The paper introduces an efficient method for computing exact test statistics in linear mixed models (LMMs) for genome-wide association studies (GWAS). The method, called Genome-wide Efficient Mixed Model Association (GEMMA), is designed to address the computational challenges posed by existing exact methods, such as EMMA, which are impractical for large-scale GWAS. GEMMA is roughly \( n \) times faster than EMMA, where \( n \) is the sample size, making exact genome-wide association analysis computationally feasible for large numbers of individuals. The authors compare GEMMA with other methods, including EMMA, EMMAX, and GRAMMAR, using two examples: a mouse GWAS for high-density lipoprotein cholesterol (HDL-C) levels and a human GWAS for Crohn’s disease. They find that GEMMA provides exact results while being significantly faster, and that approximation methods like EMMA and GRAMMAR can yield inaccurate p-values in certain settings, particularly when the sample structure is strong or the marker effect size is large. The paper also discusses the implications of using different relatedness matrices and the potential for further improvements in computational efficiency and statistical power. Overall, GEMMA offers a powerful and efficient tool for exact genome-wide association analysis, particularly for large-scale studies.The paper introduces an efficient method for computing exact test statistics in linear mixed models (LMMs) for genome-wide association studies (GWAS). The method, called Genome-wide Efficient Mixed Model Association (GEMMA), is designed to address the computational challenges posed by existing exact methods, such as EMMA, which are impractical for large-scale GWAS. GEMMA is roughly \( n \) times faster than EMMA, where \( n \) is the sample size, making exact genome-wide association analysis computationally feasible for large numbers of individuals. The authors compare GEMMA with other methods, including EMMA, EMMAX, and GRAMMAR, using two examples: a mouse GWAS for high-density lipoprotein cholesterol (HDL-C) levels and a human GWAS for Crohn’s disease. They find that GEMMA provides exact results while being significantly faster, and that approximation methods like EMMA and GRAMMAR can yield inaccurate p-values in certain settings, particularly when the sample structure is strong or the marker effect size is large. The paper also discusses the implications of using different relatedness matrices and the potential for further improvements in computational efficiency and statistical power. Overall, GEMMA offers a powerful and efficient tool for exact genome-wide association analysis, particularly for large-scale studies.