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Illustration of the maternal animal model used for genetic evaluation of beef cattle¹

D. H. Crews, Jr^*,,2 and Z. Wang

^* Agriculture and Agri-Food Canada Research Centre, Lethbridge, AB T1J 4B1 Canada; and and Department of Agricultural, Food, and Nutritional Science, University of Alberta, Edmonton, AB T6G 2E1 Canada

	Abstract

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National cattle evaluation programs for weaning weight in most beef breed associations involve implementation of the maternal animal model to predict direct and maternal EPD. With this model, direct breeding values are predicted for all animals with records or pedigree ties to animals with records, or both. Even though maternal genetic value is expressed only in animals that become dams, these effects are transmitted by all parents and inherited from parents by all animals, leading to maternal breeding values being predicted for all animals as well. A small example data set was simulated involving 12 parents, 8 nonparents, and 13 animals with weaning weight records. The pedigree was developed to include paternal and maternal half-sib families, full-sibs, and some inbreeding, similar to field populations of beef cattle. Assembly of the mixed model equations and solutions for the maternal animal model are illustrated explicitly to assist animal breeding students in their understanding of the properties of the maternal animal model and to explicitly implement the model. Model parameters and moments, fixed contemporary group solutions, adjustment of breeding values for merit of mates, interpretation of maternal permanent environmental effect solutions, and alternatives for the assembly of the equations are shown. This example should lead to increased student and producer understanding of genetic improvement programs for weaning weight in beef cattle.

Key Words: beef cattle • genetic prediction • maternal animal model • teaching

	INTRODUCTION

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Most breed associations for beef cattle participate in national cattle evaluation programs, in which breeding values are predicted for economically important traits (Golden et al., 2000). Several traits are considered to be influenced by genetic and nongenetic effects of the dam, the most familiar of which is weaning weight. Breeding values are typically predicted for weaning weight in cattle using the maternal animal model, which partitions phenotypic records into fixed (e.g., contemporary group), direct genetic, maternal genetic, and maternal permanent environmental effects (BIF, 2002). Solutions to the maternal animal model equations, therefore, contain estimates of direct and maternal genetic effects, which correspond to direct and maternal breeding values and lead directly to EPD for direct and maternal weaning weight.

Students trained in animal breeding should be familiar with the properties of the maternal animal model. Pedagogical illustration of the model, however, is problematic because the number of the equations generally precludes solution without a computer, even for small examples. Numerous computing tools and software packages have been developed to solve linear equation systems in animal breeding (e.g., Boldman et al., 1995; Gilmour et al., 2002; Meyer, 2006), but few of these allow the student to visualize building and solving equation systems.

It is preferable for students to gain experience in assembling equations more directly. To illustrate the properties of the maternal animal model, and to show an approach for setting up and solving equation systems for students in animal breeding, a small example involving 20 animals was developed. For this example, estimates of fixed contemporary group, direct genetic, maternal genetic, and maternal permanent environmental effects are obtained from setting up and solving the usual mixed model equations (e.g., Henderson, 1984).

	MATERIALS AND METHODS

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The usual assumption with a maternally influenced trait is that a record on an individual animal can be partitioned into: 1) fixed contemporary group, 2) direct (animal) genetic, 3) maternal genetic, and 4) maternal permanent environmental effects, which can be represented in matrix notation as

where known incidence matrices (X, Z₁, Z₂, and Z₃) relate the unknown fixed contemporary group (b), random direct genetic (u), random maternal genetic (m), and random maternal permanent environmental (c) effects, respectively, to observations in vector y. The vector e contains random residual effects, specific to animals with records. For this maternal animal model, expectations of the random vectors are E(u) = E(m) = E(c) = E(e) = 0, which leads to E(y) = Xb. Variances for the random effects are typically unstructured, written as

where ²_g is additive direct genetic variance, ²_m is maternal genetic variance, ²_c is maternal permanent environmental variance, ²_e is residual variance, A is the additive relationship matrix (e.g., Quaas, 1976), I_c is an identity matrix of order equal to the number of dams with progeny with records, and I_n is an identity matrix of order equal to the number of animals with records. Maternal permanent environmental and residual effects are assumed independent from the direct and maternal genetic effects, but the direct x maternal genetic covariance (_g,m) accounts for any correlation between direct and maternal genetic effects. This matrix is symmetric, and therefore only the diagonal and lower off-diagonal elements are shown. Predictions for the random effects are obtained by solving the usual mixed model equations (e.g., Henderson, 1984):

where the design matrices (X, Z_i) and vector of observations (y) are as previously defined, and the _i,j are elements in the ith row and jth column from

For this example, values of 188.80, 83.67, 112.10, 500.20, and –47.32 kg² were assumed for direct genetic, maternal genetic, maternal permanent environmental, and residual variances, and the direct x maternal genetic covariance, respectively (Crews et al., 2004).

Design matrices leading to the full coefficient matrix and the vector of right-hand sides for the mixed model equations were assembled and solved using Octave, an interpreted matrix language component of the Linux (Red Hat Enterprise Linux Ver. 4.2, Research Triangle Park, NC) operating system. Example code statements are listed in the Appendix. The inverse of the relationship matrix was constructed for the 20 animals using the Animal Breeder’s Tool Kit (ABTK Ver. 2.1.2. Colorado State University, Fort Collins). Once assembled, solutions to the mixed model equations were obtained by direct inversion. For example, if the complete system of equations is represented as

where C is the coefficient matrix and r is the vector of right-hand sides (e.g., Van Vleck, 1992), then the vector s, containing solutions for fixed contemporary group and random effects, was obtained as

The example pedigree and data file used in this example is summarized in Table 1. Among the 20 animals, there were 12 parents and 8 nonparents, and 5 animals with nonzero inbreeding coefficients ranging from 6.25 to 25%. The 5 sires and 7 dams had from 1 to 4 progeny, resulting in 4 paternal half-sib families and 4 maternal half-sib families. Maternal half-sib families are required to properly separate maternal genetic and maternal permanent environmental effects. Animals 6 and 8 were full-sibs. To fit the maternal animal model, only animals with nonmissing maternal parentage had adjusted weaning weight (WT205) records. The 5 contemporary group designations corresponded approximately to birth order or generation and sex subclasses. Each group included 2 or 5 animals with WT205 records. Pedigree ties among animals and across contemporary groups ranged from low to high, which is typical of beef cattle field data.

	RESULTS

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Fixed and random effect incidence matrices were constructed using the convention that incidence matrices have dimension equal to "records x effects". Because the example contains 13 animals with records, the number of rows of X, Z₁, Z₂, and Z₃ was 13. Given 5 contemporary groups, X had 5 columns. The maternal animal model predicts direct and maternal breeding values for all animals. Therefore, the column dimension for Z₁ and Z₂ was the number of animals (20). Only 7 females (animals 1, 4, 5, 6, 9, 10, and 11; Table 1) were dams with progeny with WT205 records, resulting in a column dimension for Z₃ of 7. The incidence matrices were initialized as matrices of appropriate dimension with all elements equal to zero, and then 1’s were added to elements corresponding to the intersection of a record with an effect (Appendix). The inverse of the numerator relationship matrix (A^–1) was computed and stored in a format compatible with Octave. To construct the full coefficient matrix, some of the relevant submatrix products were

with the remaining submatrix products (e.g., Z'_i Z_i, Z'_iZ_j, X'Z_i, Z_i'X) equal to coefficients of the usual linear model equations (e.g., Searle, 1982), which can be identified in the programming statements listed in the Appendix. The complete coefficient matrix, which is square and symmetric, was of order 52. Solutions, obtained by direct inversion, are listed in Table 2. Direct breeding values had a mean, minimum, and maximum of 0.44, –1.80, and 1.97 kg, respectively, whereas the corresponding estimates for maternal breeding values were 0.07, –0.77, and 0.91 kg, respectively. Solutions for maternal permanent environmental effects had a null mean and ranged from –1.43 to 2.05 kg. Contemporary group solutions ranged from 245.5 (group 1) to 357.6 (group 4) kg.

	DISCUSSION

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Direct Breeding Values
Twenty equations are included in this system to obtain BLUP of direct breeding values (e.g., Henderson, 1984; Van Vleck, 1992). For example, reordering of the coefficients for the direct breeding value equation for animal 3 yields

This equation illustrates several properties of the maternal animal model. First, animal 3 is a base population or foundation sire, and as such has unknown parentage and no WT205 record, so this equation does not contain contemporary group solutions, and all coefficients for these fixed effects were zero. Secondly, the direct breeding value for animal 3 is adjusted for the merit of mates (animals 1, 5, 9, and 11) and for the merit of progeny (animals 4, 5, 15, 17, and 20). Adjustments are in fact made for the direct and maternal breeding values of mates and progeny. The coefficients can be further simplified to show that the mate adjustment is one-half the coefficients of the mates’ progeny. Thrift (2000) similarly illustrated adjustment for merit of mates in national beef cattle evaluation programs. Adjustment for the maternal breeding values of mates or progeny would not be included in these equations if maternal genetic effects were not included in the model or if the direct x maternal genetic correlation was assumed to equal zero. This latter result can be verified by changing the program statements for defining in the Appendix such that the direct x maternal genetic covariance is zero rather than –47.32. The relative magnitude of the weightings placed on the various sources of information in this prediction (i.e., the coefficients) reflects that, in general, records on progeny are more informative than records on more distant relatives or from mates.

The mean direct breeding value for animals in this example was 0.44 kg. As noted above, E(u) = 0, but this applies specifically to the base population of animals. Therefore, in field populations with national cattle evaluation, all EPD or breeding values will not generally have a simple average of zero. Rather, only base population animals will have a mean breeding value and EPD of zero. However, for direct breeding values alone, 1'A^–1û= 0, with 1 defined as a column vector of ones with dimension equal to 20, to conform for multiplication with A^–1.

Maternal Breeding Values
With beef cattle, the dam provides a pre- and postnatal environment for her offspring, and the variation in the female ability to provide this environment has a genetic basis proportional to maternal heritability (e.g., Van Vleck, 1992). Maternal genetic breeding value is expressed only in females that become dams, but genes underlying this value are transmitted from both parents, and inherited by all animals, so the maternal animal model predicts maternal breeding values for all animals. The incidence matrix for maternal breeding value (Z₂) differs from that for direct breeding values (Z₁) in that nonzero elements correspond to records and the dams of calves making those records rather than records and the calves making the records directly.

Again using animal 3 as an example, reordering the coefficients for the maternal breeding value equation yields

Similar to the direct breeding value equation, there is no adjustment for ancestors or contemporary group effects because animal 3 is a foundation sire with no WT205 record. Adjustments are made for progeny (animals 4, 5, 15, 17, and 20) and for mates (animals 1, 9, and 11) in predicting maternal breeding value for animal 3, with information from direct and maternal breeding values contributing to the prediction. Again, the joint prediction of direct and maternal breeding values depends on the model equation and a nonzero direct x maternal genetic correlation.

Also similar to the direct breeding value, the expectation for base population maternal breeding values is zero (i.e., E(m) = 0), but maternal breeding values for the entire population will not generally have a null mean. The mean maternal breeding value in this example was 0.07 kg. Also, because maternal heritability (0.10) was considerably lower than direct heritability (0.23), the range and variance of maternal breeding values were less than for direct breeding values.

In most national cattle evaluation programs for WT205, breed associations report total maternal (TMAT) breeding values or EPD. Defined as TMAT_i = _i + 0.5û_i for animal i, the total maternal value is not equivalent to maternal producing ability (e.g., Van Vleck, 1992) because it contains information from the direct and maternal breeding value predictions, but not maternal permanent environmental effects. This definition clearly illustrates that no new information contributes to TMAT values beyond the predictions for direct and maternal breeding values.

Maternal Permanent Environmental Effects
Maternal permanent environmental effects are non-genetic effects of the maternal ability of females that are dams. In the mixed model equations, these effects are assumed uncorrelated to genetic effects and are typically uncorrelated and dispersed using a simple identity matrix (i.e., var(c) = I_c²_c). The equations for maternal permanent environmental effects are generally more sparse than equations for direct and maternal genetic effects because the relationship matrix does not create covariances among maternal permanent environmental effects.

For example, animal 1 is a base dam with 4 progeny with records. The maternal permanent environmental effect equation for this animal is

This equation illustrates the adjustments for the 2 progeny in contemporary group 1 and the single progeny in each of contemporary groups 2 and 3, but also adjustments for direct breeding values of all progeny of animal 1, as well as the maternal breeding value of animal 1. Because maternal permanent environmental effects are assumed uncorrelated to genetic effects, and are not inherited through additive relationship, fewer sources of information contribute to these predictions, which are computed only for dams with progeny with records. And, unlike direct and maternal breeding values, maternal permanent environmental effects are not adjusted for merit of mates.

Because maternal permanent environmental effects are nonresidual deviations, E(c) = 0, and these predictions for all dams will have a mean of zero. In this example, the maternal permanent environmental effect solutions ranged from –1.43 to 2.05 kg.

Genetic prediction with beef cattle field data leads to EPD, which is the primary tool for selection on economically important traits such as weaning weight. Given that the maternal animal model is the standard for predicting weaning weight EPD, students in animal breeding should be familiar with its properties and implementation. A small example is sufficient to illustrate computational strategies for implementing the maternal animal model as well as the most important properties of the solutions. Undergraduate-and graduate-level students can benefit from explicit examples such as used here and thereby increase their familiarity with genetic prediction and genetic improvement of beef cattle. This approach has been well received by students in graduate level animal breeding classes and could increase student and producer understanding of the maternal animal model used in genetic evaluation of weaning weight of beef cattle.

	APPENDIX

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Although numerous computing strategies and software packages are capable of implementing the maternal animal model for this example, Octave was chosen because it is a high-level, interpreted, matrix language component of the Linux operating system, allows for direct construction of relevant matrices, and is readily available by download for several operating system platforms (http://www.octave.org; last accessed 20 March 2007).

Begin by defining the incidence matrices X, Z₁, Z₂, and Z₃, and the observation vector y. For clarity, Octave code is denoted with alternate typeface, and code lines are numbered:

1 > y = [247.9 245.1 335.5 261.1 266.2 325.8 271.2 255.7 248.9 358.4 357.6 319.2 318.5]';

2 > x = zeros(13,5); x(1,1)=x(2,1)= x(3,2)=x(4,3)=x(5,3)=x(6,2)= x(7,3)=x(8,3)=x(9,3) =x(10,4)= x(11,4)=x(12,5)=x(13,5)=1;

3 > z1 = zeros(13,20); z1(1,4)=z1(2,6)= z1(3,8)=z1(4,10)=z1(5,11)= z1(6,12)=z1(7,13) =z1(8,14)= z1(9,15)=z1(10,17)=z1(11,18)= z1(12,19)=z1(13,20)=1;

4 > z2 = zeros(13,20); z2(1,1)=z2(2,1)= z2(3,1)=z2(4,5)=z2(5,6)= z2(6,4)=z2(7,6) =z2(8,1)= z2(9,9)=z2(10,5)=z2(11,10)= z2(12,10)=z2(13,11)=1;

5 > z3 = zeros(13,7); z3(1,1)=z3(2,1)= z3(3,1)=z3(4,3)=z3(5,4)= z3(6,2)=z3(7,4)=z3(8,1) = z3(9,5)=z3(10,3)=z3(11,6)= z3(12,6)=z3(13,7)=1;

Next, define the inverse matrix of genetic (co)variances () multiplied by residual variance:

6 > k = inv([188.80 –47.32 0;–47.32 83.67 0;0 0 112.10])*500.20.

Assuming that the inverse numerator relationship matrix was previously created (see text), load the square, symmetric A^–1 directly:

7 > load Ainv;

Now, assemble the mixed model equations, including the coefficient matrix and right-hand sides:

8 > c = [x'*x x'*z1 x'*z2 x'*z3; z1'*x z1'*z1+Ainv*k(1,1) z1'*z2+Ainv*k(1,2) z1'*z3; z2'*x z2'*z1+Ainv*k(2,1) z2'*z2+Ainv*k(2,2) z2'*z3; z3'*x z3'*z1 z3'*z2 z3'*z3+eye(rows(z3'))*k(3,3)];

9 > r = [x'*y; z1'*y; z2'*y; z3'*y];

Next, solve the equations by direct inversion and partition into relevant subvectors corresponding to , û, , and .

10 > s = inv(c)*r;

11 > bhat = s(1:5);

12 > uhat = s(6:25);

13 > mhat = s(26:45);

14 > chat = s(46:rows(s));

	Footnotes

 
¹ AAFC-LRC contribution number 38706064. Comments provided by L. D. Van Vleck are greatly appreciated. This work was supported as part of the University of Alberta AAFC Chair Professorship in Beef Genomics, held by the senior author.

² Corresponding author: dcrews{at}agr.gc.ca

Received for publication October 26, 2006. Accepted for publication March 16, 2007.

	LITERATURE CITED

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BIF. 2002. Guidelines for Uniform Beef Improvement Programs, 8th Ed. Department of Animal and Dairy Science, University of Georgia, Athens.

Boldman, K. G., L. A. Kriese, L. D. Van Vleck, C. P. Van Tassell, and S. D. Kachman. 1995. A Manual for Use of MTDFREML. A Set of Programs to Obtain Estimates of Variances and Covariances [Draft]. USDA, ARS.

Crews, D. H., Jr., M. Lowerison, N. Caron, and R. A. Kemp. 2004. Genetic parameters among growth and carcass traits of Canadian Charolais cattle. Can. J. Anim. Sci. 84:589–597.

Gilmour, A. R., B. J. Gogel, B. R. Cullis, S. J. Welham, and R. Thompson. 2002. ASReml User Guide, Release 1.10. VSN International, Ltd., Hemel Hempstead, UK.

Golden, B. L., D. J. Garrick, S. Newman, and R. M. Enns. 2000. A framework for the next generation of EPDs. Proc. 32nd Beef Improv. Fed. Annu. Res. Symp. Meet., Wichita, KS.

Henderson, C. R. 1984. Applications of Linear Models in Animal Breeding. University of Guelph, Guelph, Ontario, Canada.

Meyer, K. 2006. "WOMBAT" – Digging deep for quantitative genetic analyses by restricted maximum likelihood. Proc. 8th World Congress on Genetics Applied to Livestock Production. CD-ROM Comm. 27-14.

Quaas, R. L. 1976. Computing the diagonal elements of a large numerator relationship matrix. Biometrics 32:949–953.[CrossRef]

Searle, S. R. 1982. Matrix Algebra Useful for Statistics. John Wiley and Sons, New York, NY.

Thrift, F. A. 2000. Simple example illustrating adjustment for merit of mates in a national cattle evaluation program. J. Anim. Sci. 78:2475–2478.[Abstract/Free Full Text]

Van Vleck, L. D. 1992. Selection Index and Introduction to Mixed Model Methods. CRC Press, Boca Raton, FL.

This Article Abstract Full Text (PDF) All Versions of this Article: jas.2006-705v1 85/7/1842    most recent Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Crews, D. H. Articles by Wang, Z. Search for Related Content PubMed PubMed Citation Articles by Crews, D. H., Jr Articles by Wang, Z.