# How many traits can a multivariate breeders equation handle?

The multivariate breeders equation (MBE) by Lande predicts the change in a trait $\Delta \bar z$ (response) as

$\Delta \bar z = G \beta$

where $G$ is a genetic variance-covariance matrix and $\beta$ is a vector of the selection coefficients. What are the limitations on the number of traits the MBE can tolerate? Is there a theoretical limit on the number of traits $G$ and $\beta$ could contain? Or does the calculation lose power with increasing complexity?

Theoretically as an example, could I take the expression of 10000 genes, compare them to fitness to to make a 10000 row $\beta$ vector, and generate a 10000 * 10000 G-matrix from estimates of male and female gene expression in some lines, and then put them through the MBE?

• (this may be better on or cross posted on maths SE) Oct 13 '14 at 15:54
• I guess there is no theoretical limit on how many dimensions you should have. But it makes sense to reduce the dimensionality of the model if possible. Having too many variables makes a) the system look complicated (and clumsy) b) consumes more computational resources c) in some cases makes the dynamical model stiff (makes it difficult to solve) Oct 14 '14 at 4:46
• This is still fine because it is a linear equation. And 10k x 10k matrix is not that huge for computers these days. But you would end up storing more information than needed. Oct 14 '14 at 4:52
• @WYSIWYG Thanks, I guess there is a trade off to be had, on one hand we want to include as many traits as possible to ensure we have all traits affecting fitness (which is almost impossible, particularly in natural populations, and especially if we consider all ontogenetic and environmental variants). But on the other it becomes rapidly more complex, conceptually difficult and less general in its meaning. Oct 14 '14 at 7:36

Noting that the right hand side can also be written $\mathbf{G}\mathbf{P}^{-1}\mathbf{s}$, we see that the only theoretical limitation of the multivariate breeders equation is the invertibility of the matrix $\mathbf{P}$. This happens when the measures of two traits are the same for all individuals, making two columns of the matrix the same I.e. It becomes singular.