Simple multivariate animal model

Model structure

The multivariate animal extends the univariate animal model to account for genetic and environmental associations among multiple traits. In the simplest case, the model predicts $t$ phenotypic trait values $z_1,…,z_t$ for individual $i$ as a function of global intercepts $\mu_1,…,\mu_t$ and multivariate normal additive genetic $a_1,…,a_t$ and residual environmental $e_1,…,e_t$ values.

$$\begin{bmatrix} z_{1i} \ \vdots \ z_{ti} \end{bmatrix}

\begin{bmatrix} \mu_1 + a_{1i} + e_{1i} \ \vdots \ \mu_t + a_{ti} + e_{ti} \end{bmatrix}$$

Individual values are expressed as zero-centered deviations from the trait-specific global intercepts. The (co)variances of genetic and environmental values are estimated by $\mathbf{G}$ and $\mathbf{\Sigma}$ matrices, respectively.

$$\begin{bmatrix} \mathbf{a_1} \ \vdots \ \mathbf{a_t} \end{bmatrix} \sim N(\mathbf{0},\mathbf{G}\otimes \mathbf{A}), \begin{bmatrix} \mathbf{e_1} \ \vdots \ \mathbf{e_t} \end{bmatrix} \sim N(\mathbf{0},\mathbf{\Sigma} \otimes \mathbf{I}) $$

The Kronecker product $\otimes$ of $\mathbf{G}$ with the relatedness matrix $\mathbf{A}$ accounts for the expected similarity in additive genetic values among individuals. Residual environmental effects are assumed to be independently distributed, which is denoted by the identity matrix $\mathbf{I}$. There are, of course, many further directions to take the model, such as the inclusion of additional fixed and random effects.

Written by: Jordan S. Martin