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* Department of Animal Science, Agricultural University of Norway, P.O. Box 5025, N-1432 Ås, Norway
Department of Animal Breeding and Genetics, Danish Institute of Agricultural Sciences, Research Centre Foulum, DK-8830 Tjele, Denmark
Corresponding author: Jørgen Ødegård; e-mail: jorgen.odegard{at}ihf.nlh.no.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Key Words: dairy cattle • model comparison • random regression model • somatic cell score
Abbreviation key: 2lnRL = 2 ln restricted likelihood, AIC = Akaike information criterion, AS = Ali-Schaeffer curve, BIC = Bayesian information criterion, CM = clinical mastitis, L = Legendre polynomial, LM = Lidauer-Mäntysaari curve, LSCS = lactation average somatic cell score, MSEP = mean squared error of predictions, NRF = Norwegian Cattle, PE = permanent environmental, REP = repeatability model, RR = random regression, RRM = random regression model, W = Wilmink curve
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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A genetic analysis based on LSCS does not utilize all information in the data, as it does not allow simultaneous estimation of stage of lactation effects. In addition, cows may have different numbers of records per lactation, which is not accounted for in most models for genetic evaluation based on LSCS. However, a test-day model has the potential of inducing more precise adjustment for temporary environmental effects (Jensen, 2001) and for stage of lactation. Test-day models are therefore expected to result in more accurate genetic evaluations than models based on LSCS.
Test-day SCS can be modeled as repeated measurements of the same trait, but this may be suboptimal because genetic correlations between SCS at different stages of lactation are less than unity (Reents et al., 1994; Mrode et al., 1998; Haile Mariam et al., 2001). Alternatively, SCS at all different DIM could be regarded as separate traits. This assumption may lead to a nonparsimonious model and require estimation of (co)variance components for all DIM, which is computationally difficult. Random regression models (RRM) are a compromise between a simple repeatability (REP) model and a model regarding all SCS at different DIM as separate traits. Such models may be well suited for test-day analysis of longitudinal traits because the random regression (RR) coefficients induce a covariance structure along a given trajectory (Van der Werf et al., 1998). A suitable specification of a RRM may lead to a parsimonious account of the (co)variance structure across DIM.
The objectives of this study were to evaluate different test-day models, such as a REP model and RRM using orthogonal polynomials and standard linear lactation functions for modelling sire and permanent environmental (PE) effects, and to estimate genetic parameters for test-day SCS in Norwegian Cattle (NRF).
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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To reduce computational burden, data were restricted to herds from three counties (Akershus, Østfold, and Vestfold) in the central eastern part of Norway, resulting in a dataset of 341,736 test-day observations from 77,110 cows. Descriptive statistics of the dataset are presented in Table 1
. The pedigrees of the 1965 sires in the dataset were traced back as far as possible through sires and maternal grandsires, resulting in a pedigree with 2275 males.
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where:
| Yijklmpq | = | observation of SCS for daughter p of sire q, with age at calving i, calving month j, DIM l, herd-year class k, and herd test-day m;
| Ai | = | fixed effect of age i at first calving, in 15 classes; where <20 mo is the first class, >32 mo the last class, and the other classes are in single months;
| Mj | = | fixed effect of month j of first calving in 12 classes;
| HYk | = | fixed effect of herd-year class k;
| DIMl | = | fixed effect of DIM l in 300 classes where 6 d is the first class and, 305 d the last class;
| htdm | = | random effect of herd test-day m;
| Zln | = | polynomial n for DIM l, where n = {0, ..., r} for permanent environmental effects and n = {0, ..., t} for sire effects;
| rpn | = | random regression coefficient on Zln, for the permanent environmental effect p;
| rqn | = | random regression coefficient on Zln, for the transmitting ability of sire q; and
| eijklmpq | = | random residual.
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The following polynomials were used to model PE and sire effects:
An overview of the models is given in Table 2. All models included an intercept (Zl0 = 1) for both PE and sire effects (if included). For the sire models, the PE variance will contain three quarters of the additive genetic variance.
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, both PE and sire effects were fitted and evaluated, using the DMU-package (Madsen and Jensen, 2000) assuming homogeneous residual variance. The DMU-package did not allow for heterogeneous residual variances, but allowed running the most complex models with our data. In the third step, results were also obtained for the preferred model from the second step, assuming heterogeneous residual variance, using the ASREML-package. The REP model is a special case of the general RRM where only an intercept is included in the PE and sire effects (Zl0 = 1, and Zln = 0 for n > 0). The LM44 model (Lidauer and Mäntysaari, 1999) is a combination of Legendre polynomials and the curve described by Wilmink (1987). For this model, higher order of fit was not used due to convergence problems. For 7 of the RRM, the same functions were used to model both sire and PE effects, whereas 5 RRM used Legendre polynomials of lower order of fit for the sire effect than for the PE effect.
Sire variance 
, PE variance 
, and heritability 
at DIM l were calculated as:
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where:
| Zl | = | vector of polynomials in the model for DIM l;
| G | = | (co)variance matrix for sire RR coefficients;
| P | = | (co)variance matrix for PE RR coefficients; and
|  | = | residual variance.
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If the PE curve has a lower order of fit than the genetic curve, the larger flexibility of the genetic curve may capture changes in the PE curve (Pool and Meuwissen, 1999; Kettunen et al., 2000), overestimating heritability. To examine the opposite, to what degree an increased order of fit for the PE effect relative to the sire effect could affect (co)variance components, models with fourth-order Legendre polynomials for PE effect and reduced order of fit for sire effect were tested (L15, L25, L35, and L45).
Model Comparison
Models were compared in terms of the residual variance, 2 ln restricted likelihood (2lnRL), and mean square error of prediction (MSEP) for excluded observations. Values of 2lnRL for all models including sire effects were given relative to the base model (REP). Nested models (i.e., one can go from one model to another by zeroing out some parameter(s)) were compared using likelihood-ratio tests. As the likelihood tends to favor complex models with many parameters (Jensen, 2001), the more conservative Bayesian information criterion (BIC) (Schwarz, 1978) and Akaike information criterion (AIC) (Akaike, 1973) were also used, penalizing models with many parameters.
The likelihood-ratio test statistic for two models i and j, where i is nested within j, is given by:
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L(i) and L(j) are the restricted likelihoods of the models to be compared, and 
i andj are the corresponding number of parameters of those models. Related is BIC and AIC, defined as:
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where: N is the number of observations, p is the rank of fixed effects incidence matrix, and L(0) and 0 are the restricted likelihood and number of parameters of the base model (e.g., REP), respectively. For the likelihood-based criterions, models with the largest values have the best fit.
Models were also compared by their ability to predict randomly excluded observations. These observations were sampled from PE and herd test-day classes with at least 2 observations each, respectively, and from herd-year classes with at least 4 cows. Random sampling was carried out twice, generating datasets of 14,010 and 13,702 observations, respectively. For both datasets MSEP was calculated as:
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where: yi is an observation, 
i is the predicted value of the observation, and n is the total number of observations in the sample. Finally, a weighted mean of MSEP in the 2 datasets was derived. In the calculation, (co)variance components estimated on the basis of the entire dataset were used as the true parameters.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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. For the heteroscedastic models, residual variances were largest in beginning of lactation and lowest in late lactation, ranging from 0.28 to 0.50. Model comparison indicated that although the models assuming heterogenous residual variances were favored on basis of likelihood-based criteria, there were no differences in predictive ability (MSEP) for models assuming either heterogeneous or homogeneous residual variance. For both variance structures the most complex model (L05) was preferred over the simpler model (L04) based on 2lnRL, BIC, and AIC, while the models ranked approximately equal based on predictive ability. The relative difference in 2lnRL between L05 and L04 was larger when homogeneous residual variance was assumed. This may be explained by the PE effect "absorbing" some of the heterogeneity of the residual variances, thus favoring more complex modeling of PE effects for likelihood-based criteria. Absorption is also seen from Figure 1, depicting the PE variance and the residual variance by DIM for homogeneous and heterogeneous residual variance. For the models assuming homogeneous residual variance, the slope of the PE variance trajectory is obviously affected by change in residual variance throughout lactation, demonstrating the ability of the PE effects to absorb heterogeneity of the residuals. Hence, it seems appropriate to model test-day SCS with a homogeneous residual variance structure.
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. For all these RRM the sire variance was lower in early lactation (0.020 to 0.025) than later (DIM 305; 0.028 to 0.037). This is in agreement with results reported by Haile Mariam et al. (2001) and Mrode and Swanson (2003). Compared with REP, the RRM gave slightly smaller sire variance in the first weeks of lactation, and larger variance later, resulting in a larger average genetic variance for RRM than for REP.
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. With the exception of L22, the PE variances were highest in the beginning of the lactation (0.92 to 1.17), and with a wavy pattern for the first 2 mo. After 2 mo, the RRM with at least 4 polynomials for both sire and PE effects showed more stable PE variances (0.78 to 0.86 at DIM 305), whereas models with lower order of fit showed a wavier pattern. The PE variance was consistently higher for RRM than for REP. This increase, as well as the increase for sire variance (above), is probably due to improved model flexibility, allowing separation of effects that would otherwise be captured by the error term. As sire models were fitted, the PE effect is expected to contain three quarters of the additive genetic variance. This may affect the trajectory of the PE variance by DIM, but is not expected to affect the trajectory of heritability. Mrode and Swanson (2001) reported that the PE variance curve by DIM for SCS had similar shape for animal and sire models, but, as expected, that the PE variance was slightly higher for the sire model.
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. Heritability increased with stage of lactation from 0.05 to 0.07 for DIM 6 to 0.08 to 0.12 for DIM 305. This is in agreement with heritabilities for SCS reported by Haile Mariam et al. (2001) and Mrode and Swanson (2003).
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. As expected, for all models the highest correlations were found for consecutive DIM, while the lowest were between SCS at DIM early and late in lactation. Random regression models with Legendre polynomials showed consistently higher genetic correlations for the most distant DIM compared to the other models. For L22, L33, L44, and L55 the genetic correlations between DIM 6 and DIM 305 were 0.71, 0.67, 0.58, and 0.57, respectively, whereas W33, LM44, and AS44 showed lower genetic correlations (0.38, 0.50, and 0.46, respectively). Haile-Mariam et al. (2001) reported smaller genetic correlation (0.33) between SCS at the most distant DIM, using RR of second order for the sire effect.
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shows sire variance by DIM for the models REP, L15, L22, L25, L33, L35, L44, and L45. Sire variance trajectories for models L15 and L25 were systematically lower than the corresponding trajectories for the models REP and L22, respectively. However, sire variance trajectories were similar in the models L35 and L45 relative to L33 and L44, respectively. Hence, lower order of fit for the sire effect compared with the PE effect probably results in underestimation of the genetic variance, most pronounced when sire effect is fitted only using an intercept (zeroth order) or linear regression (first order).
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. Generally, the residual variances decreased and 2lnRL increased as the number of parameters in the model increased. Based on likelihood-ratio tests, BIC, and AIC, all RRM were improvements relative to REP. In addition, introducing RR coefficients to the models reduced residual variance and MSEP. The models with 3 RR coefficients for PE and sire effects were ranked over L22 for all criteria. Within these models, L33 had the smallest residual variance and MSEP, and ranked highest based on the likelihood-based criteria (2lnRL, BIC, and AIC). L33 was therefore preferred over the W33 model.
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Increasing the number of RR coefficients from 4 to 5 (from L44 to L55) resulted in reduced residual variance, a highly significant increase of the likelihood, as well as increased BIC and AIC. However, based on residual variance and MSEP, L55 and the models with 5 RR coefficients for PE effect and reduced order of fit for sire effect (L15, L25, L35, and L45) were ranked approximately equal. The latter models, except L15, ranked over L55 based on BIC, with L25 ranking highest of all models. However, as previously mentioned, the L25 model probably underestimates sire variance. Hence, sire effects should be modeled with at least second order of fit in models with higher order of fit for PE effects.
Based on MSEP, the models with 5 RR coefficients for PE (e.g., L55) were ranked approximately equal to some of the models with 4 RR coefficients (e.g., L44), indicating that higher order of fit did not substantially improve predictive ability, which could be explained by the low number of records per cow (Table 1). However, by predicting randomly excluded observations, number of observations per cow used for prediction was slightly reduced (4.19 vs. 4.43), which may have an effect on the relative difference in predictive ability. Contrary, the likelihood-based criteria indicated that the models with the most complex modeling of PE effects should be preferred. This is in accordance with the results from the comparison of heteroscedastic and homoscedastic models, indicating that, based on likelihood, more complex modeling of PE effects was preferred for models assuming homogeneous residual variance, while predictive abilities of the models were not affected by this assumption. Hence, as some of the models with 4 and 5 polynomials used to fit the PE effect showed approximately the same predictive ability, the simpler models were preferred.
The model comparisons lead forward to a model with second-order Legendre polynomials for sire effects and third-order Legendre polynomials for PE effects (L34), which are most easily tested against the L44 model (Table 4). A likelihood ratio test indicated a moderate significant (P < 0.1) reduction in likelihood going from L44 to L34. AIC, residual variance and MSEP for the 2 models were similar, while BIC ranked L34 over L44. Therefore, since L34 has fewer parameters, it should have preference over L44. Trajectories of sire and PE variances by DIM for L34 (Figure 7) were close to the trajectories presented for the models L35 (Figure 6) and L44 (Figure 3), respectively. The heritability of SCS based on L34 was 0.07 in the beginning of lactation, and 0.09 to 0.10 towards the end of the lactation.
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, the trajectories of sire, PE, and residual variances by DIM are presented for the L34 model fitted with either homogeneous or heterogeneous residual variance structure. For the PE effect, the same pattern was seen as that in Figure 1, demonstrating absorption of heterogeneity of the residual variance into the PE effect, when residual variance was assumed homogeneous. Sire variance trajectories (Figure 7) were identical for both residual variance structures, leading forward to rather similar trajectories of heritability by DIM (Figure 8). Consequently, it is unlikely that assumptions of homogeneous/heterogeneous residual variance will affect the genetic evaluation of sires.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Received for publication April 13, 2003. Accepted for publication August 4, 2003.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
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, i = lactation period, being 1 for the homoschedastic models, and 1 to 10 for the heteroschedastic models) from 2 models (L04 and L05), with either homogeneous or heterogeneous residual variance structure. The values of 2lnRL are presented as deviates from values of the L04 model with homogeneous residual variance.
), and mean squared error of predictions (MSEP) from a repeatability test-day model (REP) and 12 random regression test-day models (W33, L33, L44, L55, LM44, AS44, L15, L25, L35, L45, and L34). The values of 2lnRL are presented as deviates from values of the REP model.
