A New Method for Assessing Connectedness Between Herds

P.K. Mathur, B. Sullivan and J. Chesnais
Canadian Centre for Swine Improvement
Ottawa, Canada


The Concept of Connectedness

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 () + 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.

w'w (w'w)- 1 = I
therefore, w'w (w'w)i-1 = Ii (4)

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.

Number of Herds
Average connectedness rating
Lean growth
Sow productivity

Duroc

Hampshire

Landrace

Yorkshire

Landrace

Yorkshire
0-1
0
0
5
13
3
6
1-2
18
0
41
45
17
15
2-3
19
2
34
31
24
24
3-4
13
3
8
12
14
26
4-5
11
0
2
6
6
8
5-6
8
4
1
1
5
4
6-7
0
2
1
1
1
3
7-8
3
1
0
0
0
0
>8
0
4
0
0
0
0

Total

72

16

92

109

70

86

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

1) Minimum requirements

2) Increasing connectedness

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.

3) Creation of a genetic pool

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.