Here, we’ll take the GEBVs obtained with the RR and TP approaches and back solve to get the marker effects.

Recall that GEBVs (\(\mathbf{\hat{g} }\)) can be parameterized as \(\mathbf{\hat{g} } = \boldsymbol{\hat{\beta} W_{sc} }\), where \(\mathbf{W_{sc}}\) is a matrix of marker genotypes, as defined above, and \(\boldsymbol{\hat{\beta}}\) is a vector of allele substitution effects. \(\boldsymbol{\hat{\beta}}\) can be obtained using best linear unbiased prediction (BLUP) by

\[BLUP(\boldsymbol{\beta}) = \mathbf{W'_{sc}}(\mathbf{W_{sc}W'_{sc}})^{-1} \left [\mathbf{I} + \mathbf{G}^{-1} \frac{\sigma^2_e}{\sigma^2_g} \right ]^{-1} \mathbf{y}\] where \(\sigma^2_g\) and \(\sigma^2_e\) are genetic and residual variances, respectively.

Given BLUP of GEBVs is

\[BLUP(\mathbf{g}) = \left [ \mathbf{I} + \mathbf{G}^{-1} \frac{\sigma^2_e}{\sigma^2_g} \right ]^{-1} \mathbf{y}\]

BLUP of marker effects can be obtained using the following linear transformation

\[BLUP(\boldsymbol{\beta}) = \mathbf{W_{sc}'}(\mathbf{W_{sc}W_{sc}'})^{-1}BLUP(\mathbf{g})\]