Table 1.

Comparison of six methods and their softwares for GWAS

Case	FASTmrEMMA	E-BAYES	EMMA	CMLM	ECMLM	SUPER
Model	Multi-locus model	Multi-locus model	Single-locus model	Single-locus model	Single-locus model	Single-locus model
QTN effect	Random	Random	Fixed	Fixed	Fixed	Fixed
Polygenic background control	Yes	No	Yes	Yes	Yes	Yes
Population structure control	Yes	No	Yes	Yes	Yes	Yes
Number of variance components	Three	No. of effects	Two	Two	Two	Two
Polygenic-to-residual variance ratio	Fixed	NA	NA	Fixed	Fixed	NA
Significant critical value	LOD (logarithm of odds)=3	P-value=0.05	P-value=0.05/p, where p is no. of markers	P-value=0.05/p	P-value=0.05/p	P-value=0.05/p
Transformation matrix and performances	${\mathbf{Q}}_1{\mathbf{\Lambda }}_r^{-\frac{1}{2}}{\mathbf{Q}}_1^T$ where $\left({\mathbf{Q}}_1{\mathbf{\Lambda }}_r^{\frac{1}{2}}{\mathbf{Q}}_1^T\right)\left({\mathbf{Q}}_1{\mathbf{\Lambda }}_r^{\frac{1}{2}}{\mathbf{Q}}_1^T\right)={\widehat{\lambda }}_g\mathbf{Z}\mathbf{K}{\mathbf{Z}}^T+{\mathbf{I}}_n$ Covariance matrix of the polygenic matrix K and environmental noise are whitened.Number of nonzero eigenvalues is specified as one.	Shrinkage is selective. Large effects subject to virtually no shrinkage while small effects are shrunken to zero.	$\begin{array}{l}{\mathbf{U}}_R^T\text{ where }\\ \mathbf{SHS}={\mathbf{U}}_R\text{diag}\left({\xi }_1+\delta ,\cdots ,{\xi }_n+\delta \right){\mathbf{U}}_R^T\\ \mathbf{H}=\mathbf{ZK}{\mathbf{Z}}^T+\delta \mathbf{I}\text{ and }\mathbf{S}=\mathbf{I}-\mathbf{X}{\left({\mathbf{X}}^T\mathbf{X}\right)}^{-1}{\mathbf{X}}^T\end{array}$ One-dimensional optimization by deriving the likelihood as a function of QTN-to-residual variance ratio.	Kinship among individuals is replaced by the kinship among groups.Fit the groups as the random effect, and estimates population parameters only once and then fixes them to test genetic markers.	Kinship among individuals is replaced by the kinship among groups.Chooses the best combination between kinship algorithms and grouping algorithms.	Dramatically reduces the number of markers used to define individual relationships, and uses them in FaST-LMM.
Running time	Fast	Depend on the number of effects.	Slow	Fast	Fast	Moderate
Software Web site	https://cran.r-project.org/web/packages/mrMLM/index.html	http://statgen.ucr.edu/software.html	http://mouse.cs.ucla.edu/emma/	http://zzlab.net/GAPIT	http://zzlab.net/GAPIT	http://zzlab.net/GAPIT