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Often animals are observed more than once for a particular trait. Examples might be
Besides an animal's additive genetic value for a trait, there is a
common permanent environmental (PE) effect which is a non-genetic
effect that is common to all observations on the same animal. The
model is written as
Repeatability is a measure of the likelihood of an animal to make
a similar record as a previous record,
and is defined as a ratio of variances as
Simulating some records may help to understand this type of model.
|Animal||Sire||Dam||Year 1||Year 2||Year 3|
A PE effect can be generated for all 12 animals. Because relationships
have no bearing on PE effects, then just generate a RND and multiply
These are shown in the table below. The records
are generated according to the model equation,
|Year 1||Year 2||Year 3|
An interesting point to observe from the simulation is that the PE effects are present even for animals with only one record. Also, the same PE value is present in all records. Is this what happens in real life? Probably not. The PE effects probably accumulate as the animal ages. Something happens to the animal between the first and second records, for example, and this affects all subsequent records on that animal, but did not affect the first record.
Another assumption is that the records have a genetic correlation of one, which is true in the way that the above records were simulated because the same TBV was used for each record. However, in real life one might anticipate that the genes affecting the trait might change as the animal ages, and therefore, the genetic correlation between records could be less than unity. This is handled later when we look at multiple trait models.
We will concentrate mainly on Henderson's Mixed Model equations and
BLUP for the remainder of the course. Let
HMME are therefore,
The full HMME are too large to present here as a whole, so parts of
the matrix are given as follows.
The solutions for animals are given in the table below. Solutions
for year effects were
Estimation of Variances
The necessary items for REML estimation of variances are
The procedures are similar to that given in Lesson 11. Prior
distributions must be assumed for all random variables.
In addition to those already stated,
Computationally, the scheme is virtually the same as with the simple animal model except for the additional variance component for PE effects. There is no need to illustrate the calculations for this model.
In practical animal breeding, industry personnel are very interested
in Estimated Breeding Values (EBV), and the main question is about its
reliability or accuracy. The variance-covariance matrix of prediction
errors is given by
R, of the ith animal is defined as
Note that animals with records have a higher reliability than animals that have only progeny. Also, animal 7 had a higher reliability because it had three records while animals 11 and 12 had only one record. The Reliability reflects the years in which the records were made and the number of contemporaries within a year, and specifically who the contemporaries actually were. Reliability also includes the fact that animals were related.
In the analysis of very large numbers of animals, the calculation of is virtually impossible. Thus, animal breeders have devised many ways of approximating the diagonals of . The following method is due to Schaeffer and Jansen (1997).
For animal 1, the parents are unknown
Reliabilities are required by industry people to determine the 'official' status of an animal's EBV. The approximations that are used should be on the conservative side for safety reasons.
There may also be a method of determining approximate reliabilities by using Gibbs sampling, but not allowing the variances to change in each round. The starting values would be the solutions to the MME and the known variances. Then about 200 rounds of sampling should give a good estimate of the prediction error variance of the EBVs for each animal, which can then be used to arrive at reliability.
Often with a repeated records animal model, the records are taken over time and based on the first or second record an animal may be culled. Thus, selection could affect the mean and variance of later repeated records. This type of selection is taken into account in HMME provided that all animals have a first record before any culling has occurred and that all pedigrees are known for all animals. If these provisions are not met, then EBV and estimates of fixed effects from a repeated records analysis could be biased by culling, with the magnitude determined by the severity of the culling.
This LaTeX document is available as postscript or asAdobe PDF.Larry Schaeffer