Table 2.

A summary of the PRS test and its modifications without sample splitting. T is for a test statistic and P is for its p-value.

Data	D ={(Yi, Xi)|i = 1, …, n} = D1 υ D2 split to two parts D1 and D2.
Model	Logit [Pr(Yi = 1)] = βM,0 + XijβM,j, i ∈ D1.
Output	P-values pj(D1)’s for H0: βm,j = 0.
Model	$\text{Logit}\left[P_r(Y_i=1)\right]={\beta }_0+{\Sigma }_{j=1}^k{X}_{ij}{\beta }_j$,
Output	$U_{\left(D\right)}={(U_1\left(D\right),\dots ,U_k\left(D\right))}^{\prime }={\Sigma }_{i∊D}{X}_i(Y_i-\stackrel{‒}{Y}),\stackrel{‒}{Y}={\Sigma }_{i∊D}{Y}_i∕\mid D\mid$,
	$U\left(D_m\right)={(U_1\left(D_m\right),\dots ,U_k\left(D_m\right))}^{\prime }={\Sigma }_{i∊D_m}{X}_i(Y_i-{\stackrel{‒}{Y}}_{\left(m\right)}),{\stackrel{‒}{Y}}_{\left(m\right)}={\Sigma }_{i∊D_m}∕\mid D_m\mid \text{for}m=1,2$.
Tests	$T_{PRS}\approx c{\Sigma }_{j=1}^k{U}_j\left(D_1\right)U_j\left(D_2\right)I\left(p_j\left(D_1\right)<P_T\right)∕\text{Var}\left(U_j\left(D_1\right)\right)$.
	$T_{tSSUw\left(P_T\right)}={\Sigma }_{j=1}^k{U}_j^2\left(D\right)I\left(p_j\left(D\right)<P_T\right)∕\text{Var}\left(U_j\left(D\right)\right)$.
	$T_{atSSUw\left(\Omega \right)}={\text{min}}_{P_T∊\Omega }{P}_{tSSUw\left(P_T\right)}$.
	$T_{tSSU\left(P_T\right)}={\Sigma }_{j=1}^k{U}_j^2\left(D\right)I\left(p_j\left(D\right)<P_T\right)$.
	$T_{atSSU\left(\Omega \right)}={\text{min}}_{P_T∊\Omega }{P}_{tSSU\left(P_T\right)}$.
	$T_{SSU}={\Sigma }_{j=1}^k{U}_j^2\left(D\right)$.
	$T_{SPU\left(\gamma \right)}={\Sigma }_{j=1}^k{U}_j^{\gamma }\left(D\right)$.
	$T_{aSPU\left(\Gamma \right)}={\text{min}}_{\gamma ∊\Gamma }{P}_{SPU\left(\gamma \right)}$.
	$T_{tSPU(\gamma ,P_T)}={\Sigma }_{j=1}^k{U}_j^{\gamma }\left(D\right)I\left(p_j\left(D\right)<P_T\right)$.
	$T_{atSPU(\Gamma ,\Omega )}={\text{min}}_{\gamma ∊\Gamma ,P_T∊\Omega }{P}_{tSPU(\gamma ,P_T)}$.