A computationally efficient algorithm to leverage average information REML for (co)variance component estimation in the genomic era

Methods for estimating variance components (VC) using restricted maximum likelihood (REML) typically require elements from the inverse of the coefficient matrix of the mixed model equations (MME). As genomic information becomes more prevalent, the coefficient matrix of the MME becomes denser, presenting a challenge for analyzing large datasets. Thus, computational algorithms based on iterative solving and Monte Carlo approximation of the inverse of the coefficient matrix become appealing. While the standard average information REML (AI-REML) is known for its rapid convergence, its computational intensity imposes limitations. In particular, the standard AI-REML requires solving the MME for each VC, which can be computationally demanding, especially when dealing with complex models with many VC. To bridge this gap, here we (1) present a computationally efficient and tractable algorithm, named the augmented AI-REML, which facilitates the AI-REML by solving an augmented MME only once within each REML iteration; and (2) implement this approach for VC estimation in a general framework of a multi-trait GBLUP model. VC estimation was investigated based on the number of VC in the model, including a two-trait, three-trait, four-trait, and five-trait GBLUP model. We compared the augmented AI-REML with the standard AI-REML in terms of computing time per REML iteration. Direct and iterative solving methods were used to assess the advances of the augmented AI-REML. When using the direct solving method, the augmented AI-REML and the standard AI-REML required similar computing times for models with a small number of VC (the two- and three-trait GBLUP model), while the augmented AI-REML demonstrated more notable reductions in computing time as the number of VC in the model increased. When using the iterative solving method, the augmented AI-REML demonstrated substantial improvements in computational efficiency compared to the standard AI-REML. The elapsed time of each REML iteration was reduced by 75%, 84%, and 86% for the two-, three-, and four-trait GBLUP models, respectively. The augmented AI-REML can considerably reduce the computing time within each REML iteration, particularly when using an iterative solver. Our results demonstrate the potential of the augmented AI-REML as an appealing approach for large-scale VC estimation in the genomic era.