Covariance components and prediction for additive and nonadditive preweaning growth genetic effects in an Angus-Brahman multibreed herd M. A. Elzo and D. L. Wakeman J ANIM SCI 1998, 76:1290-1302. The online version of this article, along with updated information and services, is located on the World Wide Web at: http://jas.fass.org/content/76/5/1290 www.asas.org Downloaded from jas.fass.org by guest on January 26, 2012 Covariance Components and Prediction for Additive and Nonadditive Preweaning Growth Genetic Effects in an Angus-Brahman Multibreed Herd1 M. A. Elzo2 and D. L. Wakeman Animal Science Department, University of Florida, Gainesville 32611 ABSTRACT: Estimates of covariances and sire phenotypic variances) and nonadditive correlations expected progeny differences of additive and nonaddi- were somewhat smaller than heritabilities and addi- tive direct and maternal genetic effects for birth and tive genetic correlations. Sire additive and total direct weaning weights were obtained using records from and maternal genetic predictions for birth and wean- 1,581 straightbred and crossbred calves from the ing weight tended to increase with the fraction of Angus-Brahman multibreed herd at the University of Brahman alleles, whereas nonadditive direct and Florida. Covariances were estimated by Restricted maternal genetic predictions were similar for sires of Maximum Likelihood, using a Generalized Expecta- all Angus and Brahman fractions. These results tion-Maximization algorithm applied to multibreed showed that it is feasible to evaluate sires for additive populations. Estimates of heritabilities and additive and nonadditive genetic effects in a structured mul- genetic correlations for straightbred and crossbred groups were within the ranges of values found in the tibreed population. Data from purebred breeders and literature for these traits. Maximum values of interac- commercialproducerswillbeneededtoaccomplishthe tibilities (ratios of nonadditive genetic variances to same goal at a national level. Key Words: Beef Cattle, REML, Genetic Variance, Growth, Populations ª 1998 American Society of Animal Science. All rights reserved. J. Anim. Sci. 1998. 76:1290–1302 Introduction of specific breed groups (multibreed nonadditive expected progeny differences). Currently, national Mostofthebeefin theUnitedStates is produced by beef cattle sire evaluations use intrabreed genetic crossbred animals. The parents of these crossbred prediction procedures, and comparisons of sires across animals can be of different breeds or they may be breedsarecomputedusinga table of correction factors crossbreds themselves. A population composed of obtained using experimental data from the Meat straightbred and crossbred animals that interbreed Animal Research Center (Notter and Cundiff, 1991; constitutes a multibreed population (Elzo, 1983, Van Vleck and Cundiff, 1996). The goal of these 1990b; Elzo and Famula, 1985). In a multibreed intrabreed and interbreed predictions is to compare population additive and nonadditive genetic effects sires for additive genetic effects. However, if animals should be accounted for in the genetic evaluation ofvariousbreedsandcrossbredgroupsparticipateina model. Thus, it seems appropriate that sires be multibreed mating scheme, then they need to be evaluated for their general combining ability (mul- evaluated for additive and nonadditive genetic effects. tibreed additive expected progeny difference) and Ideally a multibreed national cattle evaluation that their specific combining abilities when mated to dams uses field data and accounts for group and random additive and nonadditive genetic effects would need to be implemented (Elzo, 1996a). To help in the development and validation of genetic evaluation 1Florida Agric. Exp. Sta., Journal series no. R-05806. The procedures for multibreed populations, an Angus- authorsthankL.D.VanVleckfor manyusefuldiscussionsandM. Brahman multibreed experimental herd was estab- A. DeLorenzo, T. A. Olson, and J. Rosales for reviewing the manuscript. The authors gratefully acknowledge the personnel at lished at the University of Florida in 1988. Thus, the thePineAcresResearchStation,andtheBeefResearchUnitofthe objectives of this study were to estimate genetic University of Florida, for their efforts in collection data and covariancesandpredictsiregeneticvaluesforadditive maintaining the multibreed experimental herd. andnonadditivedirectandmaternalgeneticeffectsfor 2To whom correspondence should be addressed. birth weight and weaning weight in the Angus- Received June 25, 1997. Accepted November 17, 1997. Brahman herd using multibreed procedures. 1290 Downloaded from jas.fass.org by guest on January 26, 2012 GROWTH COVARIANCES AND GENETIC PREDICTION 1291 Table 1. Numbers of sires, maternal grandsires, dams, and calves by breed-group-of-sire · breed-group-of-dam combination Breed group of sire Breed group of dam Angus (A) ãA ÔB ØA ØB ÔA ãB Brahman (B) Brangus Angus 16a 7 9 10 15 16 31b 2 4 7 6 11 69c 24 22 28 40 40 117d 25 24 31 45 51 ãA ÔB 13 9 9 9 17 13 5 6 2 4 5 6 13 20 23 22 24 29 29 21 25 24 27 32 ØA ØB 16 11 9 11 18 15 12 6 3 6 7 12 50 36 38 47 54 50 62 41 46 57 65 66 ÔA ãB 11 6 7 7 12 10 3 1 2 2 1 1 21 16 23 16 25 24 24 20 24 19 32 28 Brahman 13 11 9 11 20 16 10 4 6 5 38 9 45 40 36 43 107 44 53 44 39 49 195 50 Brangus 10 7 8 10 12 16 3 3 2 4 5 14 21 15 19 23 23 66 23 16 19 26 25 106 Total 16 11 10 11 20 17 35 12 9 10 38 21 229 150 162 178 271 250 308 167 179 206 388 333 aNumber of sires. bNumber of known maternal grandsires. cNumber of dams. dNumber of calves. MaterialsandMethods ing cattle producers. Semen was also collected from cleanup sires. A few crossbred AI and(or) cleanup Animals, Mating Strategy, and Records sires were produced in the multibreed herd. Similarly, initial sets of dams were those available from other The data set used consisted of 1,581 birth weight breeding experiments or donated by cooperating (BW) and 1,449 weaning weight (WW) records from producers. Between two and five sires per breed group 1,581 straightbred and crossbred calves born between of sire were used in the mating program per year. 1989 and 1996 in the Angus (A)-Brahman (B) Sireswereusedfor2 yrtocreateconnectednessacross multibreed herd of the University of Florida. Calves years.Thenumberofdams mated per breed group per weretheproductofa diallel mating strategy involving year ranged from 14 (ãA ÔB in 1990) to 74 (B in 16 A, 11 ãA ÔB, 10 ØA ØB, 11 ÔA ãB, 20 B, and 17 1995). Table 1 shows the distribution of sires, Brangus (æA ÆB) sires mated to 124 A, 78 ãA ÔB, maternal grandsires, dams, and calves per breed- 127 ØAØB,68ÔAãB,160 B, and 94 Brangus dams. group-of-sire · breed-group-of-dam subclass. The total The representation of the different breed groups of number of bulls represented in the data set was 144. sires and dams in the multibreed herd was primarily Therewere29 bulls representedas sires only, 60 bulls the result of availability of animals. A conscious effort appeared as maternal grandsires only, and 55 bulls was made to use semen and sires of different sources were sires and maternal grandsires. andpartsofthecountry,buttheresultingsets of sires within breed groups were not random samples. Semen Cow-Calf Management and Contemporary Groups was either donated by, or purchased from, AI organi- zations. Only semen from inexpensive bulls(less than Cows were maintained on bahiagrass (Paspalum $20 per straw) was purchased. Most cleanup sires notatum) pastures throughout the year with mineral were either donated by, or purchased from, cooperat- supplementation. In winter (mid-December to Downloaded from jas.fass.org by guest on January 26, 2012 1292 ELZO AND WAKEMAN March), as part of a nutrition study, second-calf and were allowed in a contemporary group. Thus, mul- oldercowswithin a breed-group-of-dam· breed-group- tibreed contemporary groups were defined as follows: of-sire subclass were allocated to six replicated forage 1) birth contemporary group: group of calves that supplementation regimens and one control (13 sup- were born in the same calving year (1989 to 1996), plementation groups). Supplementation was ber- within a periodof3 mo,andwere of the same sex (1 = mudagrass (Cynodon dactylon) hay wilted to several bull, 2 =heifer), and 2) weaning contemporary group: percentagesofDM,urea,andmolasses(Odenya etal., group of calves that were born in the same calving 1992). Heifers were not part of the winter supplemen- year (1989 to 1996), were of the same sex (1 =bull, 2 tationstudy.Becausenoneofthewintersupplementa- = heifer, 3 = steer), whose dams were in the same tion regimens was stressful to cows, it was assumed winter management group (13 winter supplementa- that their impact on maternal effects was negligible. tion groups), and were born, and also weaned, within Thus, dam winter supplementation group was only a 3-mo period. included as part of the definition of calf weaning contemporary groups. This was done to account for Multibreed Covariance Component Estimation potential differences in weaning weight that might be and Genetic Prediction Procedures attributable to differential calf consumption of winter supplemental feed supplied to their dams (e.g., Covariance components were estimated by Res- molasses). tricted Maximum Likelihood procedures (Harville, Estrus was synchronized in cows with PGF in 1977) that used a Generalized Expectation-Maximiza- 2a tion (GEM) algorithm (Dempster et al., 1977) March (A and crossbred AB) and April (B). Cows appliedtomultibreedpopulations(MREMLEM,Elzo, were artificially inseminated twice then assigned to 1994). Computations were performed using an in- one of six cleanup herds (one cleanup herd per breed house FORTRAN program compiled using XL FOR- group of sire) and exposed to a cleanup sire for 60 d. TRAN for AIX, and run in an IBM RS6000 worksta- Estrus was synchronized in heifers, which were tion, model 580. To ensure that estimates of covari- inseminated 2 wk earlier than cows. Braham dams ance matrices were positive definite, the MREMLEM were bred later than A and AB dams because of procedure computed the Cholesky elements of each concerns with calf mortality of straightbred B calves. covariance matrix first, and then each Cholesky Cows that calved less than 45 d before the synchroni- matrix was multiplied by its transpose to obtain the zation date were assigned to a second synchronization matrices of covariance estimates (Elzo, 1996b). group (one for A and AB dams, and another for B dams). Thus, the insemination dates of A and AB Multibreed Model. The model used was a two-trait dams overlapped with that of B dams. The cleanup (BW and WW) multibreed sire-maternal grandsire periods for A and AB and for B dams also overlapped, model. Each trait was assumed to have both direct and this caused the calving dates for all breed groups (D) and maternal (M) genetic effects. of dams to overlap as well. The staggered estrus Fixed environmental effects in the model were synchronization and insemination (AI and natural contemporary group and a covariate for age of dam service) systemusedinthismultibreedherdcreateda withinsexofcalfandbreedgroupofdam,wherebreed continuous, albeit long, mating season, and conse- groupwasmodeledasa regressiononthefractionofA quently a single long calving season. in the dam. The fixed regression group genetic effects Calves were born from late December to March and were as follows: 1) intrabreed additive direct (as a weaned in September (calves from A and crossbred functionoftheexpectedfractionofAallelesinallsires AB dams) and October (calves from B dams). Calves plus .5 the expected fraction of A alleles in all from B dams were weaned later to give them the maternal grandsires), 2) intrabreed additive mater- opportunity to remain with their dams for approxi- nal (as a function of the expected fraction of A alleles mately the same length of time as calves of A and AB inall maternalgrandsires), 3) interbreedABadditive dams. It should be emphasized that sires of all breed direct (as a function of the probability of A and B groups that were used as AI sires, cleanup sires, or as allelesintheparentsofallsiresplus.5theprobability AI and cleanup sires in a breeding season were mated of A and B alleles in the parents of all maternal to dams of all breed groups (A, AB crossbred, and B grandsires), 4) interbreedABadditivematernal(as a dams). This mating strategy created connections function of the probability of A and B alleles in the among sires across all breed groups of dams within a parents of all maternal grandsires), 5) intralocus breeding season. Because calves were born and interbreedA/Bnonadditivedirect(as a functionof the weaned within a 3-mo period, and sires were well probability of A and B alleles at one locus of the connected across all breed groups of dams within a progeny of all sires), 6) intralocus interbreed A/B breeding season, comparisons among sires could be nonadditive maternal (as a function of the probability fairly made across all breed groups of dams. ofAandBalleles in onelocusofthefemaleprogenyof Multibreed contemporary groups were defined simi- allmaternalgrandsires,i.e., damsofcalves), and7) a larly to intrabreed contemporary groups (BIF, 1996), combination of direct and maternal group genetic except that calves of all breed group combinations effects due to all maternal granddams (as a function Downloaded from jas.fass.org by guest on January 26, 2012 GROWTH COVARIANCES AND GENETIC PREDICTION 1293 of the expected fraction of A alleles in the maternal s = vector of sire nonadditive direct (s ) and n nd granddam of the calf). Intrabreed additive genetic maternal (s ) genetic effects, nm regression effects estimated the deviation between A v = vector of residuals, and B group additive genetic effects. Interbreed X = matrix that relates calf records to 1) to additivegeneticregressioneffectsestimatedthedevia- elements of bcg (1s and 0s), and 2) to ele- tion between interbreed AB additive genetic group ments of badx through the age of the dam and the expected fraction of A alleles in the effects and intrabreed AA and BB as a function of the dam of the calf, probability of A and B alleles being present in the Z = matrix that relates calf records to 1) ele- parents of sires and maternal grandsires. Intralocus ga ments of g through the expected fraction interbreed A/B nonadditive genetic regression effects Aad of A alleles in the sire and the maternal estimated the difference between interbreed A/B and grandsire of the calf (p + .5p ), 2) ele- intrabreed A/A and B/B group genetic effects as a As Am ments of g through the expected fraction function of the probability of A and B alleles from Aam of A alleles in the maternal grandsire of the different parents being paired at one locus in the calf (p ), 3) elements of g through Am ABad progeny of sires and maternal grandsires. the probability of A and B alleles in the Random effects in the model were as follows: 1) parents of the sire and of the maternal direct additive sire genetic effect, 2) direct additive grandsireofthecalf[(p p +p p ) Ass Bss Ads Bds maternal grandsire genetic effect, 3) maternal addi- + .5(p p + p p )], and 4) ele- Asd Bsd Add Bdd tive maternal grandsire genetic effect, 4) direct ments of g through the probability of A ABam nonadditive sire genetic effect (as a function of and B alleles in the parents of the maternal intralocus interbreed A/B interactions in the progeny grandsire of the calf (p p + p Asd Bsd Add of a sire), 4) maternal nonadditive maternal grand- p ), where p = probability, and the sub- Bdd sire genetic effect (as a function of intralocus inter- scripts A =Angus, B =Brahman, s =sire, m breed A/B interactions in the dam), and 5) residual. = maternal grandsire, ss = sire of sire, ds = In matrix notation, the multibreed mixed model dam of sire, sd = sire of dam (= maternal was as follows: grandsire), dd = dam of dam, Z = matrix that relates calf records to 1) ele- gn y = Xb +Zgaga +Zgngn +Zmgdgmgd +Zasa +Znsn +v ments of gnd through the probability of in- tralocus A and B alleles in the calf (p p As Bd ŒØ y øœ +pBs pAd), and 2) elements of gnm through s the probability of intralocus A and B alleles a ~ MVN sn in the dam of the calf (pAsd pBdd + pBsd º v ß pAdd), Z = matrix that relates calf records to 1) ele- a (cid:239)(cid:236) ØŒ Xbøœ ØŒ ZaGaZa¢ + ZnGnZn¢ + R ZaGa ZnGn Røœ (cid:252)(cid:239) ments of sad through the sire (1) and the 0 G Z ¢ G 0 0 maternalgrandsire(.5), and2) elementsof (cid:237) , a a a (cid:253) , (cid:239) 0 GnZn¢ 0 Gn 0 (cid:239) sam through the maternal grandsire (1), (cid:238) º 0 ß º R 0 0 Rß (cid:254) Zn = matrix that relates calf records to 1) ele- ments of s through the probability of in- [1] nd tralocus A and B alleles in the calf (p p As Bd + p p ), and 2) elements of s through where Bs Ad nm the probability of intralocus A and B alleles in the dam of the calf (p p + p y = vector of BW and WW calf records ordered Asd Bdd Bsd p ), and by traits within calves, Add Z = matrix that relates calf records to elements b = vector of contemporary groups (b ) and mgd cg of g through the expected fraction of A age of dam within sex of calf (b ) effects, mgd adx alleles in the maternal granddam. g = vector of intrabreed A direct (g ), in- a Aad trabreed A maternal (g ), interbreed AB Aam The multibreed model accounted for covariances direct (g ), and interbreed AB maternal ABad among direct and maternal sire additive genetic (g ) additive genetic group effects, ABam effects. Similarly, covariances among direct and g = vector of interbreed A/B nonadditive direct n maternal sire nonadditive genetic effects were al- (g ) and maternal (g ) genetic A/Bnd A/Bnm lowed. Thus, neither G nor G was block-diagonal. group effects, a n However, the residual covariance matrix R was block- g = vector of maternal granddam genetic group mgd diagonal, with 2 · 2 blocks, where 2 is the number of effects, traits. s = vector of sire additive direct (s ) and a ad Multibreed Mixed Model Equations (MMME). The maternal (s ) genetic effects, am MMME for model [1] were as follows: Downloaded from jas.fass.org by guest on January 26, 2012 1294 ELZO AND WAKEMAN ØŒº ZZZZZmXaggng¢na¢R¢Rd¢¢RRR-¢-1R-1-1-11-XX1XXXX ZZZZmZXaggngna¢¢Rd¢¢R¢RRR¢-R-1-1-11-1-ZZ1ZZZggZggagagaaaa ZZZmZZXggnagna¢¢¢Rd¢R¢RRR¢-1R–-1-1-11-ZZZ1ZZgggZggnnngnnn ZZZmZZXggnagna¢¢¢Rd¢R¢RRR¢-1R---1-111-ZZZ1ZZmmmZmmgmggggdddgddd Za¢ZRZZmZ-Xggn1gna¢¢RZd¢¢RRR¢a-R--1-111-Z+Z1ZZaZaaaaGa–1 ZnZ¢ZRZXmZgg-¢agRna1¢d¢R¢Z-RR¢n1R--1-1Z1-+Z1ZZnZnnnGnn- 1øœß ŒºØ gmggssbnnaagdøœß = ØŒº ZZZZmZXggnagna¢¢¢Rd¢R¢RRR¢-R---1-1111-yyy1yyyœßø [2] To facilitate computations, 1) equations for en- where w, z = two traits (BW maternal, WW mater- vironmental effects were ordered by trait (BW, WW) nal), the subscripts n = nonadditive, m = maternal within each environmental effect (b), and 2) equa- grandsire, A =Angus, B =Brahman, sd = sire of dam, tions for genetic effects were ordered by trait and dd =dam of dam, and p = expected fraction of A in Axy effect (i.e., BW direct, WW direct, BW maternal, WW animal xy, xy = ss, sd, and (s ) = intralocus nwz AB maternal) within each group (g , g , g ) and sire interbreed A/B environmental covariance between a n mgd (s , s ) genetic effect. traits w and z. a n Inverses of Covariance Matrices of Random Effects. Multibreed environmental covariances were ob- TheMMMErequiretheinversesofG , G , andR.The tained by formula 2 (Elzo, 1994) for the case of two a n inverse of G was obtained using the computational breeds; thus, a rulesdescribedinElzo(1990b). Themultibreeddirect and maternal additive genetic covariances between cov3(w,z)i = pAi(sewz)A + pBi(sewz)B BW and WW needed to obtain the coefficients used by + (pAspBs + pAdpBd)(sewz)AB [5] the computational rules were computed using formula 1 in Elzo (1994). For the case of two breeds this where w, z =two traits (BW, WW), the subscripts e = formula becomes: environmental, A = Angus, B = Brahman, i = calf, s = sire of i, d =dam of i, and p =expected fraction of A Ax cova(w,z)i = pAi(sawz)A + pBi(sawz)B inanimalx,x=i,s,d,(sewz)X=intrabreed(X =A,B) + (p p + p p )(s ) [3] environmental covariance between w and z, and As Bs Ad Bd awz AB (s ) = interbreed AB environmental covariance ewz AB where w, z =two traits direct and(or) maternal (e.g., between traits w and z. BW direct and BW direct, BW direct and WW The formula for the residual covariance between maternal), the subscripts a =additive, A =Angus, B = two traits (w and z) is a linear combination of direct Brahman,i=animal,s=sireofi,d=damofi,andp and maternal additive(formula [3]), maternal nonad- Ax ditive (formula [4]), and environmental (formula [5]) =expectedfractionofAinanimalx,x=i,s,d,(s ) awz X multibreed covariances; thus, = intrabreed (X = A, B) additive genetic covariance between w and z, (s ) = interbreed AB additive awz AB cov (w,z) = cov (w,z) - (d ).25cov (w,z) - genetic covariance between traits w and z. v i a i s a s (d ).0625cov (w,z) TheinverseofG wascomputedusingtherecursive m a m n + (1 - d )cov (w,z) procedure for regression models(Elzo, 1990a) applied m n m + cov (w,z) [6] to the one locus case. The covariances used for the e i computation of G- 1 were direct and maternal intralo- n where the superscripts i, s, and m refer to a calf, its cus interbreed A/B genetic covariances. sire,anditsmaternalgrandsire,thesubscripts v, a, n, TheinverseofRwascomputeddirectlybecauseRis anderepresent residual, additive genetic, nonadditive a block-diagonal, with 2 · 2 matrices of multibreed genetic and environmental, and residual covariances for BW and WW. Multibreed residual covariances were computed as linear combi- d = 1 if animal x is identified, and zero x nations of multibreed additive direct and maternal, otherwise, for x = s, m, nonadditive maternal (if the maternal grandsire was cov (w,z) = cov (w ,z ) + cov (w ,z ) + .5 a i a D D i a M M d unknown), and multibreed environmental covariances cov (w ,z ) + .5 cov (w ,z ) , a D M d a M D d (Elzo, 1994, 1996a). where the subscripts D =direct, and M Nonadditive maternal covariances included in the = maternal, residual covariance due to unknown maternal grand- cov (w,z) = cov (w ,z ) , a s a D D s sires were computed using formula 5 (Elzo, 1996a) cov (w,z) = cov (w ,z ) + 4 cov (w ,z ) + 2 a m a D D m a M M m applied to the case of two breeds; thus, cov (w ,z ) + 2 cov (w ,z ) , a D M m a M D m cov (w,z) = cov (w ,z ) , n m n M M sd cov (w,z) = (p p + p p )(s ) n m Asd Bdd Bsd Add nwz A/B [4] and cov (w,z) is computed by formula [5]. e i Downloaded from jas.fass.org by guest on January 26, 2012 GROWTH COVARIANCES AND GENETIC PREDICTION 1295 Computational Strategy. The MREMLEM algorithm upper triangular elements of the following base (Elzo, 1994, 1996b) assumed that covariances among genetic and environmental covariance matrices at sire additive genetic effects, and among sire nonaddi- each GEM iteration: 1) three 4· 4 additive direct and tive genetic effects, were zero. This time, however, maternal (intrabreed A, intrabreed B, and interbreed additive and nonadditive genetic covariances among AB), 2) one 4 · 4 nonadditive direct and maternal sires were permitted in the multibreed model. Thus, (interbreed A/B), and 3) three 2 · 2 environmental instead of sire additive and nonadditive genetic (intrabreedA,intrabreedB,andinterbreedAB). Base predictions, residual sire additive and nonadditive intrabreed and interbreed environmental contained genetic predictions were used to estimate covariance direct and maternal environmental effects. The ele- components. Because covariances among residual sire ments of each estimated base additive and nonaddi- additive and nonadditive genetic effects across sires tive genetic covariance matrix were var(BWD), arezerotheMREMLEMalgorithm could now be used. cov(BWD, WWD), cov(BWD, BWM), cov(BWD, Consequently, an additional set of computations WWM), var(WWD), cov(WWD, WWD), cov(WWD, was added to the expectation step of the MREMLEM BWM), cov(WWD, WWM), var(BWM), cov(BWM, algorithm. First, as previously done (Elzo, 1994, WWM), and var(WWM), where D = direct, and M = 1996b), theinverseoftheleft-handsideoftheMMME maternal. The elements of each estimated base (Eq. [2]), and the predicted values of s and s , were a n environmental covariance matrix were var(BWE), obtained by sparse matrix techniques(FSPAK, Perez- cov(BWE, WWE), and var(WWE), where E = en- Enciso et al., 1994). Second, the residual sire additive vironmental. and nonadditive genetic predictions and their cor- Estimates of base covariance matrices were used to responding variances of prediction errors were com- compute the multibreed covariance matrices needed puted as follows: by the MMME at every GEM iteration. Multibreed genetic and environmental covariance matrices for ŒØ sˆs øœ animals of all breed groups were computed as wˆs = [1 –.5 –.25] sˆss weighted sums of intrabreed and interbreed base º sˆmsß covariances (formulas [3] and [5] above). Intrabreed weights were the expected breed composition of an animal, and interbreed weights were the sum of the ØŒ cs,s cs,ss cs,ms øœ ØŒ 1 œø products of the expected fractions of A and B in the var(wˆ - w )=[1 - .5 - .25] c c c - .5 parents of an animal. For example, if the additive s s ss,s ss,ss ss,ms intrabreed A, intrabreed B, and interbreed AB covari- º cms,scms,sscms,msß º - .25ß ances for direct BW genetic effects are 9, 12, and 4 [7] kg2, then, by formula [3], the multibreed additive genetic covariance for a ãA ÔB sire will be equal to where the subscripts s = sire, ss = sire of sire, ms = [.75](9) + [.25](12) + [(1)(0) + (.5)(.5)](4) = 10.75 maternal grandsire of sire, sˆx = expected progeny kg2. A similar set of computations was done to obtain difference of x, x = s, ss, and ms, for a particular trait multibreed environmental covariances (formula [5]) and genetic effect (e.g., BW additive direct, WW for each animal with records. nonadditive maternal), and cx,x¢ = x, x¢th element of Random nonadditive genetic effects due to sire · the inverse of the left-hand side of the MMME [2], for breed-group-of-dam interactions were assumed to be a x,x¢ =s,ss,andms,foragiventraitandgeneticeffect. function of intralocus interbreed A/B interactions Predicted multibreed residuals and their error vari- occurring in all sire · breed-group-of-dam subclasses ancesofpredictionwerecomputedasindicatedin Elzo from a particular sire. Consequently, the nonadditive (1994). The remainder of the computations in the covariance matrix used for all sires was equal to the MREMLEM program followed the Cholesky maximi- base nonadditive covariance matrix. Had a subclass zation strategy outlined in Elzo (1996b). approach been used to model sire · breed-group-of- Priors. Initial values used to start the two-trait dam interactions, multibreed nonadditive covariances MREMLEM iterations were single-trait MREMLEM would have been computed as a weighted sum of base estimates of covariances between direct and maternal nonadditive covariances (Elzo, 1996b). In general, genetic effects for BW and WW, and zeroes for all multibreed covariances (additive, nonadditive, and direct and maternal covariances between BW and WW. Convergence was reached when the ratio of the environmental) would be computed as weighted sums difference between the sum of squares of the absolute of base covariances for random subclass effects, and values between two successive GEM iterations rela- they would be equal to base covariances for random tive to the sum of squares of the covariances of the regression effects. previous GEM iteration was less than 10- 4 in two Genetic Parameters. Ratios of additive genetic vari- consecutive GEM iterations. ances to phenotypic variances (heritabilities), ratios Estimates of Genetic, Environmental, and Phenotypic of nonadditive genetic variances to phenotypic vari- Covariances. The MREMLEM procedure estimated the ances (interactibilities), additive and nonadditive Downloaded from jas.fass.org by guest on January 26, 2012 1296 ELZO AND WAKEMAN genetic correlations, environmental correlations, and cov (w ,z ) = [(1.0)(1.0) + (0)(0)](6) = 6.0 kg2, n M M m phenotypic correlations were computed for five paren- 6) cov (w ,z ) = cov (w ,z ) = [(1)(0) + n D M m n M D m talbreedgroupcombinations(A · A,B· B,A· B,ØA (0)(1)](0) = 0 kg2, and 7) cov (w,z) = [.75](18) + e ØB · A, and ØA ØB · ãA ÔB). The required [.25](22) + [(1)(0) + (.5)(.5)](16) = 23.0 kg2. Thus, multibreedadditivedirectandmaternalgeneticcovar- by formula [8], the phenotypic variance of BW for the iances were computed as described above. ãA ÔB calf is equal to [10.75 + 10.0 + 2(.5)(- 1.0) + Nonadditive covariances for specific sire · breed- 3.5 + 6.0 + 2(.5)(0) + 23.0] = 52.25 kg2. group-of-dam combinations were computed as the Multibreed Genetic Predictions. Additive and nonad- product of the probability of intralocus interbreed ditive direct and maternal multibreed expected combinations in this mating times the base nonaddi- progeny differences (MEPD) for sires were obtained tive covariance matrix (formula [4]). For example, if by solving the MMME with the estimates of covari- the base nonadditive intralocus interbreed covariance ances obtained at convergence. for BW direct were 7 kg2, then by formula [4], the Additive and nonadditive direct and maternal sire nonadditivecovarianceforBWdirectin theprogenyof MEPD were obtained as the sum of a genetic group A· Bmatingswouldbe[(1)(1) +(0)(0)](7) =7 kg2. component (fixed) and a deviation from it (random). Phenotypic covariances were computed by adding Genetic groups were defined in terms of the genetic the appropriate additive, and nonadditive direct and components contained in the additive and nonadditive maternal variances and covariances, and environmen- direct and maternal random genetic effects. Sire tal covariances; thus, additive genetic groups were defined as a linear function of intrabreed (g ) and interbreed (g ) Aa ABa cov (w,z) = cov (w ,z ) + cov (w ,z ) + .5 additive genetic group effects. Similarly, sire nonaddi- p a D D i a M M d cov (w ,z ) + .5 cov (w ,z ) + tivegeneticgroupsweredefinedasalinearfunctionof a D M d a M D d cov (w ,z ) + cov (w ,z ) + .5 intralocus interbreed nonadditive (g ) genetic group n D D s n M M m n cov (w ,z ) + .5 cov (w ,z ) + effects. Thus, additive and nonadditive genetic groups n D M m n M D m cov (w,z) were computed as linear functions of the estimates of e [8] regression genetic group effects from the MMME. However, sire additive (s ) and nonadditive (s ) a n where w and z are two traits(e.g., BW and WW), and deviationswereobtained directly from the solutions to the subscripts p = phenotypic, a = additive, n = the MMME. nonadditive, e = environmental, i = calf, d = dam, s = The linear function used to compute sire MEPD for sire, m = maternal grandsire, D = direct, M = additive direct and maternal genetic effects was a maternal.Directandmaternaladditivegeneticcovari- weighted sum of direct and maternal sire additive ancesinformula[8]areobtainedbyformula[3],direct intrabreed and interbreed group genetic effects, plus and maternal nonadditive genetic covariances by sire additive random genetic effects. Intrabreed addi- formula [4], and environmental covariances by for- tive weights were the expected fractions of A alleles in mula [5]. As an example, consider the computation of thesiresthemselves.Interbreedadditiveweightswere the phenotypic variance of BW for calves of breed the sum of the products of expected A and B fractions group ãA ÔB, which are progeny of unrelated A sires in the parents of each sire. Thus, and ØA ØB (F ) dams. Assume that 1) the additive 1 intrabreed A, intrabreed B, and interbreed AB covari- uˆ = p gˆ + (p p + p P )gˆ + sˆ ai Ai Aa As Bs Ad Bd ABa ai ances a) for direct BW genetic effects are 9 kg2, 12 [9] kg2, and 4 kg2, b) for maternal BW genetic effects are 6, 14, and 2 kg2, and c) between BW direct and where uˆ =direct or maternal additive MEPD for sire ai maternal genetic effects are 0, - 2, and - 4 kg2, 2) the i, p =expected fraction of breed X (X =A, B) in the Xk nonadditive interbreed A/B covariance a) for direct breed group of animal k, k = i, s (sire of sire i), d BW genetic effects is 7 kg2, b) for maternal BW (dam of sire i), gˆ = generalized least squares Aa genetic effects is 6 kg2, and c) between direct and estimateofg - g , gˆ =generalized least squares Aa Ba ABa maternal BW genetic effects is 0 kg2, and 3) the estimate of g - .5(g + g ), and sˆ = best ABa AAa BBa ai environmental intrabreed A, intrabreed B, and inter- linear unbiased predictor of s . ai breed AB variances for BW are 18, 22, and 16 kg2, Nonadditive direct and maternal sire MEPD were respectively. The values of the covariances needed to computed as the sum of their direct and maternal compute the phenotypic variance of BW for the ãA nonadditive intrabreed intralocus group and random ÔB calf are as follows: 1) cov (w ,z ) = [.75](9) + genetic effects. Thus, a D D i [.25](12) + [(1)(0) + (.5)(.5)](4) = 10.75 kg2, 2) cov (w ,z ) =[.5](6) +[.5](14) +[0](2) =10.0 kg2, uˆ = (p p + p p )(gˆ + sˆ ) [10] a M M d ni Ai Bd Bi Ad n ni 3) cov (w ,z ) = cov (w ,z ) = [[.5](0) + [.5](- 2) a D M d a M D d +[(1)(0) +(0)(1)](- 4)] =- 1.0 kg2, 4) cov (w ,z ) where uˆ = direct or maternal nonadditive MEPD for n D D s ni = [(1.0)(.5) + (.0)(.5)](7) = 3.5 kg2, 5) sire i, p =expected fraction of breed X (X =A, B) in Xi Downloaded from jas.fass.org by guest on January 26, 2012 GROWTH COVARIANCES AND GENETIC PREDICTION 1297 Table 2. Estimates of base additive and nonadditive genetic covariances for birth weight and weaning weight Genetic covariances, kg2 Additive Additive Additive Additive Trait paira intrabreed A intrabreed B interbreed AB interbreed A/B BWD, BWD 6.41 7.60 .21 5.47 BWD, WWD 7.09 10.50 1.35 4.97 BWD, BWM .03 - .32 - 2.66 .14 BWD, WWM - .90 1.62 - 8.26 - .36 WWD, WWD 138.01 208.53 8.64 138.78 WWD, BWM .89 2.80 - 17.05 1.10 WWD, WWM - 32.70 - 39.91 - 51.52 3.11 BWM, BWM 4.90 5.87 33.74 5.95 BWM, WWM 1.09 2.82 105.61 3.52 WWM, WWM 100.44 151.69 751.69 154.82 aD = direct; M = maternal. sire i, p =expected fraction of breed X (X =A, B) in 11.7 min for the MREMLEM program to achieve Xd damsofbreedgroupdmatedtosirei, gˆ =generalized convergence. Tables 4 and 5 contain genetic n least squares estimate of g - .5(g + g ), parameters for straightbred and crossbred groups. A/Bn A/An B/Bn and sˆ = best linear unbiased predictor of s . Table 4 contains heritabilities, and additive genetic, ni ni Total sire MEPD were computed as the sum of sire environmental, and phenotypic correlations for five additive MEPD and sire nonadditive MEPD: parental breed group combinations (A · A, B · B, A · B, ØA ØB · A, and ØA ØB · ãA ÔB). These breed uˆti = uˆai + uˆni [11] group combinations were chosen to illustrate esti- mates of genetic parameters in straightbred animals Nonadditive and total MEPD were computed here (progeny of A · A, and B · B parents), crossbred assuming that sires had been mated to ØA ØB dams. animalswithoutinterbreedadditivebutwithnonaddi- Dams of breed group ØA ØB were chosen because the tive direct genetic effects (progeny of A · B parents), probabilities of intralocus interbreed A/B interactions crossbredanimalswithinterbreedadditiveandnonad- are the same (.5) for sires of any A and B fractions. ditive direct genetic effects (progeny of ØA ØB · A Thus, sires of any A and B fractions can be compared parents), and crossbred animals with interbreed on an equal basis for all direct and maternal genetic additive and nonadditive direct and maternal genetic effects (additive, nonadditive, and total). effects(progeny ofØAØB· ãAÔBparents). Table5 To illustrate the computation of additive, nonaddi- contains the ratios of nonadditive genetic variances to tive, and total MEPD, assume that 1) the additive phenotypic covariances (interactibilities) and nonad- intrabreed and interbreed, and the nonadditive inter- ditive genetic correlations for three parental breed breed genetic values for WW are - 8, - 14, and 20 kg, group combinations whose progeny are expected to 2) the intragroup additive and nonadditive predic- showintralocusinterbreedA/Binteractioneffects(A · tions of a ÔA ãB sire are 6 and 9 kg, and 3) the ÔA B, ØA ØB · A, and ØA ØB · ãA ÔB). ãB sire will be mated to A dams. This sire’s additive Intrabreed heritability estimates were of medium direct MEPD (formula [9]) is equal to [.25](- 8) + size for direct and maternal effects of BW and WW, [(.5)(.5) + (0)(0)](- 14) + (6) = .5 kg, its nonaddi- although they were somewhat larger for direct genetic tive MEPD (formula [10]) is [(.25)(0) + effects (.22 (A) and .23 (B) for BWD, and .25 (A) (.75)(1)](20) +[(.25)(0) +(.75)(1)](9) =21.75 kg, and .29 (B) for WWD) than for maternal genetic and its total MEPD (formula [11]) is .5 + 21.75 = effects (.17 (A) and .18 (B) for BWM, and .18 (A) 22.25 kg. ResultsandDiscussion Table 3. Estimates of base environmental covariances for birth weight and weaning weight Covariance Components and Genetic Parameters Environmental covariances, kg2 The MREMLEM estimates of base intrabreed and interbreed additive, and interbreed nonadditive Trait pair Intrabreed A Intrabreed B Interbreed AB genetic covariances are shown in Table 2, and those of BW, BW 17.51 19.40 8.19 base intrabreed and interbreed environmental covari- BW, WW 21.61 23.79 - 5.75 ances are in Table 3. It took 5 GEM iterations and WW, WW 356.17 408.24 12.56 Downloaded from jas.fass.org by guest on January 26, 2012 1298 ELZO AND WAKEMAN Table 4. Estimates of heritabilities, and additive genetic, environmental, and phenotypic correlations for birth weight (BW) and weaning weight (WW) Breed group combination Parametera A · A B · B A · B ØA ØB · A ØA ØB · ãA ÔB h2(BWD) .22 .23 .19 .16 .13 r (BWD, WWD) .24 .26 .25 .25 .26 A r (BWD, BWM) .01 - .05 - .02 - .08 - .11 A r (BWD, WWM) - .04 .05 .01 - .05 - .06 A h2(WWD) .25 .29 .22 .18 .15 r (WWD, BWM) .03 .08 .06 - .06 - .10 A r (WWD, WWM) - .28 - .22 - .25 - .22 - .21 A h2(BWM) .17 .18 .15 .32 .38 r (BWM, WWM) .05 .09 .07 .44 .51 A h2(WWM) .18 .21 .16 .35 .41 r (BW, WW) .27 .27 .27 .24 .22 E r (BW, WW) .23 .27 .24 .28 .30 P ah2 = heritability; r = additive genetic correlation; r = environmental correlation; r = phenotypic A E P correlation; D = direct; M = maternal. and .21 (B) for WWM). Estimates of additive genetic tive variances, 1) heritability estimates for direct correlations were close to zero (- .05 to .09), except for genetic effects were smaller in progeny groups with those between BWD and WWD (.24 (A) and .26 interbreed additive variation (e.g., .13 for BWD, and (B)), and between WWD and WWM (-.28 (A) and .15 for WWD, in progeny from ØA ØB · ãA ÔB - .22 (B)). Estimates of intrabreed additive genetic parents) than those in progeny groups without covariances, heritabilities and genetic correlations interbreed additive variation (e.g., .19 for BWD, and obtained here were similar to those estimated with .22 for WWD, in progeny from A · B parents), and 2) field data for these two breeds by researchers at the heritability estimates for maternal genetic effects University of Georgia (Kriese et al., 1991; Pollak et were larger in progeny groups with interbreed addi- al., 1994, reporting covariances estimated at the tivevariation(e.g., .38forBWM,and.41forWWM,in University of Georgia; A. Nelson, personal communi- progenyfromØAØB· ãAÔBparents) thanthoseof cation). Thus, it seems that the animals in the AB progeny groups without interbreed additive variation multibreed herd had a reasonable representation of (e.g., .15forBWM,and.16 forWWM,in progenyfrom the A and B populations in the country. A · B parents). Similarly, estimates of additive The pattern of values of interbreed additive genetic genetic correlations tended to be larger in progeny covarianceestimates were substantially different from groups in which interbreed variation was expected to intrabreed covariances (Table 2). Additive direct be present (Table 4). interbreed genetic variances were much smaller than Estimates of intralocus interbreed A/B nonadditive intrabreed ones. Additive maternal genetic variances, covariances due to sire · breed-group-of-dam interac- however, were much larger than intrabreed ones. Additive interbreed covariances also tended to be of larger absolute value than intrabreed additive covari- ances. These patterns were observed to a much lesser Table 5. Estimates of interactibilities and nonadditive extent in the covariance estimates from the single- genetic correlations for birth weight (BW) trait runs used as priors. Thus, a factor that must and weaning weight (WW) have contributed to these differences is the small size of the data set relative to the large number of ØA ØB ØA ØB · covariances(49) being estimated in the two-trait run. Parametera A · B · A ãA ÔB Another factor that might be relevant is the fact that i2(BWD) .15 .06 .05 only data from progeny of crossbred parents help r (BWD, WWD) .18 .18 .18 estimate interbreed additive covariances. Thus, a N r (BWD, BWM) .03 .03 .03 N larger variety of crossbred parents in the multibreed r (BWD, WWM) - .01 - .01 - .01 N population should yield better estimates of interbreed i2(WWD) .18 .08 .06 additive covariances. rN(WWD, BWM) .04 .04 .04 Despite seemingly poor estimates of interbreed rN(WWD, WWM) .02 .02 .02 i2(BWM) .16 .07 .06 additive covariances, estimates of genetic parameters r (BWM, WWM) .12 .12 .12 in progeny groups from parental mating groups with N i (WWM) .20 .09 .07 2 interbreed additive variation (e.g., progeny from ØA ØB · ãA ÔB parents, Table 4) were reasonable. ai2 = interactibility (ratio of intralocus interbreed genetic vari- ance to phenotypic variance); r =nonadditive genetic correlation; N Because of the values estimated for interbreed addi- D = direct; M = maternal. Downloaded from jas.fass.org by guest on January 26, 2012