mirror of
https://github.com/MillironX/beefblup.git
synced 2024-12-22 09:08:16 +00:00
Convert wiki to docs
This commit is contained in:
parent
b4e0591ed2
commit
97bd43323d
GPG key ID:
139C07724802BC5D
3 changed files with 666 additions and 0 deletions
3
docs/src/.markdownlint.json
Normal file
3
docs/src/.markdownlint.json
Normal file
|
|
@ -0,0 +1,3 @@
|
|||
{
|
||||
"MD041": false
|
||||
}
|
||||
105
docs/src/beefblup-cli.md
Normal file
105
docs/src/beefblup-cli.md
Normal file
|
|
@ -0,0 +1,105 @@
|
|||
```@meta
|
||||
CurrentModule = BeefBLUP
|
||||
```
|
||||
beefblup Command Line Interface (CLI) documentation
|
||||
|
||||
> _A work in progress_
|
||||
|
||||
**Notice:** This document is a draft for what the command-line interface for
|
||||
beefblup would look like as of version 1.0, if beefblup was even a command-line
|
||||
application to begin with (it's not). It is modeled (loosely) after the man page
|
||||
format. It is not intended to be taken seriously, but instead to serve as a
|
||||
useful thought experiment and brainstorming ground on the future of beefblup.
|
||||
Please use it if it clarifies things for you. If it doesn't, ignore it.
|
||||
|
||||
## Input file
|
||||
|
||||
beefblup requires a very specific format of input file. The format may be in
|
||||
comma-separated values (CSV) or Excel 2007+ (XLSX) format. CSV files should not
|
||||
be quoted (and therefore cannot have commas within cell values). Other formats
|
||||
may be forthcoming.
|
||||
|
||||
A beefblup data file must have at least six columns appearing in this order:
|
||||
|
||||
- ID
|
||||
- Sire ID
|
||||
- Dam ID
|
||||
- Birthdate
|
||||
- Fixed effect(s)
|
||||
- Response variable(s)
|
||||
|
||||
The first row always contains column names. The values of column names are
|
||||
unimportant for the first four columns, as they will always be treated the same
|
||||
regardless of the name. The generated report will use the column names
|
||||
of fixed effects and response variables as given.
|
||||
|
||||
Each fixed effect should have its own column, to as many as are needed. There is
|
||||
no limit to the number of fixed effects as defined by beefblup, however its
|
||||
dependencies might have some. The same rules apply to response variables.
|
||||
|
||||
Unknown values should be left blank (`,,`). Do not substitute null placeholders
|
||||
(e.g. `NULL`, `NA`, `0`, `nothing`, `undefined`, etc.) for unknown values.
|
||||
|
||||
An example spreadsheet might have the following format
|
||||
|
||||
| ID | Sire ID | Dam ID | Birthdate | Sex | Weaning Weight |
|
||||
| --- | ------- | ------ | --------- | ------ | -------------- |
|
||||
| 1 | | | 1/1/1990 | Male | 354 |
|
||||
| 2 | 1 | | 1/1/1990 | Female | 251 |
|
||||
| 3 | 1 | | 1/1/1991 | Male | 327 |
|
||||
| 4 | 1 | 2 | 1/1/1991 | Female | 328 |
|
||||
| 5 | | 2 | 1/1/1991 | Male | 301 |
|
||||
| 6 | | | 1/1/1991 | Female | 270 |
|
||||
| 7 | | | 1/1/1992 | Male | 330 |
|
||||
|
||||
## Synopsis
|
||||
|
||||
```bash
|
||||
beefblup [-G SNPs_file] [-M num_response_vars] [-o report_spreadsheet]
|
||||
[--no-aod] [--no-year] [--no-season] [--no-autob] [--maternal] input_file
|
||||
[report_file]
|
||||
```
|
||||
|
||||
## Command line basic syntax
|
||||
|
||||
The most basic input is to simply pass the input file name to the program.
|
||||
|
||||
```bash
|
||||
beefblup filename.csv
|
||||
```
|
||||
|
||||
In this case beefblup will insert fixed-effects for age-of-dam, year, and
|
||||
season, and will calculate the EBVs for the response variable in the final
|
||||
column. The report will then be saved as `filename_report.txt`.
|
||||
|
||||
## Suppressing automatically-calculated fixed effects
|
||||
|
||||
If you don't wish to include one of the automatically calculated fixed-effects
|
||||
from your model, you can pass arguments to suppress them.
|
||||
|
||||
### Suppress Age-of-dam
|
||||
|
||||
```bash
|
||||
beefblup --no-aod filename.csv
|
||||
```
|
||||
|
||||
### Suppress year
|
||||
|
||||
```bash
|
||||
beefblup --no-year filename.csv
|
||||
```
|
||||
|
||||
### Suppress season
|
||||
|
||||
```bash
|
||||
beefblup --no-season filename.csv
|
||||
```
|
||||
|
||||
### Suppress all calculated fixed effects
|
||||
|
||||
```bash
|
||||
beefblup --no-autob filename.csv
|
||||
```
|
||||
|
||||
The argument `--no-autob` comes from the nomenclature of assigning fixed-effects
|
||||
to the matrix _b_ in Henderson's mixed-model equations.
|
||||
558
docs/src/how-to-calculate-epds.md
Normal file
558
docs/src/how-to-calculate-epds.md
Normal file
|
|
@ -0,0 +1,558 @@
|
|||
```@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
|
||||
|
||||
|
||||
|
||||
```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:
|
||||
|
||||
|
||||
|
||||
```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
|
||||
Loading…
Reference in a new issue