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. lmt accepts plain marker data and will calculate all necessary derivatives required by the model. Genotypes can be scaled using average or marker specific allele content variance (2pq), and allele frequencies are either calculated from the data, or read from user specified input files. Futher, if requested $$G$$ can be adjusted to fit $$A$$, even for models where $$G$$ is not build(e.g. single step SNP-BLUP).

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 SNP-BLUP 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 $$M$$ is a centered and scaled matrix of marker counts, $$\zeta = \Omega_{m}(\Gamma_m \otimes I)\Omega_{m}^{'}$$, $$\xi = A_{n,g}A_{g,g}^{-1}$$, $$\Sigma_g$$ is the global genetic co-variance matrix, $$\lambda$$ is the polygenic weight and $$\gamma$$ is the genomic weight. In most applications $$\gamma \zeta $$ is set to $$ \gamma I\otimes \Sigma_g$$. For that case it can be shown that the above variance structure can be transferred into an ordinary single step $$H$$ matrix which uses $$G$$. Note that in the instruction file $$\zeta$$ is parametrized independently of $$\Sigma_g$$, but it is advisable to ensure that the following equality holds $$\gamma 1'\zeta 1==\gamma \Sigma_g$$.