# Genomic data in lmt

lmt can account for genomic data via genomic relationship matrices $$G$$ and single step relationship matrices $$H$$ where ssGBLUP^{[1]}, ssGTBLUP^{[2]}, and ssSNPBLUP^{[3]} 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 $$A$$^{[4]}, even for models where $$G$$ is not build(e.g. ssSNP-BLUP).

## Genetic Groups and Meta-founders

If the pedigree constituting $$A$$ contains phantom parents^{[5]}, genetic groups are automatically fitted for all different Single-Step co-variance structures. Alternatively, genetic groups can be fitted as an extra factor.

Meta-founders^{[6]} are fitted automatically if a meta-founder co-variance matrix has been supplied.

## Single-Step SNP-BLUP

lmt's Single-Step SNP-BLUP model uses the approach described in Liu and Goddard 2014^{[3]}, 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 $$\gamma 1'\zeta 1==\gamma \Sigma_g$$.

## References

- ↑ Christensen et al.; Genomic prediction when some animals are not genotyped; Genetics Selection Evolution; 2010
- ↑ Mäntysaari et al.;Efficient single-step genomic evaluation for a multibreed beef cattle population having many genotyped animals; Journal of Animal Science;2017
- ↑
^{3.0}^{3.1}Liu et al.;A single-step genomic model with direct estimation of marker effects; Journal of Dairy Science;2014 - ↑ O.F. Christensen; Compatibility of pedigree-based and marker-based relationship matrices for single-step genetic evaluation; Genetics Selection Evolution; 2012
- ↑ Westell et al.; Genetic Groups in an Animal Model; Journal of Dairy Science; 1988
- ↑ Garcia et al.; Metafounders are related to F st fixation indices and reduce bias in single-step genomic evaluations; Genetics Selection Evolution; 2017