FaST-LMM: Factored Spectrally Transformed Linear Mixed Models

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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