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GENETICS |
* Division of Genetics and Genomics, Roslin Institute, EH25 9PS Midlothian, UK; and Department of Genetics and Biotechnology, Danish Institute of Agricultural Sciences, DK-8830 Tjele, Denmark
1 Corresponding author: andreas.kranis{at}bbsrc.ac.uk or a.kranis{at}sms.ed.ac.uk
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Key Words: turkey • egg production • random regression • longitudinal model • model comparison
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Daily milk and egg production in turkeys are analogous traits in that both change over time in a broadly comparable way, first increasing to a peak and then declining over time, and thus, when modeling the average production as a function of time, curves with similar shapes are generated. The success in the application of RR models in dairy cattle has attracted the attention of poultry breeders. Anang et al. (2002) reported that a RR model appeared the most favorable model for analyzing egg production data when compared with other longitudinal models, including a multitrait (MT) model. Other studies investigated the efficiency of RR for the genetic evaluation of other egg-related traits, such as fertility and hatchability (Sapp et al., 2004).
The benefits of RR models are that the partition of variation in egg laying among different sources, such as genetic or permanent environment, is not assumed constant during the whole laying period. Changes over time for each source of variation can be expressed as time-dependent components for all birds in the population, whereas changes in the mean over time may be described by a fixed regression model. With these benefits, RR models offer more accurate modeling of variance-covariance structures, in turn leading to more accurate predictions of breeding values (Huisman et al., 2002). Furthermore, the modeling of components over time provides an opportunity for identifying optimum points for recording and hence to maximize the cost-effectiveness of selection (Sapp et al., 2004) or even alter the shape of the longitudinal response through genetic means (Schaeffer, 2004).
These features of RR are of great importance for turkey breeders for improving the output of selection. Although the emphasis is placed on improving BW and conformation, maintaining a satisfactory egg production is also a key objective. This can be difficult to achieve due to the negative genetic correlation with growth traits (Kranis et al., 2006). The application of RR models could provide alternative approaches to deal with this constraint, but research has focused to date only on egg-laying chickens. Therefore, the objective of the current study was to investigate the application of RR models in the genetic analysis of egg production of turkeys. Furthermore, the results from this analysis were compared with analyses of the same data set using MT and repeatability (REP) models. To assess the goodness of fit, the effectiveness of predicting missing values was examined in reduced data sets and then compared with observed values for the 3 alternative models.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Hens laid in individual trap nests, and production was individually recorded on a daily basis for 20 wk. So, the data set consisted of a sequence of 140 binary records (0 = no egg; 1 = egg laid), with each corresponding to a specific day of the whole production period. If a hen died during the laying period, the rest of its records after the date of the death were treated as missing values. Cracked eggs were not counted as laid eggs, because they could not be hatched. Eggs laid on the floor were also excluded, because they could not be assigned to a hen.
Data Analysis
The trait analyzed in the current study was the cumulative production of hatchable eggs laid in trap nests over 5 consecutive periods of 28 d, covering the whole 140-d laying period. In this way, each period corresponded approximately to egg production over 1 mo. The data were analyzed using the REP, MT, and RR models. All the analyses were performed using the average information restricted maximum likelihood algorithm with the package DMU (Jensen and Madsen, 1994).
The REP and RR models included a fixed regression to account for the phenotypic trajectory for the mean egg production over the different periods. This trajectory was modeled using the family of curves described by Ali and Schaeffer (1987). Although this model was initially introduced to describe the milking production curve, it was found to be useful for modeling the egg production data. Thus, the expected egg production (y) of the population at time t was described by the following formula:
 | ([1]) |
where b0, b1, b2, b3, and b4 = the regression coefficients and t = each 1 of the 5 periods (t = 1, ... 5). For all models, a super factor with levels for every combination of year, hatch, and pen was fitted as an additional fixed effect.
REP Model
The REP model treated the 5-period summary measurements as repeated records. The model included a permanent environmental and an additive genetic effect and a fixed regression to model the phenotypic trajectory. Hence, the model to describe the egg production of period t (yijt) was as follows:
 | ([2]) |
where Si = the ith combined fixed effect; FRt = the fixed regression terms given by equation 1; cj and aj = the random effects of the permanent environment and additive genetic effect, respectively, for the bird j [with c ~N(0, 
2cI) and a ~N(0, 2aA), respectively]; and eijt = the residual (e ~N(0, 2eI)). This model assumes a genetic correlation of unity and independence of residuals across all periods; A = the relationship matrix among the birds.
RR Model
An extension of the REP model was to include time functions in the random part of the model. There were 2 candidates for the functional form of the random regression: the Ali-Schaeffer functions, following the suggestion by Anang et al. (2001), and Legendre polynomials. Difficulties were encountered in convergence with the Ali-Schaeffer function, so Legendre polynomials were used. These are a family of orthogonal polynomials suited to use in RR models (Pool et al., 2000).
The Legendre polynomials of order m were denoted as 
m(w), where wi = the period standardized to lie between –1 and 1, using the following formula (Schaeffer, 2004):
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where, ti = 1, ... 5; and tmin and tmax corresponded to the earliest and latest period, respectively (tmin = 1 and tmax = 5).The RR model used for the egg production yij in period t was
 | ([4]) |
where Si = the ith combined fixed effect; FRt = the fixed regression for month t given by equation 1; the third term represented the permanent environmental effect; the fourth term represented the additive genetic effect of the jth bird; and eijt = the residual term. Terms cjm and ajm = the RR coefficients for the Legendre polynomial of order m.
The maximum degree of the orthogonal polynomials fitted was tested to determine the most appropriate combination. The likelihood of each model was compared with a log-likelihood test using the appropriate d.f., determined by the difference between numbers of model parameters (for each effect, the d.f. were as follows: 
(m + 1)(m + 2), where m corresponded to the order of polynomials). Calculation of nonzero eigenvalues of the corresponding eigenfunction of the covariance matrix provided further evidence for the necessary polynomial order (Meyer and Hill, 1997). Based on the log-likelihood test, the RR model using third-order polynomials was the best (RR3), but the corresponding eigenvalue to the cubic regression was close to zero. So, the analysis was repeated for the RR model using second-order polynomials (RR2) to compare the results.
By letting matrix G be a 5 x 5 matrix of the estimates of variance for each period (in the diagonal) and the covariance between different periods (off-diagonal elements), it can be calculated by the covariance function (Kirkpatrick et al., 1990)
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where, for RR3, 
= a 4 x 5 matrix of the time covariates and V = a 4 x 4 matrix containing the covariance components of the intercept and the RR coefficients for the additive genetic effect (matrices and V were 3 x 5 and 3 x 3, respectively, when using the second-order polynomial). Likewise, a covariance matrix was computed for the permanent environmental effect (C). The residual covariance matrix (R) was the 5 x 5 identity matrix multiplied by the homogenous residual variance component. The total phenotypic covariance matrix (P) was the sum of the additive genetic, permanent environmental and residual covariance matrices (P = G + C + R).
The heritability (h2) for time i and the genetic correlation (
) between time points j and k were defined as the following:
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and
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where gi,i and pi,i = the diagonal elements of matrices G and P corresponding to the genetic and phenotypic variance for period i and gj,k = the element of the G matrix corresponding to the genetic covariance between periods j and k.
The SE of heritability was calculated by extending the methodology proposed by Fischer et al. (2004), adapted to accommodate the output of the DMU package. The formula used to estimate the variance of the heritability estimate for the ith period was based on a Taylor series expansion and it was given by the following equation:
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where gi,i and pi,i and vgi,i and vpi,i = the diagonal elements of matrices G and P and VG and VP, respectively. Matrices VG and VP correspond to the variance of G and P.
MT Model
A MT model was also fitted as a contrast with the regression models. Here, the egg numbers of the 5 subperiods were treated as different traits and analyzed simultaneously. Hence, the egg production yij with period k was as follows:
 | ([9]) |
where Sik = the ith combined fixed effect; ajk = the random additive genetic effect for the bird j [a N(0,2aA)]; and eijk = the residual term [e ~ N(0,2eI)]. The additive genetic effect was given as the direct product of matrices G, the 5 x 5 matrix that describes the genetic variance-covariance among the 5 periods, and A, the relationship matrix among the birds. The residual across all 5 periods was defined as the direct product between E and I.
Model Comparison
To compare the predictive ability of the various models, 2 cross-validation strategies were used. In both, the data were divided into 2 parts. The first part, corresponding to 80% of the total data, was used to estimate parameters for the different models, which were then used to predict the second part, the remaining 20% of the data. Goodness of fit was then assessed by measuring MS errors of prediction.
The first strategy created a reduced data set for estimation as follows: within each generation, the first period was deleted for the first bird, second period for the second bird, and so on. This was repeated for each group of 5 birds as they appeared in the data set after sorting on generation, hatches, and pen. Therefore, 2,400 periods were deleted, near balanced in relation to the fixed effects. Predictions based on model parameters were straightforward for REP, RR2, and RR3 models. However, in contrast to the others, the MT model does not include a permanent environment as a second random effect, because it is contained in the residual for each period, but it should be accounted for when assessing goodness of fit. Therefore, the missing residual, conditional on the observed residuals of the other periods for an individual, was estimated via a multiple regression. So, for the MT model, the adjusted prediction was the sum of the fixed and the additive genetic effect plus the missing residual, estimated by the regression.
The second strategy was only used for comparison of RR2 and RR3 models, and in this strategy, the entire last period was deleted. The objective of this second comparison was to detect if a model overfits the data. It was not possible to include the MT model in this comparison, because it would not be possible to predict the deleted record from the 4 remaining ones.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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. For the REP model, the estimates were assumed to be constant across the periods, whereas for the MT and RR models, estimates were available for each of the 5 time points. The observed trend was that phenotypic variance increased along with the periods.
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for all 3 models. The RR3 model had the higher estimates, with the exception of the fourth period, but the differences were small compared with the estimation errors. The heritability estimates from the RR2 and MT methods were very close for all the periods and for the total production period. The REP model had the lowest estimates from all models. In Table 3, the estimates of the ratio of the permanent environmental variance to the total variance are also presented. The estimates of the ratio of the permanent environmental variance to the total variance estimated from the RR models were close to each other, whereas the REP estimates were lower.
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presents the genetic correlations between egg numbers of different periods, estimated based on MT and on RR2 models. Although the values differed to some extent between the 2 models, the same pattern was observed. First, a weak correlation between the first and the middle stages of the production was observed. Second, the correlation between consecutive periods was strong. Third, the first and the last period were positively correlated. Phenotypic covariances are presented in Table 5. The phenotypic correlations were positive between all periods.
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). This plot allows the visualization of the dynamics of the genetic variance over time.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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The use of the Ali-Schaeffer equation as the function for the fixed regression provided a robust tool to describe the trajectory of the average egg production (Anang et al., 2001). In the current data set, the fit was perfect, because a 5-term equation was used to model 5 time points. However, a very good fit was also obtained when the same equation with the 5 time points was tested to fit for 10 or more time points (results not shown). This result provides evidence that the Ali-Schaeffer equation can be used satisfactorily to model the average egg production, with the benefit of being simpler than other models proposed, such as the Grossman et al. (2000) persistency model.
The REP model used in the current analysis was similar to the test-day models introduced by Ptak and Schaeffer (1993); however, it was unsatisfactory for the present data set, because more detailed analyses showed that the major assumption of the REP model did not hold. Results from the RR and MT models illustrated that the genetic correlations between egg productions of different periods varied from 0.1 to 0.9, whereas the REP model assumes a value of 1. Another assumption of the REP model is that genetic variance remains constant between periods, and this was not supported by the estimates derived from the RR and MT models. In brief, the REP model offers a quick and simple approach for the genetic analysis of longitudinal data and has been used for the genetic evaluation of the egg production in poultry (Anang et al., 2001), but the limitations stemming from the assumptions of the model makes it less preferable than other options.
The RR models offer an improvement over the REP model, because they allow the modeling of the genetic covariance between periods and are a development of covariance functions described by Kirkpatrick et al. (1994). Anang et al. (2002) concluded that RR was the preferred model for the genetic evaluation of egg production of laying chickens, and the current data extend this observation to turkeys when compared against the MT model, which represents a more traditional approach to modeling repeated records over time. The comparison of genetic parameter estimates from the RR2 and MT models showed that both models were equally effective to describe the dynamics of the genetic variance over time. The general shape of the heritability profile obtained from the 3 models agreed with results from Anang et al. (2000, 2002). Similar trends are also observed for heritability of milk production using test-day models in dairy cattle (Olori et al., 1999).
The RR models can deal with a large number of production periods with few parameters. In the present study, the total number of covariances for the MT model was 30, compared with 13 and 21 for the RR2 and RR3 models, respectively. The model comparison using the first cross-validation strategy showed that both RR models had a lower MSE than the MT model. The lowest MSE was obtained with the RR3 model.
The superior prediction ability of the RR3 model over RR2 could be associated with a larger number of explanation variables. Therefore, a second cross-validation strategy was used to discriminate between RR2 and RR3. The second strategy was used to assess the ability of the 2 RR models to predict the egg production beyond the observed period, rather than predicting an internal missing value. Using the second strategy, the RR2 model gave the best fit, suggesting that the advantage of the RR3 model in the initial model comparison was a consequence of the larger number of explanatory variables associated with the RR3 model.
Further indications for rejecting the RR3 model were provided by the substantially different heritability profile when compared with RR2 and MT models and the larger SE. The latter suggests that the RR3 model overfits the data, and further evidence of overfitting was implied by the eigenvalue of the third-order regression coefficient being close to zero, although the RR3 model had a lower log-likelihood value. Olori et al. (1999) used similar arguments when considering the appropriate order of RR when modeling lactation curves. Therefore, a tradeoff seems to exist between the number of parameters and the model efficiency, and so determining the polynomial order requires consideration. It was concluded that the RR2 model was the most appropriate model in this data set.
Apart from the appropriate polynomial order, the number of the time points that a RR model will fit is also crucial. Initially, a 10-period model was considered, but it failed to converge. Possibly, the underlying biological mechanism, involving overlapping ovipositions, interfered with the separation into 10 periods. The egg number distribution within each period was more erratic and the approximation via the normal distribution less satisfactory.
One approach not followed in the current study was the use of transformations to reduce deviations from normality, even though this was considered by Kranis et al. (2006) and others in studies of the egg production for the whole laying period. The justification for excluding this approach was that simple data screening showed that a separate transformation would be required for each period. This would result in cumbersome evaluation procedures and would obscure inferences.
In conclusion, the application of RR in egg production of turkeys appears to be promising. It can effectively model the laying procedure even when missing values exist, as highlighted in the current study. The implementation of RR allows the genetic evaluation of egg production on a monthly basis and could provide helpful information for breeders to optimize selection strategies. Nevertheless, in view of the rapidly changing heritability over the initial period, use of more time points may be warranted, to derive full benefits of modeling the genetic variation over time and to provide a more reliable framework for breeders.
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ACKNOWLEDGMENTS |
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Received for publication August 9, 2006. Accepted for publication November 9, 2006.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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