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Estimation of correlation between maternal permanent environmental effects of related dams in beef cattle¹

H. Iwaisaki^*, S. Tsuruta^,2, I. Misztal and J. K. Bertrand

^* Department of Agro-biology, Niigata University, Niigata 950-2181, Japan; and and Department of Animal and Dairy Science, University of Georgia, Athens 30602-2771

	Abstract

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Weaning weights from Gelbvieh (GV; n = 82,138) and Limousin (LM; n = 88,639) calves were used to estimate genetic and environmental variance components with models that included different values for the correlation () between permanent environmental effects of dams and their daughters. Each analysis included fixed discrete effects of contemporary group, sex of calf, age of dam at calving, and month of calving, a fixed continuous effect of age of calf, random direct and maternal additive genetic effects, permanent environmental effects due to dams, and residual effects. The REML procedure was employed with a "grid search," in which the likelihood was computed for a series of values for . For both breeds, models that included a nonzero value for fitted the data significantly better than the model that did not include . The maximum restricted likelihood was obtained for of approximately –0.2 for both breeds. Estimates of residual and direct genetic variances were similar for all values of , including zero; however, estimates of maternal genetic variance and maternal heritability increased slightly, and maternal permanent environmental variance and the proportion of the maternal variance to the total (phenotypic) variance decreased slightly, when the correlated structure for permanent environmental effects was assumed. As the value of became more negative, absolute values of the direct-maternal genetic covariance and direct-maternal correlation estimates were decreased. Pearson and rank correlations for direct genetic, maternal genetic, and maternal environmental effects estimated with and without were very high (>0.99). These results indicated that the linear relationship between maternal permanent environmental effects of dams and their daughters for weaning weight is negative but low in both breeds. Considering this relationship in the operational model did not significantly affect estimated breeding values, and thus, it may not be important in genetic evaluations.

Key Words: Beef Cattle • Genetic Parameters • Maternal Permanent Environmental Correlation • Weaning Weight

	Introduction

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For maternally influenced early-growth traits in beef cattle, biometrical models have been well established (Willham, 1963, 1972; Koch, 1972); however, it would not be easy to estimate all the parameters using these models for field data (Thompson, 1976; Meyer, 1992a). Thus, maternal animal models commonly used in analyses of field data assume no environmental covariance between dams and offspring (e.g., Meyer, 1992b; Waldron et al., 1993; Robinson, 1996a).

Recently, more sophisticated maternal animal models were presented to at least partially account for the nongenetic relationship between maternal effects of two subsequent generations (Koerhuis and Thompson, 1997; Dodenhoff et al., 1998; Quintanilla et al., 1999). Several studies in beef cattle have investigated the influence of a regression on maternal phenotype and grandmaternal effects fit in the model on parameter estimates or genetic evaluations (Meyer, 1997; Dodenhoff et al., 1998, 1999a, Dodenhoff et al., b). The model of Quintanilla et al. (1999) takes into account the correlation between maternal permanent environmental effects of two adjacent generations. Presenting the computational aspects using a Bayesian method, they analyzed a small-sized data set of Spanish beef cattle; however, the effect of using such a model in a national genetic evaluation is unknown. For possible increased accuracy of genetic evaluation and selection, it is necessary to determine the importance of such an environmental correlation in various breeds and to evaluate its influences on predicted direct and maternal effects.

The objective of this study was to compare estimates of direct and maternal genetic parameters and predicted genetic values for weaning weight in large data sets of beef cattle using models that included various correlations between maternal permanent environmental effects with REML procedures (Patterson and Thompson, 1971).

	Materials and Methods

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Data used in the current analyses, provided by the American Gelbvieh (GV) Association and the North American Limousin (LM) Foundation, consisted of 82,138 and 88,639 records of weaning weight for GV and LM breeds, respectively. These data sets included 7,441 sires and 39,678 dams for GV and 11,845 sires and 52,012 dams for LM, respectively. The numbers of animals in the pedigree files were 121,390 for GV and 155,101 for LM. All dams of animals with records were known. Characteristics of the data are summarized in Table 1.

where y was an N x 1 vector of observations, b was a vector of fixed effects, u_d and u_m were q x 1 vectors of random direct and maternal additive genetic effects, respectively, with q representing the number of animals in the pedigree, u_p was an r x 1 vector of random maternal permanent environmental effects with r standing for the number of dams of animals with observations, e was an N x 1 vector of random residual effects, and X, Z_d, Z_m, and Z_p were known incidence matrices relating the observations to b, u_d, u_m, and u_p, respectively. Fixed effects in the model were contemporary group (13,101 and 16,752 levels for GV and LM, respectively), sex of calf (two levels), age (yr) of dam at calving classes (six levels), and month of calving (12 levels). Age (d) of calf at weaning also was considered as a linear covariate. Contemporary groups were defined as herd-year combinations.

The model assumed that:

where A was the additive relationship matrix among all animals, E was a correlation matrix among maternal permanent environmental effects contributed by dams, as given by Quintanilla et al. (1999), I was an identity matrix, was the direct additive genetic variance, _dm was the additive direct-maternal genetic covariance, was the maternal additive genetic variance, was the maternal permanent environmental variance, and was the residual variance.

Following the methodology described by Quintanilla et al. (1999), the E matrix was expressed as E = {g()}, where was the correlation between maternal permanent environmental effects of dams and their daughters, assuming that the dam-daughter environmental relationship was constant across generations. In this matrix, the diagonal elements were ones, the off-diagonal elements corresponding to females related in the maternal pathway were powers of , and the remaining off-diagonal elements were zeros. The inverse of the E matrix was constructed directly from the maternal pedigree. Its nonzero elements were functions of , and they were located on the diagonal and off diagonal between dams and daughters. As pointed out by Quintanilla et al. (1999), including the correlation matrix E provides a computationally feasible method to account for the nongenetic relationship that may exist between the maternal effects of dams and their daughters.

Two models, or a model without considering and a model comprising , were denoted as Models 1 and 2, respectively.

In this study, (co)variance estimation was performed using REML for models that considered values of = 0 ( not fitted in the model), –0.10, –0.20, and –0.30. Applying a numerical strategy of "grid search" for a series of values of , the global maximum of the likelihood with respect to the parameters of the (co) variance components was located during the maximization step of the average information algorithm (Johnson and Thompson, 1995). Convergence was assumed when the Euclidian norm of the vector of first derivatives was less than 10^–10. Computations were conducted using the AIREMLF90 program (Misztal et al., 2002).

Models with various values of including zero were compared by the likelihood ratio test that uses a ² critical value with 1 df.

	Results and Discussion

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The computational scheme using the REML procedure applied to models using nonzero values for performed well for both data sets, and the successful convergence was attained under the stopping criterion given. Quintanilla et al. (1999) performed Bayesian estimation of parameters using Gibbs sampling. Because the model contained the parameter, whose full conditional posterior distribution is not proportional to the known standard ones, in addition to the (co)variance components in the so-called reduced Willham model (Koerhuis and Thompson, 1997), Quintanilla et al. (1999) employed the Gibbs sampling algorithm, carrying out Metropolis-Hasting steps within Gibbs, as described by Tanner (1996). Bayesian analyses could suitably handle the estimation of parameters contained in unconventional models even in very large data sets, although long computation times would likely be required. Consequently, a REML procedure was employed here with a "grid search" in which the restricted log likelihood was calculated for a series of values for . From a Bayesian point of view, given a value of , REML estimates are the elements of the mode of the joint posterior density of all (co)variance components, when flat priors are used for all parameters associated with fixed effects (or nuisance parameters), and nuisance parameters are integrated out from the joint posterior density (e.g., Harville, 1974). An approach similar to the current one was used to estimate the parameters in the so-called integrated Falconer-Willham model, which includes (co)variance components due to direct and maternal effects as well as a regression on maternal phenotype (Koerhuis and Thompson, 1997; Meyer, 1997). All these approaches are nested, two-step procedures that have the advantage of avoiding differentiation of the log likelihood with respect to . In contrast to the Bayesian approach of Quintanilla et al. (1999), the REML procedure using the AI algorithm would be applicable to data sets of limited size; however, it might be computationally more efficient.

Quintanilla et al. (1999) conducted a simulation study where the covariance structure between all maternal effects expressed during lactation and the maternal environment provided to the progeny were included, and investigated the ability of the correlation matrix E to describe the biological model and to estimate the relevant genetic parameters. They found that assuming independence between maternal environmental effects of two generations resulted in biased estimates of all the (co)variance components. The model that included the correlation structure between permanent environmental effects was a simplification of the biological and simulation model, and it was observed that including yielded point estimates of (co)variance components and genetic parameters that were closer to the simulated values.

Parameter estimates for direct and maternal weaning weights from models that used various values of are presented in Tables 2 and 3 for GV and LM, respectively. The likelihood ratio test showed that models with nonzero values for fitted the data significantly better than the model with = 0 for both breeds. The parameter structure and the number of parameters to be estimated were the same for these models. The REML value for of approximately –0.2 was obtained for both breeds. The negative and sizable estimate of here agreed very well with the estimate for a Spanish local beef breed obtained by Quintanilla et al. (1999), who reported the posterior mean for of –0.19 with a posterior probability of 76% that this value was negative. These results indicate that the relationship between maternal permanent environmental effects of two subsequent generations for weaning weight in beef cattle could be negative but low. In their simulation study, Quintanilla et al. (1999) commented that models containing could at least partially explain the existence of a nongenetic relationship between maternal effects in two subsequent generations.

When maternal permanent environmental effects were considered uncorrelated, the direct heritability estimate for weaning weight was moderate in GV, high in LM, and similar to, or slightly higher than, the values obtained by Dodenhoff et al. (1999a). Maternal heritability estimates for weaning weight were smaller than those for direct heritabilities in both breeds. Several studies have observed that estimated maternal heritability was similar in magnitude, or larger than, estimated direct heritability (e.g., Meyer, 1992b; Waldron et al., 1993; Koch et al., 1994; Dodenhoff et al., 1999a). In the model without , the proportion of maternal permanent environmental variance to total (phenotypic) variance was 0.10 and 0.13 for GV and LM, respectively, which were comparable to maternal heritability estimates. Negative estimates with sizable to moderate values were obtained for genetic correlations between direct and maternal effects (–0.26 and –0.59 for GV and LM, respectively). For LM and Angus breeds, Meyer (1997) obtained estimates of approximately –0.5 for the direct-maternal genetic correlation, indicating that the direct-maternal genetic covariance is much more important than its environmental counterpart. Results here for direct and maternal parameters in GV, obtained using a larger-sized data set, support the previous findings by Duangjinda et al. (2001).

Changes in estimates of (co)variance components and genetic parameters showed similar trends for both breeds when the correlated structure between the maternal permanent environment effects was added to the model. Estimates of residual and direct genetic variances did not seem to be appreciably influenced regardless of whether was considered in the model. However, estimates of maternal genetic variance and maternal heritability increased slightly, and maternal permanent environmental variance and the proportion of the maternal variance to the total variance decreased slightly, when a correlated structure for the permanent environmental effects was assumed. Furthermore, as the value of became more negative, the direct-maternal genetic covariance and direct-maternal correlation estimates were decreased. Although the difference in estimates between models with = 0 and < 0 was not large, the tendency of the changes was consistent with that in the simulation study of Quintanilla et al. (1999).

Correlations between estimates of direct genetic, maternal genetic, and maternal permanent environmental effects and most probable producing abilities from two models with and without were examined for three groups of animals with relatively high, moderate, and low predicted values. Pearson and rank correlations were all very high (>0.99) for both breeds (data not shown). These findings suggest that for either breed, considering the correlation between maternal permanent environmental effects of two adjacent generations in operational maternal animal models would not be important in the genetic evaluations of animals. Nonetheless, further investigation using data from beef breeds with relatively high milking ability (which should have the stronger relationships between dams and daughters) might be useful to confirm the current findings. If the number of contemporary groups is high, it could affect estimates of variance components. In addition, we may need more detailed and careful investigation, probably by simulation or a larger data set if possible, on the effect of amount of additive relationships among dams.

Analyzing Australian Angus data, Robinson (1996a) suggested that the negative estimates of direct-maternal genetic correlations were more likely to be a consequence of additional variation between sires or sire x year variation than evidence of a true negative genetic relationship. Also, using simulated data, Robinson (1996b) mentioned the possibility that biased estimates of direct-maternal genetic correlation may be obtained with models that ignore additional sire or sire x year variation. Estimates of ratios of sire x year and sire x herd variances to phenotypic variances have been found to be lower than 10%, and usually are approximately 5% for early growth traits in beef cattle (e.g., Notter et al., 1992; Lee and Pollak, 1997; Dodenhoff et al., 1998). A relatively small fraction of the variance due to the sire x herd interaction accounted for much of the direct-maternal covariance estimated ignoring this interaction effect (Baschnagel et al., 1999). Especially for Angus and LM, Meyer (1997) found that, although estimates of the regression on maternal phenotype were little affected, fitting a sire x herd-year interaction substantially decreased estimates of the direct-maternal genetic covariance and thereby the genetic correlation in absolute value. Using nine sets of field data for weaning weight in Angus cattle, Dodenhoff et al. (1999b) compared certain models to account for the environmental dam-offspring covariance and to investigate the sire x herd-year interaction on direct and maternal parameters. They also reported that the interaction effects were more important than the influence of a dam’s phenotype on her daughter’s maternal ability, or the regression on maternal phenotype and grandmaternal genetic and grandmaternal permanent environmental effects. In those studies where sire x year or sire x herd interaction was fitted as an additional random effect, estimates of direct and maternal variances decreased and even the sign of the estimates for direct-maternal genetic covariances changed. A variety of possible reasons for these findings, including factors such as variation among sires of different origins, heterogeneous variances between herds, preferential treatment, or nonrandom mating among herds, have been reviewed and discussed (e.g., Baschnagel et al., 1999; Dodenhoff et al., 1999b). If the interaction between nonadditive genetic effects and breeds exists, the interpretation of direct-maternal covariances would be more obscure. Further investigation will be needed to reveal the underlying causes of sire x herd or sire x year interaction so that functional models can be constructed to provide better estimates of (co)variances for direct-maternal and maternal permanent environmental effects.

	Implications

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Results indicated that a sizable antagonistic relationship might exist for weaning weight in beef cattle between maternal permanent environments provided by dams and their daughters. Ignoring such a relationship in the statistical model may lead to underestimating maternal heritability and overestimating both the fraction of phenotypic variance due to maternal permanent environmental effects and the magnitude of the genetic correlation between direct and maternal effects. However, biases may not be substantial in genetic evaluations of both direct and maternal effects. Further research may be necessary to confirm these results using data from other beef cattle breeds with relatively strong relationships between dams and daughters.

	Footnotes

 
¹ Appreciation is expressed to the American Gelbvieh Association and the North American Limousin Foundation for providing the data.

² Correspondence—phone: 706-583-0017; fax: 706-583-0274; e-mail:shogo{at}uga.edu.

Received for publication September 13, 2004. Accepted for publication December 10, 2004.

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