Animal Model - Intraherd Evaluation

The discussion starts out with three examples of relationship matrices and their corresponding inverses. This illustrates two points: 1) any complete relationship inverse can be formed from the inverse of the relationships among parents; and 2) the only non-zero elements in the relationship inverse are between parents, parents and progeny, and between mates. Three equivalent models are developed to arrive at breeding value predictions for all animals using all known sources of relative information. A genotypic model, gametic model and finally a combination of the two leads to the reduced animal model which affords the opportunity to solve a set of equations as large as the number of parents in the herd. The equations to be solved are more amenable to iterative solution procedures because most of the progeny information is added to the RHS of the equations instead of through the relationship inverse. Outline A. Relationship Matrix and Inverse 1. Structure non-zero elements of inverse a. parents b. parents and progeny c. mates 2. Examples to illustrate formation of a complete relationship inverse from an inverse of relationships among the parents. B. Three Equivalent Models 1. Equivalent model theory 2. Genotypic model 3. Introduction to gametic model and development of equivalent models for records of a single animal 4. Reduced animal model a. genotypic for parents b. gametic for non-parents 5. Review of equivalent models with example to show equivalence C. Reduced Animal Model 1. Properties of solutions 2. Back solutions for non-parents a. average breeding value for future sibs b. a prediction of the difference between future sibs and non-parents currently in the herd 3. How relatives influence animal prediction 4. Representation of phenotypic, genetic, and environmental trends Animal Science 562 Animal Model 2 What is an animal model? A statistical model which can be applied to data obtained from dairy cattle, or any other species for that matter, which describes biological processes and effects quantitatively. Concept of a breeding value Sire 1⁄2 Breeding Value of Animal 1⁄2 Dam You will need to understand the concept of a breeding value as an expression of relative genetic merit for an individual animal. The sire and dam each contribute 1⁄2 of their genes to a young animal. The breeding value of this young animal is the genetic effect of this sample of genes from the sire and dam expressed in the units of measure for the trait (i.e. kg of milk, kg of weight, etc.). There are many animal models. The appropriate model is determined by the goals for economic improvement, species, trait, and the kind and amount of genetic variation. Some examples are: 1. First lactation milk yield, additive genetic effects. 2. All lactation records for milk yield, additive genetic effects, simple repeatability model. 3. Multiple traits, as in first lactation milk and fat yield. 4. Direct and maternal additive genetic effects, single trait, single records. What must be known before conducting a genetic evaluation? The management conditions which affect our records are unknown and these must be estimated so we can make adjustments for the conditions which change from herd to herd. We don't have perfect data to predict the breeding value of all animals. We use all the data available to find the best linear unbiased predictor of the breeding value for all animals. The following information must be known before conducting a genetic evaluation with an animal model. 1. Genetic theory and statistical methods. a. Methods enable us to follow what someone else has done. b. Theory enables us to manipulate mathematical expressions and chart a new direction. 2. Definition of a correct model. This will depend on the breed, traits, and management conditions under which records are made. Animal Science 562 Animal Model 3 3. Complete identification of all ancestors in pedigree. 4. The kind and amount of genetic variation for each trait and breed. How do you conduct a genetic evaluation? 1. Collect data and pedigree information. 2. Standardize records for age at calving, days in milk, and month of calving. 3. Conduct pilot studies. a. Define model appropriate for conditions affecting records. b. Demonstrate feasibility. 4. Develop computational procedures. c. Develop new software. d. Import technology and software from other countries. e. Modify computing strategies. 1) Set up equations and solve iteratively. 2) Adopt indirect procedures which iterate on data. Properties of solutions: The animal model has become the foremost approach to genetic evaluation of animals in the world. The solutions have properties which are defensible by the scientific community (Kennedy et al. 1988) and breeders have found it provides them with a way to improve the genetic merit of their cattle. We don't know how the different management conditions from herd to herd affect each record, therefore these are estimated. The estimates are Best Linear Unbiased Estimates. The true breeding value of every animal is unknown, in fact we will never know the true breeding value. We can, however, obtain a prediction of the breeding value based on the information available. The predictor is the Best Linear Unbiased Predictor. In practical terms, the animal model allows us to use all available information to estimate management conditions and predict the breeding value. Sources of information which can be included are: 1) The animals own record (i.e. milk and fat yield, weight at fixed age). 2) All progeny information on individual, sire, dam and other relatives. In addition, breeding values are: 1) Adjusted for heritability and repeatability. 2) Directly comparable across herds and over time. 3) Less subject to change from first to later evaluations as more information on the individual animal becomes available. 4) Adjusted for genetic competition due to genetic merit of mates. 5) The best procedure for ranking animals. Animal Science 562 Animal Model 4 A. Relationship Matrix and Inverse All animal models use the additive genetic relationship matrix among animals (A) in one form or another. Originally defined by Wright (1922) as the coefficient of parentage among animals, elements of the relationship matrix are twice the probability of identical genes by descent occurring in two individuals. When multiplied by the additive genetic variance (σa), Aσa is the covariance among breeding values. The inverse of A, called A, has very special properties. Knowledge of these properties is helpful when defining alternative animal models and will be used later to explain the composition of an animal’s evaluation. We will begin by assuming we have ordered all animals with progeny in a manner such that they will precede those with no progeny. Second, we will take note of the fact that the structure of the relationship inverse is such that the only non-zero elements are between parents, parents and progeny, and mates. Therefore, A is a sparse matrix, although A may not be a sparse matrix. A general representation of the relationship inverse is ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − + = − − N N M N M NM M A A sd ' 5 . ' 5 . ' 25 . 1 1 where Asd -1 is the inverse of the relationship among parents, M is an incidence matrix relating parents to progeny, and N is a diagonal matrix which accounts for the number of known parents (e.g., values of 0 < N ≤ 2 are typically found) plus any inbreeding in the non-parent animals. Later, these properties will be used to illustrate how solutions for a non-parent depend only on its own record, the solution(s) for fixed effects and the solution for its parents. For example, consider the sire, dam and offspring relationship A sire dam offspring = ⎡