Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

Abstract

Background Linear mixed-effects models (LMM) are a leading method in conducting genome-wide association studies (GWAS) but require residual maximum likelihood (REML) estimation of variance components, which is computationally demanding. Previous work has reduced the computational burden of variance component estimation by replacing direct matrix operations with iterative and stochastic methods and by employing loose tolerances to limit the number of iterations in the REML optimization procedure. Here, we introduce two novel algorithms, stochastic Lanczos derivative-free REML (SLDF_REML) and Lanczos first-order Monte Carlo REML (L_FOMC_REML), that exploit problem structure via the principle of Krylov subspace shift-invariance to speed computation beyond existing methods. Both novel algorithms only require a single round of computation involving iterative matrix operations, after which their respective objectives can be repeatedly evaluated using vector operations. Further, in contrast to existing stochastic methods, SLDF_REML can exploit precomputed genomic relatedness matrices (GRMs), when available, to further speed computation. Results Results of numerical experiments are congruent with theory and demonstrate that interpreted-language implementations of both algorithms match or exceed existing compiled-language software packages in speed, accuracy, and flexibility. Conclusions Both the SLDF_REML and L_FOMC_REML algorithms outperform existing methods for REML estimation of variance components for LMM and are suitable for incorporation into existing GWAS LMM software implementations. Electronic supplementary material The online version of this article (10.1186/s12859-019-2978-z) contains supplementary material, which is available to authorized users.

Assortative mating biases marker-based heritability estimators

“…We used GCTA v1.91.3b 20 to construct genomic related matrices and perform HE regression. We obtained using BOLT-LMM v2.3.4 51 for computational efficiency; though BOLT-LMM uses a randomized algorithm, its numerical accuracy is comparable to that of the exact algorithm implemented GCTA 52 .…”

“…We therefore utilized a modified Python implementation of the REML algorithm presented in ref. 52 (available at https://github.com/rborder/SL_REML 56 ). We used LDAK v5.0 to obtain adjusted HE regression estimates 53 .…”

SLEMM: million-scale genomic predictions with window-based SNP weighting

“…This article presents a new software tool to address the challenge, which we refer to as SLEMM (short for Stochastic-Lanczos-Expedited Mixed Models). SLEMM can efficiently perform Stochastic Lanczos Restricted Maximum Likelihood (REML) analysis ( Border and Becker, 2019 ) for millions of genotyped individuals and weight SNPs based on proximity and minor allele frequency (MAF). Because of linkage disequilibrium (LD) between SNPs, effect sizes of SNPs that are close to each other tend to be similar in model fitting ( Yang and Tempelman 2012 ; Zeng et al 2018 ), which stimulates proximity-based SNP weighting.…”

Hybrid of Restricted and Penalized Maximum Likelihood Method for Efficient Genome-Wide Association Study

“…On the other hand, the inversion of the covariance matrix is required to perform on each computation of likelihood, an operation that is proportional to the cube of individual number. As a result, improving the computational efficiency of REML for estimating variance components has become one of the research hotspots [8][9][10][11][12].…”

“…Ganjgahi et al [4] proposed a weighted least squares REML (WLS-REML) using a noniterative one-step random effect estimator to decrease the computational cost. Border and Becker [12] developed stochastic Lanczos derivative-free REML and Lanczos first-order Monte Carlo REML to further improve the computing speed. However, these existing methods for REML variance components estimation are mainly aimed at single-locus LMM, which is not effective enough to handle a complex genetic structure.…”