MULTIBREED EVALUATION -
THEORY AND APPLICATION[1]
M. A. Elzo
Animal Science Department
University of Florida,
Gainesville 32611
Introduction
The vast majority of the
beef produced in the United States comes from crossbred animals. If all animals involved in beef cattle
production systems in the United States were included in a gigantic genetic
evaluation scheme, they would constitute a complex multibreed population. Even if we considered only partial sections
of the United States multibreed population, these are likely to be reasonably
complete multibreed subpopulations. In
other words, multibreed populations in the United States are the rule rather
than the exception. If multibreed
populations are so abundant, then why are there no multibreed genetic
evaluation systems to evaluate animals, especially sires, for additive,
nonadditive, and total multibreed expected progeny differences? What are the problems that need to be solved
to produce national multibreed genetic evaluations?
This presentation discusses
some theoretical and applied aspects of multibreed genetic evaluation and
estimation of covariance component procedures used at the University of Florida
that may be useful for the development of national multibreed genetic
evaluation systems. Problems that need
to be solved to generate national, and perhaps international, multibreed
genetic evaluations are subsequently discussed in a general context.
Multibreed Populations
Multibreed populations are
those composed of straightbred and crossbred animals that interbreed. In other words, crossbred animals are not
just the end product of some straightbred × straightbred, straightbred ×
crossbred, and crossbred × crossbred matings, but they also actively participate
as parents of the next generation if chosen by some selection process. For the purposes of this discussion,
multibreed population will be classified as either complete or incomplete.
Complete multibreed
populations are those whose mating pattern follow a diallel design, i.e., they
have the same breed groups of sires and dams, and sires are mated across breed
groups of dams. The Angus-Brahman Herd
of the University of Florida has a complete multibreed design; Table 1 shows
the number of sires used per breed group of sire × breed group of dam in this
herd. In Table 1, BGS (BGD) = breed
group of sire (dam), A = Angus, B = Brahman, and Br = Brangus.
Incomplete multibreed
populations are those that have different numbers and (or) kinds of mating
groups of sires and dams. In addition,
sires of some mating groups may be preferentially mated to certain breed groups
of dams. The mating design of the
Sanmartinero-Brahman Multibreed Herd of La Libertad, Colombia, was incomplete;
Table 2 contains the numbers of bulls per breed group of sire × breed group of
dam combination; BGS (BGD) = breed group of sire (dam), S = Sanmartinero, and B
= Brahman.
The structure of the overall
multibreed population and the various multibreed subpopulations (e.g.,
Angus-Brangus-Brahman, Simmental-Canadian Simmental-Simbrah-Brahman) in North
America, because of the practice of grading up to a single breed and (or) their
goal of producing composite breeds, is likely to be more complete to the design
shown in Table 2. Multibreed Expected
Progeny Differences (MEPD) were computed for the Angus-Brahman Herd (Elzo and
Wakeman, 1998; Elzo et al., 1998b) and for the Sanmartinero-Brahman Herd (Elzo
et al., 1999). Considering the much
larger data sets available for the various multibreed subpopulations in North
America, it seems reasonable to assume that at least from a number of data
viewpoint, such evaluations are feasible.
The main issue here is likely to be connectedness among multibreed
contemporary groups. Cornell University
has had a Multibreed Genetic Evaluation program for the Simmental-Canadian
Simmental-Simbrah population that accounts for group nonadditive genetic
effects since 1997 (Klei and Quaas, 1995; Pollak and Quaas, 1998). Thus, at least in this multibreed population,
connectedness among multibreed contemporary groups was not a problem. This aspect needs to be investigated in all
multibreed subpopulations in the United States.
Genetic and Environmental
Effects
The type of genetic and
environmental effects present in multibreed populations can be classified in
almost exactly the same form as it is done in genetic analyses within
populations. The sole exception is the
existence of interbreed genetic effects.
Multibreed populations have additive and nonadditive, direct and
maternal, intrabreed and interbreed genetic effects.
Figure 1 shows a simplified
flowchart of genetic effects considered in a multibreed model. If all interbreed genetic effects were
omitted this chart will reflect an intrabreed analysis. A similar chart can be constructed for environmental
effects. However, environmental effects
associated with specific genetic effects are of less interest, thus they are
usually lumped together in less specific residual terms. The larger the number of base breeds in a
multibreed population the larger the need for this type of simplifying
assumptions. For example, there could
be intra and interbreed residual terms, or, a different residual term per breed
group of calf could be defined.
Whichever assumption is made relative to residual effects, it will be a
compromise between a desire for accuracy and computational feasibility.
Modeling Strategies
Additive and nonadditive
genetic effects in a multibreed prediction model can be viewed from a subclass
(i.e., 1 and 0 in the design matrix) or a regression viewpoint. Explaining a genetic effect in terms of
regression is done to simplify computations, but regression predictions are
usually approximations to the complete (subclass) effect. On the other hand, prediction of genetic
effects not represented in the data (e.g., future crossbred matings) can be
easily obtained. The author has
considered both strategies when modeling multibreed additive and nonadditive
genetic effects.
The subclass approach to
multibreed additive genetic effects is similar to the intrabreed situation,
except that the matrix of covariances among animals depends on at least two
sets of additive intrabreed and, if parents are crossbred, on one set of
additive interbreed genetic covariances.
The regression approach for additive genetic effects aim at predicting
all intra and interbreed genetic effects present in every animal. The usefulness of this approach in a
national animal evaluation seems unclear.
The subclass approach to
multibreed nonadditive genetic effects, where these effects are predicted
within sire × breed group of dam subclasses, can quickly become an extremely
large problem if breed groups of dams are defined in terms of their expected
breed compositions. One way to simplify
the problem is to redefine breed groups of dams in terms of ranges rather than
individual breed expectations. However,
the problem could still be too large from a practical standpoint. Another approach is to use regression to
approximate nonadditive genetic effects within sire × breed group of dam
subclasses (Elzo, 1983). If the number
of factors used to predict nonadditive genetic effects is small (e.g.,
intralocus interbreed), it can yield substantial computational savings. If, on the other hand, the number of
nonadditive prediction factors is large, computations can be even more involved
than using a subclass approach based on ranges of breed groups of dams. One of the main advantages of the regression
approach is that it permits the prediction of nonadditive genetic effects for
future matings. Thus, even if sires are
initially tested with a fraction of the future spectrum of possible matings,
predictions of nonadditive genetic effects for all of them can be easily made.
Multibreed Linear Model
Figure 2 shows, as an
example, a generic multibreed sire-maternal grandsire linear model. An animal model would substitute calf for
sire and dam for maternal grandsire.
The capital letters in parenthesis in Figure 2 indicate additive (A),
nonadditive (N), direct (D), and maternal (M) genetic effects. Fixed effects are similar to intrabreed
models except for exclusion (or inclusion) 
of breed group effects. For example, multibreed contemporary groups
would follow the usual Beef Improvement Federation recommendations (BIF, 1996)
excluding breed group of calf. In fact,
calves of several breed groups must be reared together to create a
multibreed comparison group. Similarly,
age of dam would consider dam breed composition in some form, in addition of
sex of calf. Additive and nonadditive
genetic groups have similar meaning to that of intrabreed genetic models, and
can follow a subclass or a regression approach. Similarly, random additive genetic effects can have a subclass or
regression form. Preferably all
additive and nonadditive genetic effects should be deviated from a single base
(e.g., a base breed). The active
options of the current version of the MREMLEM program are regression additive
and nonadditive groups, subclass additive random effects, and regression
nonadditive random effects.
The type of multibreed
expected progeny differences (MEPD) produced by the multibreed model in Figure
2 are: 1) additive direct (AD), 2) additive maternal (AM), 3) nonadditive
direct (ND), 4) nonadditive maternal (NM), 5) total direct (TD = AD + ND), and
6) total maternal (TM = AM + NM). A
regression approach yields predictions of maximum values of genetic
effects. Thus, if regression was used
to predict nonadditive regression effects (as in MREMLEM), nonadditive genetic
predictions for specific crossbred matings need to be computed by weighing the
predicted values obtained from the solutions to the mixed model equations by
their probability of occurrence in these matings.
Estimation of Multibreed
Covariances
The largest problem to be
solved in multibreed genetic evaluation systems by far is that of estimating
the large number of multibreed covariances needed. However, because additive, nonadditive, and environmental
multibreed covariances are linear combinations of a much smaller set of base
additive, nonadditive, and environmental covariances this covariance estimation
problem is drastically reduced.
Computations can be further decreased if during a prediction or a
covariance estimation run, only multibreed covariances corresponding to breed
combinations present in the evaluation are computed.
Figure 3 shows the
computational procedure used by the MREMLEM program to estimate covariance
components. The computational procedure
uses a Generalized Expectation-Maximization algorithm (GEM, Dempster et al.,
1977) to first compute the elements of the Cholesky Decomposition of the base
additive, nonadditive, and environmental covariance matrices, then the base
covariance matrices are computed by multiplication by their transposes, and
finally multibreed covariance matrices are computed as linear combinations of
base covariance matrices. Subsequently,
genetic parameters (heritabilities, interactibilities, correlations) are
computed for a predetermined number of breed groups.
The MREMLEM program has been
used to estimate additive and nonadditive genetic, and environmental base and
multibreed covariances in several small experimental multibreed data sets (Elzo
and Wakeman, 1998; Elzo et al., 1998a,b, 1999). The main problem encountered when estimating covariances in these
data sets was the presence of multicollinearity when solving the set of
equations in the maximization step of the GEM algorithm. This multicollinearity problem seemed to get
worse when interbreed additive covariances (segregation covariances in the
terminology of Wright (1968), Lande (1981), and Lo et al., 1993) needed to be
computed. Adding a small number to the
diagonal of these equations (Elzo, 1996) helped somewhat. Partial steps (Jennrich and Schluchter,
1986) in the maximization step of the GEM algorithm were usually needed to
increase the value of the log-likelihood of the complete data. Whether much larger number of data will
decrease multicollinearity needs to be investigated. However, because multibreed covariances are linear combinations
of base covariances, if these base covariances are too similar, then
some degree of multicollinearity in the set of equations of the maximization
step seems inevitable. Adding a
computational rule of equality of estimated base covariances (or base
covariance matrices) in the multibreed GEM algorithm could be an alternative to
obtain approximate estimates.
Analyses of Multibreed Data
Sets
Three multibreed herds have
been analyzed using multibreed genetic evaluation procedures: 1) the Angus-Brahman
herd of the University of Florida for preweaning growth traits (Elzo and
Wakeman, 1998), and carcass traits (Elzo et al., 1998b), 2) the
Romosinuano-Brahman herd of Turipaná, Colombia for pre and postweaning growth
traits (Elzo et al., 1998a), and 3) the Sanmartinero-Brahman herd of La
Libertad, Colombia for pre and postweaning growth traits (Elzo et al.,
1999). The preweaning growth traits
were birth and weaning weights, and the postweaning growth trait was weight
gain between weaning and 16 mo of age.
The carcass traits were carcass weight, area of the longissimus muscle,
fat over the longissimus muscle, kidney-pelvic-heart fat, marbling score, and
Warner-Bratzler shear force. The goal
in all these studies was the characterization of these multibreed herds for the
analyzed traits. Thus, base and
multibreed additive, nonadditive, environmental, and phenotypic covariance
components and genetic parameters were estimated, and additive, nonadditive,
and total genetic values were predicted.
The multibreed model used in
these analyses was a sire-maternal grandsire model (Figure 2) and covariances
and genetic parameters were estimated as shown in Figure 3. As examples of the parameters obtained in
these analyses, Tables 3 contains estimates of heritabilities and
interactibilities (ratios of intralocus interbreed nonadditive to phenotypic
variances) for preweaning growth traits, and Table 4, estimates of the same
parameters for carcass traits, in the Angus-Brahman multibreed herd of the
University of Florida. The most salient
aspects of these tables is that heritability estimates were similar in Angus
and Brahman (except for Warner-Bratzler shear force), interactibility values
were somewhat lower (but not zero), except for fat over the longissimus dorsi
(steers slaughtered at similar fat endpoints).
A similar pattern of heritability-interactibility estimates for growth
traits was obtained in the Romosinuano-Brahman and Sanmartinero-Brahman
multibreed herds.
To illustrate predictions
results for growth traits, Figure 4 shows direct additive, nonadditive, and
total MEPD in the Angus-Brahman herd (left column) and the Sanmartinero-Brahman
herd (right column), and Figure 5 shows the same arrangement for maternal
additive, nonadditive, and total MEPD in these two herds. The meaning of the numbers in the abcisa of
the Angus-Brahman graphs is: 1 = Angus, 2 = ¾A ¼B, 3 = ½A ½B, 4 = ¼A ¾B, 5 =
Brahman, and 6 = Brangus. The letters
in the abcisa of the Sanmartinero-Brahman graphs mean: S = Sanmartinero, X = ½S
½B, C = Brahman. Sires were ordered by
date of entry into the stud within breed group of sire; thus, they show
an unweighted trend in sire usage over time by breed group in these herds.
The main points in these
graphs are: 1) the samples of Angus and Sanmartinero sires behaved in similar
fashion relative to the samples of Brahman sires in each herd for direct
additive MEPD, but not for maternal additive MEPD, where Sanmartinero sires
were clearly superior to Brahman sires for maternal additive MEPD for weaning
weight, 2) there was nonadditive direct and maternal variation among sires of
all breed groups for nonadditive MEPD, 3) total MEPD tended to increase toward
Brahman sires for direct and maternal genetic weaning weight effects in the
Angus-Brahman herd, but only for direct genetic effects in the
Sanmartinero-Brahman herd; the total maternal MEPD of Sanmartinero sires was
larger than those of Brahman sires.
Although some of the sires used in the Colombian herds were the same as
those used in the Angus-Brahman herd, a joint analysis has not been done yet,
thus only indirect comments, as the ones above, are possible at this time.
Figures 6 and 7 illustrate
multibreed predictions for carcass traits in the Angus-Brahman herd of the
University of Florida. Direct additive,
nonadditive, and total MEPD for carcass weight and longissimus muscle area are
shown in Figure 6, whereas Figure 7 contains the same MEPD for marbling score
and Warner-Bratzler shear force. Recall
that steers were slaughtered at approximately the same fat endpoint. These graphs reaffirm visually the existence
of intralocus interbreed nonadditive genetic variability in the Angus-Brahman
multibreed herd, as well as show in detail differences in direct additive MEPD
among sires of the six breed groups.
For example, Angus sires had the smallest additive and total MEPD for
carcass weight, among the largest ones for area of the longissimus muscle, the
largest ones for marbling score, and the lowest ones for Warner-Bratzler shear
force. According to these graphs,
intralocus interbreed nonadditive MEPD would increase most sire's total MEPD
values for carcass weight and longissimus muscle area, but it would decrease
most sire's total MEPD for marbling score and Warner-Bratzler shear force.
The ranking of sires by
additive and nonadditive MEPD for all growth and carcass traits was
substantially different (correlation between .04 and .41). The correlation between additive and total
MEPD yielded values between .81 and 1.00 among the growth and carcass traits
analyzed in these herds. Thus,
accounting for intralocus interbreed nonadditive genetic effects will be more
useful to help identify the best sires based on total MEPD for some traits
(e.g., weight traits) than for others (e.g., kidney-pelvic-heart fat).
Estimates of covariance
components and genetic predictions indicated that there was substantial
variability for intralocus interbreed nonadditive genetic effects in the three
analyzed multibreed herds. Thus, selection
of animals for both additive and nonadditive genetic effects is feasible in
these herds. Furthermore, when sires
were ordered by their additive, nonadditive, and total MEPD, sires from all
breed groups could be found in the top, middle, and bottom tiers. A conservative selection approach would be
to first select animals for their additive MEPD, and then select the best ones
for their total MEPD among those in the preselected group. This would ensure additive genetic progress
and increased combining ability of the selected sires in the next generation.
National Multibreed
Evaluations
The main problem to be
solved to have a unified system of national multibreed genetic evaluations is
to agree on the definition of the national beef cattle multibreed
population(s). Elzo (1995)
discussed three possible alternatives, two of which would produce national
multibreed genetic evaluations. The
three definitions were: 1) a single multibreed population including all
straightbred and crossbred beef cattle in the country, 2) several overlapping
multibreed subpopulations, each composed by several breeds and all crossbred
groups of these breeds, and 3) several extended breeds which included
straightbred sires and their mates of various breed compositions. The first two definitions will yield
national multibreed genetic evaluations.
Furthermore, the inclusion of international data in USA intrabreed
genetic evaluations (e.g., Canadian Simmental, Uruguayan Hereford) has opened
the door for continental, and perhaps world genetic evaluations in the future.
The simplest situation would
be to have a single national multibreed population. Although this may not be hitherto possible, it should be
seriously considered for the future.
The second definition is more realistic at this time. The data set used in the Cornell Multibreed
Evaluation which includes Simmental, Canadian Simmental, and Simbrah, is the
closest to one to meet the criteria of this definition. It would be complete if all Brahman data
were included as well. Other multibreed
subpopulations could be Angus-Brangus-Brahman, Hereford-Braford-Brahman,
Limousin-Bramousin-Brahman, etc. Since
all these multibreed subpopulations have Brahman in common, connectedness among
them would be created by using reference Brahman sires in (some) of their
contemporary groups.
Some breed associations are
currently accepting straightbred and crossbred records from other breeds (E. J.
Pollak, personal communication).
Consequently, ties among overlapping multibreed populations will become
stronger over time, thus facilitating the generation of national multibreed
additive, nonadditive, and total MEPD.
A second problem is
methodological. Multibreed procedures
are computationally more demanding than intrabreed ones. Given the complexity of the national
multibreed data sets in terms of the numbers of breeds and crossbred groups,
efficient linear and nonlinear multibreed computational procedures need to be
developed.
A third problem is
publication of information. The large number
of MEPD (additive, nonadditive, total)
generated per animal in a multibreed evaluation makes it unfeasible to
completely publish them on paper. Thus,
electronic publication should be considered the primary source. Furthermore, publication of nonadditive and
total MEPD for every potential crossbred mating of an animal is clearly
inadvisable. Perhaps a mating program
should be part of the electronic service, which would provide a list of
potential mates for a dam with their additive, nonadditive, and total
MEPD. These mates could be chosen
within a single breed or among several breeds specified by the user. In short, with electronic distribution of
MEPD, additional services to facilitate and expand their usability will need to
be considered.
The problems outlined above
are simply a chapter in the ongoing evolution of national genetic evaluation
methodology. Solutions to these
problems will require an even larger level of communication and cooperation
among producers, breed associations, and university researchers.
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Figure 4. Direct Additive, Nonadditive, and Total
Multibreed Expected Progeny Differences for Weaning Weight in the Angus-Brahman
Herd of the University of Florida (left column) and in the Sanmartinero-Brahman
Herd of La Libertad, Colombia (right column).
Figure 5. Maternal Additive, Nonadditive, and
Total Multibreed Expected Progeny Differences for Weaning Weight in the
Angus-Brahman Herd of the University of Florida (left column) and in the
Sanmartinero-Brahman Herd of La Libertad, Colombia (right column).
Figure 6. Direct Additive, Nonadditive, and Total
Multibreed Expected Progeny Differences for Carcass Weight and Longissimus
Muscle Area in the Angus-Brahman Multibreed Herd of the University of Florida.
Figure 7. Direct Additive, Nonadditive, and Total
Multibreed Expected Progeny Differences for Marbling Score and Warner-Bratzler
Shear Force in the Angus-Brahman Multibreed Herd of the University of Florida.