# Simple univariate animal model

Fitting a simple univariate model in R.

## What does this model do?

We assume that the phenotypic trait value of an individual is the sum of an intercept (that is, a baseline value, which in simple cases can be thought of as the mean trait value), a breeding value (that is, a contribution of genetic differences) and an environmental deviation.

The phenotypic variance can be decomposed into an additive genetic variance (the variance of breeding values) and an environmental variance (the variance of environmental deviations).

We may be interested in the narrow-sense heritability, which is the additive genetic variance divided by the phenotypic variance.

A simple univariate animal model let’s us estimate all those parameters.

## More formally?

Here it is not too difficult to describe the model in plain English, but for more complex models a mathematical equation may prove easier, shorter and less ambiguous.

Here we could write the model as

$$y_i = \mu + a_i + \epsilon_i$$

We have:

• $y_i$, the trait value for individual $i$.
• $\epsilon_i$, the residual or environmental deviation for individual $i$. We assume the $\epsilon$ of all individuals follow a normal distribution of mean zero and variance $\sigma_E^2$, which we can write as $\mathbf{\epsilon} \sim N(0,\mathbf{I}\sigma_E^2)$, where $\mathbf{I}$ is an identity matrix.
• $a_i$, the breeding value of invidual $i$. We assume $\mathbf{a} \sim N(0,\mathbf{A} \sigma_A^2$, where $\mathbf{A}$ is the pairwise additive genetic relatedness matrix. $\mathbf{A}$ can be computed from a pedigree or from individual genetic data.

Often we will compute the heritabiltiy ($h^2$), which is $h^2 = \frac{\sigma_A^2}{\sigma_A^2 + \sigma_E^2}$

Writen by: Timothée Bonnet

##### Univariate animal model using MCMCglmm

Univariate animal model using MCMCglmm.

##### Univariate animal model using brms

Univariate animal model using brms