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Effects of Non-Random Mating
L. R. Schaeffer, April 1999
Updated April 2000
The models that have been discussed in this course have assumed an
infinitesimal animal model, and that animals
are mating randomly. What are the
effects when these assumptions are not true.
There are many forms of selection that can nullify or
render meaningless any statistical analysis if it is ignored.
In this set of notes, only selection of
animals to produce the next generation, or non-random mating,
will be considered.
Non-random mating produces the following consequences.
- Causes changes to gene frequencies at all loci.
- Changes in gene frequencies cause a change in
genetic variance. Recall from quantitative genetics that
- In finite populations, non-random mating causes a
reduction in the effective population size, which subsequently
causes an increase in levels of inbreeding.
- Joint equilibrium becomes joint disequilibrium, and therefore,
non-zero covariances between additive and dominance genetic
effects are created.
- If the pedigrees of animals are not complete nor traceable to
the base generation, then non-random mating causes genetic
evaluations by BLUP to be biased, and causes estimates of
genetic variances, by any method, to be biased.
- Because the genetic variance decreases due to joint disequilibrium
and inbreeding, response to selection is generally lower than
expected by selection index over several generations.
Effect on Inbreeding
Recall that under random mating and finite population size that the
change in inbreeding per generation was defined as
where Ne is the effective population size, or effective number
of breeding individuals.
Wright (1940) and Crow (1954) showed that if the average family size,
was v, then
Ne = 4N/(v+2),
where N is the number of mating males and females. An idealized
population that is able to maintain itself has v=2, thus,
Ne=N. Belonsky and Kennedy (1985) simulated a population of
100 females and 5 males per year over ten generations with
a single record per animal and discrete generations.
Parents were either randomly selected, selected on the basis of
phenotype, or selected on the basis of BLUP EBVs. The change
in inbreeding under the different selection criteria are given in
the table below.
Change in Inbreeding
Under random selection of mates, the rate of inbreeding over 10 years is
the same for all heritabilities, and is lower than the other forms
of mate selection. Inbreeding accumulates due simply to the small
Phenotypic selection of mates gives higher rates
of inbreeding which increase with increasing heritabilities. Selection
of mates using BLUP EBV created the highest rates of inbreeding.
Note that there was a decline in inbreeding rate at heritability
equal to .6. Why? BLUP EBVs make use of information from relatives.
At low heritabilities an animal's EBV is more heavily influenced by
the parent average, and many progeny are needed to overcome this
influence. Selection on the parent average results in
more related animals being selected together, and hence more
At the higher heritability of .6, the animal's own record
has more influence relative to the parent average, resulting in
fewer half-sibs and full-sibs being selected as parents of the next
generation, and consequently lower inbreeding levels.
Effect on Genetic Evaluation
The effects of non-random mating on genetic evaluation are minimal
If the above conditions hold, then application of BLUP does not lead
to bias in EBVs, but selection increases the variance of prediction
error over populations that are randomly mating. However, in
animal breeding, the practical situation is that complete pedigrees
seldom exist. Thus, bias can creep into estimates of fixed effects
- Complete (no missing parent information) pedigrees are known
back to a common base population which was mating randomly,
- Data on all candidates for selection are available, and
- Genetic parameters from the base population are known.
Recall that HMME for a simple animal model are
A generalized inverse of the coefficient matrix can be represented
Then remember that
These results indicate that HMME forces the covariance between
estimates of the fixed effects and estimates of additive genetic
effects to be null. However, there is a non-zero covariance
between estimates of the fixed effects and the true additive
genetic values of animals. Hence, any problem with the true additive
genetic values, and there will be problems with estimates of
Consider the equation for
and the expectation of this vector is
The fixed effects solution vector contains a function of the
expectation of the additive genetic solution vector.
Normally, because the BLUP methodology requires
then the fixed effects solution vector is also unbiased.
Due to selection, however,
and therefore, the expectation of the fixed effects solution
vector contains a function of
consequently biased. If
is biased, then this
will cause a bias in
Re-state the model (in general terms) as
To simplify, assume that
and that neither is drastically affected by
The prediction problem is the same as before. Predict a function
by a linear function of
the observation vector,
and such that
is minimized. Form the variance of prediction errors and add
a LaGrange multiplier to ensure the unbiasedness condition, then
differentiate with respect to the unknown
matrix of LaGrange multipliers and equate to zero. The solution gives
the following equations.
it can be shown that the following equations give the exact same
solutions as the previous equations.
If a generalized inverse to the above coefficient matrix is
then some properties of these equations are
Firstly, these results suggest that if non-random mating has
occurred and has changed the expectation of the random vector, then
an appropriate set of equations is the generalized least squares
equations. However, we have seen earlier that such equations give
a lower correlation with true values and large mean squared errors
(when matings are at random). Secondly, the estimates of the
fixed effects have null covariances with the true random effects,
and the covariances between estimates of the fixed effects and
estimates of the random effects are non-zero, which is opposite to
the results from BLUP. With the least squares solutions, application
of the regressed least squares procedure could be subsequently used
to give EBVs.
There is another problem with these equations. If
as in an animal model, then
and the generalized
least squares equations do not have a solution unless
This is not very useful for genetic evaluation purposes.
An Alternative Model
Earlier in these notes, the Mendelian sampling variance was
assumed to be unaffected by non-random mating, but could be
reduced by the accumulation of inbreeding.
The animal model equation is
The animal additive genetic effect can be written as
are matrices of ones and zeros,
such that each row has an element that is 1 and all others are 0,
and these indicate the sire and dam of the animal, respectively,
is the Mendelian sampling effect. Due to non-random
which is not a null vector, in general. Let
then the model becomes
If all animals were non inbred then all of the diagonals of would be equal to .5.
Note that the matrix
or its inverse are not necessary in
this model, and that sires and dams (resulting from selection) are
fixed effects in this model. The equations to solve are
Thus, for each animal with a record we need to know both parents,
but we do not need to be able to follow pedigrees back to the
base generation, except for calculating inbreeding coefficients
of all animals.
This model was applied to the example data which was simulated
in Lesson 9. There were 12 animals with records, four sires and
four dams. The solutions for the sires and dams are shown
below, after forcing their sum to be zero.
The solutions for sires and dams represent estimated transmitting
abilities and should be multiplied by 2 to give EBV. The estimates
of the Mendelian sampling effects for animals 5 through 16 were
These solutions sum to zero, automatically, but the general
property would be
example all of the diagonal elements of
to 2, but with inbred individuals this would not be the case.
EBV are created by summing sire and dam solutions with the Mendelian
sampling estimates. The results for animals 5 through 16 were
The correlation of EBV with the true breeding values (shown in
Lesson 9) was .8009 which is greater than the correlation obtained
with BLUP (.7547). This model appears to be superior to the
simple animal model (based on only one example). However, from this
model it is possible to have two EBV for some animals. Animal 5,
for example, had an EBV of -6.60 based on its own record plus its
sire and dam solutions, and based on its progeny as a sire has an
EBV of -14.16. Which evaluation is correct or better to use?
If the progeny of animal 5 are random progeny, and if animal 5 has
a chance to have many more progeny, then EBV=-14.16 is probably
the better result to use. If the progeny are not a random sample
of progeny, then the other EBV may be better. The correlation of
sire and dam solutions with their true breeding values for animals
1 to 8 was .5573. For animals 1 to 4, the sire and dam solutions
are the only information available for these animals because they
did not have records.
The simple animal model combines information from data, from
the parent average, and from progeny. The above model computes
estimated breeding values based on progeny only, and based on
parent average plus data. This difference in concept is due to
the fact that parents are obtained by selection. Non-random mating
is taken into account because the sire and dam of each animal with
a record is included in the model. The solutions for sires and
dams from this model are valid estimates of transmitting abilities
provided that the progeny are a random sample of their progeny.
The Mendelian sampling estimates for animals provides a means of
estimating the additive genetic variance. Inbreeding should still
be taken into account in the matrix .
This model also avoids the problem of forming phantom parent groups
for animals with parent information. If an animal with a record
has an unknown dam (or sire), then a phantom dam (sire) can be
created which has this animal as its only progeny. If both parents
are unknown, then both a phantom sire and phantom dam need to be
assumed, with this animal as their only progeny.
Further study on this model is warranted.
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