Kang et al. (2008) describe an efficient mixed model formulation for the special case of only one random effect besides the error, which avoids any matrix computation in the REML estimation of variance components. Piepho et al. (2011) re-parametrize their formulation to allow for a fixed residual variance. This re-parametrization might be especially useful in a plant breeding context. Here, the phenotypes used for estimation of marker effects are commonly the adjusted (for all other random and fixed effects) entry means, obtained beforehand from a one- or two-step adjustment procedure, most likely a mixed-model analysis (Moehring and Piepho, 2009). From this analysis, good estimates of the residual variance are usually available, so that it is not necessary and even counterproductive to re-estimate this parameter in RR-BLUP (Moehring and Piepho, 2009). Please see Piepho et al. (2011) for details.
The method is restricted to the case where $R = I\sigma^2$, where $R$ is the error variance-covariance matrix and $\sigma^2$ is the error variance. An independent estimate of $R$ is often available from the analysis that yielded adjusted means. In case $R$ does not meet this assumption, a linear transformation (rotation) can always be applied to ensure $R = I\sigma^2$ (Piepho et al., 2011, Schulz-Streeck et al., 2012), provided that $R$ is known. Hence, we replace $y$ by $L_R y$ and $Z$ by $L_R Z$, where $y$ is the vector with the adjusted means, $R^{-1} = (L_R)^2$ such that $L_R$ is square and symmetric and $Z$ is the matrix with marker information. $L_R$ is easily obtained from a spectral decomposition of $R^{-1}$. With these replacements, analysis can proceed assuming that $R = I\sigma^2$ with $\sigma^2 = 1$.
The package
Moehring, J., Piepho, H. P. (2009): Comparison of weighting in two-stage analyses of series of experiments. Crop Science 49, 1977-1988
Piepho HP, Schulz-Streeck T, Ogutu JO (2011): A stage-wise approach for analysis of multi-environment trials. Biuletyn Oceny Odmian 33:7-20
Schulz-Streeck T, Ogutu JO, Piepho HP (2012) Comparisons of single-stage and two-stage approaches to genomic selection. Submitted
rrBlupM6, rrBlupRotation