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J. Anim. Sci. 2003. 81:927-932
© 2003 American Society of Animal Science

Genetic evaluation of growth in Nellore cattle by multiple-trait and random regression models

P. R. C. Nobre1,{dagger}, I. Misztal2,*, S. Tsuruta*, J. K. Bertrand*, L. O. C. Silva{ddagger} and P. S. Lopes{dagger}

* Department of Animal and Dairy Science, University of Georgia, Athens 30602-2771; and {dagger} University of Vicçosa, Vicçosa MG 36571-000, Brazil; and and {ddagger} Embrapa Beef Cattle, Campo Grande-MS, CEP 79106-970, Brazil

2 Correspondence:
phone: 706-542-0951; E-mail:
ignacy{at}uga.edu.


   Abstract
Top
 Abstract
Introduction
 Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 
The objective of this study was to identify issues in genetic evaluation of beef cattle for growth by a random regression model (RRM). Genetic evaluation data included 2,946,847 records of up to nine sequential weights of 812,393 Nellore cattle measured at ages ranging from birth to 733 d. Models considered were a five-trait multiple-trait model (MTM) and a cubic RRM. The MTM included the effects of contemporary group, age of dam class, additive direct, additive maternal, and maternal permanent environment. Both additive effects were assumed correlated. The RRM included the same effects as MTM, with the addition of permanent and random error effects. The purpose of the random error effect, which was in addition to a residual effect with constant variance, was to model heterogeneous residual variances. All effects in RRM were modeled as cubic Legendre polynomials. Expected progeny differences (EPD) were obtained iteratively using a preconditioned conjugate gradient algorithm. Numerically accurate solutions with RRM were not obtained until the random regressions were orthogonalized. Computing requirements of RRM were reduced by more than 50%, without affecting the accuracy by removing regressions corresponding to very low eigenvalues and by replacing the random error effects with weights. Afterward, the correlations between EPD from RRM and from MTM for EPD on selected weights were between 0.84 and 0.89. For sires with at least 50 progeny, these correlations increased to 0.92 to 0.97. Low correlations were caused by differences in parameters. The RRM applied to growth is prone to numerical problems. Estimates of EPD with RRM may be more accurate than those with MTM only if accurate parameters are applied.

Key Words: Beef Cattle • Genetic Analysis • Regression Analysis


   Introduction
Top
 Abstract
 Introduction
Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 
Currently, the genetic evaluation of beef cattle for growth uses a multiple-trait model (MTM) with traits defined at certain ages of animals (e.g., at birth, 205 and 365 d; BIF, 1996). Because ages at actual weights rarely fit the defined traits, weights obtained at ages close to those defined are preadjusted, and other weights are discarded. The preadjustment is only for the mean and not for variances.

Today there is an interest in longitudinal models for beef, where weights at all ages can be accommodated, and expected progeny differences (EPD) can be obtained at any age of life. Varona et al. (1997) fitted a growth curve to a Bayesian model. Their procedure was suitable only for small populations. Meyer (2000) analyzed continuous growth using random regression models (RRM). Albuquerque and Meyer (2001) estimated parameters of RRM for the same breed as that in this study. The analyses of weights as a longitudinal trait may result in increased accuracy of evaluation by eliminating the need for preadjustment by its ability to incorporate all weights with appropriate covariances. Meyer (2002) estimated that RRM increased accuracy of EPD up to 6% using simulated data. However, actual gains with field data sets are unknown; RRM may result in lower accuracy than MTM if parameters for RRM are poor or computations are inaccurate.

Computer programs used for RRM in dairy are complicated and extensively optimized (Jamrozik and Schaeffer, 2000). Models in beef cattle may be more complicated than in dairy because of correlated direct and maternal effects. Tsuruta et al. (2001) developed a computer program that supports large data sets by using an iteration on data technique with the preconditioned conjugate gradient (PCG) algorithm. That program has sufficiently low memory requirements to support national genetic evaluations.

The objectives of this study were to implement the genetic evaluation of weights for a large population of beef cattle using a RRM and to compare EPD from RRM and MTM.


   Materials and Methods
Top
 Abstract
 Introduction
 Materials and Methods
Results and Discussion
 Implications
 Literature Cited
 
Data
Data were collected by the Brazilian Zebu Breeders Association (ABCZ) and provided by the Brazilian Agricultural Research Corporation (EMBRAPA). The data consisted of records on 619,989 Nellore animals, the progeny of 11,847 sires and 273,263 dams raised under Brazilian pasture conditions. The records were collected from 1975 to 1999. Traits considered were birth weight (BW), and as many as eight sequential weights (W1 to W8) up to 733 d. Records were missing sequentially; all animals had records on BW, 51% had records on W4, and 6% had records on W8. The data set is summarized in Table 1Go; Nobre et al. (2003) provides more detailed description of the data.


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  		Table 1. Characteristics of the data set by traits
 
Models
The two models (MTM and RRM) were the same as that described by Nobre et al. (2003). The MTM was:


where yijklt = tth weight preadjusted for age in contemporary group i, age-of-dam class j of animal k and dam l; cgit = fixed effects of contemporary group i and weight t; aodjt = fixed effects of age of dam class j and weight t; dkt = random direct genetic effect of animal k and weight t; mlt and mpelt = random maternal genetic and maternal permanent environment effect of dam l and weight t; and eijklt = random residual effect. The direct and maternal genetic effects were assumed correlated. This model can be written in matrix notation as:


where y = vector of records preadjusted to fixed age; ß = vector of fixed effects (contemporary group and age of dam class); d = vector of additive direct genetic random effects of the animal; m = vector of additive maternal genetic random effects; mp = vector of random effects of maternal permanent environment; e = vector of residual random effects X was the incidence matrix for fixed effects; and Z1, Z2, and Z3 were incidence matrices for animal, maternal, and maternal permanent environmental effects, respectively.

The variances and covariances were defined as follows:


where G0d was a covariance matrix of random direct genetic effects; G0m was a covariance matrix of random maternal genetic effects; G0dm was a matrix of genetic covariances between direct and maternal effects; G0mp was a covariance matrix of random maternal permanent environmental effects; R0 was a covariance matrix of random residual effects; A was the additive genetic relationship matrix; Ic was an identity matrix whose order was the number of dams; In was an identity matrix whose order was the number of animals; and {otimes} was the direct product operator.

(Co)variance components were estimated for five traits at a time. Parameters presented here were based on averages from analyses of models that contained that particular parameter.

The RRM was defined as follows:


where yijklm = observation in contemporary group i, age of dam class j, animal k, dam l, and record m; agej = fixed regression coefficient for age of animal; cgdi = fixed regression coefficient for contemporary group i; caddj = fixed regression coefficient d for age of dam class j; ddk and pdk = random regression coefficients d for additive direct and permanent environmental effects of animal k; mdl and mpdl = random regression coefficients d for additive maternal and maternal permanent environmental effects of dam l; rdm = random regression coefficient d for error effects of record m; zdm = dth coefficient of Legendre polynomial for age at observation m; and {varepsilon}ijklm = the residual effect. The purpose of the error effect was to indirectly model heterogeneous residual variance (Van der Werf and Schaeffer, 1997); the available software did not allow modeling this directly.

The random regression model could be written in matrix notation as:


where yR was the vector of records; ßR was the vector of fixed effects; dR, p, mR, mpR, and r were vectors for additive direct genetic, permanent environment, additive maternal genetic, maternal permanent environmental, and error effects, respectively; XR was the incidence matrix for fixed effects; ZRd, ZRm, and ZRr were incidence covariate matrices for direct, maternal, and error effects, respectively; and eR was a vector of residual effects.

The variances were defined as follows:



where GR0d, GR0m, and GR0dm were 4 x 4 covariance matrices of random regression for direct effect, genetic effect, and their covariances, respectively; G0p, GR0mp, and RR0 were 4 x 4 covariance matrices of random regression for permanent environment, maternal permanent environment, and error effects, respectively; was assumed constant residual variance; A was an additive genetic relationship matrix; Ik was an identity matrix whose order was the number of animals; Il was an identity matrix whose order was the number of dams; and In was an identity matrix whose order was the number of records.

For observation m containing trait t and with Legendre polynomials corresponding to age for trait t, the residual effect in MTM are approximately equivalent to the sum of residual, permanent environment and error effects in RRM:


Diagonalization
Consider the random regression part of a model for one effect and specific age m:


where zdm = dth covariable for at age m and adi = dth regression coefficient for random effect i. In general, these coefficients are correlated and G0i is not diagonal. For better numerical properties of mixed model equations, the coefficients can be orthogonalized and their variances diagonalized. Compute Vi and Di as eigenvectors and eigenvalues of Goi:


Then, random regressions can be transformed as:


where zmi* = Vi'zm are the new covariables; ai* = Vi'ai is the new regression vector for effect i. The covariance structure for ai* is diagonal: Var (ai*) = Di. Such a transformation can be done separately for each effect. In this case, covariables zmi* are no longer identical for each effect because they depend on Go for a particular effect. After the diagonalization, solutions are computed on a new scale. The solutions on the original scale can be obtained as ai = Via*i.

In the transformation above, some eigenvalues may be very close to zero. Regression coefficients corresponding to those covariables have values close to zero. Consequently, these coefficients can be dropped from the model with a negligible decrease in accuracy but at noticeable savings in computations.

Heterogeneous Residual Variances as Weights
In the RRM, the error was modeled as a fixed residual and a variable effect r. The combined value of the residual + error variance for observation i was:



where R = (co)variance matrix of error effects; = residual variance; and z was the vector of covariables. If the combined residual variance is computed for each observation as above, the same variance can be presented as:



where wi can be called the weight of observation i. When BLUP software supports weights, as was the case in this study, the effect r can be eliminated at a considerable saving in computations.

Parameters
Parameters for both models were as reported by Nobre et al. (2003) for a sample with missing traits. Shapes of variances from RRM showed more fluctuations than those from MTM, especially at later ages; the estimates of direct variance for BW were more than two times lower and for W8 more than two times higher in RRM. Also, the direct-maternal correlations were more negative with RRM.

Parameters for the RRM could be converted from MTM. However, such a conversion is not obvious and is a topic for a separate study. For example, estimation procedures for such a conversion would require putting appropriate weights on the accuracy of matching a wide range of variances and covariances. Also, the RRM includes a permanent environmental effect whereas MTM does not.

Computing Procedures
The EPD were obtained by program BLUPF90 (Misztal, 1999), with solutions obtained by the sparse-matrix factorization package FSPAK90 (Misztal and Perez-Enciso, 1998) and by BLUP90IOD, which uses an iteration on data with the PCG solver (Tsuruta et al., 2001). The first program computed exact solutions in the absence of numerical errors, but it required much higher computing resources. The second program was iterative and computed increasingly more accurate solutions as the iteration progressed. The convergence criterion for that program was defined as the relative average squared differences between consecutive solutions; two criteria were used: 10-10 (called lower accuracy) and 10-12 (called higher accuracy).

Computations
Initially, EPD were obtained by programs BLUPF90 via FSPAK90 and BLUP90IOD via PCG with MTM and RRM using a sample of the data as mentioned above. Due to computing limitations, the MTM was a five-trait model using W1, W2, W3, W5, and W7. The RRM used all weights available. Solutions by RRM were calculated before and after diagonalization and with lower and higher accuracy for BLUP90IOD. Subsequently, the computations were repeated for the complete data set, but only with program BLUP90IOD because the computing requirements for BLUPF90 were excessive. Correlations between solutions from various runs were computed for five traits and direct and maternal effects separately.


   Results and Discussion
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 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
Implications
 Literature Cited
 
With the sample data set, the correlations between MTM for direct and maternal effects and various traits obtained with BLUPF90 and BLUP90IOD were higher than 0.99, but the same correlations for the original RRM were lower than 0.27 when the convergence criterion was set to 10-10, and lower than 0.43 when set to 10-12. Thus, the PCG iteration with the diagonal preconditioner was very accurate for the MTM but not accurate for the RRM. In general, the RRM may have a very slow convergence rate due to a large number of equations and subsequently poor numerical properties, especially when the polynomials are far from orthogonal and corresponding (co)variance matrices are ill conditioned. Lidauer et al. (1999) reported that estimation of breeding values with RRM required greater accuracy in the solutions of mixed-model equations than with the single-trait animal model.

Table 2Go shows the values of eigenvalues corresponding to covariances in each effect of the RRM. For the genetic effects, the first two eigenvalues explained 84% of the genetic variance, and the last eigenvalue was close to zero. Also, for permanent environmental effects, the first two eigenvalues explained 98% of variance. For maternal permanent environmental effects and residual effects, those eigenvalues explained 97 and 90% of variances, respectively. Small eigenvalues indicated that parameters of RRM were indeed poorly conditioned and also indicated potential of reducing the number of effects in the model.


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  		Table 2. Eigenvalues for genetic (G), permanent environmental (Pe), maternal permanent environmental (MPe), and residual (R) effects with a random regression model
 
When the diagonalization was done for all the effects, low correlations remained until the covariance matrices were recreated as VD*V', where D* was like D but with small eigenvalues set to zero. Afterwards, the correlations for RRM computed with iterative and finite algorithms increased to 0.979 with convergence criterion set to 10-10 and to 0.998 when set to 10-12. This indicated that the diagonalization and stricter convergence criterion were essential in obtaining accurate EPD from RRM by iteration. When the error effect was replaced by weights and effects corresponding to very small eigenvalues were dropped from the model, the correlations remained the same, but memory and computing time requirements decreased more than two times (Table 3Go); computing times refer to a 550-MHz class personal computer.


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  		Table 3. Number of rounds until convergence, memory, and time requirements for random regression models for the complete model (31 effects) after replacing the error effect by weights (27 effects), and additionally after removing effects associated with small eigenvalues (22 effects)a
 
Estimated correlations between EPD by MTM and RRM obtained with the complete data by BLUP90IOD with convergence criterion set to 10-12 are reported in Table 4Go. The correlations were less than 0.90 for all animals and increased up to 0.97 for sires with >=100 progeny. Higher correlations were expected, and the correlation close to 1.0 was expected for W1, for which the definition in the MTM is accurate.


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  		Table 4. Estimated correlations between expected progeny differences with multiple trait and random regression models for all animals and for sires with >=20, >=50, and >=100 progenies
 
The MTM used only up to five weights per animal, whereas the RRM used up to nine. This could decrease correlations between RRM and MTM especially for later ages. However, the absence of some traits in MTM would make a small difference to animals with a large number of records. However, the direct (maternal) correlations were as low as 0.95 (0.92) for sires with >=100 progeny. This suggested that important differences in EPD between RRM and MTM existed due to factors beyond fewer traits used in MTM.

To exclude the effects of unequal amounts of information in comparisons, solutions by MTM and RRM were also obtained using only data for BW. Also, there is no age variability for BW; therefore, given numerically accurate solutions and functionally identical parameters, the MTM and RRM should provide identical results for BW. The genetic correlations between MTM and RRM were very similar to those obtained before. Differences for BW could be due to larger numbers of fixed effects to estimate in RRM. However, dropping all but constant terms for covariables in RRM did not change the correlations. Therefore, assuming numerically accurate computations, differences between RRM and MTM were largely due to differences in parameter estimates used in both models.

Estimated genetic parameters by RRM can be inaccurate for various reasons: data set size, data selection, model and methodology applied. However, the parameters can be estimated more accurately after improvements in methodologies, making computations more reliable and less expensive (Misztal et al., 2000; Pool et al., 2000).


   Implications
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
 Implications
Literature Cited
 
Random regression models could be useful in beef cattle genetic evaluation because weights at any age can be used, and expected progeny differences can be estimated for any age. For growth in beef cattle, expected progeny differences by random regression models can be inaccurate because of the poor numerical properties of these models. These properties can be improved by orthogonalizing covariables for each random effect. Thereafter, a computing package using the preconditioning conjugate gradient iteration can compute expected progeny differences for national data sets within a reasonable time. Even if the numerical properties of the random regression model are adequate, expected progeny differences may not be better than those from a multiple-trait model if the parameter estimates used in the random regression model are poor.


   Footnotes
 
1 Current address: Embrapa Beef Cattle Breeding Program—Geneplus, Brazil. Back

Received for publication July 3, 2002. Accepted for publication December 24, 2002.


   Literature Cited
Top
 Abstract
 Introduction
 Materials and Methods
 Results and Discussion
 Implications
 Literature Cited
 


Albuquerque, L. G., and K. Meyer. 2001. Estimates of covariance functions for growth from birth to 630 days of age in Nellore cattle. J. Anim. Sci. 79:2776–2789.[Abstract/Free Full Text]

BIF. 2002. Guidelines for Uniform Beef Improvement Programs. Beef Improv. Fed., Athens, GA.

Jamrozik, J., and L. R. Schaeffer. 2000. Comparison of two computing algorithms for solving mixed model equations for multiple trait random regression test day models. Livest. Prod. Sci. 67:143–153.[Medline]

Lidauer, M., I. Stradén, E. A. Mäntysaari, J. Pösö, and A. Kettunen. 1999. Solving large test-day models by iteration on data and preconditioned conjugate gradient. J. Dairy Sci. 82:2788–2796.[Abstract]

Meyer, K. 2000. Random regression to model phenotypic variation in monthly weights of Australian beef cows. Livest. Prod. Sci. 65:19–38.[Medline]

Meyer, K. 2002. Accuracy of genetic evaluation of beef cattle for growth fitting a random regression model in genetic evaluation. J. Anim. Sci. 80(Suppl. 1):49.

Misztal, I. 1999. Complex models, more data: Simpler programming? Proc. Inter. Workshop Comput. Cattle Breed, Tuusala, Finland. Interbull Bul. 20:33–42.

Misztal, I., and M. Perez-Enciso. 1998. A Fortran 90 interface to sparse matrix package FSPAK with dynamic memory allocation and sparse matrix structure. Proc. 6th World Cong. Genet. Appl. Livest. Prod., Armidale, Australia22:77–78.

Misztal, I., T. Strabel, J. Jamrozik, E. A. Mäntysaari, and T. H. E. Meuwissen. 2000. Strategies for estimating the parameters needed for different test-day models. J. Dairy Sci. 83:1125–1134.[Abstract]

Nobre, P. R. C., I. Misztal, S. Tsuruta, J. K. Bertrand, L. O. C. Silva, and P. S. Lopes. 2003. Analyses of Growth Curves of Nellore Cattle by Multiple Trait and Random Regression Models. J. Anim. Sci. (In press)

Pool, M. H., and T. H. E. Meuwissen. 2000. Reduction of the number of parameters needed for a polynomial random regression test day model. Livest. Prod. Sci. 64:133–145.[Medline]

Tsuruta, S., I. Misztal, and I. Strandén. 2001. Use of preconditioned conjugate gradient algorithm as a generic solver for mixed-model equations in animal breeding applications. J. Anim. Sci. 79:116–1172.

Varona, L., C. Moreno, L. A. Garcia Cortes, and J. Altarriba. 1997. Multiple trait genetic analysis of underlying biological variables of production functions. Livest. Prod. Sci. 47:201–209.


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