Genetic Groups in an Animal Model

Rules are presented for assigning coefficients to the genetic group portion(s) of the mixed model equations after transformation to solve directly for total genetic value (group plus animal solutions) simultaneously for sires and cows using an animal model. Inclusion of all known relationships seems to reduce the need for groups to account for genetic selection and genetic trend. Migration of animals into a population, however, results in a need for grouping to account for genetic merit of the migrants. Selection of parents on which records are not available also creates a need for grouping. Group solutions represent the average genetic merit of phantom (unidentified, or represented by only one descendant) animals selected to be parents that do not have records available. Groups can be crossclassified with time and the genetic path of selection. The total genetic value for every animal includes a function of genetic groups. The function of genetic groups is specific for each individual animal and depends on the number of generations to the base phantom ancestors and on the genetic groups to which those phantom ancestors are assigned. The group coefficients presented account for genetic selection that cannot be defined by known genetic relationships. INTRODUCTION Historically, animals have been allocated to genetic groups to account for selection that cannot be accounted for by known genetic Received May 13, 1987. Accepted November 4, 1987. relationships. Sires have been assigned to groups based on arbitrary criteria such as year of birth, stud, year of entry into service, geographical region, or pedigree information (7). Pollak and Quaas (6) demonstrated that the need for grouping decreases as the genetic relationships among animals become more complete. Simultaneous sire and cow evaluation including all known relationships minimizes the necessity of grouping (3, 6, 10). Nevertheless, some animals entering the population (e.g., from herds starting test or from other countries) could have extensive pedigrees but would not have recognized parents that contribute ties and records to the data. Group effects can be thought of as accounting for selection not accounted for by records of relatives. Under this concept, groups would be assigned only if animals were missing genetic relationships. The genetic merit of all descendants of any animal that has a missing parent would then include a function of the genetic group of the missing ancestor. Thompson (10) suggested a similar approach to genetic grouping in the accumulated groups model. Inclusion of groups in the animal model has proved computationally challenging (8, 11). The purpose of this paper is to describe simple rules for calculating the coefficients associated with group effects of the mixed model equations for an animal model. THE ANIMAL MODEL WITH GENETIC G ROUPS The mixed model equations (MME) for the animal model provide direct solutions for animals with records (4). Genetic values of animals without records, such as sires or dams for which there are no records, are predicted by augmenting the MME with a function of the inverse of the relationship matrix (5). In most models for animal evaluation previously used, only a single group effect was included as the total genetic group effect for an 1988 J Dairy Sci 71:1310-1318 1310 GENETIC GROUPS IN AN ANIMAL MODEL 1311 animal. If, however, groups are to account for unknown genetic selection, consideration of precisely what information is unknown is necessary. An alternative to including a single group effect for each animal is to define groups only for phantom (unknown, with only a single descendant) parents that do not have a record. The use of the term phantom is to emphasize that those animals are not themselves of interest. Phantom animals are considered only to facilitate computing solutions to the MME for the animal model. The group genetic effects represent the average genetic contr ibution of phantom animals selected to be parents to their descendants that do have records. Under the assumption that knowing relationships accounts for the effect of selection on the related ancestors, there is no need to assign groups when both parents are known. Genetic groups must be assigned if one or more parents are unknown by assigning phantom parents to replace each of the unknown animals. The phantom parents are assumed to be average representatives of the genetic groups of similar animals selected to be parents at the same time. Selection differentials may be different for phantom males and females. Thus, two parallel sets of groups could represent phantom sires and phantom dams selected to be parents. Four parallel sets of groups could represent phantom sires of sires, phantom sires of cows, phantom dams of sires, and phantom dams of cows corresponding to the four paths of selection allowing for each path to have different genetic selection differentials. Genetic groups could be cross-classified by sex of animal and sex of phantom parent to account for four selection paths (9). If only two sets of groups are used, phantom sires and phantom dams, then consideration should be taken of the different generation intervals for each of the four paths of selection when allocating genetic groups by t ime and sex of phantom parent. A phantom dam of a bull would not necessarily be a representative of the same time group as a phantom dam of a cow even though the bull and cow were born in the same year because the dam of the bull is likely to have been born before the dam of the cow (14). The time period for the genetic group could be defined by subtracting the average genetic interval from the year of birth of the animal with the missing parent to estimate the year of birth of the missing parent (11, 14). EXAMPLE OF A L L O C A T I O N TO G ROUPS Consider the pedigree in Figure 1. Different amounts of pedigree information are available on the identified animals. Both parents are unknown for $1, D1, and Dz; one parent is unknown for each of D3, D4, Ds, and Sz; and both parents of $3 and $4 are known. Figure 2 shows the phantom parents that would be assigned for the example of Figure 1. A projected year (or t ime period) of birth for each phantom parent can be estimated from average generation intervals. One-half of the effect of the phantom parent genetic group is at t r ibuted to its progeny. EQUIVALENT MIXED MODEL EQUATIONS WITH GROUPS I N C L U D E D To develop notat ion for the MME including equations for phantom parents, let: y be a vector of records, X be an incidence matr ix associating records with fixed effects represented in h, h be a vector of fixed effects, e.g., herd-yearseason (HYS) effects, g be the vector of order n of effects of groups to which phantom parents have been assigned, Z be an incidence matrix associating records with elements in al (if an identified animal does not have a record, e.g., a bull with daughters with records, the corresponding row in Z is null), a0 be a vector of genetic values of the phantom parents, al be a vector of genetic values corresponding to identified animals with the variance of (a~ ' ) ' 2 al = AOa, e be a vector of residual effects with variance Roe 2, which in this development is assumed to be lOe 2, (20 be the incidence matrix assigning phantom animals to groups, (21 be the coefficient matrix relating identified animals to group effects, A~0 be the submatrix of the numerator relationship matrix corresponding to relationships Journal of Dairy Science Vol. 71, No. 5, 1988