Linear Model Theory

A statistical model is a way in which we can describe reality using observed variables. There are 3 parts to every model.

1. An equation where the trait observations are described as being influenced by factors

   $y_{ijkl}=A_i+B_j+c_k+…+e_{ijkl}$

where

• $y_{ijkl}$ - observation on a trait,
• $μ$ - mean of population
• $A_i$ - effect of factor $A$, level $i$, on the trait
• $B_j$ - effect of factor $B$, level $j$, on the trait
• $C_k$ - effect of factor $C$, level $k$, on the trait
• $e_{ijkl}$ - residual effect of all factors not observed

2. An indication of whether a factor is fixed or random
3. List of all implied or explicit assumptions or limitation of the first two parts

​​ Regression Variables

Also known as a covariate, is a variable that has a particular relationship with the observations. For example the relationship between height and weight of an animal.

​​ Animal Models

An animal model is one where there is one or more observation per animal, and factors affecting the observation are described along with an additive genetic effect.

​​ Multiple Trait Models

Animals are typically observed for more than one trait as multiple traits affect the profitability of the animal.

Multiple Trait (MT) analyses advantages:

• Low Heritability Traits
  • When a there is a trait with a higher heritability than another, that other trait gains more in accuracy than the high trait.
• Culling
  • Gives a better unbiased estimate of economic value on animal

Disadvantages

• Estimation of Correlations - MT analysis needs accurate genetic and residual correlations.
• Computing Cost

​​ Random Regression Model

As an animal ages different genes turn on and off causing changes in the animal. It is important to measure an animal at different times because of this, as traits can change. These traits are called infinitely dimensional traits.

​​ Phantom Parent Groups

In populations there can be animals with unknown parents. It is assumed that these animals came from a large random mating population, with no inbreeding. This assumption is hazy when it comes to two separate animals with unknown parents born 10 years apart. The genetic averages of the two groups 10 years apart are likely to be different, and are likely to be from two different populations. We can overcome this problem by creating Phantom Parent Groups

​​ Creating Phantom Groups

Phantom groups can follow the four pathways of selection and year of birth. For instance, an female born in 2005 with unknown parents would have a male parent be Sire of Dam group-2005 (SD-2005) and a female parent be Dam of Dam group-2005 (DD-2005). All other females born in 2005 would have parents assigned to the same group.

Males would go through the same process, having parents assigned to DS and SS paths respectively.

What this is trying to achieve is to identify potentially different populations. There are equations for each phantom parent groups, which can estimate genetic differences between groups. One prerequisite is that the groups are of moderate size. If the groups are too small, some can be combined.