MA Elzo and DL Wakeman growth genetic effects in an Angus-Brahman multibreed herd

15 Pages·1998·0.27 MB·English

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.
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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
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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
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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
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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:
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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
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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
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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
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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
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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.
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