Table 2.

Relationships between the existing and new tests.

Method	Model or test statistic	Relation to the new tests
The Average method (Shen et al 2011)	${\sum }_{j=1}^k{Y}_{ij}={\alpha }_0+{\alpha }_1{x}_i+e_i$, Applying the Score test on H0: α1 = 0.	Average = GEE-SPU(1).
TATES (van der Sluis et al 2013)	Yij = β0,j + β1,jxi + eij for j = 1, 2, ..., k. Testing for H0: β1,1 = ... = β1,k = 0 with analytical approximations to calculate a p-value.	TATES ≈ GEE-UminP ≈ GEE-SPUw(∞).
CCA=MANOVA (Ferreira & Purcell 2009; Yang & Wang 2012)	CCA seeks to maximize the correlation between a linear combination of (Yi1, ..., Yik) and xi. Test statistic: $B=S_{XX}^{-1∕2}{S}_{XY}{S}_{YY}^{-1}{S}_{YX}{S}_{XX}^{-1∕2}$	CCA=MANOVA=GEE-Score.
MDMR (Zapala & Schork 2012)	$D_{ij}=d(Y_i,Y_j),A=(-D_{ij}^2∕2)$, G = (I – 11′/n)A(I – 11′/n), H = X(X′X)–1X′, Test statistic: F = tr(HGH)/tr[(I – H)G(I – H)]	MDMR=GEE-SPU(2) if d(,) is Euclidean.
KMR (Maity et al 2012)	Test statistic: $T_{KMR}={(Y-\stackrel{‒}{Y})}^{\prime }{V}_0^{-1}K{V}_0^{-1}(Y-\stackrel{‒}{Y})$	KMR = GEE-SPU(2) if K = XX′ and Rw = Corr(Yi|H0).
MultiPhen (O'Reilly et al., 2012)	${\pi }_j\left(y\right)=Pr(x_i=j\mid Y_i=y),{\kappa }_j\left(y\right)={\sum }_{m=0}^j{\pi }_m\left(y\right)$ for j = 0, 1, 2, $\log \frac{{\kappa }_j\left(y\right)}{1-{\kappa }_j\left(y\right)}={\alpha }_j-y^{\prime }\beta$ for j = 0 and 1. Applying the Score (or likelihood ratio) test on H0: β = 0.	MultiPhen ≈ GEE-Score.
Generalized Kendall's tau (Zhang et al 2010)	uij = (Yi1 – Yj1, ..., Yiq – Yjq)′, ${\stackrel{‒}{u}}_i={\sum }_{j=1}^n{u}_{ij}∕n,\tau ={\sum }_{i=1}^n{x}_i{\stackrel{‒}{u}}_i$. Test statistic: $T={\tau }^{\prime }{V}_0^{-1}\tau$.	GK-tau = GEE-Score.