J. Anim Sci. 2008. 86:1057-1066. doi:10.2527/jas.2007-0398
© 2008 American Society of Animal Science
Evaluation of methods for computing approximate accuracies of predicted breeding values in maternal random regression models for growth traits in beef cattle
J. P. Sánchez*,
,1,
I. Misztal* and
J. K. Bertrand*
* Animal and Dairy Science Department, University of Georgia, 425 River Road, Athens 30602; and
Departamento de Producción Animal, Facultad de Veterinaria, Universidad de León, Campus de Vegazana, León, 24071, Spain
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Abstract
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The objective of this study was to determine the suitability of 2 methods for computing approximate accuracies of predicted breeding values, in which accuracy was defined as the squared correlation between the predicted and true breeding value, when modeling growth traits in beef cattle using random regression (RR) models. The first method (Strabel et al., S-M-B) was designed for use with multitrait models; thus, its use with RR models requires the clustering of measurements into different traits. The second method (Tier and Meyer, T-M) was more general, because it accounted for random coefficients other than zeros and ones and thus it could be used directly when fitting RR models. To investigate the performance of both methods, their results were compared with the true accuracies using a balanced simulated data set. The largest difference between approximate and true average accuracies for direct effects was observed at 205 d when S-M-B was used (4.6% males and 8.8% females). With regard to maternal effects, the largest differences in average accuracies were observed at 205 d in males when S-M-B was used (31.8%) and at the same age in females but when using T-M (33.3%). In general, bias increased for direct effect accuracies in males at the tails of the accuracy range, but for females and for maternal effect accuracies in both sexes, bias increased as accuracy increased. When a population was simulated to create large numbers of progeny for base females that did not have individual records, much greater errors were observed in the regression of approximate values on the true ones. When both approximate methods were compared using a real beef cattle data set, a good agreement was observed, particularly for direct effect accuracies in sires [i.e., at 205 d, the regressions were 0.98 (direct) and 0.95 (maternal) with r2 over 0.99]. The largest discrepancies for sires between the methods were observed at 205 d for direct (2.7%) and maternal (16.3%) effect accuracies. For dams, the largest differences between methods were also observed at 205 d, 9.3% (direct), and 15.2% (maternal). The differences between methods for nonparent cattle were greater than for dams for maternal effect accuracies but intermediate between sires and dams for direct effect accuracies. In spite of the less biased results provided by T-M, its use could be problematic when employed in evaluations of large populations due to its greater memory and computation requirements (e.g., 170 and 478% more than S-M-B for a population of 11 million).
Key Words: accuracy • beef cattle • growth • linear spline • random regression
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INTRODUCTION
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For genetic evaluation of growth in beef cattle, there has been interest to move from the traditional multitrait (MT) approach to longitudinal models. Several alternatives have been proposed to account for the effect of age either on the averages, variances, or both, of weight in such models. Originally Legendre polynomials were employed (Albuquerque and Meyer, 2001
; Meyer, 2005a), but recently, spline functions have been suggested as an alternative (Meyer, 2005b). Linear spline functions (Misztal, 2006) have received much attention (Bohmanova et al., 2005; Robbins et al., 2005; Sánchez et al., 2008). The major reasons for using such functions are their good numerical properties; smaller number of estimation artifacts, especially at the tails of the fitted periods; and the possibility to use the same (co)variances as in the traditional MT approach, if knots are placed at ages that define the traits (Misztal, 2006).
Accuracies for breeding values are computed to provide an assessment of risk associated to selection decisions. However, the computation of true accuracies is usually too expensive. Several approximations have been proposed. Tier and Meyer (2004, T-M) provided a method for MT or random regression (RR) models; their approach considers the correlation between traits in the MT model or coefficients in the RR model during the computation of equivalent number of records, which allows for the approximation of prediction error variances and covariances. An alternative way to calculate accuracies for RR models is to use approaches that have been proposed for MT models, such as the method of Strabel et al. (2001, S-M-B). This is possible if the observations are clustered into different traits with a covariance structure.
The first objective of this study was to compare by simulation the approximate accuracies from T-M and S-M-B to those obtained after sparse inversion of the mixed models equation (MME). The second was to compare the approximate methods using a real beef cattle data set.
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MATERIALS AND METHODS
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Animal Care and Use Committee approval was not obtained for this study because part of the data used in this study were obtained from a computer-simulated population and the other part was provided by the American Gelbvieh Association (AGA) and included the information used to produce the June 2006 AGA Sire Summary.
Data
Three data sets comprising beef cattle growth traits were used. The first one (D1) was a simulated population of 20,000 cattle with 3 measurements distributed in 4 generations. To generate the first generation of measured cattle, 5,000 dams were randomly mated to 500 sires. In subsequent generations, 10% of the males and 40% of the females were randomly selected and mated to produce the next generation of 5,000 cattle. The average (SD) numbers of offspring per sire and dam were 16 (6.25) and 3.28 (2.4), respectively. Measures were assumed to occur at d 1 without variation and then at 205 and 365 d with SD of 20 d for both traits. The simulation model included a fixed contemporary group effect, which was the combination of generation and a factor indicating when the measurement was taken (i.e., at 1, 205, or 365 d). A linear spline with 3 knots to account for the effect of age on the mean was considered. The model also included direct genetic effects, maternal genetic effects, permanent environmental effects, and maternal permanent environmental effects. These last 4 were random effects that were assumed to be 3-knot linear spline functions of age. The residual variance was assumed to be heterogeneous and was described by a linear spline function of age with 3 knots, again at 1, 205, and 365 d.
A slightly different version of this data set (D1a) was generated using 100 and 400 founder males and females. On average, the base males had 50 offspring, and the base females had 12.5 offspring. The subsequent 4 generations were produced in the same way as D1.
The second data set (D2) was also a simulated population that was generated using the same model and structure as D1, but 5 generations, each having 2 million cattle with records, were produced. The objective when simulating and analyzing this data set was to examine computational performances of the methods; thus, no results of the obtained accuracies are presented. In both simulated data sets, mating was restricted to avoid full-sibs. The parameters used during the simulations and subsequent analyses were those reported by Legarra et al. (2004), and they are summarized in Table 1.
The third data set (D3) belonged to the US Gelbvieh population, and it was the data set used to produce the July 2006 sire summary. Table 2 shows the basic statistics describing the records in this data set. The pedigree associated with this data set contained 1,154,905 cattle. The average (SD) number of offspring per sire was 24.9 (120.2) and for females was 2.3 (2.2). In the pedigree file, 57% of the cattle were nonparents. This was a multibreed data set in which more than 50 breeds were represented, either as purebred or cross-bred cattle (mainly crossbred). In the multibreed analyses of this data set, up to 70 unknown parent groups were considered to accommodate the additive genetic x generational effect of breeds or groups of breeds (Legarra et al., 2007; Sánchez et al., 2008).
Methods
The algorithm of Strabel et al. (2001), (S-M-B) is an extension of a single-trait algorithm (Misztal and Wiggans, 1988) to MT and maternal models. In this procedure, during the computation of the direct effect accuracies, the maternal variances are added to the residual variance. Similarly, during the computation of the accuracies of the maternal effects, the direct variance is added to the residual variance. Direct and maternal accuracies are computed separately, indirectly assuming that these effects are uncorrelated. The first step of this algorithm is the computation of a single-trait effective number of observations and a single-trait amount of information from the relationships between cattle; this is done iteratively, as described by Misztal and Wiggans (1988). The next step is to modify the single-trait information by accounting for the information from the residual and genetic covariance structure between traits. This method was applied to D1, D1a, D2, and D3, considering them always via a MT model. Thus, the design matrices for the random effects were assumed to be formed by zeros and ones. In our case, the time points defining the traits to be used with this algorithm were 1, 205, and 365 d. Note that in spite of this definition, the prediction of breeding values could be done independently using a RR model.
The algorithm of Tier and Meyer (2004), (T-M) was designed to compute accuracies under either the RR or MT models. This method considers the (co)variance structure among traits (or coefficients in its RR version) during the computation of the equivalent number of records; thus, these equivalent numbers of records are matrices with dimensions equal to the number of traits (coefficients in the RR version). Once these matrices are known, they are modified for offspring and parent contributions; this is done by tracing twice through the pedigree, the first time (backward) accounting for offspring information contribution and the second time (forward) accounting for parent contributions. For this algorithm to be efficient, the pedigree must be sorted by year of birth. With this method, the maternal and direct genetic effects also are assumed to be uncorre-lated. In the implementation of T-M for this study, some aspects of the original algorithm were modified. Reduction of the record contribution due to repeated measurement for 1 animal in the same contemporary group in the RR model and discounting of records due to large sizes of sire half-sib families within contemporary group were not considered due to high memory requirements, particularly in the case of D2. Also, with the maternal models, better results were observed when the direct (maternal) genetic variance was moved to the residual variance instead of to the permanent environmental (maternal environmental) variance, as was suggested in the original study (Tier and Meyer, 2004). For random effects in the RR model, 2 basic functions were used in the analysis: linear splines (RR-ls), as used during the simulation, and cubic Legendre polynomials (RR-lp). In both cases, the residual variance was assumed to be heterogeneous and described by a linear spline function, with knots at 1, 205, and 365 d. For both methods, the only fixed effect considered during the computation of the equivalent number of records was the contemporary group.
Data set D1 was also used to compute the exact accuracies after a generalized sparse inversion of the MME (GI) from MT, linear spline RR models, and Legendre Polynomial RR models.
Accuracy was defined as the squared correlation coefficient between the predicted and true breeding values
where acci,t = the accuracy of the predicted breeding value for the trait or coefficient t of the animal i; Var(ui,t – ûi,t) = the prediction error variance for the animal i and trait or coefficient t (this quantity is computed approximately in the nonexact methods); and 
2at = the additive variance of the trait or coefficient t.
In this study, T-M was applied either assuming a linear spline (T-M_ls) or a Legendre polynomial (T-M_lp) model, S-M-B was applied assuming a MT model, and GI was applied assuming the corresponding model used in the approximated method.
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RESULTS
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Simulated Data Set D1
The averages for the actual and computed accuracies calculated using the different approaches are shown in Table 3. In general, except for direct and maternal birth weight effect accuracies from T-M for males, all approximate methods overestimated the accuracies with the bias, defined as the difference between the means of exact and approximated accuracies, being stronger for the maternal effect. For direct effects, the bias on the average computed accuracy was in general greater with S-M-B, reaching a maximum of 4.6% at 205 d in males and of 8.9% at 205 d in females. For maternal effects, the greatest bias was observed at 365 d for males when using S-M-B (31.8%) and at the same age for females using T-M_lp (33.3%). The standard deviations of the computed and exact accuracies were very similar across traits in all the methods and models, ranging from around 0.07 for maternal effects in males to around 0.20 for direct effects in females. For direct effects, the average accuracies in S-M-B had values between 2.3 and 4.1% greater than both T-M models. For maternal effects, larger variations in the differences between methods were observed, ranging from 4.7% (greater in T-M) at 365 d to 11.9% (greater in S-M-B) at birth.
Table 4 presents the regression statistics of computed accuracy values from T-M and S-M-B on the real ones using D1. The regression coefficients for direct effect accuracies in males at all considered ages were lower than 1, and the intercepts were slightly greater than 0, meaning that at extremely high and low accuracy values, the bias will be larger than for intermediate accuracy values. For female direct effect accuracies, the regression coefficients were greater than 1, meaning that the greater the true accuracies, the larger the bias. With regard to maternal effects, in general except for T-M in males, regression coefficients were greater than 1 but intercepts were 0. The r2 was high (0.97 to 0.99) for all fitted regression lines, with maternal effects having the lowest values. These regression results are graphically shown in Figure 1 for the case of weight at 205 d using T-M_ls.
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Table 4. Statistics from the regression of accuracies from the different approximated methods and models on the exact ones obtained after generalized inversion of the mixed models equations for the respective models, using D1
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Figure 1. Accuracies (ACC) of direct and maternal genetic effects at 205 d (weaning weight) obtained using generalized inversion (GI) vs. those obtained using the approximated method of Tier and Meyer (TM; 2004) as applied to D1 and considering a linear spline random regression model, shown by sex. int. = intercept; Ge. = generation.
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The approximate accuracy values for both T-M and S-M-B obtained for the alternative simulation structure in D1a greatly overestimated the accuracies for the direct effect of females, especially for those without their own individual records. This is illustrated for T-M_ls at 205 d in Figure 2, which presents the results for direct and maternal effects accuracies in both males and females. This problem was not observed when the dams had a small number of offspring, as in D1, or the base dams had their own individual records (results not shown), or when no maternal effect was included (results not shown).
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Figure 2. Accuracies (ACC) of direct and maternal genetic effects at 205 d (weaning weight) obtained using generalized inversion (GI) vs. those obtained using the approximated method of Tier and Meyer (TM; 2004) as applied to a simulated data set in which the base generation (Ge.) was formed by 400 females and 100 males (D1a), shown by sex. int. = intercept.
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Gelbvieh Data Set D3
Table 5 shows the averages for the computed accuracies for direct and maternal effects by sex for cattle with and without offspring from D3. In this case, the average differences between direct effect accuracies between S-M-B and T-M were on average, across traits and sexes, similar to the differences obtained with D2 (3.8%); however, the range in the differences for D3 across traits and sexes was greater. For maternal effect, S-M-B produced accuracy values that were on average across traits and sexes, 15.9% greater than those from T-M; in the case of D1, they were only around 5.0% greater. As expected, traits showing large average discrepancies between methods were those with variation at the measurement ages [i.e., weaning weight (205 d) and yearling weight (365 d)]. For cattle without offspring, the differences between S-M-B and T-M methods in direct effect accuracies were less than for males with progeny but greater than for females with progeny. However, for maternal effect accuracies, cattle with no offspring had the largest discrepancy between the 2 approximate methods. In addition, it should be noted that, on average, nonparental cattle had greater accuracy value than sires. This is a consequence of the large proportion of sires that had only a few offspring and had no individual record (i.e., 54% of the sires had no individual record). Most of these sires are phantom cattle needed for tracing the pedigree of crossbreed cattle back to the purebred founders; note that the US Gelbvieh population is considered to be multibreed (Legarra et al., 2007; Sánchez et al., 2008).
Regression statistics of T-M accuracies on S-M-B are presented in Table 6. All the regression coefficients are slightly lower than 1 and the intercepts nearly 0 for cattle with offspring; thus, S-M-B produced larger values of computed accuracies than T-M not only on average, as indicated in Table 5
, but throughout the entire range of accuracy values. For cattle with offspring, the r2 coefficients of the regression lines of the T-M methods on S-M-B were always greater than 0.99 and 0.98 for direct and maternal effect accuracies, respectively. As expected, cattle without offspring were observed to have a worse fit than cattle with progeny, and their r2 values were as low as 0.86. The regression results of T-M_ls on S-M-B for direct weaning weight (205 d) are graphically shown in Figure 3.
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Table 6. Statistics from the regression of accuracies from the approximated Tier and Meyer (2004) method, assuming random regression (RR) models on those approximated by Strabel et al. (2001) assuming multitrait models, when the Gelbvieh data set, D3, was used
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A nearly perfect match (intercept equal to 0 and slope equal to 1) was observed between computed accuracies assuming T-M_lp and T-M_ls. Although the regressions are not shown, it is important to note that the worse case was for the direct yearling weight accuracies for all types of cattle but particularly for cattle without offspring (r2 = 0.98). This could be a consequence of some artifact in the fit of random effects using Legendre polynomials at older ages.
Computation and Memory Requirements
Table 7 shows times and memory requirements for the different approximation methods and models used for all the data sets including D2. As was expected given its greater dimension, T-M_lp was the approach requiring the greatest resources. It was found that S-M-B clearly exhibited better performance in terms of central processing unit usage and memory requirement than T-M_ls. The memory requirement and the central processing unit usage for T-M_ls when analyzing D3 were approximately 170 and 478% greater compared with S-M-B. However, it should be noted that the dimensions in T-M_ls and S-M-B models were different, because the former included a direct permanent environmental effect but the latter did not.
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Table 7. Random access memory usage and central processing unit (CPU) time for the different approximated methods and data sets
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DISCUSSION
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Computed accuracies for direct effects from an algorithm assuming a MT model, defining traits at birth, weaning, and yearling, were on average greater than those from an algorithm specifically designed for RR models and evaluated at 1, 205, and 365 d of age. This was observed both under the balanced conditions of a simulated example and with a real data set; in this real case, the differences between the methods were greater. It can also be seen from the results that with respect to the true accuracies (i.e., obtained by GI of the MME), bias is slightly greater in the case of the method assuming a MT model. It needs to be noted that the largest bias occurred in unusual situations (females that had no individual record but had a large number of offspring) or for animals for which an exact accuracy value is less important (females). For sires, both approximated procedures performed similarly. Even though there were some differences between the methods, the high memory and computation requirements for the method assuming a RR model were high, which could prevent its usage in very large populations.
The differences between approximate and exact computed accuracies were greater for maternal effect accuracies than for direct effect accuracies. Even under conditions of the simulated data set, the greater the value of the true accuracies, the greater the bias, because the regression was greater than 1 and the intercept nearly 0 (Figures 1 and 2). Both the differences in performance between maternal and direct effect approximate accu racy computations, as measured by the regression of the computed values on the true ones, and the lower performance of the approximate methods in the case of females with no individual record or large number of progeny were probably due to the fact that both approximation methods compute direct and maternal effect accuracies separately by adding to the residual other sources of variation [i.e., maternal genetic and environmental (co)variances when computing direct effect accuracies]. Thus, neither the relationships between residuals after amalgamating the genetic and environmental (co)variances nor the negative correlations between direct and maternal effects, which in the real data set was previously estimated to be around –0.2 (Legarra et al., 2004) and the same was assumed during the simulation, were considered. However, the nonzero correlations would not be expected to be the critical factor, because in both approximated and exact computations, they were assumed to be zero. An attempt was made to address differences between exact and approximated direct effect accuracies in females when base dams had large offspring groups and no individual records, which could be the consequence of ignoring the residual correlation between records after amalgamating genetic (co)variances into the residuals. Before combining the genetic effects into the residuals, an approximate diagonalization of genetic (co)variance matrix was implemented using the same formulas as used by T-M during the computation of contributions from the relationships. However, after this implementation, the improvement in the performance for direct effect accuracies in females was marginal. Problems encountered when dealing with maternal models could be overcome by considering both direct and maternal genetic effects, as previously suggested by Strabel et al. (2001). It could be relatively straightforward to extend the formulas to account for the relationship contributions; however, it would not be so simple to jointly discount for the con temporary group structure for both maternal and direct effects.
The main advantage of the T-M algorithm is that it allows for proper computation of approximate accuracies for any desired selection index, because approximate prediction error covariances are also available; this is not the case for S-M-B algorithm. However, memory and computational requirements to implement T-M are greatly increased.
It should be noted that there was generally good agreement between T-M_lp and S-M-B, indicating that both procedures provided almost the same effective number of records. Although, in T-M_lp, they were computed on the Legendre polynomials coefficients basis, and in S-M-B, it was done directly on the age scale. Because both approaches treat the relationship between animals in an equivalent way, it can be concluded that the differences between the design matrices for the random effects were not very important, because very similar final results for the accuracies at the specific times (1, 205, and 365 d) were observed.
Jamrozik et al. (2000) suggested another approximate method for computing accuracies under RR models that relies on the concept of equivalent number of progeny (Koots et al., 1997). This method could be considered a simplified version of Tier and Meyer’s algorithm (Tier and Meyer, 2004), which does not allow for an easy way of considering prediction error covariance between coefficients. The equivalent number of progenies is the number of offspring, as a unique source of information, that one animal needs to have for reaching the same degree of accuracy as it has due to its own records. Afterwards, selection index theory is applied to account for the relationships between animals. Tier and Meyer (2004) observed a slightly better performance in terms of less bias for T-M compared with the method of Jamrozik et al. (2000) when both were applied to beef cattle field data. Tier and Meyer (2004) also observed that a method proposed by Harville (1999) for approximating the inverse of 1 matrix using a Gibbs sampler was the approximated methodology providing the least bias.
As suggested by Misztal and Wiggans (1988), the general requirements for an algorithm computing approximate accuracies are ease of implementation and the computational requirements should not be greater than those for the evaluation. Both methods investigated in this study provided satisfactory accuracies for the most relevant animals (i.e., sires with progeny), and both used a reasonable amount of computational resources. A relatively straightforward improvement could be, as done by Tier and Meyer (2004), to implement a discount due to single-sired contemporary groups; however, this could be very expensive in terms of memory in large populations. Implementing other capabilities that could jointly approximate maternal and direct accuracies would provide better results; however, the animals most affected by this improvement would be those with a small amount of information for which having a very precise accuracy computation is less important.
In conclusion, this study provided an assessment of the performance of 2 approximate methods for computing accuracy for RR models for growth. In general for the most relevant animals (sires), both algorithms performed satisfactorily, but in some particular situations of information structure and animals with little information, greater errors were observed between exact accuracies and those obtained by both approximate methods. Also, it was observed that the method employing a MT model had slightly greater bias compared with methods assuming RR models. When the results from both approximate methods were compared with each other, the computed accuracies were close for direct effect accuracies; however, the differences for maternal effects accuracies were slightly greater. In spite of all the differences between methods, a practical implication of the study was that using an algorithm assuming a MT model to compute accuracies after a RR evaluation could provide quite satisfactory results with much less computational usage of resources.
1 Corresponding author: jpsans{at}unileon.es
Received for publication July 3, 2007.
Accepted for publication January 12, 2008.
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LITERATURE CITED
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Robbins, K. R., I. Misztal, and J. K. Bertrand. 2005. A practical longitudinal model for evaluating growth in Gelbvieh cattle. J. Anim. Sci. 83:29–33.[Abstract/Free Full Text]
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Abstract
INTRODUCTION
