mirror of
https://github.com/MillironX/beefblup.git
synced 2024-09-20 21:12:03 +00:00
559 lines
16 KiB
Markdown
559 lines
16 KiB
Markdown
|
```@meta
|
|||
|
CurrentModule = BeefBLUP
|
|||
|
```
|
|||
|
|
|||
|
# How to Calculate EPDs
|
|||
|
|
|||
|
Not to exclude our Australian comrades or our dairy friends, this guide could
|
|||
|
alternately be called
|
|||
|
|
|||
|
- How to Calculate Expected Breeding Values (EBVs)
|
|||
|
- How to Calculate Predicted Transmitting Abilities (PTAs)
|
|||
|
- How to Calculate Expected Progeny Differences (EPDs)
|
|||
|
|
|||
|
Since I'm mostly talking to American beef producers, though, we'll stick with
|
|||
|
EPDs for most of this discussion.
|
|||
|
|
|||
|
Expected Breeding Values (EBVs) (which are more often halved and published as
|
|||
|
Expected Progeny Differences [EPDs] or Predicted Transmitting Abilities [PTAs]
|
|||
|
in the United States) are generally found using Charles Henderson's linear
|
|||
|
mixed-model equations. Great, you say, what is that? I'm glad you asked...
|
|||
|
|
|||
|
## The mathematical model
|
|||
|
|
|||
|
Every genetics textbook starts with the following equation
|
|||
|
|
|||
|
```math
|
|||
|
P = G + E
|
|||
|
```
|
|||
|
|
|||
|
Where:
|
|||
|
|
|||
|
- _P_ = phenotype
|
|||
|
- _G_ = genotype (think: breeding value)
|
|||
|
- _E_ = environmental factors
|
|||
|
|
|||
|
Now, we can't identify _every_ environmental factor that affects phenotype, but
|
|||
|
we can identify some of them, so let's substitute _E_ with some absolutes. A
|
|||
|
good place to start is the "contemporary group" listings for the trait of
|
|||
|
interest in the [BIF Guidelines], though for the purposes of this example, I'm
|
|||
|
only going to consider sex, and birth year.
|
|||
|
|
|||
|
```math
|
|||
|
P = G + E_{year} + E_{sex}
|
|||
|
```
|
|||
|
|
|||
|
Where:
|
|||
|
|
|||
|
- _E<sub>n</sub>_ is the effect of _n_ on the phenotype
|
|||
|
|
|||
|
Now let's say I want to find the weaning weight breeding value (_G_) of my
|
|||
|
favorite herd bull. I compile his stats, and then plug them into the equation
|
|||
|
and solve for G, right? Let's try that.
|
|||
|
|
|||
|
### Calf Records
|
|||
|
|
|||
|
ID | Birth Year | Sex | YW (kg)
|
|||
|
-- | - | - | -
|
|||
|
1 | 1990 | Male | 354
|
|||
|
|
|||
|
```math
|
|||
|
354 \textup{kg} &= G_1 + E_{1990} + E_{male}
|
|||
|
```
|
|||
|
|
|||
|
Hmm. I just realized I don't know any of those _E_ values. Come to think of it,
|
|||
|
I remember from math class that I will need as many equations as I have
|
|||
|
unknowns, so I will add equations for other animals that I have records for.
|
|||
|
|
|||
|
### Calf Records
|
|||
|
|
|||
|
ID | Birth Year | Sex | YW (kg)
|
|||
|
-- | - | - | -
|
|||
|
1 | 1990 | Male | 354
|
|||
|
2 | 1990 | Female | 251
|
|||
|
3 | 1991 | Male | 327
|
|||
|
4 | 1991 | Female | 328
|
|||
|
5 | 1991 | Male | 301
|
|||
|
6 | 1991 | Female | 270
|
|||
|
7 | 1992 | Male | 330
|
|||
|
|
|||
|
```math
|
|||
|
\begin{aligned}
|
|||
|
251 \textup{kg} &= G_2 + E_{1990} + E_{female} \\
|
|||
|
327 \textup{kg} &= G_3 + E_{1991} + E_{male} \\
|
|||
|
328 \textup{kg} &= G_4 + E_{1991} + E_{female} \\
|
|||
|
301 \textup{kg} &= G_5 + E_{1991} + E_{male} \\
|
|||
|
270 \textup{kg} &= G_6 + E_{1991} + E_{female} \\
|
|||
|
330 \textup{kg} &= G_7 + E_{1992} + E_{male}
|
|||
|
\end{aligned}
|
|||
|
```
|
|||
|
|
|||
|
Drat! Every animal I added brings more variables into the system than it
|
|||
|
eliminates! In fact, since each cow brings in _at least_ one term
|
|||
|
(_G<sub>n</sub>_), I will never be able to write enough equations to solve for
|
|||
|
_G_ numerically. I will have to use a different approach.
|
|||
|
|
|||
|
## The statistical model: the setup
|
|||
|
|
|||
|
Since I can never solve for _G_ directly, I will have to find some way to
|
|||
|
estimate it. I can switch to a statistical model and solve for _G_ that way. The
|
|||
|
caveat with a statistical model is that there will be some level of error, but
|
|||
|
so long as we know and can control the level of error, that will be better than
|
|||
|
not knowing _G_ at all.
|
|||
|
|
|||
|
Since we're switching into a statistical space, we should also switch the
|
|||
|
variables we're using. I'll rewrite the first equation as
|
|||
|
|
|||
|
```math
|
|||
|
y = b + u + e
|
|||
|
```
|
|||
|
|
|||
|
Where:
|
|||
|
|
|||
|
- _y_ = Phenotype
|
|||
|
- _b_ = Environment
|
|||
|
- _u_ = Genotype
|
|||
|
- _e_ = Error
|
|||
|
|
|||
|
It's not as easy as simply substituting _b_ for every _E_ that we had above,
|
|||
|
however. The reason for that is that we must make the assumption that
|
|||
|
environment is a **fixed effect** and that genotype is a **random effect**. I'll
|
|||
|
go over why that is later, but for now, understand that we need to transform the
|
|||
|
environment terms and genotype terms separately.
|
|||
|
|
|||
|
We'll start with the environment terms.
|
|||
|
|
|||
|
## The statistical model: environment as fixed effects
|
|||
|
|
|||
|
To properly transform the equations, I will have to introduce
|
|||
|
_b<sub>mean</sub>_ terms in each animal's equation. This is part of the fixed
|
|||
|
effect statistical assumption, and it will let us obtain a solution.
|
|||
|
|
|||
|
Here are the transformed equations:
|
|||
|
|
|||
|
```math
|
|||
|
\begin{aligned}
|
|||
|
354 \textup{kg} &= u_1 + b_{mean} + b_{1990} + b_{male} + e_1 \\
|
|||
|
251 \textup{kg} &= u_2 + b_{mean} + b_{1990} + b_{female} + e_2 \\
|
|||
|
327 \textup{kg} &= u_3 + b_{mean} + b_{1991} + b_{male} + e_3 \\
|
|||
|
328 \textup{kg} &= u_4 + b_{mean} + b_{1991} + b_{female} +e_4 \\
|
|||
|
301 \textup{kg} &= u_5 + b_{mean} + b_{1991} + b_{male} + e_5 \\
|
|||
|
270 \textup{kg} &= u_6 + b_{mean} + b_{1991} + b_{female} + e_6 \\
|
|||
|
330 \textup{kg} &= u_7 + b_{mean} + b_{1992} + b_{male} + e_7
|
|||
|
\end{aligned}
|
|||
|
```
|
|||
|
|
|||
|
Statistical methods work best in matrix form, so I'm going to convert the set of
|
|||
|
equations above to a single matrix equation that means the exact same thing.
|
|||
|
|
|||
|
```math
|
|||
|
\begin{bmatrix}
|
|||
|
354 \textup{kg} \\
|
|||
|
251 \textup{kg} \\
|
|||
|
327 \textup{kg} \\
|
|||
|
328 \textup{kg} \\
|
|||
|
301 \textup{kg} \\
|
|||
|
270 \textup{kg} \\
|
|||
|
330 \textup{kg}
|
|||
|
\end{bmatrix}
|
|||
|
=
|
|||
|
\begin{bmatrix}
|
|||
|
u_1 \\
|
|||
|
u_2 \\
|
|||
|
u_3 \\
|
|||
|
u_4 \\
|
|||
|
u_5 \\
|
|||
|
u_6 \\
|
|||
|
u_7
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
b_{mean}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
b_{1990} \\
|
|||
|
b_{1990} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{1992}
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
b_{male} \\
|
|||
|
b_{female} \\
|
|||
|
b_{male} \\
|
|||
|
b_{female} \\
|
|||
|
b_{male} \\
|
|||
|
b_{female} \\
|
|||
|
b_{male}
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
e_1 \\
|
|||
|
e_2 \\
|
|||
|
e_3 \\
|
|||
|
e_4 \\
|
|||
|
e_5 \\
|
|||
|
e_6 \\
|
|||
|
e_7
|
|||
|
\end{bmatrix}
|
|||
|
```
|
|||
|
|
|||
|
That's a nice equation, but now my hand is getting tired writing all those _b_
|
|||
|
terms over and over again, so I'm going to use [the dot product] to condense
|
|||
|
this down.
|
|||
|
|
|||
|
```math
|
|||
|
\begin{bmatrix}
|
|||
|
354 \textup{kg} \\
|
|||
|
251 \textup{kg} \\
|
|||
|
327 \textup{kg} \\
|
|||
|
328 \textup{kg} \\
|
|||
|
301 \textup{kg} \\
|
|||
|
270 \textup{kg} \\
|
|||
|
330 \textup{kg}
|
|||
|
\end{bmatrix}
|
|||
|
=
|
|||
|
\begin{bmatrix}
|
|||
|
u_1 \\
|
|||
|
u_2 \\
|
|||
|
u_3 \\
|
|||
|
u_4 \\
|
|||
|
u_5 \\
|
|||
|
u_6 \\
|
|||
|
u_7
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
1 & 1 & 0 & 0 & 1 & 0 \\
|
|||
|
1 & 1 & 0 & 0 & 0 & 1 \\
|
|||
|
1 & 0 & 1 & 0 & 1 & 0 \\
|
|||
|
1 & 0 & 1 & 0 & 0 & 1 \\
|
|||
|
1 & 0 & 1 & 0 & 1 & 0 \\
|
|||
|
1 & 0 & 0 & 1 & 1 & 0
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
b_{mean} \\
|
|||
|
b_{1990} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{1992} \\
|
|||
|
b_{male} \\
|
|||
|
b_{female}
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
e_1 \\
|
|||
|
e_2 \\
|
|||
|
e_3 \\
|
|||
|
e_4 \\
|
|||
|
e_5 \\
|
|||
|
e_6 \\
|
|||
|
e_7
|
|||
|
\end{bmatrix}
|
|||
|
```
|
|||
|
|
|||
|
That matrix in the middle with all the zeros and ones is called the **incidence
|
|||
|
matrix**, and essentially reads like a table with each row corresponding to an
|
|||
|
animal, and each column corresponding to a fixed effect. For brevity, we'll just
|
|||
|
call it _**X**_, though. One indicates that the animal and effect go together,
|
|||
|
and zero means they don't. For our record, we could write a table to go with
|
|||
|
_**X**_, and it would look like this:
|
|||
|
|
|||
|
Animal | mean | 1990 | 1991 | 1992 | male | female
|
|||
|
-- | - | - | - | - | - | -
|
|||
|
1 | yes | yes | no | no | yes | no
|
|||
|
2 | yes | yes | no | no | no | yes
|
|||
|
3 | yes | no | yes | no | yes | no
|
|||
|
4 | yes | no | yes | no | no | yes
|
|||
|
5 | yes | no | yes | no | yes | no
|
|||
|
6 | yes | no | yes | no | no | yes
|
|||
|
7 | yes | no | no | yes | yes | no
|
|||
|
|
|||
|
Now that we have _**X**_, we have the ability to start making changes to allow
|
|||
|
us to solve for _u_. Immediately, we see that _**X**_ is **singular**, meaning
|
|||
|
it can't be solved directly. We kind of already knew that, but now we can
|
|||
|
quantify it. We calculate the [rank of _**X**_], and find that there is only
|
|||
|
enough information contained in it to solve for 4 variables, which means we need
|
|||
|
to eliminate two columns.
|
|||
|
|
|||
|
There are several ways to effectively eliminate fixed effects in this type of
|
|||
|
system, but one of the simplest and the most common methods is to declare a
|
|||
|
**base population**, and lump the fixed effects of animals within the base
|
|||
|
population into the mean fixed effect. Note that it is possible to declare a
|
|||
|
base population that has no animals in it, but that gives weird results. For
|
|||
|
this example, we'll follow the convention built into `beefblup` and pick the
|
|||
|
last occuring form of each variable.
|
|||
|
|
|||
|
### Base population
|
|||
|
|
|||
|
<dl>
|
|||
|
<dt>Year</dt>
|
|||
|
<dd>1992</dd>
|
|||
|
|
|||
|
<dt>Sex</dt>
|
|||
|
<dd>Male</dd>
|
|||
|
</dl>
|
|||
|
|
|||
|
Now in order to use the base population, we simply drop the columns representing
|
|||
|
conformity with the traits in the base population from _**X**_. Our new
|
|||
|
equation looks like
|
|||
|
|
|||
|
```math
|
|||
|
\begin{bmatrix}
|
|||
|
354 \textup{kg} \\
|
|||
|
251 \textup{kg} \\
|
|||
|
327 \textup{kg} \\
|
|||
|
328 \textup{kg} \\
|
|||
|
301 \textup{kg} \\
|
|||
|
270 \textup{kg} \\
|
|||
|
330 \textup{kg}
|
|||
|
\end{bmatrix}
|
|||
|
=
|
|||
|
\begin{bmatrix}
|
|||
|
u_1 \\
|
|||
|
u_2 \\
|
|||
|
u_3 \\
|
|||
|
u_4 \\
|
|||
|
u_5 \\
|
|||
|
u_6 \\
|
|||
|
u_7
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
1 & 1 & 0 1 \\
|
|||
|
1 & 1 & 0 0 \\
|
|||
|
1 & 0 & 1 1 \\
|
|||
|
1 & 0 & 1 0 \\
|
|||
|
1 & 0 & 1 1 \\
|
|||
|
1 & 0 & 0 1
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
b_{mean} \\
|
|||
|
b_{1990} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{male} \\
|
|||
|
\end{bmatrix}
|
|||
|
+
|
|||
|
\begin{bmatrix}
|
|||
|
e_1 \\
|
|||
|
e_2 \\
|
|||
|
e_3 \\
|
|||
|
e_4 \\
|
|||
|
e_5 \\
|
|||
|
e_6 \\
|
|||
|
e_7
|
|||
|
\end{bmatrix}
|
|||
|
```
|
|||
|
|
|||
|
And the table for humans to understand:
|
|||
|
|
|||
|
Animal | mean | 1990 | 1991 | female
|
|||
|
-- | - | - | - | -
|
|||
|
1 | yes | yes | no | no
|
|||
|
2 | yes | yes | no | yes
|
|||
|
3 | yes | no | yes | no
|
|||
|
4 | yes | no | yes | yes
|
|||
|
5 | yes | no | yes | no
|
|||
|
6 | yes | no | yes | yes
|
|||
|
7 | yes | no | no | no
|
|||
|
|
|||
|
Even though each animal is said to participate in the mean, the result for the
|
|||
|
mean will now actually be the average of the base population. Math is weird
|
|||
|
sometimes.
|
|||
|
|
|||
|
Double-checking, the rank of _**X**_ is still 4, so we can solve for the average
|
|||
|
of the base population, and the effect of being born in 1990, the effect of
|
|||
|
being born in 1991, and the effect of being female (although I think [Calvin
|
|||
|
already has an idea about that one]).
|
|||
|
|
|||
|
Whew! That was some transformation. We still haven't constrained this model
|
|||
|
enough to solve it, though. Now on to the genotype.
|
|||
|
|
|||
|
## The statistical model: genotype as random effect
|
|||
|
|
|||
|
Remember I said above that genotype was a **random effect**? Statisticians say
|
|||
|
"_a random effect is an effect that influences the variance and not the mean of
|
|||
|
the observation in question._" I'm not sure exactly what that means or how that
|
|||
|
is applicable to genotype, but it does let us add an additional constraint to
|
|||
|
our model.
|
|||
|
|
|||
|
The basic gist of genetics is that organisms that are related to one another are
|
|||
|
similar to one another. Based on a pedigree, we can even say how related to one
|
|||
|
another animals are, and quantify that as the amount that the genotype terms
|
|||
|
should be allowed to vary between related animals.
|
|||
|
|
|||
|
We'll need a pedigree for our animals:
|
|||
|
|
|||
|
### Calf Records
|
|||
|
|
|||
|
ID | Sire | Dam | Birth Year | Sex | YW (kg)
|
|||
|
-- | - | - | - | - | -
|
|||
|
1 | NA | NA | 1990 | Male | 354
|
|||
|
2 | NA | NA | 1990 | Female | 251
|
|||
|
3 | 1 | NA | 1991 | Male | 327
|
|||
|
4 | 1 | NA | 1991 | Female | 328
|
|||
|
5 | 1 | 2 | 1991 | Male | 301
|
|||
|
6 | NA | 2 | 1991 | Female | 270
|
|||
|
7 | NA | NA | 1992 | Male | 330
|
|||
|
|
|||
|
Now, because cows sexually reproduce, the genotype of one animal is halfway the
|
|||
|
same as that of either parent.<sup>[a](#a)</sup> It should go without saying
|
|||
|
that each animal's genotype is identical to that of itself. From this we can
|
|||
|
then find the numerical multiplier for any relative (grandparent = 1/4, full
|
|||
|
sibling = 1, half sibling = 1/2, etc.). Let's write those values down in a
|
|||
|
table.
|
|||
|
|
|||
|
ID | 1 | 2 | 3 | 4 | 5 | 6 | 7
|
|||
|
-- | - | - | - | - | - | - | -
|
|||
|
1 | 1 | 0 | 1/2 | 1/2 | 1/2 | 0 | 0
|
|||
|
2 | 0 | 1 | 0 | 0 | 1/2 | 1/2 | 0
|
|||
|
3 | 1/2 | 0 | 1 | 1/4 | 1/4 | 0 | 0
|
|||
|
4 | 1/2 | 0 | 1/4 | 1 | 1/4 | 0 | 0
|
|||
|
5 | 1/2 | 1/2 | 1/4 | 1/4 | 1 | 1/4 | 0
|
|||
|
6 | 0 | 1/2 | 0 | 0 | 1/4 | 1 | 0
|
|||
|
7 | 0 | 0 | 0 | 0 | 0 | 0 | 1
|
|||
|
|
|||
|
Hmm. All those numbers look suspiciously like a matrix. Why don't I put them
|
|||
|
into a matrix called _**A**_?
|
|||
|
|
|||
|
```math
|
|||
|
\begin{bmatrix}
|
|||
|
1 & 0 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
|
|||
|
0 & 1 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\
|
|||
|
\frac{1}{2} & 0 & 1 & \frac{1}{4} & \frac{1}{4} & 0 & 0 \\
|
|||
|
\frac{1}{2} & 0 & \frac{1}{4} & 1 & \frac{1}{4} & 0 & 0 \\
|
|||
|
\frac{1}{2} & \frac{1}{2} & \frac{1}{4} & \frac{1}{4} & 1 & \frac{1}{4} & 0 \\
|
|||
|
0 & \frac{1}{2} & 0 & 0 & \frac{1}{4} & 1 & 0 \\
|
|||
|
0 & 0 & 0 & 0 & 0 & 0 & 1
|
|||
|
\end{bmatrix}
|
|||
|
```
|
|||
|
|
|||
|
Now I'm going to take the matrix with all of the _u_ values, and call it
|
|||
|
_**μ**_. To quantify the idea of genetic relationship, I will then say that
|
|||
|
|
|||
|
```math
|
|||
|
\textup{var}(μ) = \mathbf{A}σ_μ^2
|
|||
|
```
|
|||
|
|
|||
|
Where:
|
|||
|
|
|||
|
- _**A**_ = the relationship matrix defined above
|
|||
|
- _σ<sub>μ</sub><sup>2</sup>_ = the standard deviation of all the genotypes
|
|||
|
|
|||
|
To fully constrain the system, I have to make two more assumptions: 1) that the
|
|||
|
error term in each animal's equation is independent from all other error terms,
|
|||
|
and 2) that the error term for each animal is independent from the value of the
|
|||
|
genotype. I will call the matrix holding the _e_ values _**ε**_ and then say
|
|||
|
|
|||
|
```math
|
|||
|
\textup{var}(ϵ) = \mathbf{I}σ_ϵ^2
|
|||
|
```
|
|||
|
|
|||
|
```math
|
|||
|
\textup{cov}(μ, ϵ) = \textup{cov}(ϵ, μ) = 0
|
|||
|
```
|
|||
|
|
|||
|
Substituting in the matrix names, our equation now looks like
|
|||
|
|
|||
|
![\Large Figure 25. Nearly complete mixed-model
|
|||
|
equation](https://latex.codecogs.com/svg.latex?%5Cinline%20%5Cbegin%7Bbmatrix%7D%20354%5Ctextup%7B%20kg%7D%5C%5C%20251%5Ctextup%7B%20kg%7D%5C%5C%20327%5Ctextup%7B%20kg%7D%5C%5C%20328%5Ctextup%7B%20kg%7D%5C%5C%20301%5Ctextup%7B%20kg%7D%5C%5C%20270%5Ctextup%7B%20kg%7D%5C%5C%20330%5Ctextup%7B%20kg%7D%20%5Cend%7Bbmatrix%7D%20%3D%20%5Cmu%20+%20X%20%5Cbegin%7Bbmatrix%7D%20b_%7Bmean%7D%5C%5C%20b_%7B1990%7D%5C%5C%20b_%7B1991%7D%5C%5C%20b_%7Bfemale%7D%5C%5C%20%5Cend%7Bbmatrix%7D%20+%20%5Cvarepsilon)
|
|||
|
|
|||
|
```math
|
|||
|
\begin{bmatrix}
|
|||
|
354 \textup{kg} \\
|
|||
|
251 \textup{kg} \\
|
|||
|
327 \textup{kg} \\
|
|||
|
328 \textup{kg} \\
|
|||
|
301 \textup{kg} \\
|
|||
|
270 \textup{kg} \\
|
|||
|
330 \textup{kg}
|
|||
|
\end{bmatrix}
|
|||
|
= μ + X
|
|||
|
\begin{bmatrix}
|
|||
|
b_{mean} \\
|
|||
|
b_{1990} \\
|
|||
|
b_{1991} \\
|
|||
|
b_{male} \\
|
|||
|
\end{bmatrix}
|
|||
|
+ ϵ
|
|||
|
```
|
|||
|
|
|||
|
We are going to make three changes to this equation before we are ready to solve
|
|||
|
it, but they are cosmetic details for this example.
|
|||
|
|
|||
|
1. Call the matrix on the left side of the equation _**Y**_ (sometimes it's
|
|||
|
called the **matrix of observations**)
|
|||
|
2. Multiply _**μ**_ by an identity matrix called _**Z**_. Multiplying by the
|
|||
|
identity matrix is the matrix form of multiplying by one, so nothing changes,
|
|||
|
but if we later want to find one animal's genetic effect on another animal's
|
|||
|
performance (e.g. a **maternal effects model**), we can alter _**Z**_ to
|
|||
|
allow that/
|
|||
|
3. Call the matrix with all the _b_ values _**β**_.
|
|||
|
|
|||
|
With all these changes, we now have
|
|||
|
|
|||
|
```math
|
|||
|
Y = Z μ + X β + ϵ
|
|||
|
```
|
|||
|
|
|||
|
This is the canonical form of the mixed-model equation, and the form that
|
|||
|
Charles Henderson used to first predict breeding values of livestock.
|
|||
|
|
|||
|
## Solving the equations
|
|||
|
|
|||
|
Henderson proved that the mixed-model equation can be solved by the following:
|
|||
|
|
|||
|
![\Large Figure 27. Solution to mixed-model
|
|||
|
equation](https://latex.codecogs.com/svg.latex?%5Cinline%20%5Cbegin%7Bbmatrix%7D%20%5Chat%7B%5Cbeta%7D%5C%5C%20%5Chat%7B%5Cmu%7D%20%5Cend%7Bbmatrix%7D%20%3D%5Cbegin%7Bbmatrix%7D%20X%27X%26X%27Z%5C%5C%20Z%27X%26Z%27Z+A%5E%7B-1%7D%5Clambda%20%5Cend%7Bbmatrix%7D%5E%7B-1%7D%20%5Cbegin%7Bbmatrix%7D%20X%27Y%5C%5C%20Z%27Y%20%5Cend%7Bbmatrix%7D)
|
|||
|
|
|||
|
```math
|
|||
|
\begin{bmatrix}
|
|||
|
\hat{β} \\
|
|||
|
\hat{μ}
|
|||
|
\end{bmatrix}
|
|||
|
=
|
|||
|
\begin{bmatrix}
|
|||
|
X'X & X'Z \\
|
|||
|
Z'X & Z'Z+A^{-1}\lambda
|
|||
|
\end{bmatrix}^{-1}
|
|||
|
\begin{bmatrix}
|
|||
|
X'Y \\
|
|||
|
Z'Y
|
|||
|
\end{bmatrix}
|
|||
|
```
|
|||
|
|
|||
|
Where
|
|||
|
|
|||
|
- The variables with hats are the statistical estimates of their mixed-model
|
|||
|
counterparts
|
|||
|
- The predicted value of _**μ**_ is called the _Best Linear Unbiased
|
|||
|
Predictor_ or _BLUP_
|
|||
|
- The estimated value of _**β**_ is called the _Best Linear Unbiased Estimate_
|
|||
|
or _BLUE_
|
|||
|
- ' is the transpose operator
|
|||
|
- λ is a single real number that is a function of the heritability for the trait
|
|||
|
being predicted. It can be left out in many cases (λ = 1).
|
|||
|
- λ = (1-h<sup>2</sup>)/h<sup>2</sup>
|
|||
|
|
|||
|
What happened to
|
|||
|
|
|||
|
## Footnotes
|
|||
|
|
|||
|
### a
|
|||
|
|
|||
|
An animal **can** share its genome with itself by a factor of more than one:
|
|||
|
that's called inbreeding! We can account for this, and `beefblup` does as it
|
|||
|
calculates _**A**_. This is an area that actually merits a good deal of study:
|
|||
|
see chapter 2 of _Linear Models for the Prediction of Animal Breeding Values_ by
|
|||
|
Raphael A. Mrode (ISBN 978 1 78064 391 5).
|
|||
|
|
|||
|
[BIF Guidelines]:
|
|||
|
https://beefimprovement.org/wp-content/uploads/2018/03/BIFGuidelinesFinal_updated0318.pdf
|
|||
|
[The Dot Product]:
|
|||
|
https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:multiplying-matrices-by-matrices/v/matrix-multiplication-intro
|
|||
|
[rank of _**X**_]: https://math.stackexchange.com/a/2080577 [Calvin already has
|
|||
|
an idea about that one]: https://www.gocomics.com/calvinandhobbes/1992/12/02
|