lme4breeding: a fast linear mixed model for multi-trait/multi-environment experiments

Mixed models are a cornerstone in quantitative genetics to study the genetics of complex traits. A standard quantitative genetic model assumes that the effects of some random factors (e.g., individuals) are correlated based on their identity by descent and state. In addition, other relationships arise in the genotype by environment interactions (i.e., covariance structures). Open-source mixed model routines are available but do not account for complex covariance structures or are extremely slow to fit big/dense genomic models. The lme4breeding R package was developed as an extension of the lme4 package and allows correlated random effects and complex covariance structures to be fitted for responses with different distributions. The correlation between levels of the random effect (e.g., individuals) is accounted for by eigen-decomposing the relationship matrix and post-multiplying the incidence matrix of the levels of this random factor by the Cholesky decomposition of the corresponding (co)variance matrix (eigen values) and the response by the eigen vectors. For big genomic models the eigen decomposition of relationship matrices coupled with sparse matrix solvers presents a massive increase in speed compared to dense formulations used in other popular software and the same level of accuracy is achieved.