Table 1.

Summary of GWAS association statistics

Association statistic	Statistical test$\left({\mathit{H}}_{\mathbf{0}}\right)$	Assumptions on$\mathit{\beta }$	Ancestry-related covariates	Degrees of freedom
ADM	$e_l=0$	–	$\alpha$	1
Standard GWAS	$\beta =0$	$\beta ={\beta }_1={\beta }_2$	$\alpha$	1
SNP1	$\beta =0$	$\beta ={\beta }_1={\beta }_2$	$l,\alpha$	1
MIX	$e_l\circ \beta =0$	$\beta ={\beta }_1={\beta }_2$	$\alpha$	1
SUM	$\beta =0$ and $e_l=0$	$\beta ={\beta }_1={\beta }_2$	$\alpha$	2
Tractor	${\beta }_1=0$ and ${\beta }_2=0$	–	$l,\alpha$	2

All tests adjust for global ancestry and can be used on binary traits, and all tests except MIX can be implemented with adjustment for additional covariates and use on quantitative traits. For more information on the comparison of standard GWAS, ADM, SUM, and MIX, see Pasaniuc et al.¹⁴ and Seldin et al.²² We note that while additional methods exist,^36,37,38,39 we do not focus on them in this work because they do not directly relate to Equation 1.