Several approximation methods have been proposed to make genome-wide analysis using linear mixed models possible. Probably the simplest and fastest of these approximations, GRAMMAR (Genome-wide Rapid Association using Mixed Model And Regression), implemented in the software GenABEL9, first estimates the residuals from the LMM under the null model, and then treats these residuals as phenotypes for further genome-wide analysis by a standard linear model10. This substantially reduces per-SNP computation time, making it linear in the number of individuals. More recently two more-sophisticated approximate approaches have been suggested. Zhang et al7 use P3D (Population Parameters Previously Determined) which avoids repeatedly estimating variance components when performing each test by simply using the pre-estimated variance components from the null model; their method is implemented in the software TASSEL. Kang et al1 also avoid repeatedly estimating variance components by a slightly different strategy, which keeps the heritability estimated from the null model fixed when testing individual SNPs. Their approach is implemented in the software EMMAX (EMMA eXpedited). (This approximation, and related ideas, was also considered by previous authors, including10,11.) Both these last two approximations
