# [R-sig-ME] estimating variance components for arbitrarily defined var/covar matrices

Matthew Keller mckellercran at gmail.com
Thu Feb 26 00:42:32 CET 2015

```Hi all,

This is a typical problem in genetics and I'm trying to figure out whether
there's any way to solve it using lmer or similar, and if not, why it isn't
possible.

Often in genetics, we have an n-by-n matrix (n=sample size) of genetic
relationships, where the diagonal is how related you are to yourself (~1,
depending on inbreeding) and off-diagonals each pairwise relationship. I'd
like to be able to use lmer or some other function in R to estimate the
variance attributable to this genetic relationship matrix. Thus:
y = b0 + b*X + g*Z + error
where y is a vector of observations, b is a vector of fixed covariate
effects and g is a vector of random genetic effects. X and Z are incidence
matrices for b & g respectively, and we assume g ~ N(0, VG). The variance
of y is therefore
var(y) = Z*Z' * VG + I*var(e)

Z*Z' is the observed n-by-n genetic relationship matrix. Given an observed
Z*Z' genetic relationship matrix, is there a way to estimate VG?

I guess this boils down to, if we have an observed n-by-n matrix of
similarities, can we use mixed models in R to get the variance in y that is
explained by that similarity?