(1)
Across-herd comparisons are important to ensure selection from
a large genetic base. However, the accuracy of the comparisons
between EBVs from different herds depends upon the degree of connectedness
between them.
If there are few or no genetic exchanges between a herd and the
remainder of the tested population, the EBVs of animals in that
herd cannot be compared accurately to the EBVs estimated in other
herds, even when they have a high repeatability, because of the
lower degree of connectedness.
Connectedness can be defined as a measure of the relationships
between herds or contemporary groups as they affect the accuracy
of comparing the genetic values of animals from one herd or group
to the other. Contemporary groups can be similar time periods
in different herds, different periods or management groups within
a herd, etc. The higher the connectedness the more accurate the
comparisons of EBVs across groups or herds. Therefore, it is
important to measure the degree of connectedness and, if necessary,
bring it to a level that allows comparison of EBVs with reasonable
accuracy.
There is no well-established procedure for measuring the degree
of connectedness. Therefore, research was carried out at CCSI
to develop an accurate and practical method of measuring the degree
of connectedness, develop recommendations to ensure a minimal
level of connectedness, and apply the method to all herds in the
national genetic evaluation program.
The Method
The accuracy or error in estimating the difference between the EBVs of two animals from different herds is measured as the prediction error variance (PEV) of the difference between the two estimates. This depends upon the PEV (or repeatbilities) of the individual EBVs, the degree of relationship between them, the PEV of the difference between herd effects and the covariance between herd effects and EBVs. The average of all such pairwise comparisons of EBVs across two herds depends mainly upon the PEV of the difference between herd effects. Therefore, the PEV of difference between herd effects can be used to measure the degree of connectedness between two herds.
Kennedy and Trus (1993) compared several methods
including those based on genetic links (e.g. Banos and Cady, 1988)
and confirmed through simulation that the PEV of the difference
between herd effects is very highly correlated (0.995) with the
average PEV of the differences between EBVs, and proposed it as
the method of choice to measure connectedness. However, they
felt it would be difficult to apply to large populations with
many management units because of the amount of computation involved.
In addition to the computational challenge, the PEV of the difference
between estimates of herd effects has another drawback. It depends
on the size and structure of the two herds as well as on the nature
of the "connections" between them. If the connectedness
rating is the PEV itself, a rating of 0.8 mm2 for backfat,
for example, may correspond to either two large herds that are
not well connected, or two smaller herds that are well connected.
To separate the notion of connectedness from the effects of herd
size and structure, the connectedness rating between two herds
was defined as the correlation between the estimates of their
herd effects, i.e.,
(1)
In this manner, any reduction in accuracy associated with insufficient
connectedness can be more effectively separated from that associated
with insufficient herd or management group size. The joint effect
of these factors on the accuracy of the comparisons between EBVs
from different herds can be formulated as follows:
= var (
) +
var (
) - 2 cov (
)
(2)
) + var (
)
- 2 CRij 
(3)
One of the major problems in using this approach with large data
sets is that a direct inverse of the entire set of mixed model
equations is very difficult to compute. Hence, the following
procedure was used, based on L. Schaeffer's suggestion.
where,
w'w = coefficient matrix of mixed model equations,
I = identity matrix,
Ii = a vector of the identity matrix corresponding to contemporary
group i (a vector with 1 for the contemporary group and zeros
otherwise),
(w'w)i-1= a vector of inverse elements for
the ith contemporary group.
The vector (w'w)i-1 was obtained for one contemporary group at a time through interation (1000 rounds). These vectors were combined and the block of inverse elements corresponding to the most recent contemporary groups was extracted out. The inverse elements were the prediction error variances and covariances for the herd effects of interest.
The method can be used to obtain inverse elements for some rows
or columns of any large matrix for which a direct inverse is not
possible. It can be used to compute the exact prediction error
variances and covariances for herds, litters, animals, or any
other effect included in the model, despite the very large total
number of equations.
Application to Herds in the Canadian
Swine Improvement Program
Using the data from the national database, the above method was
applied to the Duroc, Hampshire, Yorkshire, and Landrace breeds
for backfat and age, and to the Yorkshire and Landrace breeds
for litter size. Connectedness ratings were computed for all
herds with at least 10 pigs probed in the second half of 1997,
separately for each breed.
Generally, most herds had high connectedness ratings with few
herds and low to moderate connectedness ratings with other herds.
The ratings were usually higher for herds in the same region,
but there were some good connections across herds in different
regions as well.
An average connectedness rating was also calculated for each herd.
It is the average of a herd's connectedness ratings with all
other herds in the program. The average rating gives an indication
of the accuracy of comparing EBVs from one herd to all others.
For reporting purposes, connectedness ratings between individual
herds are considered confidential information (each breeder has
received a report showing how well his herd is connected to every
other herd on the program, for each breed), while average connectedness
ratings are considered public information.
The number of herds on the national program according to average
connectedness rating is given in the following table.
|
| ||||||
|
| ||||||
Connectedness ratings were usually higher for highly heritable
traits such as backfat and age and lower for traits with low heritability
such as sow productivity traits.
Connectedness and Accuracy of
Comparing EBVs Across Herds
The PEV of differences between herd effects was estimated for
all pairs of herds. The square roots of these estimates were
taken as the standard errors of prediction (SEP) between herd
estimates. Its relationship with connectedness rating was studied.
There were 71 Duroc herds. Therefore, there were 2485 pairs of
herds. The relationship between the connectedness ratings of
these pairs and the SEP of differences between herd effects is
shown in Figure 1.
Figure 1. Relationship between the connectedness
rating and the standard error of differences between pairs of
herd effects for backfat evaluations.
As expected, the SEP decreases as the connectedness rating increases.
The SEP is very high when connectedness rating is low.
The connectedness rating was very low for some pairs of herds.
Their standard error of prediction ranged between 0.5 and 2.0
mm. If we convert these values to a confidence range (95%), one
may expect an error ranging from ±1 to ±4 mm when comparing
the backfat EBVs of animals across herds.
The error is especially large when connectedness is below 5%.
There is little gain in accuracy when the connectedness rating
increases above that point.
The relationship between the average connectedness rating of each
herd with the average standard error of differences with all other
herds showed a similar trend as for pairwise comparisons except
that the extreme values were averaged out. The average values
were lower than for pairwise comparisons. Therefore it is important
to have at least 3% average connectedness rating for reasonably
accurate comparisons for backfat and age EBVs.
There were a few herds that did not follow the pattern of a decline
in SEP when the connectedness rating decreased. The main reason
for this was that the SEP of herd differences was affected by
the size of the herds and their structure, since the SEP of individual
herd effects depends strongly on contemporary group size. The
SEP was very large when contemporary group size was low, especially
when the size was below 10.
This analysis suggests that the accuracy of comparing EBVs from
two contemporary groups depends not only on connectedness but
also on the size and structure of the groups. To make accurate
comparisons of EBVs across groups, it is very important to have
a reasonable contemporary group size in addition to connectedness.
Using Common Sires to Increase
Connectedness
A simple approach to increase connectedness between a pair of
herds is to use common sires. This increases the level of connectedness
between them. The magnitude of this increase depends upon the
magnitude of their use in both herds.
This can be illustrated by a three dimensional diagram where the
percentages of litters from common sires in each of the two herds
are given on the two horizontal axes and the connectedness rating
on the vertical axis (Figure 2). The figure shows that connectedness
increases as the proportion of litters from the common sire increases.
It is very high if 100% of the litters in both herds are from
common sires. If the proportion of litters from the common sires
is low in one of the herds, the other herd has to have a larger
proportion of litters to reach a higher level of connectedness.
There are different optimum levels of connectedness ratings that
can be achieved through the use of common sires depending upon
the percentage of litters they have in other herds. If the sires
have less than 25% of litters in another herd, the maximum level
of connectedness rating possible is about 15% even if more than
50% of the litters in the original herd are produced from the
common sires. However, if the sires have more than 75% of the
litters in the other herd, the connectedness rating increases
with the percentage of litters in the original herd. Even higher
degrees of connectedness may be possible as a result of connections
through common ancestors such as grandparents or as a result of
placing pigs from the same litters in two herds.
Figure 2. Relationship between connectedness rating
and percentage of litters from common sires in two Hampshire herds.
Recommendations
The simplest way to increase connectedness is to use AI boars
that have progeny in well connected herds. A list of such AI
boars has been provided along with the connectedness reports for
each herd. For each AI boar, the number of progeny, the number
of herds in which these progeny are found, and a weighted connectedness
rating are given. Usually about 15% of progeny from well-connected
sires in a herd are enough to reach the minimum level of connectedness.
One may also use young AI boars that do not have any progeny,
provided they come from well-connected herds.
In order to have well-connected boars of sufficient
genetic merit, it is necessary that breeders participating in
the program make some of their superior genetics available. The
National Genetics Committee and CCSI's Board of Directors have
recommended that a national genetic pool of AI boars be established
with co-operating AI Centres. Participation in the pool would
be voluntary, but only those who participate would be allowed
to draw from the pool. The participating breeders would make
available some of their top genetics for potential use in the
pool. Breeders participating in the pool become part of a super-nucleus
structure which allows them to compete effectively at the international
level.
References
Kennedy, B.W., and D. Trus. 1993. Considerations
on genetic connectedness between management units under an animal
model. J. Anim. Sci. 71:2341-2352.
Banos, G., and R.A. Cady. 1988. Genetic relationship
between the United States and Canadian Holstein bull populations.
J. Dairy Sci. 71:1346-1354.