Asymptotically exact fit for linear mixed model in genetic association studies.

The linear mixed model (LMM) has become a standard in genetic association studies to account for population stratification and relatedness in the samples to reduce false positives. Much recent progresses in LMM focused on approximate computations. Exact methods remained computationally demanding and without theoretical assurance. The computation is particularly challenging for multiomics studies where tens of thousands of phenotypes are tested for association with millions of genetic markers. We present IDUL and IDUL† that use iterative dispersion updates to fit LMMs, where IDUL† is a modified version of IDUL that guarantees likelihood increase between updates. Practically, IDUL and IDUL† produced identical results, both are markedly more efficient than the state-of-the-art Newton-Raphson method, and in particular, both are highly efficient for additional phenotypes, making them ideal to study genetic determinants of multiomics phenotypes. Theoretically, the LMM likelihood is asymptotically unimodal, and therefore the gradient ascent algorithm IDUL† is asymptotically exact. A software package implementing IDUL and IDUL† for genetic association studies is freely available at https://github.com/haplotype/IDUL.