (Philipson et al., 1994). A genetic evaluation of sires arose as a request by dairy farmers, with the aim of reducing the frequency of calving problems. Genetic evaluation of sires for CE has been conducted routinely since 1995 (Alday and Ugarte, 1997). To improve the evaluation system, data collection and the model for evaluation were modified; and a sire-maternal grandsire threshold model (Van Tassell et al., 2003) was implemented in 2003.
Female fertility is another important functional trait in dairy cattle because it affects direct reproduction costs and influences calving interval, calving season, and involuntary culling rate (Boichard et al., 1997), thereby decreasing the profitability of the herd. Moreover, fertility and yield have an unfavorable relationship in Holstein cattle (Pryce et al., 2004), and reproductive performance has been found to deteriorate as milk yield increases.
Effects of dystocia on fertility measures have not been thoroughly studied, but there seems to be a negative effect of dystocia on days open (DO) and number of services, which worsens fertility (Dematawewa and Berger, 1997, 1998). Moreover, a moderate and positive genetic relationship between CE and days open has been shown (Lee et al., 2003), which would imply that difficult calving leads to delay estrus and conception.
Neither of these studies considers the fact that a complex relationship exists between CE and fertility traits. Phenotypically, CE affects the cow’s fertility in the next reproductive cycle by a cause-and-effect relationship. For example, a difficult parturition might stress the cow and delay the appearance or the intensity of the heat. Retentions of placental membranes or delays in uterine involution are other consequences of a difficult calving (Rajala and Grohn, 1998). On the other hand, CE is not a phenomenon external to the cow, as is age of cow or herd. Calving ease is a trait influenced by both a calf direct genetic effect (the effect attributable 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). Obviously, fertility traits are also affected by environmental effects and genetic effects. These considerations complicate the analyses. Considering CE only as a fixed effect would ignore the fact that both CE and fertility are (partially) determined genetically and that they may be genetically correlated. On the other hand, considering only the genetic correlation would ignore the cause-and-effect relationships between CE and fertility traits. Gianola and Sorensen (2004) outlined a solution for this kind of biological system, describing the use of recursive multiple-trait models in a quantitative genetic context. These authors extended quantitative genetic theory to suit situations in which recursiveness or simultaneity exists between the phenotypes involved in a multivariate system, assuming an infinitesimal and additive model of inheritance.
A recursive multiple-trait model is a particular case of the more general "structural equation models" (Johnson and Wichern, 2002). Loosely speaking, these are models in which phenotypes interact among themselves; effects are observed with noise (and have to be inferred); and models produce different residual effects. According to Johnson and Wichern (2002, p. 524) "structural equation models are sets of linear equations used to specify phenomena in terms of their presumed cause and effect relationship variables. In their most general form, the models allow for variables that cannot be measured directly." A typical example is a multivariate model that includes covariates. It is possible that the covariates themselves are not directly observed, but they can be predicted (to some degree of accuracy) from one or several other measured variables. Moreover, covariates included in the model may be related in some way (say, correlated). The same happens with the quantities of interest (the "true" phenotypes), which may have to be predicted from others (the "observed" phenotypes). A typical example would be the prediction of feed efficiency from consumption and growth. The recursive multiple-trait model is a particular case in which phenotypes (i.e., different traits) interact among themselves and, moreover, in which trait 1 affects trait 2 but not the opposite. The case in which trait 2 also affects trait 1 is called simultaneity. An example is the relationship between milk yield and SCC (Wiggans and Shook, 1987). High SCC, which are usually associated with subclinical mastitis, may inhibit milk yield; milk yield in turn has a dilution effect on SCC.
The relationship between CE (corresponding to first calving) and fertility is a recursive one, because CE might affect fertility in the following reproductive cycle, but obviously fertility cannot affect CE, which preceded it. The model for fertility traits, as we show later, includes a term for CE. The recursive multiple-trait methodology can handle all the information properly and provide estimates for genetic parameters and estimates of the effect of CE on fertility traits.
Another problem related to fertility traits is the existence of censored and missing data. Censoring in fertility (Schneider et al., 2005) comes from the fact that cows left the study (milk recording) with some fertility information, before reaching the next parturition, because of culling, death, or sale. As a consequence, we ignore the outcome of the reproductive cycle. This might cause bias if culling is due to poor fertility. If we plainly discard censored data, and dystocia causes a dramatic loss of fertility leading to culling, its effect will be underestimated. This information can be analyzed by a survival analysis (Schneider et al., 2005) or by a data augmentation technique (Donoghue et al., 2004).
The objective of this work was to study the phenotypic and genetic relationships between CE and fertility traits by using recursive multiple-trait models (Gianola and Sorensen, 2004) and handling censored and categorical data. Estimates of the effect of the different levels of CE on fertility measures, and estimates of the genetic relationships between fertility and CE are shown.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Female Fertility.
Female fertility is a trait that can be defined in many different ways. For example, it can be described as the ability to produce a live offspring during an economically and physiologically approved period (Hyppänen and Juga, 1998). De Jong (1997) defined good cow fertility as an animal in lactation that shows her heat in time and gets pregnant after the first insemination. Both definitions show the subjectivity of this trait.
Fertility measures chosen for this study were DO (measured in days), days to first service (DFS, measured in days), number of inseminations per conception (NINS), and outcome of first insemination (OFI). Outcome of first insemination was coded as 1 when females calved after the first insemination, whereas cows that failed to calve were coded as 0. Calving interval was not used because of its similarity to DO.
Data Characteristics
In the Basque Country Autonomous Community (Spain), CE data collection has been ongoing since 1992. Data are collected monthly at the same time as milk recording (officially recognized Basque milk recording scheme). The recording technician asks the farmer how the births have occurred since the last visit. The technician then scores each calving with the criterion described in Table 1
. Calving ease information was matched with the reproductive data collected in the AI recording system, which includes dates of all services, and also with data from the milk recording system. Calving records between 1995 and 2002 and data from the first 8 lactations were considered in the analysis. Moreover, every cow was required to have first lactation records to avoid culling bias and because dystocia mainly affects heifers. Calving ease records were edited as described in Alday and Ugarte (1997). In addition, multiple parities and parities showing a CE score of 5 were discarded; these performances are considered to lack a CE genetic component, because they are originated by pathologies such as abnormal presentation of the fetus and have little relationship to causes (pelvic opening, calf size) of the other CE score.
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The distribution of CE scores is summarized separately by parity in Table 2. The editing procedure that required cows to have the first parity record could introduce some bias in the analysis attributable to selection, but the frequencies of CE scores were very similar to those of the general population of the Basque Country. As given in Table 2, dystocia has a higher incidence in primiparous cows than in later parities, as was expected (Ducrocq, 2000; Steinbock et al., 2003). Because scores in the fourth category were very infrequent, they were combined with the third category to simplify the analysis and to minimize the problems with extreme categories (Moreno et al., 1997).
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. Features of the fertility data are described in Tables 4 and 5.
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Data Analysis
Recursive Multiple-Trait Model.
Recursive multiple-trait models were described in the Introduction. Gianola and Sorensen (2004) described the implications of these models and methods used in quantitative genetics. They proposed a Bayesian strategy for implementation through Markov chain Monte Carlo methods. However, the method is neither straightforward nor obvious to implement in a general manner. The Appendix shows that only recursive models, but not simultaneous models, can be implemented using software usually applied by geneticists to address genetic analyses by fitting (in this case) CE as another fixed effect.
Censored Data.
Censored data for DO and NINS were handled by data augmentation (Tanner and Wong, 1987; Sorensen and Gianola, 2002). The methodology applied to this specific problem (right censoring) was explained by Korsgaard et al. (2003) and consists in generating, within a Gibbs sampling scheme, samples from the predictive distributions of the censored records. The predictive distributions are truncated normal distributions, whose truncation point was the observed phenotype (largest DO or NINS corresponding to the observation period), the mean was the sum of (fixed and random) effects affecting that record, and the variance was the residual variance. This methodology has already been used by Donoghue et al. (2004) in the analysis of reproductive data.
Categorical Data.
Outcome at first insemination and CE were analyzed by a threshold model. These traits have 2 and 3 ordered categories, respectively. The threshold model assumes that the observed category is determined by the value of an underlying continuous random variable, called liability (Wright, 1934). A data augmentation technique was used within a Gibbs sampling scheme, as described in detail by Albert and Chib (1993), Sorensen et al. (1995), and Sorensen and Gianola (2002). The first threshold for both traits was arbitrarily set to 0. The second threshold for CE was set to 1. For reasons of identifiability, the residual variance for OFI was set to 1, because the threshold and the residual variance cannot be estimated simultaneously for binary traits. Thus, we used the algorithm of Korsgaard et al. (2003) to sample residual variances with some diagonal elements equal to 1. The number of inseminations, although of a categorical nature, was considered as a continuous trait because it had a high number of categories (5) that were distributed following a symmetric, centered distribution considered to be normal. Continuous and threshold univariate analyses were carried out to study this trait, and similar results were obtained (E. López de Maturana, unpublished data). Therefore, for simplicity we chose to consider NINS as a continuous trait.
Models.
A sire model was used for fertility measures and a sire-maternal grandsire model was used for CE. Animal models including maternal effects were not adopted because reaching convergence of the Gibbs sampler for CE was difficult. This result was reported previously by Luo et al. (2001) and extensively described for a threshold animal model with maternal effects. Linear models were applied to DO, DFS, and NINS. As discussed, threshold models were adopted for OFI and CE.
Four bivariate analyses were carried out: 1) DO-CE, 2) DFS-CE, 3) NINS-CE, and 4) OFI-CE because of the computational requirements. Indeed, a multivariate analysis considering all traits was not feasible because the correlation between NINS and OFI tends to be 1: When NINS is equal to 1, OFI is also equal to 1 (both traits give the same information).
In matrix notation, the models were
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where UF is a vector of fertility traits (liabilities for OFI), UCE is a vector of liabilities for CE, and bF and bCE are vectors of systematic effects. In all fertility models, bF includes the "fixed" effect of CE, following the analysis of recursive models shown in the Appendix 1 (Gianola and Sorensen, 2004).
For DO and NINS, systematic effects included in bF are the contemporary group (HYS, with 4,600 levels for DO and NINS), the month of successful service (12, plus 1 level for unknown month) and parity-age (9 levels). For DFS, systematic effects are the same as for DO and NINS, except that month of successful service is not included; in that case, the number of levels for HYS is also 4,600. For OFI, bF includes HYS (4,600 levels), parity-age, the month of first service (12 plus 1 level for unknown month), and the technician who inseminates at first service (127 levels).
For CE, bCE contains the effects included in the routine genetic evaluation: contemporary group (interaction between herd, year of calving, and technician who collects CE data, with 3,501 levels), the month of calving (12 levels), the interaction between number of parity and sex of calf (4 levels), and the breed of service sire (2 levels, Holstein and no Holstein). Table 6 gives the effects included in each model.
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I, where I is the identity matrix and H0 is the 4 x 4 permanent environmental covariance matrix for fertility traits. Because no multivariate analysis was carried out for fertility traits, permanent environmental covariances between fertility traits were not estimated. A permanent environmental effect was not considered for OFI, because it leads to problems in the behavior of the Markov chains (see Figures 1 and 2). The Markov chain did not converge to the posterior marginal distribution (Figure 1). We tried several models, but none of them converged. The complexity of the bivariate analysis between 2 threshold traits (CE and OFI) might make the cow permanent effect difficult to estimate.
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A, where G0 is
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and A is the numerator relationship matrix among sires.
However, because bivariate analyses were run, the following matrix was estimated in each analysis:
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The term 
si2 is the genetic sire variance of the trait I, and term mgsCE 2 corresponds to genetic variance of the maternal grandsire effect for CE. Covariances sFmgsCE, sCEmgsCE, and sFsCE are the genetic covariances between sire effect for fertility traits and maternal grandsire effect for CE, between sire effect and the maternal grandsire effect for CE, and between sires effects for fertility traits and CE, respectively.
eF and eCE are the vectors of residual terms for the 2 traits. They are jointly distributed following a multivariate normal distribution of mean 0 and covariance R0, where
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ei2 corresponds to residual variance for trait i, whereas eF,CE corresponds to residual covariance between CE and fertility traits. XF, XCE, WF, ZsCE, ZsF, ZmgsCE are known incidence matrices with the appropriate dimensions for each trait. Fixed effects are those included in bF and bCE; random effects are pF, usF, umgsCE, and the residual terms, represented by eF and eCE.
Implementation.
Estimation was carried out by Bayesian inference using Gibbs sampling. The reader is referred to the previous paragraphs for details related to censored data (for some records of NINS and DO), threshold models (for CE and OFI), and recursive models (considering that CE affects fertility traits). The rest of the Gibbs sampling scheme was similar to a multiple-trait, maternal-effects analysis (see, for example, Sorensen and Gianola, 2002). The missing values were also handled by data augmentation, as described by Sorensen and Gianola (2002), by generating a sample of the missing phenotype as the sum of effects, plus a random residual. The prior distributions for the fixed effects and (co)variance parameters were uniform, bounded between 106 and – 106. The full posterior conditional distributions for the location parameters (fixed and random effects) were univariate normal. The full posterior distributions for variance components were inverted Wishart distributions.
In each analysis, Gibbs sampling was carried out through a unique chain of 300,000 iterations, discarding the first 50,000 samples and retaining one every 50 samples. Thus, 5,000 samples were used to compute posterior means and standard deviations. The length of burn-in period was assessed by visual examination of trace plots of the estimates of genetic parameters. Features of the marginal posterior distributions were obtained using the Bayesian Output Analysis package (available at http://www.public-health.uiowa.edu/boa).
Simpler Analyses.
For the sake of comparison, simpler analyses for fertility traits were carried out to estimate the effect of CE. These analyses included CE as a fixed effect, but they did not consider CE as a trait. Therefore, the existence of genetic correlations between CE on fertility traits was ignored.
Posterior Analysis of the Effect of CE on Fertility Traits.
Estimates and confidence intervals of the difference between estimates of the effect of CE = 3 (dystocia) and CE = 2 (moderate difficulty) with respect to the effect of CE = 1 (normal parturition) were obtained from the marginal posterior distributions of these effects. The significance level was calculated as the percentage of samples higher than 0.
The effect of CE on OFI, as obtained in the Gibbs sampling, was estimated on the underlying scale and its meaning is not obvious. We transformed each sample in the underlying scale to the visible scale (i.e., percentage of success). This method involves transforming the estimates on the underlying scale to the probability scale. The method is similar to the one used for presentation of PTA for CE (Van Tassell et al., 2003) and can be summarized as follows:
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where effect (CE = 2) is the effect estimated for CE = 2 on the underlying scale, 
(x, 
) is the cumulative normal distribution of mean x and residual variance , and µ is the average of the population. As mentioned, the residual variance is equal to 1. The value on the underlying scale corresponding to the mean on the observable scale is 0.0258. This value was obtained from tables of the normal distribution considering the population mean of OFI reported in Table 4
. From the samples transformed to the observed scale, effects and P-values were computed as described previously.
Estimation of Genetic Correlations
Because of the complexity of the models, Figure 3 displays the relationships between all genetic effects of fertility measures and CE. Because sire and sire-maternal grandsire models were used, the estimates of the additive genetic covariances among sire effects were transformed to an animal model using the following formulas (Kriese et al., 1991). Genetic covariances between direct effects of the ith fertility trait and CE were obtained from
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where DFiDCE corresponds to the covariance between direct effects of the ith fertility trait and CE and SFiSCE is the covariance between sire effects of the ith fertility trait and CE.
Thus, genetic correlations between direct effects of the ith fertility trait and CE were obtained from
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where
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and
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where SFi2 and SCE2 correspond to sire variances of the ith fertility trait and CE, respectively. DFi2 and DCE2 correspond to direct genetic variances of the traits.
Estimates of maternal genetic variance were calculated following
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and
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Genetic correlations between direct genetic effects of fertility traits and the maternal effect of CE were generated from these expressions:
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where DFiMCE corresponds to covariance between the direct genetic effect of the ith fertility trait and maternal effect of CE. sFimgsCE corresponds to covariance between sire effect of the ith fertility trait and maternal grandsire effect of CE, obtained from the analyses.
Thus, the genetic correlations of direct effects of fertility traits with the maternal effect of CE were calculated as
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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has the features of posterior distributions of CE scores on fertility traits obtained using recursive methodology, with estimates of first score set to 0. Because calvings with a CE of 3 (dystocia) cause the most problems for farmers, only estimates of the effect of this level on fertility measures is discussed here. Analyses of the features of posterior distributions (means, SD) of CE scores on fertility traits indicated statistical evidence proving that the effect of CE = 3 was significantly different from the effect of CE = 1. The level of significance was different in each case.
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DO.
The estimate of the effect of score-3 calving was statistically significant (P < 0.05), implying a delay in the pregnancy of the cow by 31 d. This result is consistent with those provided by Dematawewa and Berger (1997), who reported an increase of DO of 34 d, although they are not directly comparable because of the differences in data and applied models.
NINS.
The effect of dystocia was statistically significant (P < 0.05). This fact indicates that 0.5 more inseminations are needed to impregnate the cow in the next reproductive cycle after a difficult calving. These results are similar to those provided by Dematawewa and Berger (1997), who reported that 0.22 more services were necessary after a difficult calving.
OFI.
The effect of dystocia was statistically significant (P < 0.01). However, as explained, the value of the estimates for the effect of CE on OFI does not make biological sense, because estimates are expressed on the underlying (unobservable) scale. Therefore, we transformed the differences back to the observed scale. Thus, parities scored 3 for calving difficulty, in comparison with score-1 parities, impaired fertility, because they decreased the incidence of success at first insemination by 12% (P < 0.01).
As for the simpler analysis, Table 8 shows the estimates of CE scores on fertility traits obtained without considering the recursive methodology, that is, by analyzing only fertility traits without considering that CE has a genetic background and may have a genetic correlation with the fertility traits. In general, the results indicated that the effects of dystocia on fertility measures were underestimated (except for OFI, because similar estimates have been obtained from both analyses), compared with the results obtained by applying the recursive methodology. The recursive model gives a better description of the biological relationships, although the estimation might be difficult and a good data set might be required to estimate all the unknowns in the model.
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provides features of the genetic parameters after applying the transformations explained in the Materials and Methods section. As has been observed in previous studies (e.g., Dematawewa and Berger, 1998), posterior means of heritabilities for DFS, DO, and NINS were low. Heritability for OFI (0.12) was larger than that found by González-Recio and Alenda (2005) using data originating from the same population. It is necessary to point out that the application of the recursive methodology in this study might have affected the results. Posterior means of heritabilities for DCE and maternal CE were low (0.07 and 0.05), although they are within the range of values reported in other studies (Djemali et al., 1987; Van Tassell et al., 2003). The estimated genetic correlation between direct and maternal components for CE was negative (– 0.31 ± 0.28). This value is similar to that used in the genetic evaluation for CE (– 0.41; E. López de Maturana, unpublished data). A negative association has also been reported by other authors (Djemali et al., 1987; Van Tassell et al., 2003) and reflects the genetic antagonism between direct and maternal effects.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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All these results may be used to help quantify the cost of dystocia in production systems. Moreover, this study shows that the effect of CE might be included in fertility models.
The estimate of a genetic correlation between direct effects of DO and CE was the only correlation significantly different from 0. This result indicates that genes of the calf related to difficult births are moderately genetically correlated with reduced reproductive success of the cow.
The use of the recursive model methodology allowed us to distinguish between the effect of CE on fertility and the genetic relationship between CE and fertility traits. Estimates of effects and parameters are more realistic and give a better explanation of biological phenomena. Carrying out the analysis is not difficult, but a great number of parameters must be estimated.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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where bi, ui, and ei are fixed, random, and residual terms for the ith record, and
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is a matrix containing ones in the diagonal and regression coefficients 
i,j in the off-diagonals, with the sign changed. These coefficients describe how trait i is affected by trait j, that is, the regression coefficient of trait i on trait j. However, an equivalent model can be described as
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where W = – (
– I) is a matrix that contains 0 in the diagonals and the coefficients. This can be further transformed in
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where
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is a matrix of records for each trait, ordered conveniently. The last representation can be regarded as a model in which traits are "fixed effects" for other traits, and bw is a vector of solutions for those "fixed effects."
This form of presenting the simultaneous model is equivalent to that presented by Gianola and Sorensen (2004; equation [68]), although perhaps easier to recognize. However, the problem in solving for bw or, equivalently, , is that, for statistical analysis, the determinant of – 1 is involved in the likelihood function (equation [37] in Gianola and Sorensen, 2004). Intuitively, the reason is that if trait 1 influences trait 2 and the opposite, certain conditions have to be fulfilled for the system to be possible. For a Gibbs sampling scheme, there is no closed form for the fully conditional posterior distribution; Gianola and Sorensen (2004) propose the use of a Metropolis-Hastings algorithm. However, as stated by Gianola and Sorensen (2004), for recursive models (where trait 2 does not influence trait 1), will be a triangular matrix with the diagonal elements equal to 1, for example,
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whose determinant is always 1. Therefore, its elements can be obtained as the solutions of bw in the system, which is implemented like any other fixed effect. This is the strategy we used in this work. However, it cannot be applied for simultaneous models, where there are pairs of traits in which trait 1 affects trait 2 and trait 2 affects trait 1.
Example
This example is inspired by Veerkamp and Thompson (1999). Suppose a cow’s record, for a given test day, is 30 kg for milk yield and 500 kg for live weight. A typical model to estimate the genetic correlation between milk yield and live weight would be (model for milk yield):
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(model for live weight)
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in matrix form,
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where m, u, and e are general population means, additive genetic effects of the cow, and residuals for that record, respectively. The terms u and e follow bivariate normal distributions.
Recursive Model.
If we consider that live weight influences milk yield (a recursive model), and following Gianola and Sorensen (2004), the models are in matrix algebraic form:
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where
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The term 2,1 is set to zero because it is assumed that milk yield does not influence live weight. As discussed before, this is equivalent to:
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or to
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These models are equivalent to (model for milk yield):
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(model for live weight):
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This is straightforward to fit into any common genetic parameter estimation software (e.g., VCE or MTDFREML) just by including live weight as a covariate for milk yield and analyzing both traits simultaneously. As explained before, inferences are correct because likelihood functions for both representations ([1] and [2]) of the model are equivalent.
Simultaneous Model.
If we consider that milk yield also influences live weight, an appropriate model would be
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where
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We may also write this model as:
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However, the likelihood function of model [4] is not the same as that of model [3] (and therefore not correct). The reason is that the determinant of – 1, which is involved in the likelihood function for the first model, is ignored in the second. An example would be an estimate of
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which could be legitimate under the second model. This estimate is not admissible in the first model because its likelihood is null (the determinant of – 1 is zero).
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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Received for publication July 4, 2005. Accepted for publication November 28, 2006.
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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OFIDIR-CEDIR), considering a permanent effect in the model for the outcome of first insemination.
