Variance component estimation for mixed model with 2 loci

[openmendel]

Variance component estimation for mixed model with 2 loci

[Hua-Zhou]

@fboehm, there is a chance you can just feed the transpose of your data into functions and read off the estimates.

In your model, Y is n x d, X is nd x pd, B is pd x 1, Sigma1, Sigma2 are n x n, and V1, V2 are d x d. You assume
vec(Y) ~ N(XB, Sigma1 \otimes V1 + \Sigma2 \otimes V2).
Let Kdn be the nd x nd commutation matrix
https://en.wikipedia.org/wiki/Commutation_matrix
Then
vec(Y') = Knd vec(Y) ~ N(Knd X B, Knd (Sigma1 \otimes V1 + Sigma2 \otimes V2) Knd') = N(Knd X B, V1 \otimes Sigma1 + V2 \otimes Sigma2).

So feeding the transformed data Y', Knd X, Sigma1, Sigma2 into functions, you may read off the estimates of V1, V2 directly. The catch is current functions assume the mean structure vec(XB). Is there any structure in your X matrix? Or if the primary interest is estimating V1, V2, you can use REML: first obtain residuals from ordinary least squares then fit the variance component model using just residuals. If none of these apply, it's also straightforward modify current functions to accommodate your X.

[fboehm]

Thanks, @Hua-Zhou! Your explanation is very helpful. I think that I can proceed based on your detailed reply. Thanks again!