By Karl Broman
(This article was originally published at The stupidest thing… » R, and syndicated at StatsBlogs.)
Linear mixed models (LMMs) have become widely used for dealing with population structure in human GWAS, and they’re becoming increasing important for QTL mapping in model organisms, particularly for the analysis of advanced intercross lines (AIL), which often exhibit variation in the relationships among individuals.
In my efforts on R/qtl2, a reimplementation R/qtl to better handle high-dimensional data and more complex cross designs, it was clear that I’d need to figure out LMMs. But while papers explaining the fit of LMMs seem quite explicit and clear, I’d never quite turned the corner to actually seeing how I’d implement it. In both reading papers and studying code (e.g., lme4), I’d be going along fine and then get completely lost part-way through.
But I now finally understand LMMs, or at least a particular, simple LMM, and I’ve been able to write an implementation: the R package lmmlite.
It seemed worthwhile to write down some of the details.
The model I want to fit is y = X b + e, where var(e) = sK + tI, where K is a known kinship matrix and I is the identity matrix. Think of y as a vector of phenotypes and X as a matrix of covariates. Let v = s+t be the residual variance, and let h = s/(s+t) = s/v be the heritability.
First, a shout to Artem Tarasov, who wrote a series of blog posts walking through and explaining the source code for FaST-LMM and pylmm, and to Nick Furlotte, whose pylmm code is especially clear and easy-to-read. Only by reading their work did I come to understand these LMMs.
Back to the model fit:
- For a fixed value of the heritability, h, we have var(e) = v[hK + (1-h)I] = vV where V is …read more
Source:: statsblogs.com