Genomic data in lmt

lmt can account for genomic data via genomic relationship matrices $$G$$ and single step relationship matrices $$H$$ where single step G-BLUP, single step GT-BLUP, and single step SNP-BLUP are supported

Single step SNP-BLUP

lmt's Single step SNP-BLUP model uses the approach described in Liu and Goddard 2014, where the adverse of the singe step SN-PBLUP variance structure can be written as

$$ \left( \begin{array}{ccc} (A^{n,n})^{-1}\otimes \Sigma_g +\xi (A_{g,g}\otimes \Sigma_g \lambda + M\zeta M^{'}\gamma)\xi^{'} & \xi (A_{g,g}\otimes \Sigma_g \lambda + M\zeta M^{'}\gamma) & \xi M\zeta\gamma \\ (A_{g,g}\otimes \Sigma_g \lambda + M\zeta M^{'}\gamma)\xi^{'} & A_{g,g}\otimes \Sigma_g \lambda + M\zeta M^{'}\gamma & M\zeta\gamma \\ \gamma \zeta M^{'}\xi^{'} & \gamma \zeta M^{'}& \zeta\gamma \end{array} \right) $$

where $$\zeta = \Omega_{m}(\Gamma_m \otimes I)\Omega_{m}^{'}$$ and $$\xi = A_{n,g}A_{g,g}^{-1}$$, $$\Sigma_g$$ is the global genetic co-variance matrix, $$\lambda$$ is the polygenic weight and $$\gamma$$ is the genomic weight. Note that when parametrizing this variance structure the following equality should hold $$\gamma 1'\zeta 1==\gamma \Sigma_g$$