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Maternal Effects Model
L. R. Schaeffer, March 1999
Updated March 16, 2000
In some species of livestock, such as beef cattle, sheep or swine,
provides an environment for its offspring to survive and grow. Females
vary in their ability to provide a good environment for their offspring,
and this variability has a genetic basis. The offspring inherit
directly an ability to grow (or survive) from both parents, and
environmentally do better or poorer depending on their dam's maternal
ability. Maternal ability is a genetic trait and is transmitted,
as usual, from both parents, but maternal ability is only expressed by
females when they have a calf (i.e. much like milk yield in dairy cows).
A model to account for maternal ability is
is the growth trait of a young animal,
is a vector of fixed factors influencing growth, such as
contemporary group, sex of the offspring, or age of dam,
is a vector of random additive genetic effects (i.e. direct
genetic effects) of the animals,
is a vector of random maternal genetic (dam) effects, and
in this model, is a vector of maternal permanent
environmental effects (because dams may have more than one
offspring in the data).
The expectations of the random vectors,
are all null
vectors in a model without selection, and the variance-covariance
is the additive genetic variance,
is the maternal genetic variance,
is the additive genetic by maternal genetic covariance,
is the maternal permanent environmental
In this model, a female animal, i, could have its own growth
record for estimating
The same female could later
have offspring of its own for estimating
and the offspring would also contribute towards
The maternal effects model can be more complicated
if, for example, embryo transfer is practiced. Recipient dams would
have maternal effects, but would not have direct genetic effects on
that calf, see Schaeffer and Kennedy (1989).
To better understand this model we will simulate some records. Let
Now any positive definite matrix can be partitioned into the
product of a matrix times its transpose (i.e. Cholesky decomposition),
differs from previous models in that both the additive genetic and
maternal genetic effects need to be generated simultaneously because
these effects are genetically correlated. We will generate true
breeding values for three animals, A, B, and C, where C is
an offspring of sire A and dam B, and then we will generate an
observation on animal C.
For A, we need two random normal deviates which will be pre-multiplied
We assume that A is a base population animal that is
unrelated to B. Let
Similarly for animal B,
Creating a progeny's true breeding value is similar to the scalar
version. Take the average of the parents' true breeding values
and add a random Mendelian sampling term.
Note that all animals have both a direct and maternal genetic breeding
Next for all dams a maternal permanent environmental effect should be
generated. In this case only for animal B, multiply a random normal
suppose it is -4.491.
An observation for animal C is created by following the model
The Fixed Effects contribution of 140 was arbitrarily chosen for
this example. The point of main importance is that the observation
on animal C consists of the direct genetic effect of animal Cplus the maternal genetic effect of the dam (B) plus the maternal
permanent environmental effect of the dam (B) plus a residual.
To illustrate the calculations, assume the data as given in the
CG stands for contemporary group, the only fixed effect in this
example. Assume that the appropriate variance parameters are
were used in the simulation in the previous section.
Based on the matrix formulation of the model,
the MME are
Note that these numbers are not equal to
The matrices are
The other two right hand side matrices can be easily obtained from
Thus, the order of the MME will be 24. The inverse of the
relationship matrix is
The solutions to the MME are
No correlations with true values were calculated for this small
Estimation of Variances
Let the inverse of HMME be represented as
Then the REML formulas and results for this model are
All of these values seem to be twice as large as the assumed values
at the start. The REML process is iterative and the new estimates
need to be used to re-form the MME, solve, and so forth.
These data were not very amenable to estimation of the maternal
genetic or permanent environmental components
because there were only two dams with progeny, and the estimation of
the covariance between direct and maternal components would suffer
because none of the animals appeared both as a dam and as an
offspring. That is, the only link between direct and maternal effects
is through relationships. To show that this is a problem, note that
This relationship holds for nearly all animals in the analysis,
except animals 3, 4, and 9.
Thus, the estimate of the maternal genetic component for animal 1 is
a function of the elements of
and the solution for
the direct genetic component. The correlation between
was -.961, which is a lot stronger than the
assumed prior correlation of -.196.
There needs to be female animals with records that later have their
own progeny in order to get sound REML estimates of the direct -
maternal covariance. Thus, several generations of data are needed
with good links between generations. To get a good estimate of the
maternal ability of a bull's daughters, a bull has to have daughters
which have had calves (i.e. grandprogeny). Often there are insufficient
links between generations and this has given rise to many highly
biased estimates of the direct - maternal genetic correlation.
The maternal effects model requires a slightly different approach
because of the non-zero correlation between direct and maternal
genetic effects. Let
Now look at
The prior distributions for the variance components are
where vp, ve, ,
S2p, S2e, and
are hyperparameters, p=2 is the dimension of .
The prior distributions for
are scaled inverted Chi-square distributions while that for is an inverse Wishart distribution, the multivariate equivalent of
the inverse Chi-square distribution. Thus, we will need a good
random Wishart matrix generator.
The joint posterior distribution is constructed and from this the
fully conditional distributions for each component of the unknowns
can be derived. The results are similar to the previous models.
The vectors ,
in the MME
are processed in the same manner as before with some small
modifications. For the fixed effects,
For the additive (direct) genetic effects,
For the maternal genetic effects,
For the maternal PE effects,
The computations are the same as before for
but are a little different for the genetic
variances and covariance. For the residual component,
For the maternal PE component,
and np is the number of
dams with offspring. For the genetic components, ,
calculations are first,
and perform a Cholesky decomposition on
the inverted matrix, i.e.,
then utilize a Wishart matrix generator such as
then GI is the new
to use in
re-forming the MME for the next round of sampling. Thus,
Then the next round of sampling begins. A critical point in this
scheme is the Cholesky decomposition of
check should always be made that this calculation has been
performed satisfactorily, i.e. that the routine did not encounter
a 0 or negative number on the diagonals during the decomposition.
However, the manner in which
is formed and presuming
is positive definite, then this problem should not
arise, but a check is still worthwhile.
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