J. Anim. Sci. 2005. 83:29-33
© 2005 American Society of Animal Science
A practical longitudinal model for evaluating growth in Gelbvieh cattle
K. R. Robbins1,
I. Misztal and
J. K. Bertrand
Animal and Dairy Science Department, The University of Georgia, Athens 30602-2771
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Abstract
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Genetic evaluation of growth in Gelbvieh beef cattle was examined by multiple-trait (MTM) and random regression (RRM) analysis. The data set comprised 541,108 animals with 1,120,086 records. Approximately 15% of the animals in the data set had at least one record measured outside of the accepted MTM age ranges for weaning weight (Wwt) and yearling weight (Ywt). Fourteen percent of Wwt records and 19% of Ywt records were measured outside the accepted ranges for MTM analysis, and thus were excluded from MTM evaluations. Two RRM evaluations were performed using cubic Legendre polynomials (RRML) and linear splines (RRMS) with three knots at 1, 205, and 365 d of age. Data Set 1 (d1) utilized all available records, whereas Data Set 2 (d2) included only records measured within MTM ranges (1 d, 160 to 250 d, and 320 to 410 d). The RRML models did not reach convergence until diagonalization was imposed. After diagonalization, it was found that all longitudinal models required fewer iterations to converge than the MTM. Correlations between the MTM, RRML-d2, and RRMS-d2 evaluations were 
0.99 for all three traits, indicating that these models were equivalent when predicting breeding values from data within the MTM age ranges. Correlations between MTM, RRML-d1, and RRMS-d1 were >0.99 for Bwt and >0.95 for Wwt and Ywt. The lower correlations for Wwt and Ywt indicate that the added information does affect breeding value prediction. The RRM has the capability to incorporate records measured at all ages into genetic evaluations at a computing cost similar to the MTM.
Key Words: Beef Cattle • Legendre Polynomial • Multiple Trait • Random Regression • Spline
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Introduction
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Multiple-trait models (MTM) treat all records for a trait as being measured at the same age and having the same parameters. This creates the need for age adjustments, establishment of age ranges, and the subsequent elimination of records measured outside these ranges. These characteristics of MTM have led to the development of random regression models (RRM) for growth traits (Meyer and Hill, 1997
; Albuquerque and Meyer, 2001). These models eliminate the need for MTM adjustments by modeling the changes in (co)variance over time. Meyer (2004) showed that these properties of RRM could lead to increases in the accuracy of breeding value prediction.
Although the RRM is theoretically more appealing, previous RRM evaluations using Legendre polynomials (RRML) have yielded inaccurate results (Nobre et al., 2003b). Nobre et al. (2003a) found that estimates of RRML parameters for which little data were available were unreliable. Recently, Legarra et al. (2004) attempted to eliminate unreliable RRML parameter estimates by deriving them directly from MTM parameters using methods similar to those used by Kirkpatrick et al. (1990, 1994). Bohmanova et al. (2004) looked at RRM for growth traits using simulated data and found that, in addition to reliable parameters, RRML required diagonalization for numerical accuracy.
Although much of the interest in longitudinal models has been focused on RRML, linear splines present a simple alternative. With splines, poorly estimated parameters are not a factor as they use MTM parameters. A spline fits a series of functions through control points referred to as knots. The use of linear spline functions can eliminate the numerical problems associated with RRML and decrease the cost of implementation.
The objective of this study was to compare genetic evaluations of growth data in a large beef cattle population using RRM and MTM to determine the practicality of implementing longitudinal evaluations for national data sets.
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Materials and Methods
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Evaluations were performed on Gelbvieh records spanning 1972 to 2001. The initial data set had records on 667,174 animals for birth weight (Bwt), weaning weight (Wwt), and yearling weight (Ywt). All animals that were less than 50% Gelbvieh, had dams younger than 550 d, or were the only animal in a contemporary group were removed. All records greater than four standard deviations from the mean weight were removed. Seven age-of-dam classes (AOD) were created beginning at 550 d of age. Contemporary groups (CG) included breeder defined CG, sex, and percentage Gelbvieh. In addition, Bwt CG included birth year and season. Two data sets were formed. Data Set 1 (d1) contained all available records, whereas Data Set 2 (d2) contained only the records measured at (1 d, 160 to 250 d, and 320 to 410 d).
The complete data set (d1) comprised 1,120,086 records on 541,108 animals. Approximately 15% of the animals had at least one record measured outside the accepted MTM age ranges for Wwt and Ywt. Fourteen percent of Wwt records and 19% of all Ywt records were measured outside the accepted ranges for MTM analysis. However, nearly 65% of records measured outside MTM ranges were within 20 d of age range bounds. A summary of the complete data set is found in Table 1.
All parameters used in this study were from work done by Legarra et al. (2004). Legarra et al. (2004) provides detailed descriptions of the methods used to estimate all parameters. All models and parameters used in this study were the same as those used by Bohmanova et al. (2004), with the exception of fixed effect estimation. Because contemporary groups did not remain constant across traits, a continuous regression could not be used in the longitudinal evaluations.
Fixed effects for all models were determined using the following equation:
where cgi = contemporary group i; 
lt = linear, quadratic, and cubic regression coefficients at age l and trait t; agel = age l of animal; aget = equals the reference age of trait t; and aodj = age of dam class j. In longitudinal models, the equation was nested in three dummy variables, representing the three traits. In addition, the age of dam classes and contemporary groups were renumbered for Wwt and Ywt traits. These modifications were made to ensure that the equation was equivalent for all models used in this study.
The MTM model presented in scalar notation was:
where yijklmnt = weight for trait t at age l of contemporary group i and age of dam group j; dirkt = the random direct additive effect of animal k for trait t; matmt = the random maternal effect of dam m for trait t; mpemt the maternal permanent environmental effect of dam m for trait t; and eijklmnt = random residual effect. Direct and maternal effects were assumed correlated.
The RRM using linear splines in scalar notation was:
where yijklmn = weight for trait t of contemporary group i, and age of dam group j; dirdk and pedk = spline coefficients d for additive direct and permanent environmental effects for animal k; matdm and mpedm = spline coefficients d for maternal and maternal permanent environmental effects for dam m; eijklmn = weighted heterogeneous random residual; and sdl = dth coefficient of the linear spline function for an observation taken at age l.
The RRM, constructed with cubic Legendre polynomials, was defined as:
where yijklmn = weight for trait t of contemporary group i, and age of dam group j; dirdk and pedk = random regression coefficients d for additive direct and permanent environmental effects for animal k; mdm and mpedm = random regression coefficients for maternal and maternal permanent environmental effects for dam m; eijklmn = weighted heterogeneous random residual; and zdl = dth coefficient of Legendre polynomial for observation taken at age l.
The weights for the heterogeneous residual variance were modeled using linear splines, as in Bohmanova et al. (2004) and Legarra et al. (2004), and implemented by weighting each observation. Solutions were computed by the program BLUP90IOD, which uses iteration on data with the precondition conjugate gradient iteration (Tsuruta et al., 2001).
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Results and Discussion
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In this study, MTM was fitted to d2 only, whereas RRML and RRM with linear splines (RRMS) were fit to d1 and d2. The number of iterations required for convergence of each model is reported in Table 2. Both the RRML and RRMS required fewer iterations than the MTM, indicating that longitudinal models do not have higher computational requirements than multiple-trait models. This is in agreement with earlier results showing that similar RRM required less memory and time than MTM (Nobre et al., 2003b). However, it should be noted that only the RRML was diagonalized, as convergence could not be reached otherwise. Bohmanova et al. (2004) obtained convergence of RRML without diagonalization using a smaller simulated data set. In that case, the diagonalization decreased computing more than fivefold. To achieve the most accurate results, a strict convergence criterion of 10–14 was assigned to all models. Previous work by Nobre et al. (2003b) found that strict convergence criteria were necessary to obtain accurate RRML results. Similar strict convergence criterion for the preconditioned conjugate gradient iteration was necessary in a study by Tsuruta et al. (2001).
Table 3 shows Pearson correlations of fixed effects between the MTM and longitudinal models when applied to d2. The estimation of fixed effects was equivalent for all models used in this study. Correlations for direct effects between all models can be found in Table 4. As expected, correlations for Bwt were the highest as there are no age ranges for this measurement. Due to the ability of the longitudinal model to incorporate changes in (co)variance, the correlations were lower for Wwt and Ywt. However, these decreases were relatively small in correlations that involved models utilizing d2. This suggests that the changes in variance within the MTM ranges are small. The decrease in correlations of RRML-d1 and RRMS-d1 with MTM for Wwt and Ywt were larger, indicating that the inclusion of all available records does affect breeding value predictions. However, due to the low numbers of outlying measurements in the data set, these correlations were still >0.95. Inclusion of large amounts of data measured outside the MTM age ranges could result in even lower correlations.
Correlations for maternal effects are found in Table 5. Some discrepancy existed between RRML-d2 and RRMS-d2 for weaning and yearling maternal effects. The high correlations for Bwt were expected because the models were equivalent when estimating effects at birth. The RRMS-d2 had high correlations at weaning as well. The RRML, however, could have some trouble fitting a function for maternal effects at weaning due to large biological differences between birth maternal effects and weaning maternal effects, resulting in lower correlations for Wwt maternal effects. Birth maternal effects are largely a measure of amniotic effects, whereas later maternal effects are a measure of milking. The lower correlations for both models at yearling age are relatively unimportant as maternal effects post-weaning are merely residuals of earlier maternal effects. Albuquerque and Meyer (2001) found that heritabilities for maternal effects peak around 110 to 120 d of age, meaning higher responses would be expected if selection on maternal effects in this age range were practiced. Unlike the RRML, the current MTM does not predict maternal breeding values at this age. Both the RRMS-d2 and RRML-d2 had high correlations with the MTM for maternal permanent environmental effects at all ages. The correlations, reported in Table 6, indicate that the longitudinal models and the MTM are very similar in prediction of maternal permanent environmental effects.
In Table 7, sire rank correlations between MTM and the longitudinal models are presented. These correlations are for sires with 50 or more progeny records for at least one of the three traits. The high rank correlations indicate that implementing a longitudinal evaluation will have little impact on the rank of moderate- to high-accuracy sires with greater than 50 progeny when the models use only information measured within the usual MTM age ranges. However, lower rank correlations between RRML-d1 and RRMS-d1 with MTM were observed for Wwt and Ywt. Table 8 shows that when all available data are used the number of progeny records increases considerably for some sires. Furthermore many new sires are generated by the inclusion of all available records. Table 9 shows that the increase in records does have an effect on predicted breeding values, especially Wwt breeding values. These factors contribute to the lower sire rank correlations when all available data are used.
Although no age restrictions were placed on d1, the majority of records in d1 were close to the MTM age ranges. Although the RRML and RRMS evaluations were similar when using d1, the models may not agree when large numbers of early Wwt and late Ywt records are present. A three-knot spline was used in RRMS, with the third knot at 365 d. As a result, the variance of records measured after 365 d of age was determined using an extension of the linear function between weaning and yearling knots. As an animal’s age increases beyond 365 d, the interpolation becomes less accurate. Bohmanova et al. (2004) compared variances between RRML and RRMS. Although variances and correlations were quite similar for points between Wwt and Ywt, the variances as approximated by RRMS around 100 d were visibly smaller (and correlations larger) than those obtained from the RRML, and the reverse was true for observations over 365 d. Adding a knot at 100 d strongly decreased the differences in variances and correlations. When many records are available around 100 d and over 400 d, it may be useful to add extra knots to RRMS. Furthermore, as age increases, growth approaches an asymptotic value that, in addition to little data being available, can cause high-order polynomial regressions like those used by RRML to become erratic.
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Implications
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The results of this study indicate that longitudinal models can be implemented effectively in beef cattle growth evaluations. The simplicity and easy implementation of the linear spline random regression model makes it an appealing alternative to the current multiple-trait model. The random regression model utilizing cubic Legendre polynomials requires diagonalization; however, it allows for smoother (co)variance functions. Both random regression models give practical and more flexible evaluations, while providing a more theoretically sound alternative to the multiple trait model with relatively small cost of implementation.
1 Correspondence: Rhodes Center for Animal and Dairy Science (phone: 706-542-0965; fax: 706-583-0274; e-mail: krobbin1{at}uga.edu).
Received for publication April 16, 2004.
Accepted for publication September 2, 2004.
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Literature Cited
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Bohmanova, J., I. Misztal, and J. K. Bertrand. 2004. Studies on multiple trait and random regression models for genetic evaluation of beef cattle for growth. J. Anim. Sci.
Kirkpatrick, M., D. Lofsvold, and M. Bulmer. 1990. Analysis of the inheritance, selection, and evolution of growth trajectories. Genetics 124:979–993.[Abstract]
Kirkpatrick, M., W. G. Hill, and R. Thompson. 1994. Estimating the covariance structure of traits during growth and aging, illustrated with lactation in dairy cattle. Genet. Res. Camb. 64:57–69.[Medline]
Legarra, A., I. Misztal, and J. K. Bertrand. 2004. Constructing covariance functions for random regression models for growth in Gelbvieh beef cattle. J. Anim. Sci. 82:1564–1571.[Abstract/Free Full Text]
Meyer, K. 2004. Scope for a random regression model in genetic evaluation of beef cattle for growth. Livest. Prod. Sci. 86:69–83.
Meyer, K., and W. G. Hill. 1997. Estimation of genetic and phenotypic covariance functions for longitudinal or ’repeated’ records by restricted maximum likelihood. Livest. Prod. Sci. 47:185–200.
Nobre, P. R. C., I. Misztal, S. Tsuruta, and J. K. Bertrand. 2003a. Analysis of growth curves of Nellore cattle by multiple-trait and random regression models. J. Anim. Sci. 81:918–926.[Abstract/Free Full Text]
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Abstract
Introduction
