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Estimating maternal genetic effects in livestock¹

Animal Breeding and Genetics Group, Wageningen University, Wageningen, The Netherlands

	Abstract

Top Abstract INTRODUCTION THEORY AND RESULTS DISCUSSION IMPLICATIONS LITERATURE CITED

 
This study investigates the estimation of direct and maternal genetic (co)variances, accounting for environmental covariances between direct and maternal effects. Estimated genetic correlations between direct and maternal effects presented in the literature have often been strongly negative, and their validity has been questioned. Explanations of extreme estimates have focused on the existence of environmental covariances between dam and offspring. As a solution, models including a regression on dam-phenotype have been proposed, but have yielded biased estimates. The performance of models that implement the variance structure arising from the classical model of Willham, however, has not been evaluated. This study investigated the covariance structure of the parts of the residual term that arise from Willham’s model. Results show that a correlation between the residual of the record of an individual and that of its dam is a direct consequence of combining Willham’s model with the usual assumption that phenotypic covariances between different traits are the sum of additive genetic and environmental covariances. Stochastic simulations show that fitting this structure yields unbiased estimates of the genetic (co)variances. When correlated residuals were ignored in the cases investigated, the bias in the estimated genetic correlations was approximately equal to the value of the environmental correlation. In contrast to models including a regression on dam-phenotype, there were no difficulties with interpretation of results, and the approach was consistent with standard quantitative genetic theory. The use of Willham’s model while accounting for correlated residuals is conceptually appealing and yields unbiased results, with no need for regression on dam phenotype. Inclusion of the ability to fit the residual variance structure required for maternal effects into existing software packages would be helpful to animal breeders.

Key Words: bias • environmental covariance • genetic parameter • maternal effect • model • variance component estimation

	INTRODUCTION

Top Abstract INTRODUCTION THEORY AND RESULTS DISCUSSION IMPLICATIONS LITERATURE CITED

 
Many traits of interest in livestock production are affected by maternal effects. For such traits, Dickerson (1947) and Willham (1963) presented a general quantitative genetic model in which a trait is the sum of a direct effect due to the individual with the measured phenotype and a maternal effect due to its mother. Their model is commonly used in livestock genetic improvement. Estimates of genetic correlations between direct and maternal effects, however, often have been strongly negative (e.g., Robinson, 1996; Meyer, 1997; Cheverud, 2003).

Explanations of extreme estimates have focused on the existence of environmental covariances between records of dam and offspring, and on the fixed effects structure used in statistical models for data analyses (Robinson, 1996; Koerhuis and Thompson, 1997; Meyer, 1997). The statistical model predominantly used to analyze maternal effects does not account for such environmental covariances between records of dam and offspring (Quaas and Pollak, 1980; Henderson, 1988). When ignored in the statistical analyses, environmental covariances between dam and offspring may bias the estimate of the genetic correlation between direct and maternal effects (Koch, 1972).

To account for environmental covariances between dam and offspring records, Robinson (1996) suggested a model including a regression on maternal phenotype (see also Falconer, 1965; Dodenhoff et al., 1999). Using simulated data, however, Robinson (1996) and Koerhuis and Thompson (1997) observed that such models still yielded biased estimates.

The goal of this study was to investigate the covariance structure that arises among relatives when traits follow the model of Willham (1963). Simulations were used to investigate the consequences of ignoring environmental covariances between dam and offspring records and to investigate the biases of estimates when fitting a statistical model that corresponds to Willham’s (1963) model. Finally, the interpretation of models including a regression on maternal phenotype was examined.

	THEORY AND RESULTS

Top Abstract INTRODUCTION THEORY AND RESULTS DISCUSSION IMPLICATIONS LITERATURE CITED

 
Willham’s Model
This section investigated the covariance structure that arises when traits follow Willham’s model. First consider a single offspring per dam, in which case permanent environmental effects of the dam can be ignored.

One Offspring Per Dam.
With Willham’s model, the observed phenotype of individual i (P_i) is the sum of a phenotypic direct effect due to the individual itself (P_D,i) and a phenotypic maternal effect due to its dam j (P_M,j; Willham, 1963),

(1)

The direct and maternal effects are not observed; they are the phenotypic effects of offspring and dam underlying the observed phenotype P_i. The maternal effect is a phenotypic effect; the genotype of the dam is part of the maternal phenotype. For example, the maternal effect on juvenile growth in beef cattle originates predominantly from milk yield of the dam, which has a heritability of about 30% (Meyer et al., 1994). Most of the observed variability due to the maternal effect, therefore, has a non (additive) genetic origin.

It follows from quantitative genetic theory that phenotypes for direct and maternal effects are the sum of breeding values (A) and a sum of nonadditive genetic and environmental effects usually combined as "environment," E (Willham, 1963),

(2)

Applying Eq. 2 to individual i and to its dam j shows that the phenotypic covariance between dam and offspring equals

(3)

in which ²_AD and ²_AM are the additive genetic variance for the direct and maternal effect, and _ADM is the direct-maternal additive genetic covariance. The last term of Eq. 3 is the environmental covariance between the direct effect of the dam, expressed in her own phenotype, and the maternal effect of the dam, expressed in the phenotype of her offspring. It is the nongenetic part of the full phenotypic covariance, Cov(P_D,j, P_M,j), which is a covariance between 2 distinct traits due to a single individual, but observed in different records. Thus, we may write Cov(P_D,j, P_M,j) = Cov(A_D,j, A_M,j) + Cov(E_D,j, E_M,j), as usual in quantitative genetics.

With some exceptions (e.g., Koerhuis and Thompson, 1997; Meyer, 1997; Dodenhoff et al., 1999), the last term of Eq. 3 is usually ignored in variance component estimation. When omitting Cov(E_D,j, E_M,j) from the statistical model, it is assumed implicitly that either maternal effects are fully heritable (²_EM = 0 and ²_AM 0) or that direct and maternal effects are correlated genetically but not environmentally (r_ADM [–1, +1] and r_EDM = 0). Heritabilities are, however, usually less than 1, and the common resemblance between phenotypic and genetic correlations (see Lynch and Walsh, 1998, for a review) indicates that genetic and environmental correlations may have similar magnitude. Omitting Cov(E_D,j, E_M,j) from the statistical model, therefore, represents a strong a priori assumption, which disagrees with common animal breeding knowledge. There is no reason to expect that environmental covariances between dam and offspring are restricted to special cases, such as the fatty udder syndrome in beef cattle. Environmental covariances between dam and offspring are likely to be a general phenomenon.

It follows from Eq. 3 that in a statistical model for data analyses, y_i = fixed + A_D,i + A_M,j + e_i, in which e_i covers both E_D,i and E_M,j, the residuals of an individual and its dam will be correlated, Corr(e_i, e_j) = Cov(E_D,j, E_M,j)/Var(e). Hence, a statistical model that is consistent with Eq. 2 is

(4)

in which Xb represents fixed effects; Z_Da_D and Z_Ma_M represent direct and maternal breeding values; e represents the residual, R_ii = 1, R_ij = , when i and j are a dam-offspring pair, but R_ij is zero otherwise, with _e² = ²_ED + ²_EM. Thus, residual variances are homogeneous, and residuals of dam and offspring are correlated with = _EDM/(²_ED + ²_EM). The value of needs to be estimated.

Single Litters with Multiple Offspring.
When litters consist of multiple offspring, nongenetic covariances between littermates may arise because of the environmental component of the maternal effect and because of other environmental effects common to littermates, so that

(5)

in which E_C,j represents the nonmaternal environmental effect that is common to littermates (e.g., due to physical conditions, such as light or temperature). Conditional on the breeding values, the covariance between records of littermates, say i and j, is

(6)

which is a positive value. The covariance between records of dam and offspring, say i and k, is

(7)

which may take either positive or negative values. Thus, there are 2 types of nongenetic covariances between individuals, which can be accounted for in different ways in the statistical analyses. Equation 4 can be used when the residual covariance structure is extended to:

(8)

in which ²_e = ²_ED + ²_EM + ²_EC, ₁ = (²_EM + ²_EC)/ ²_e, and ₂ = _EDM/²_e. Hence, analysis of maternal effects with data on single litters of multiple offspring involves the estimation of 2 residual correlations.

Alternatively, following Koerhuis and Thompson (1997), a random nongenetic litter effect (also referred to as common or permanent environment) can be added to the model,

(9)

in which Z_C connects observations to litters, and e_C contains an element for each litter. In Eq. 9, e_C covers both E_M and E_C, whereas e covers E_D. Consequently, the common environmental effect contributing to the phenotype of an individual will be correlated to the residual of the dam, Cov(e_C, e') = B_EDM, in which B connects litters to dams. Thus, a model with a common environmental effect (Eq. 9) requires fitting a correlation between the common environmental effect and the residual of the dam (Koerhuis and Thompson, 1997).

Multiple Litters with Multiple Offspring Per Litter.
With multiple litters of the same dam, we may distinguish between nongenetic effects that are specific to a litter, E_C,l and nongenetic effects that are common to all offspring of a dam, E_M,j (i.e., a permanent dam effect),

(10)

in which j(l) denotes the litter l of dam j. Now there are 3 residual covariances. First, the nongenetic covariance among littermates,

(11)

Second, the covariance among offspring of the same dam (maternal sibs) born in different litters,

(12)

Third, the covariance between dam and offspring,

(13)

A statistical model corresponding to this variance structure is given by Eq. 4, when the residual structure is extended to:

(14)

in which ²_e = ²_ED + ²_EM + ²_EC, ₁ = (²_EM + ²_EC)/ ²_e, ₂ = ²_EM/²_e, and ₃ = _EDM/²_e. Alternatively, Eq. 9 can be extended by including an additional random effect for each litter, distributed independently of other effects. Thus, with multiple litters of multiple offspring each, either 3 residual correlations need to be estimated, or 2 nongenetic random effects need to be estimated, one for the litter and one for the dam, combined with a single residual correlation between the nongenetic dam effect and the residual of the dam.

Correlated residuals are fitted to improve the estimated genetic variance components; usually there will be no interest in the values of the residual correlations per se. There is no need, therefore, to distinguish between E_M,j and possible other environmental effects common to all litters of a dam (e.g., the physical effect of the pen when all litters are born in the same pen.) Though existence of such effects will affect the magnitude of the residual correlations, the variance structure remains the same.

Simulations.
Simulated data were used to investigate bias of the estimated genetic variance components, when either ignoring or accounting for environmental covariances between direct and maternal effects. Simulated genetic and environmental correlations between direct and maternal effects were either zero or –0.3 (Table 1). Two statistical models were used to analyze the data. The model accounting for environmental correlations was equal to Eq. 4. The model ignoring environmental correlations had Var(e) = I²_e. Both models were implemented using ASREML (Gilmour et al., 2002). Equation 4 was implemented by fitting a moving-average time-series for the variance structure of the residual (Box and Jenkins, 1976; Gilmour et al., 2002). A first-order moving-average series is given by e_t = _t – _t–1, in which _t is distributed independently and || < 1. Thus, ²_e = (1 + ²) ², and Cov(e_t, e_t–1) = –², so that the correlation between residuals of consecutive records is = –/(1 + ²).

Estimates obtained using Eq. 4 were unbiased in all cases (Table 1). Estimates obtained assuming independent residuals were biased when the environmental correlation between direct and maternal effects deviated from zero. In the cases studied, bias of the estimated genetic correlation was approximately equal to the value of the residual correlation; estimates corresponding to true genetic correlations of 0 and –0.3 were –0.332 and –0.632, respectively, when the residual correlation was –0.3.

Regression Model
Following Falconer (1965), Robinson (1996) proposed to include a regression on dam-phenotype in the model, to account for environmental covariances between records of dam and offspring. Koerhuis and Thompson (1997), Meyer (1997), and Dodenhoff et al. (1999) further investigated the utility of this regression model for estimating genetic variance components. Little attention, however, has been given to the quantitative genetic interpretation of the regression model.

In this section, the relationship between Willham’s model and a model with a regression on dam-phenotype will be examined. In the current notation, the model with a regression on dam-phenotype is

(15)

in which dam(i) denotes the dam of individual i, and ß the linear regression of P_i on the observed phenotype of the dam.

Interpretation of Model Terms.
The meaning of the regression model becomes clearer when P_dam(i) in Eq. 15 is substituted by the model for the phenotype of the dam (i.e., Eq. 15 written for the dam), and substitution is continued to the granddam and earlier female ancestors. Thus, Eq. 15 is substituted repeatedly into itself, showing that

giving

(16)

in which k = 1 denotes the individual itself, k = 2 denotes its dam, k = 3 denotes its granddam, etc., and _i(k) = A_D,i(k) + A_M,dam(i(k)) + E_D,i(k). Considering the direct effect, Eq. 16 shows that the phenotype of an individual not only contains its own direct effect, but in addition ß times the direct effect of its dam, ß² times the direct effect of its granddam, etc.

In comparison to Willham’s model, therefore, the regression model has a different definition of the direct effect; it is no longer restricted to the effect of the individual itself but it is a combination of effects of individual, dam, granddam, etc. In other words, there is no strict separation between direct and maternal effects in the regression model. For example, when ß = –0.2, 20% of the dam’s breeding value for direct effect is subtracted from P_i, and 4% of the granddam’s breeding value for direct effect is added to P_i. Similarly, the maternal effect no longer refers strictly to the dam but contains elements due to the granddam and more distant female ancestors. Moreover, the interpretation of direct and maternal effects depends on the value of ß; there is no longer an interpretation that is valid in general.

The series

converges to 1/(1 – ß). Thus, when ß = –0.2, the phenotype contains a total of 1/1.2 0.83 direct breeding values, maternal breeding values, and direct environmental effects. With Willham’s model, in contrast, the phenotype contains the full direct and maternal breeding values, irrespective of the environmental covariance between dam and offspring records.

Inheritance of breeding values from parents to offspring follows from decomposing breeding values in Eq. 15 and 16 into half the breeding value of the sire, plus half the breeding value of the dam, plus a Mendelian sampling term, giving

other terms. This result shows that inheritance is no longer Mendelian in the regression model. For example, when ß = –0.2, the sire contributes 50% of its direct breeding value to the offspring, but the dam only 30%.

Response to Selection.
Using Eq. 15, and taking expectations of the current and previous generation, shows that response to selection in the regression model equals

(17)

Selection response, therefore, depends not only on the change in true breeding values but also on ß. Hence, though ß was introduced to account for an environmental correlation, it is a component of genetic change over generations, indicating that there is no complete separation between genetic and environmental components in the regression model.

The dependency of the selection response on ß hampers the comparison of estimates of variance components originating from different data sets, as will be shown next. Consider 2 populations with identical genetic variance components, but population A has r_EDM = 0, whereas population B has r_EDM < 0. When assuming that ß corresponds to the ratio of the direct-maternal environmental covariance over the phenotypic variance (Meyer, 1997), then ß = 0 for population A and ß < 0 for population B. Furthermore, suppose that Willham’s model is the true model. The response to mass selection is identical in both populations because it is independent of r_EDM with Willham’s model,

in which denotes the intensity of selection and _P the phenotypic standard deviation (Willham, 1963).

Despite the difference in ß, therefore, the response to mass selection calculated from Eq. 17 should be identical for both populations,

However, this expression can be true only when the variance components differ between both populations. Hence, when populations have identical genetic variance components but a different environmental covariance between direct and maternal effects (i.e., different ß), a regression model yields either different variance components for both populations or different predictions of response to mass selection, whereas both should be equal when maternal effects follow Willham’s model.

When assuming that the expression for response to selection is correct, estimates of variance components will differ between both populations, even though true values are identical. Consequently, comparison of estimates of genetic parameters among populations is meaningful only when the value of ß is identical in both populations and when there is no longer an interpretation for direct and maternal genetic (co)variance components that is generally valid.

	DISCUSSION

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Model Choice.
All models are approximations of reality; the true model underlying the data is unknown. I chose to simulate data using Willham’s model, whereas Robinson (1996) and Koerhuis and Thompson (1997) simulated data using the regression model. For genetic improvement of livestock, models are a means to an end; they are used to estimate variance components and breeding values, with the ultimate aim to facilitate genetic improvement. Although models cannot be said to be right or wrong, their contribution to our understanding of heritable variation in livestock populations and to genetic improvement can be investigated.

In Willham’s model, the observed phenotype may be due to both a direct and a maternal effect (Eq. 1), both of which may have a heritable component (Eq. 2), and there may be both a genetic and an environmental correlation between direct and maternal effects (Willham, 1963; Lynch and Walsh, 1998). Willham’s model, therefore, is a very general description for the inheritance of maternally affected traits. The variance structure described by Eq. 4 is the natural result of combining Willham’s model with standard quantitative genetic theory, in which phenotypic covariances between traits are the sum of additive genetic covariances and environmental covariances (e.g., Falconer and Mackay, 1996). There is no difficulty with interpreting Eq. 2; it can be interpreted as is usual in animal breeding. Breeding values, for example, refer directly to expected performance of offspring.

With the regression model, the regression term is a combination of breeding values and nongenetic effects of previous generations (see Theory and Results). This causes several difficulties with interpretation of model terms: i) the interpretation of direct and maternal breeding values depends on the value of the regression coefficient, ii) inheritance of breeding values is no longer Mendelian, iii) changes in breeding values are no longer sufficient to measure genetic response to selection, and iv) comparison of estimates of genetic variance components between data sets is meaningful only when the value of the regression coefficient is equal for those data sets. Essentially, a regression model introduces an additional genetic component, which is confounded with other genetic and environmental components. Moreover, there is no need for the regression model because Willham’s model implemented with correlated residuals yields unbiased results. Thus, though a regression model can in principle be used to describe the inheritance of maternal effects, its use in livestock genetic improvement raises considerable difficulty with interpretation of results.

When analyzing field data, unexpected results may be obtained for many reasons. There is no reason to assume that strongly negative estimates of the direct-maternal genetic correlation are solely due to negative environmental correlations. In beef cattle, for example, negative estimates of the direct-maternal genetic correlation have sometimes been attributed to negative environmental dam-offspring covariances but also to the fixed effects fitted (e.g., a sire x herd interaction; Meyer, 1997). For broiler chickens, Koerhuis and Thompson (1997) found that models accounting for environmental dam-offspring covariances yielded estimates of genetic correlations that were somewhat more negative than estimates obtained when ignoring environmental covariances, whereas fitting more detailed fixed effects considerably reduced absolute values of the estimates.

Accounting for environmental dam-offspring covariances, therefore, will not always prevent extreme estimates for the genetic correlation. The purpose of the present paper, however, was not to present a "true" model, but to investigate the interpretation of alternative models in the light of quantitative genetic theory and experience. The observation that genetic and phenotypic correlations are usually very similar (Lynch and Walsh, 1998) indicates that estimating direct-maternal genetic correlations while assuming zero environmental correlations represents a strong a priori assumption. Results in Table 1 indicate that, even with a moderate environmental correlation of –0.3, the estimated genetic correlation may be biased substantially. Before assuming that environmental correlations are zero, therefore, it is important to empirically validate this assumption.

Implementation.
Depending on the data structure, analyses of maternal effects while accounting for environmental dam-offspring covariances may be feasible within standard software, such as ASREML (Gilmour et al., 2002). In the simulations, dams had a single offspring only, in which case the residual of a female was correlated to the residuals of her dam and of her offspring. This residual variance structure was fitted with ASREML by using a moving average structure for the residual within dam line. When females have a dam but no offspring in the data, a simple correlation between the residuals of dam and offspring can be fitted using the CORU statement in ASREML. In another study that approach was used to estimate genetic parameters for calving ease traits in dairy cattle (our unpublished observations). There seems to be no software available that can fit the required residual variance structure for general data sets (Eq. 14). Inclusion of such capability in existing software packages would be useful to breeders in many species, although application may be restricted to data sets of limited size.

	IMPLICATIONS

Top Abstract INTRODUCTION THEORY AND RESULTS DISCUSSION IMPLICATIONS LITERATURE CITED

 
Many traits of interest in livestock production are affected by the mother of the individual on whom the trait is observed. Improvement of such traits by means of artificial selection requires knowledge of the genetic background of the maternal effect. However, genetic analysis of maternally affected traits has proven difficult, and extremely negative estimates of the genetic correlation between the direct and maternal effect have been met with skepticism. This study presents methodology to estimate genetic parameters for maternal genetic effects and also accounts for environmental covariances between dam and offspring. The models presented yielded unbiased results. Furthermore, substantial difficulties with interpretation of results from previous methods were identified. Application of models presented in this study facilitates genetic improvement of maternally affected traits in livestock.

	Footnotes

 
¹ I thank J. van der Werf for giving helpful comments on a draft version of this manuscript.

² Corresponding author: piter.bijma{at}wur.nl

Received for publication October 7, 2005. Accepted for publication December 6, 2005.

	LITERATURE CITED

Top Abstract INTRODUCTION THEORY AND RESULTS DISCUSSION IMPLICATIONS LITERATURE CITED

 


Box, G. E. P., and G. M. Jenkins. 1976. Time Series Analyses: Forecasting and Control. Holden Day, San Francisco, CA.

Cheverud, J. M. 2003. Evolution in a genetically heritable social environment. Proc. Natl. Acad. Sci. USA 100:4357–4359.[Free Full Text]

Dickerson, G. E. 1947. Composition of hog carcasses as influenced by heritable differences in rate and economy of gain. Iowa Agric. Exp. Stn. Res. Bull. 354:489–524.

Dodenhoff, J., L. D. Van Vleck, and D. E. Wilson. 1999. Comparison of models to estimate genetic effects for weaning weight of Angus cattle. J. Anim. Sci. 77:3176–3184.[Abstract/Free Full Text]

Falconer, D. S. 1965. Maternal effects and selection response. Pages 763–774 in Genetics Today: Proc. XIth Int. Congr. Genet., Vol. 3. Pergamon, Oxford, UK.

Falconer, D. S., and T. F. C. Mackay. 1996. Introduction to Quantitative Genetics. 4th ed. Addison Wesley Longman, Essex, UK.

Gilmour, A. R., B. J. Gogel, B. R. Cullis, S. J. Welham, and R. Thompson. 2002. ASReml User Guide Release 1.0. VSN Int. Ltd., Hemel Hempstead, UK.

Henderson, C. R. 1988. Theoretical basis and computational methods for a number of different animal models. J. Dairy Sci. 71:1–16.[Abstract/Free Full Text]

Koch, R. M. 1972. The role of maternal effects in animal breeding. VI. Maternal effects in beef cattle. J. Anim. Sci. 35:1316–1323.[Abstract/Free Full Text]

Koerhuis, A. N. M., and R. Thompson. 1997. Models to estimate maternal effects for juvenile body weight in broiler chickens. Genet. Sel. Evol. 29:225–249.

Lynch, M., and B. Walsh. 1998. Genetics and Analyses of Quantitative Traits. Sinauer, Sunderland, MA.

Meyer, K. 1997. Estimates of genetic parameters for weaning weight of beef cattle accounting for direct-maternal environmental covariances. Livest. Prod. Sci. 52:187–199.

Meyer, K., M. J. Carrick, and B. J. P. Donnelly. 1994. Genetic parameters for milk production of Australian beef cows and weaning weight of their calves. J. Anim. Sci. 72:1155–1165.[Abstract]

Quaas, R. L., and E. J. Pollak. 1980. Mixed model methodology for farm and ranch beef cattle testing programs. J. Anim. Sci. 51:1277–1287.[Abstract/Free Full Text]

Robinson, D. L. 1996. Models which might explain negative correlations between direct and maternal genetic effects. Livest. Prod. Sci. 45:111–122.

Willham, R. L. 1963. The covariance between relatives for characters composed of components contributed by related individuals. Biometrics 19:18–27.

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