FaST-LMM: Factored Spectrally Transformed Linear Mixed Models

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.

This implementation is my attempt to learn Julia and numerical linear algebra. The code is being tested.

Guide to the directories:

• src: Julia source code
• data: Example data for development and testing
• test: Code for testing
• docs: Notes on comparisons with other implementations