| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
* Animal and Dairy Science Department, University of Georgia, Athens 30602
Instituto Nacional de Investigación Agropecuaria, Las Brujas, Uruguay
1 Corresponding author: ignacy{at}uga.edu
 |
ABSTRACT |
|---|
 TOP ABSTRACT APPENDIXACKNOWLEDGEMENTSREFERENCES |
|---|
Key Words: inbreeding coefficient • unknown parent • recursive algorithm
VanRaden (1992) presented a method to calculate inbreeding while accounting for missing parents. Animals with missing parents were assumed to have inbreeding coefficients equal to the mean of the inbreeding coefficients for animals with known parents born during the same year.
VanRaden’s algorithm recovered most of the inbreeding caused by incomplete pedigrees when the number of missing pedigrees was not too large (Lutaaya et al., 1999). In that study, different proportions of missing information were simulated by randomly deleting 10 to 50% of dam identification in a Holstein population with almost complete pedigrees. VanRaden’s method recovered most of the inbreeding when the missing dam information was 10 to 20% (Lutaaya et al., 1999).
The algorithm by VanRaden is based on the tabular method (Emik and Terrill, 1949). For large pedigrees, this algorithm requires extracting pedigrees for each individual, which makes the algorithm complicated. In addition, for long pedigrees, the size of the table is large and computations are long. The computing cost of the tabular method is approximately n x 22p, where n is the population size and p is an average number of generations of pedigree per animal.
Different methods are currently used to compute inbreeding coefficients efficiently for large populations (Tier, 1990; Meuwissen and Luo, 1992; Sargolzaei et al., 2005). The last 2 algorithms are optimized versions of the algorithm by Quaas (1976), which has a computational cost of n2. The algorithms based on the recursive algorithms (RA) seem to be the simplest (Tier, 1990; Miglior et al., 1992), and their computing cost is approximately n x 2p. All of these methods as described assign an inbreeding coefficient of zero when at least one parent is missing. Recently, Croquet et al. (2006) applied a modification of the method of Meuwissen and Luo that assigned nonzero inbreeding to animals with unknown parents.
The objectives of this study were to investigate a simple RA to calculate inbreeding coefficients by using rules from the tabular method (Emik and Terrill, 1949) and to expand it to consider animals with missing parent information (VanRaden, 1992).
The RA assumes that animals must be renumbered based on year of birth (YOB) so that parents precede their progeny. For each animal (x) present in the pedigree, the inbreeding coefficient Fx is calculated as Fx = 0.5Rsd, where Rsd is the numerator relationship between the sire (s) and the dam (d) of the animal (x) (Emik and Terrill, 1949). Computation of Rsd is recursive and involves tracing the ancestors and computing the relationship between s and d. Three cases are considered:
 |
where animal x is younger than y, and s and d are the sire and dam of x, respectively.
In the last case, the order of animals x and y should be switched if necessary so that subsequent relationships are those of the sire and dam of the younger animal (x) to the older animal (y), which is essential if the 2 animals are in the direct line of descent (Emik and Terrill, 1949).
In the algorithm above, x = 0 or y = 0 indicates an unknown parent or parents. To extend the algorithm to nonzero inbreeding of unknown parents, let a negative code denote the YOB of their progeny. For example, x = –1998 would indicate a parent of an animal born in 1998. Let bi be an average inbreeding of all animals born in year i. Then, the first rule above (Rxy = 0) is modified as follows:
 |
In the iterative procedure, b is calculated each round. Let yob(x) be a YOB of known animal x. In the first round bi is zero. In subsequent rounds,
 |
and the algorithm is repeated until convergence. The average above includes only animals with both parents identified (VanRaden, 1992).
The convergence criterion was defined as in Lutaaya et al. (1999) as the absolute change in mean of inbreeding for all animals between 2 consecutive rounds less than 10–6. The algorithm was implemented in Fortran 90 and the source code is shown in the Appendix.
The algorithm was tested by using the pedigree of US Holsteins born from 1969 to 2002 (n = 17,094,297) obtained from the Animal Improvement Programs Laboratory, Agricultural Research Service, USDA (Beltsville, MD). Missing YOB for parents were calculated as the minimum YOB from the offspring minus 3 yr. Animal identifications were checked to avoid impossible offspring-parent relationships based on the YOB, and they were renumbered from oldest to youngest. Solutions of the RA with unknown parent information were compared for correctness to the algorithm by VanRaden (1992), as used in Lutaaya et al. (1999). After convergence, both methods gave the same estimates of inbreeding. Computing time was 550 s for one round of the VanRaden algorithm and 287 s for one round of RA. Despite differences among methods, the computing was fast for both methods given the size of the pedigree.
The algorithms based on the Quaas algorithm or the tabular method explicitly require ordering animals from the oldest to the youngest. During initial trials, pedigrees used by RA were not sorted; however, the algorithm was modified to treat an animal without parents known as y (Table 1
). For several data sets, including commercial data sets, inbreeding coefficients produced by such an algorithm were almost the same as by the original algorithm with animals sorted, although the differences increased with larger pedigrees with longer generations (Table 2). Some differences arose from pedigree errors, which were detected and eliminated when animals were sorted, and others were due to incorrect use of the recursion formula ("x younger than y"). The RA algorithm provides many correct results without explicit sorting because of implicit sorting during the recursion process. Although the modified algorithm may be useful in some situations, renumbering has an additional benefit of detecting errors in the pedigree.
|
|
shows the mean inbreeding coefficients per birth year at rounds 1, 2, and 6. Convergence was reached at 6 iterations. Mean inbreeding per year had a pattern similar to the official inbreeding for all the population of US Holsteins as calculated by the Animal Improvement Programs Laboratory (2007).
|
We demonstrated that the RA to compute inbreeding coefficients can be modified to account for nonzero inbreeding of unknown parents. The algorithm is simple, is relatively inexpensive, and can also be used for other purposes, including computing of relationships of groups of animals (e.g., sires) by using the full pedigree or the creation of nonadditive relationships (e.g., the dominance relationships). For example, most exploitable dominance relationships are due to sire x maternal grandsire classes with large numbers of members. The dominance relationship matrix for such sire combinations can be constructed by computing relationships among such sires and grandsires, and possibly of their ancestors. Because only relationships of interest are computed, the cost of computations can be reasonable.
|
APPENDIX |
|---|
|
TOPABSTRACTAPPENDIXACKNOWLEDGEMENTSREFERENCES |
|---|
|
|
ACKNOWLEDGEMENTS |
|---|
|
TOPABSTRACTAPPENDIXACKNOWLEDGEMENTSREFERENCES |
|---|
Received for publication August 3, 2007. Accepted for publication December 17, 2007.
|
REFERENCES |
|---|
|
TOPABSTRACTAPPENDIXACKNOWLEDGEMENTSREFERENCES |
|---|
This article has been cited by other articles:
|
|
 |
|
  I. Misztal, A. Legarra, and I. Aguilar Computing procedures for genetic evaluation including phenotypic, full pedigree, and genomic information J Dairy Sci, September 1, 2009; 92(9): 4648 - 4655. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Legarra, I. Aguilar, and I. Misztal A relationship matrix including full pedigree and genomic information J Dairy Sci, September 1, 2009; 92(9): 4656 - 4663. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
