Genetic analysis in structured populations used mixed linear models where the variance matrix of the error term is a linear combination of an identity matrix and a positive definite matrix.
The linear model is of the familiar form: $$y = X \beta + e$$, where
$y$: phenotype
$X$: covariates
$\beta$: fixed effects
$e$: error term
Further $V(e) = \sigma_G^2 K + \sigma_E^2 I$, where $\sigma_G^2$ is the genetic variance, $\sigma_E^2$ is the environmental variance, $K$ is the kinship matrix, and $I$ is the identity matrix.
The key idea in speeding up computations here is that by rotating the phenotypes by the eigenvectors of $K$ we can transform estimation to a weighted least squares problem.
The main purpose of the Julia package was to test the feasibility and speed of the LMM implementation in Julia. In our tests, the Julia code is almost as fast as R calling C/C++ code, and faster than native Python or R implementations of the same algorithm.
We are now working on an updated version of this package that performs LMMs for multiple traits as in an LMM eQTL scan.