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L. R. Schaeffer, March 1999

Updated March 16, 2000

In some species of livestock, such as beef cattle, sheep or swine, the female 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

where 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,
,
,
,
and
are all null
vectors in a model without selection, and the variance-covariance
structure is

where is the additive genetic variance, is the maternal genetic variance, is the additive genetic by maternal genetic covariance, and is the maternal permanent environmental variance. Also,

where

and

and

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
,
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), or

Let and . This model 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,

For *A*, we need two random normal deviates which will be pre-multiplied
by .
We assume that *A* is a base population animal that is
unrelated to *B*. Let
,
then

Similarly for animal

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 values.

Next for all dams a maternal permanent environmental effect should be
generated. In this case only for animal *B*, multiply a random normal
deviate by
,
suppose it is -4.491.

An observation for animal *C* is created by following the model
equation,

The Fixed Effects contribution of 140 was arbitrarily chosen for this example. The point of main importance is that the observation on animal

**HMME**

To illustrate the calculations, assume the data as given in the table below.

Animal | Sire | Dam | CG | Weight |

5 | 1 | 3 | 1 | 156 |

6 | 2 | 3 | 1 | 124 |

7 | 1 | 4 | 1 | 135 |

8 | 2 | 4 | 2 | 163 |

9 | 1 | 3 | 2 | 149 |

10 | 2 | 4 | 2 | 138 |

CG stands for contemporary group, the only fixed effect in this
example. Assume that the appropriate variance parameters are
those which
were used in the simulation in the previous section.
Based on the matrix formulation of the model,
the MME are

where

Note that these numbers are not equal to

Finally, .

The matrices are

The other two right hand side matrices can be easily obtained from and . 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 example.

**REML**

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 and was -.961, which is a lot stronger than the assumed prior correlation of -.196.

**Bayesian Estimation**

The maternal effects model requires a slightly different approach
because of the non-zero correlation between direct and maternal
genetic effects. Let
,
then

and then

Now look at

where

The prior distributions for the variance components are

where

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 ,
,
,
and
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
and
,
but are a little different for the genetic
variances and covariance. For the residual component,

for . For the maternal PE component,

where , and

then invert and perform a Cholesky decomposition on the inverted matrix, i.e.,

then utilize a Wishart matrix generator such as

CALL WISHRT(T,GI,ndf)where ndf, 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 . A 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 that is positive definite, then this problem should not arise, but a check is still worthwhile.

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