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Since their introduction in the 50's, variance component mixed models have been widely used in many application fields. In this context, ReML estimation is by far the most popular procedure to infer the variance components of the model. Although many implementations of the ReML procedure are readily available, there is still need for computational improvements due to the ever-increasing size of the datasets to be handled, and to the complexity of the models to be adjusted. In this paper, we present a Min-Max (MM) algorithm for ReML inference and combine it with several speed-up procedures. The ReML MM algorithm we present is compared to 5 state-of-the-art publicly available algorithms used in statistical genetics. The computational performance of the different algorithms are evaluated on several datasets representing different plant breeding experimental designs. The MM algorithm ranks among the top 2 methods in almost all settings and is more versatile than many of its competitors. The MM algorithm is a promising alternative to the classical AI-ReML algorithm in the context of variance component mixed models. It is available in the MM4LMM R-package. Author summary: Mixed models have been a cornerstone of the quantitative genetics methodology for decades. Due to the growing size of datasets, their associated computational cost is a major burden, particularly in genome-wide association studies that routinely address Millions of markers, requiring as many mixed models to be fitted. In the particular case of a 2-variance component mixed models efficient procedures such as FaST-LMM or GEMMA have been developed to analyze panels with tens of thousands of individuals. However, there is room for improvement in cases where the computational burden is due to the number of variance components in the model rather than the panel size, a classical situation in plant genetics where several variance components are required to handle various polygenic effects. We consider a "MM" (Min-Max) algorithm as an alternative to the by-default AI (Average Information) algorithm used to perform inference in mixed models. The MM algorithm can be combined to the classical tricks used to accelerate the inference process (e.g. simultaneous orthogonalization or squared iterative acceleration methods). The MM procedure is implemented in an MM4LMM package, and is competitive compared to classical algorithms used in plant genetics. This package should help geneticists handling more complex models to analyze their data than today. [ABSTRACT FROM AUTHOR]

The effective sample size (ESS) is a metric used to summarize in a single term the amount of correlation in a sample. It is of particular interest when predicting the statistical power of genome-wide association studies (GWAS) based on linear mixed models. Here, we introduce an analytical form of the ESS for mixed-model GWAS of quantitative traits and relate it to empirical estimators recently proposed. Using our framework, we derived approximations of the ESS for analyses of related and unrelated samples and for both marginal genetic and gene-environment interaction tests. We conducted simulations to validate our approximations and to provide a quantitative perspective on the statistical power of various scenarios, including power loss due to family relatedness and power gains due to conditioning on the polygenic signal. Our analyses also demonstrate that the power of gene-environment interaction GWAS in related individuals strongly depends on the family structure and exposure distribution. Finally, we performed a series of mixed-model GWAS on data from the UK Biobank and confirmed the simulation results. We notably found that the expected power drop due to family relatedness in the UK Biobank is negligible. [ABSTRACT FROM AUTHOR]