Genotype-by-environment interactions (GEIs) are important for not only a precise understanding of genotype–phenotype relationships but also improving the environmental adaptability of crops. Although many formulae have been proposed to model GEI effects, a comprehensive comparison of their efficacy in genome-wide association studies (GWASs) has not been performed. Therefore, the advantages and disadvantages of the formulae are not well recognized. In this study, linear mixed models (LMMs)

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... ting of various combinations of foreground fixed genetic and background random genetic effect terms were constructed. Next, the power to detect quantitative trait loci (QTLs) with GEI effects was compared across the LMMs by using simulation. The fixed genetic effect terms of the genotype main effects and GEI (GGE) model were preferred over those based on the additive main effects and multiplicative interaction (AMMI) model because the latter showed p-value inflation, whereas the former yielded theoretically expected p-value distribution. With regard to the background random genetic effects, inclusion of genotype-by-trial interaction effects has been recommended to achieve high power and robustness when phenotype data are obtained from multiple environments and multiple trials. The recommended form of LMM was applied to GWASs performed using real agronomic trait data of tomato F1 varieties (Solanum lycopersicum L.) that were obtained from two different cropping seasons. The GWASs detected QTLs with not only persistent effects across the cropping seasons, but also cropping season-specific effects. Thus, the application of GWAS strategy to phenotypic data from multiple environments and trials allowed the detection of more QTLs with GEI effects.