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:

$$y = X \beta + e$$

\(y\): 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.