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ANIMAL GENETICS |
,3
* Grup de Recerca en Remugants,
and
Departament de Ciència Animal i dels Aliments, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
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Abstract |
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 Top Abstract INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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y,
) as reference parameters. The bivariate model was preferable for both variables, minimizing mean square error and maximizing y,. A threshold approximation for litter size was preferable over a linear approximation. Models were also compared with a simulation study, comparing the correlation coefficient between simulated and predicted breeding values (a,â). The bivariate threshold model was favored, with a y, of 0.677 and 0.834 for litter size and days to lambing, respectively. Correlation coefficients between simulated and predicted breeding values in the bivariate linear model were reduced slightly to 0.651 and 0.831, respectively, and they were lowest with linear univariate models (0.642 and 0.802). Although the bivariate models for ewe litter size and days to lambing were more accurate than the univariate models, the threshold approaches showed a greater advantage under the bivariate model. For the purpose of genetic evaluation of litter size in sheep, use of the threshold-linear model seems justified. In the Ripollesa breed, the evaluation of litter size can benefit from recording birth weight.
Key Words: days to lambing • goodness-of-fit • litter size • predictive ability • Ripollesa breed
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INTRODUCTION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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). This has been confirmed with simulated data (Meijering and Gianola, 1985; Varona et al., 1999), but the results from field data have generated an important controversy. Threshold models have been reported as better (Varona et al., 1999), worse (Hagger and Hofer, 1989), and similar to linear approaches (Meijering, 1985; Olesen et al., 1994). Moreover, Varona et al. (1999) suggested that joint analysis of a threshold and a linear trait could increase the accuracy of prediction for the discrete trait due to the stabilizing effect of the linear trait. Unfortunately, this hypothesis has not been tested on discrete ewe reproductive traits such as litter size, which is one of the most important traits for characterizing ewe profitability.
Ewe fertility characterizes the ability of the ewe to successfully mate and lamb and is usually treated as a dichotomous trait, delivery or not delivery, after the mating season. However, the low or null heritabilities described for this trait (Fogarty, 1995) have greatly limited its inclusion in selection programs. As an alternative, Gabiña (1989a,b) expressed fertility as days to lambing, the time elapsed between the moment when the ram is placed with the ewes and the subsequent parturition, this trait being a continuous variable.
The purpose of this study was to compare bivariate Bayesian animal models for litter size and days to lambing records in the Ripollesa sheep breed, assuming linear and threshold approximations for litter size, using the methodologies described by Van Tassell and Van Vleck (1996) and Van Tassell et al. (1998), respectively. Moreover, univariate approximations were also tested. Models were compared in terms of goodness-of-fit and predictability of future records, as well as the correlation of predicted and true breeding values using simulated data.
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MATERIALS AND METHODS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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Field Data
Data were collected from a flock of Ripollesa ewes kept at the experimental farm of the Universitat Autònoma of Barcelona since 1986. The Ripollesa breed is a meat-purpose, medium-sized Catalonian autochthonous sheep breed from the Spanish entrefino group adapted to semiextensive farming conditions in the Mediterranean area (Ministerio de Agricultura Pesca y Alimentación, 1980; Guillaumet and Caja, 2001). The flock analyzed was founded from the acquisition of ewes and rams in 1986, 1990, and 1993, from 3 purebred Ripollesa farms (Torre Marimon, Caldes de Montbui, Barcelona, Spain; Torrebonica, Terrassa, Barcelona, Spain; and SEMEGA, Monells, Girona, Spain). After 1993, with the exception of 2 rams bought in 1997, the flock was closed, with all subsequent ewe and ram replacements generated within the flock, following phenotypic selection for litter size. Flock size varied from 80 to 120 ewes from 1986 to the present.
Adult ewes were managed according to a fall-lambing system (Christmas harvesting), with natural mating and registered paternity, whereas replacement ewe lambs were mated in summer and lambed in December to February. As an exception, from 1995 to 1998, AI with fresh semen was used after estrous synchronization (Chronogest, Intervet, Salamanca, Spain) and 400 IU of PMSG. Natural mating began 10 to 15 d after AI to avoid multiple paternity compatibilities. Rams were housed with the ewes after grazing for approximately 90 d.
All female and most (91.9%) male ancestors were known for animals born within the analyzed flock. The pedigree file consisted of 376 ewes with phenotypic records, 19 known female ancestors, and 20 rams with an average of 12.5 daughters per sire.
Traits Analyzed
Two ewe reproductive traits were considered, litter size (LS) and days to lambing (DL). Litter size was defined as the number of lambs at birth (sum of alive and dead), whereas DL took the value of the number of days between the date on which the rams were placed with the flock and the subsequent parturition of the ewe. After editing, the data set contained 1,598 LS records (single, 54.1%; twins, 44.1%; triplets, 1.8%; and quadruplets, 0.1%) and 1,699 DL records registered from 1986 to 2005, with 257 (15.1%) of them censored at 240 d, corresponding to ewes that failed to lamb. The numbers of observations are summarized in Table 1.
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and Matos et al. (1997b); 2) year of lambing, with 19 levels from 1986 to 2004 (Table 1
); and 3) the random permanent environmental effect of the ewe. Primiparous ewes that lambed in January and February were assigned to the preceding year. Given that the flock was founded with animals from 3 flocks without known genetic relations among them, 3 genetic groups were defined, and founders from the same flock were assigned to the same genetic group. The numerator relationship matrix was constructed through procedures described by Westell et al. (1988).
Bivariate Linear-Linear Model.
Both LS and DL were modeled as continuous traits and assumed to be sampled from the following multivariate normal distribution:
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where yLS = the vector of LS records; yDL = the data vector of DL; and X, Z1, and Z2 = the corresponding incidence matrices of systematic (b' = [b'LS b'DL), permanent environmental (p' = [p'LS p'DL]), and additive genetic (a' = [a'LS a'DL]) effects, respectively. Note that R = the residual (co)variance matrix, with dimensions of 2 x 2. Censored DL data were augmented following a data augmentation procedure (Tanner and Wong, 1987), according to the methodology described by Guo et al. (2001). Missing records were also augmented as described by Tanner and Wong (1987). In a Bayesian setting, we assumed:
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where G = the additive genetic (co)variance matrix, and A = the numerator relationship matrix between individuals. In a similar way, the permanent environmental effects were assumed to follow a bivariate normal distribution:
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where P = the permanent environmental (co)variance matrix. Flat priors were assumed for variance components and the systematic effects (b).
Bivariate Threshold-Linear Model.
Litter size was modeled as a threshold trait (Wright, 1934), as described by Sorensen et al. (1995) and Van Tassell et al. (1998). We assumed that LS was related to an unobservable continuous variable (liability), uLS being the liability vector. Thus, the joint distribution of LS liabilities and DL records was assumed to be conditionally bivariate normal:
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but the response of LS (yLS) was modeled with the following distribution:
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where t = the threshold that defined the 2 categories of response.
A priori distributions for the remaining parameters were assumed, as for the linear-linear approach, with the exception of the threshold for LS liability and the residual LS variance for LS liability, which were fitted to 0 and 1, respectively (Sorensen et al., 1995). Note that the Gibbs iterations required a data augmentation step that allowed us to include liability as an unknown parameter in the model (Tanner and Wong, 1987), generating random samples of uLS, as described by Van Tassell et al. (1998).
Univariate Linear and Threshold Models.
Litter size was analyzed separately, with linear (Gianola and Sorensen, 2002) and threshold Bayesian (Sorensen et al., 1995) methodology to compare with results from the bivariate models. Days to lambing was also analyzed under a univariate linear model. Priors were defined as for the multivariate analysis, although within the univariate framework.
Gibbs Sampler.
In this study, all models were solved through the Gibbs sampling technique (Gelfand and Smith, 1990) to obtain autocorrelated samples from the joint posterior density and subsequently from the marginal posterior densities of all of the unknowns in the model. A unique Gibbs sampler chain was launched for each model with a length of 500,000 points, and the first 50,000 were discarded as burn-in. The effective length of the burn-in period and the chain size were calculated following the methods of Raftery and Lewis (1992) and Geyer (1992), respectively (Table 2).
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y,) between phenotypic and predicted records. For the traits analyzed with the linear-linear methodology, the expectation of the predictive distribution of a given record, i, was computed as (Varona et al., 1999):
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where xi, z1i, and z2i = the ith rows of the incidence matrices that link systematic, permanent environmental, and additive genetic effects; 
,ß = the elements of R–1 for the multivariate model; and êLSi and êDLi = the residuals for the ith LS and DL records, respectively. Note that 
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, â , ê, and ,ß = posterior mode estimates. The MSE was defined as:
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Censored DL records were not used to compute MSE and y,.
Within the threshold-linear approach, the expectation of liability was computed as described by Varona et al. (1999):
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and the MSE for LS took the form (Varona et al., 1999):
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where 
(.) = the cumulative normal distribution with argument, as described within the parentheses.
Predictive Ability.
Prediction of future observations given past data is a question of concern to animal breeders that can be answered using the concept of predictive density, a notion that arises naturally in Bayesian statistics (Matos et al., 1997a). To estimate predictive ability, 2 new data sets were created, with 50% of the LS or DL records removed from each, and both were analyzed with univariate and bivariate models. All censored DL records were removed. For goodness-of-fit, MSE and y, were computed between expectations from the predictive distribution and the removed records.
Validation Study.
Twenty datasets with 1,598 LS records and 1,442 DL records were simulated for the threshold-linear model with parameters obtained from the Ripollesa data analysis under the threshold-linear model. The pedigree and incidence of systematic, permanent, and additive genetic effects were the same as for the Ripollesa data set, as was the incidence of censored records in DL. Each population was analyzed with the previously defined models using the same strategy applied to the real data set. Correlations between simulated and predicted breeding values were estimated.
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RESULTS AND DISCUSSION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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) and clearly lower than the 1.71 reported by Al-Shorepy and Notter (1996) in a composite population (50% Dorset x 25% Rambouillet x 25% Finnsheep) and the 2.2 observed by Hansen and Shrestha (2002) in Canadian meat-purpose breeds.
Days to lambing averaged 165.7 d, although values varied greatly among years (Table 1). Moreover, this average underestimated the true value, given that right-censored records were present with an incidence of 15.1%. A previous analysis of DL in the Spanish Rasa Aragonesa breed for fall deliveries (Gabiña, 1989a) resulted in an average of 164.9 d, although values oscillated between 157.7 and 169.3 d, depending on the flock and breeding season. The distribution of phenotypic DL records (Figure 1) was also comparable to that of Gabiña (1989a). Although Figure 1 shows a moderately skewed distribution, adequacy of linear models to analyze this trait was shown by Donoghue et al. (2004a, b).
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. Regarding the LS trait, the joint threshold-linear model performed better than the bivariate linear model, because MSE and y, showed differences greater than 4% (MSE = 0.213 vs. 0.222; y, = 0.666 vs. 0.640), and both models were clearly preferable to univariate models (Table 3). The greatest improvement in the MSE of LS was achieved when bivariate models were compared with single-trait models, 0.222 vs. 0.239 (difference, 7.7%) for linear approaches and 0.213 vs. 0.237 for threshold approaches (difference, 11.3%). These are the first available results comparing goodness-of-fit of threshold and linear analyses of ewe LS under a bivariate model with an additional linear trait. Although there are no comparable results using bivariate models, a slight advantage of threshold approaches has also been suggested for univariate methodology for LS (Olesen et al., 1994; Matos et al., 1997a). The univariate threshold analysis of the Ripollesa LS also reached a better goodness-of-fit, as measured by MSE, than the univariate linear model (0.239 vs. 0.240; y, = 0.626 vs. 0.615), although differences were minimal and probably irrelevant. The same tendency was observed by Olesen et al. (1994) and Matos et al. (1997a), who reported null or minimal differences in terms of MSE, whereas the estimates of y, were slightly higher for the threshold approach. The stabilizing effect provided by the correlated linear trait could explain why the threshold model has shown greater improvements in bivariate than in univariate models, as was pointed out by Varona et al. (1999).
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y, = 0.663), whereas the linear-linear approach increased MSE (198.4) and reduced y, (0.654), with differences of 0.97 and 1.38%, respectively. Discrepancies between the bivariate linear-linear and univariate linear models were minimal, lower than 0.31% (Table 3). Correlation coefficients for LS and DL were clearly similar, showing that DL can be effectively analyzed using Gaussian linear models. These results highlight the importance of adequate modeling of discrete traits, provided that it improves the goodness-of-fit of the remaining traits in multivariate models. As a whole, joint analysis of DL and LS improved the goodness-of-fit of the model for both traits, reaching its maximum when LS was treated as a threshold trait.
Predictive Ability
The MSE and y, for the different models used to predict the removed LS and DL records in the Ripollesa data set are shown in Table 4. Estimated MSE were greater, and y, was smaller than for the goodness-of-fit analyses, which is not surprising, because the analyses were performed with fewer records. The MSE for LS were close to the values reported by Matos et al. (1997a) in the Rambouillet breed and smaller than the estimates obtained by Olesen et al. (1994) in the Norwegian sheep breeds and by Matos et al. (1997a) in Finsheep ewes, although they considered more than 2 phenotypic categories for LS. Correlation coefficients varied from 0.47 to 0.54 for LS and from 0.53 to 0.55 for DL (Table 4), which are similar to those reported by Matos et al. (1997a) and clearly greater than those published by Olesen et al. (1994). It is important to note that similar predictive ability was found for both traits in terms of y,, probably because the heritability estimates were similar (results not shown), as observed by Matos et al. (1997a).
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y, than the linear-linear model for LS (0.228 vs. 0.236; y, = 0.539 vs. 0.513) and for DL (203.3 vs. 205.3; y, = 0.545 vs. 0.538). Bivariate models were preferable to univariate models (Table 4), as shown by the results for goodness-of-fit. The advantage of the threshold-linear model also has been shown for bivariate analysis of calving ease and birth weight in American Gelbvieh by Varona et al. (1999), who suggested that information provided by continuous traits could substantially improve predictive ability for threshold traits. For LS, differences between univariate approaches were smaller than for bivariate models in terms of MSE (0.42 vs. 3.51%), similar to that described by Varona et al. (1999), but were greater for y, (6.14 vs. 5.06%). In general, the results contrast with published works on LS that did not observe substantial differences in terms of predictive ability (Matos et al., 1997a) or were minimally favorable to the threshold approach (Olesen et al., 1994).
Simulation Results
Table 5 presents empirical correlations between simulated and predicted breeding values for both traits. Correlations were greater with the threshold-linear model for LS (0.68) and DL (0.83), which is not surprising, because that model was used to simulate the data. Differences in correlations between simulated and predicted data for direct genetic effects for DL were similar for both bivariate models, whereas the univariate approach showed a reduction in mean correlations (0.83 vs. 0.80). These results suggest that information provided by the categorical trait of LS to prediction of DL was not negligible, in contrast to results obtained by Varona et al. (1999) with birth weight and calving ease of American Gelbvieh. For LS, the correlation coefficients for threshold models were 1.5% greater for univariate models and 4.0% greater for bivariate models than for linear models. These greater correlations are similar to those reported by Varona et al. (1999), but smaller than differences described by Meijering and Gianola (1985) and Hoeschele (1988) with different incidence rates and heritabilities. Similar differences were found between univariate and bivariate models, which were clearly less than values observed by Janss and Foulley (1993) and Varona et al. (1999). In summary, the bivariate threshold-linear model proved to be somewhat preferable for both discrete and continuous traits.
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IMPLICATIONS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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Footnotes |
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3 Current address: Grup de Recerca en Nutrició, Maneig i Benestar Animal, Departament de Ciència Animal i dels Aliments, Universitat Autònoma de Barcelona, Bellaterra, Spain.
2 Corresponding author: joaquim.casellas{at}uab.es
Received for publication June 7, 2006. Accepted for publication September 29, 2006.
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LITERATURE CITED |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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