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Linear mixed models (LMMs) have been widely used in genome-wide association studies to control for population stratification and cryptic relatedness. However, estimating LMM parameters is computationally expensive, necessitating large-scale matrix operations to build the genetic relationship matrix (GRM). Over the past 25 years, Randomized Linear Algebra has provided alternative approaches to such matrix operations by leveragingmatrix sketching, which often results in provably accurate fast and efficient approximations. We leverage matrix sketching to develop a fast and efficient LMM method calledMatrix-Sketching LMM (MaSk-LMM) by sketching the genotype matrix to reduce its dimensions and speed up computations. Our framework comes with both theoretical guarantees and a strong empirical performance compared to the current state-of-the-art for simulated traits and complex diseases. 

Abstract BackgroundIdentifying variants associated with complex traits is a challenging task in genetic association studies due to linkage disequilibrium (LD) between genetic variants and population stratification, unrelated to the disease risk. Existing methods of population structure correction use principal component analysis or linear mixed models with a random effect when modeling associations between a trait of interest and genetic markers. However, due to stringent significance thresholds and latent interactions between the markers, these methods often fail to detect genuinely associated variants. ResultsTo overcome this, we propose CluStrat, which corrects for complex arbitrarily structured populations while leveraging the linkage disequilibrium induced distances between genetic markers. It performs an agglomerative hierarchical clustering using the Mahalanobis distance covariance matrix of the markers. In simulation studies, we show that our method outperforms existing methods in detecting true causal variants. Applying CluStrat on WTCCC2 and UK Biobank cohorts, we found biologically relevant associations in Schizophrenia and Myocardial Infarction. CluStrat was also able to correct for population structure in polygenic adaptation of height in Europeans. ConclusionsCluStrat highlights the advantages of biologically relevant distance metrics, such as the Mahalanobis distance, which captures the cryptic interactions within populations in the presence of LD better than the Euclidean distance.